OpenAI’s public math repository is not a cryptography release—but it is a strategic signal for anyone running a post-quantum migration. The catalog documents 722 manuscripts organized into 372 result families, produced while stress-testing an internal model on open research problems. Some families include Lean formalizations; many do not, and OpenAI explicitly warns that unformalized writeups may contain errors.
For security leaders, the important question is not whether every claim is final. It is whether AI-scale mathematical discovery is beginning to touch the same number theory, complexity theory, and quantum algorithmic ideas that underpin RSA, ECC, lattice schemes, and threat timelines.
Why this belongs on a PQC roadmap
Three forces converge:
- Quantum threat modeling — Families such as 279 (exact quantum factoring over a fixed finite gate set) and 284 (randomized vs quantum query separation) sit squarely in the conversation about when integer factorization and search problems shift from classical to quantum regimes—even when individual claims await independent verification.
- Classical public-key assumptions — Breakthroughs in elliptic-curve arithmetic, BSD/Selmer results, and quasi-Riemann progress do not “break TLS tomorrow,” but they reshape how quickly expert communities can analyze structured instances, special curves, and related hard problems that appear in legacy deployments.
- AI-assisted cryptanalysis velocity — QTL’s analysis of the HAWK withdrawal showed how model-assisted review can collapse multi-year evaluation windows. A 700+ manuscript drop generalizes that pattern: mathematics at machine scale becomes an input to crypto-agility, not a curiosity.
How to read the interactive map below
We classified about 50 families into seven PQC lenses (keyword rules on titles and summaries; a Lean doc link alone is not enough). The explorer lists all 372 families and all 722 manuscripts from OpenAI’s catalog—use All manuscripts (722) for every paper with abstract and PDF link, or All families (372) for grouped results (including Families 057, 071, 348, 360):
- Quantum algorithms & physics — factoring models, query complexity, quantum many-body results tied to hardware and error correction narratives
- Number theory & ECC — zeta/L-function progress, elliptic-curve structure, modularity—adjacent to classical PKI analysis
- Complexity & hardness — UGC-class results and approximation thresholds that affect how we reason about hard problems in reductions
- Factorization & arithmetic — polynomial factorization, prime statistics, totients—RSA/DSA adjacent
- Algebra & structures — Kaplansky/Mahler/factor-algebra results that influence advanced constructions and future cryptanalysis tools
- Spin & statistical physics — spin glasses, magnetization—inputs to noise, randomness, and QEC storytelling
- Formal verification — Lean-linked families that matter for high-assurance implementations
Use search to jump to a family ID (for example 279 or 102), filter by category, and expand rows for QTL’s one-line PQC lens on each entry. Featured cards highlight families security teams should discuss in quarterly risk reviews.
Interactive map: OpenAI math catalog
OpenAI’s openai/math release contains 722 manuscripts grouped into 372 result families. Expand a family to switch between Overview and each paper tab, or use All manuscripts for a flat list. PQC curated lists 50 families with direct security-math relevance (Lean formalization alone does not qualify).
Quantum algorithms & physics
Directly shapes quantum threat models, algorithm timelines, or query-complexity assumptions used in security proofs.
Complexity & hardness
Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
Number theory & ECC
Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
Algebra & structures
Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
Factorization & arithmetic
Affects polynomial or integer factorization—core to RSA/DSA analysis and to comparing classical vs quantum factoring.
Spin & statistical physics
Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
Formal verification
Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
General math (non-PQC)
Included for full-catalog browsing. No specific post-quantum migration impact is assigned by QTL.
High-signal families for security leaders
Exact quantum factoring over a fixed finite gate set.
Gives a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer with probability one. Both gate count and qubit count are polynomial in the input length, and one fixed finite gate set suffices. ([Lean](lean/docs/279.md))
PQC lens: Directly shapes quantum threat models, algorithm timelines, or query-complexity assumptions used in security proofs.
Primary manuscript on GitHub → Family index on GitHub →M562
Exact quantum factoring over a fixed finite gate set
We give a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer N ≥ 2 with probability one. A fixed finite set of bounded-arity gates suffices, and both the gate count and the number of qubits have polynomial worst-case bounds in the input length.
View manuscript on GitHub → Open PDF (raw) →The quasi-Riemann hypothesis.
Proves that every Dirichlet L-function, including `ζ(s)`, is zero-free in `Re s 7/8`, resolving the quasi-Riemann hypothesis.
PQC lens: Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
Primary manuscript on GitHub → Family index on GitHub →M005
The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane `Re s 7/8`
We prove that all finite-order Hecke L-functions over ` Q(√(-3))` and all Dirichlet L-functions are zero-free in the half-plane `Re s 7/8`, with the principal pole at s = 1…
View manuscript on GitHub → Open PDF (raw) →M006
The Quasi-Riemann Hypothesis (alternate 11/12 proof)
We establish the quasi-Riemann hypothesis by proving that every Dirichlet L-function, including Riemann's zeta function, has no zeros in the half-plane ` Re s 11/12`.
View manuscript on GitHub → Open PDF (raw) →M007
Uniform exclusion of Landau–Siegel zeros
We prove the uniform exclusion of Landau–Siegel zeros. There is an absolute constant c > 0 such that every real zero `β (0,1)` of every primitive nonprincipal real Dirichlet L-function of conductor q ≥ 3 satisfies `(1-β)log q≥ c`.
View manuscript on GitHub → Open PDF (raw) →The full BSD formula from low Selmer corank.
Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one for some prime q, including finiteness of the Tate–Shafarevich group. With result 006, this gives full BSD for a density-one set of quadratic twists of every elliptic curve over ℚ.
PQC lens: Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
Primary manuscript on GitHub → Family index on GitHub →M002
Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank
We prove the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one at some prime q. The analytic and Mordell–Weil ranks equal that corank, and the Tate–Shafarevich group is finite. The formula includes all prime factors and requires no additional hypotheses on reduction, rational torsion, isogenies, complex multiplication, or residual Galois representations.
View manuscript on GitHub → Open PDF (raw) →M003
The Selmer converse for elliptic curves at every prime
We prove the Selmer converse in coranks zero and one for every elliptic curve over ℚ and every prime p: if the full p-power Selmer group has ℤp-corank `r 0,1 `, then the analytic and Mordell–Weil ranks both equal r, and the entire Tate–Shafarevich group is finite. As an application at the additive prime 3, we prove that for every prime `ℓ≡4,7,8 pmod9`, the cubic `X³⁺Y³⁼ℓ Z³` has analytic and Mordell–Weil rank one and finite Tate–Shafarevich group. In particular, every such ℓ is a sum of two rational cubes.
View manuscript on GitHub → Open PDF (raw) →M004
The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one
We prove the two-primary Birch and Swinnerton-Dyer leading-term formula for every elliptic curve over the rationals whose two-power Selmer group has corank at most one. In this range, the algebraic rank, analytic rank, and Selmer corank are equal, and the Tate–Shafarevich group is finite. Combined with the quadratic-twist Selmer distribution, this gives the exact two-primary formula for a density-one set of signed squarefree twists of each fixed curve, ordered by absolute value; the common rank is zero or one, with each value having density one half.
View manuscript on GitHub → Open PDF (raw) →The Unique Games Conjecture and optimal approximation thresholds.
Proves Khot's Unique Games Conjecture. Independent direct reductions also establish NP-hardness, on unweighted graphs, of approximation beyond the Goemans–Williamson ratio for Max-Cut, below factor two for Vertex Cover, and within any fixed constant factor for Min-UnCut and directed feedback vertex set. These direct proofs use established PCP and Label Cover hardness results. ([Lean](lean/docs/102.md))
PQC lens: Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
Primary manuscript on GitHub → Family index on GitHub →M204
The Unique Games Theorem
We prove the Unique Games Conjecture. For every fixed `ε,δ (0,1/2)`, we give a deterministic polynomial-time reduction from 3SAT to Unique Games over a fixed finite alphabet, with completeness at least `1-ε` and soundness at most δ.
View manuscript on GitHub → Open PDF (raw) →M205
A Direct Proof of Optimal Max-Cut Hardness
We prove that approximating Max-Cut on simple unweighted graphs within any fixed factor greater than the Goemans–Williamson constant is NP-hard.
View manuscript on GitHub → Open PDF (raw) →M206
The Factor-Two Hardness Threshold for Vertex Cover
We prove that minimum Vertex Cover is NP-hard to approximate within every fixed factor below two, even on simple unweighted graphs.
View manuscript on GitHub → Open PDF (raw) →M207
Constant-factor hardness of Min-UnCut
For every fixed C > 1, approximating Min-UnCut within factor C is NP-hard, even on simple undirected unweighted graphs.
View manuscript on GitHub → Open PDF (raw) →M208
Constant-factor hardness of directed feedback vertex set
Approximating minimum directed feedback vertex set within any fixed constant factor is NP-hard, even on unweighted digraphs.
View manuscript on GitHub → Open PDF (raw) →A torsion-free group algebra that is not directly finite.
Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽2 is not directly finite, disproving Kaplansky's conjecture even without torsion. Companion examples give injective nonsurjective cellular automata on all configurations, refuting Gottschalk's surjunctivity conjecture. Another counterexample is an integral group-ring matrix, invertible over the rational group ring, with Fuglede–Kadison determinant strictly between zero and one, disproving the unrestricted Determinant Conjecture. ([Lean](lean/docs/197.md))
PQC lens: Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
Primary manuscript on GitHub → Family index on GitHub →M350
A Torsion-Free Group Algebra That Is Not Directly Finite
We construct a finitely presented torsion-free counterexample to Kaplansky's direct finiteness conjecture over the field of two elements. The group admits a finite two-dimensional classifying complex.
View manuscript on GitHub → Open PDF (raw) →M351
A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two
We disprove Kaplansky's direct-finiteness conjecture by constructing a finite field K of characteristic two, a finitely presented group G, and finite sums `a,b K[G]` with `ab=1` but `ba≠1`. The group G is nonsofic. The same elements define a cellular automaton on KG that is injective but not surjective, disproving Gottschalk's surjunctivity conjecture.
View manuscript on GitHub → Open PDF (raw) →M352
A Counterexample to the Group-Ring Determinant Conjecture
We disprove the unrestricted group-ring Determinant Conjecture. We construct a finitely generated group G and a square matrix over ` Z[G]` that is invertible over ` Q[G]` and has Fuglede–Kadison determinant strictly between zero and one. The logarithmic integral defining the determinant is finite.
View manuscript on GitHub → Open PDF (raw) →M353
A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic
We construct a counterexample to Kaplansky's direct-finiteness conjecture in odd characteristic. For one specified odd prime p, we obtain a field K of order p4, a finitely generated group G containing torsion, and finite sums `a,b K[G]` with `ab=1` but `ba≠1`. The same elements define a cellular automaton on KG that is injective but not surjective.
View manuscript on GitHub → Open PDF (raw) →Bloch's law, its lattice correction, and the spherical magnetization law.
Proves Bloch's T3/2 law with its exact coefficient for three-dimensional quantum Heisenberg ferromagnets at every positive quantum spin, allowing nonnegative symmetric finite-range couplings whose support generates ℤ3. The thermodynamic limit precedes the zero-field derivative and low-temperature limit. The family also proves spontaneous magnetization for nearest-neighbor models in every dimension d ≥ 3 and determines the first lattice correction for three-dimensional nearest-neighbor couplings.
PQC lens: Directly shapes quantum threat models, algorithm timelines, or query-complexity assumptions used in security proofs.
Primary manuscript on GitHub → Family index on GitHub →M549
Bloch's Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions
We prove Bloch's T3/2 law for the spontaneous magnetization of three-dimensional quantum Heisenberg ferromagnets at every fixed positive quantum spin. The result holds for every nonnegative symmetric finite-range interaction whose support generates ℤ3, including spatially anisotropic couplings. The leading magnetization deficit has the exact coefficient determined by the determinant of the quadratic one-magnon dispersion.
View manuscript on GitHub → Open PDF (raw) →M550
The first lattice correction to Bloch's law
We prove the first lattice correction to Bloch's law for the three-dimensional nearest-neighbor quantum Heisenberg ferromagnet at every fixed spin `S= tfrac12,1, tfrac32, `. The spontaneous-magnetization deficit agrees with the full ideal-magnon density up to `o(β⁻⁵/²)`. In addition to the leading Bloch term, this gives the correction `3ζ(5/2)(β S)⁻⁵/²/(128π³/²)`. The magnetization is the right derivative at zero field of the thermodynamic pressure; the volume limit precedes the field derivative, and the low-temperature limit is taken last.
View manuscript on GitHub → Open PDF (raw) →M551
The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet
We prove the spherical magnetization law for the three-dimensional nearest-neighbor isotropic quantum Heisenberg ferromagnet at every fixed positive quantum spin and every sufficiently low fixed positive temperature. As the even periodic cubes grow, the symmetric zero-field magnetization converges in moments to a uniform direction with a deterministic positive magnitude. This magnitude equals the right derivative at zero field of the infinite-volume pressure.
View manuscript on GitHub → Open PDF (raw) →M552
Spontaneous magnetization in the quantum Heisenberg ferromagnet
For every dimension d ≥ 3 and every spin `S frac12,1, frac32, `, we prove that the nearest-neighbor isotropic quantum Heisenberg ferromagnet has a translation-invariant, spontaneously magnetized equilibrium state at every sufficiently low positive temperature. The same state satisfies the KMS condition for the zero-field dynamics and has magnetization at least `S/4`. This resolves the low-temperature ordering problem in the spontaneous-magnetization formulation.
View manuscript on GitHub → Open PDF (raw) →The Mahler conjectures, functional inequalities and polar-product symplectic width.
Resolves the symmetric and nonsymmetric geometric Mahler conjectures in every dimension, with Hanner polytopes and simplices as the respective volume-product minimizers and all equality cases classified.
PQC lens: Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
Primary manuscript on GitHub → Family index on GitHub →M174
The symmetric Mahler conjecture and its equality cases
We resolve the symmetric Mahler conjecture positively, including its equality classification. Every origin-symmetric convex body in ℝn has volume product at least `4n/n!`, with equality exactly for invertible linear images of Hanner polytopes.
View manuscript on GitHub → Open PDF (raw) →M175
The Mahler Conjecture for General Convex Bodies
We resolve the Mahler conjecture for general convex bodies positively. For every convex body `K Rn`, n ≥ 1, with Santaló point `s(K)`, `|K| |(K-s(K))^ |≥ (n+1)n⁺¹/(n!)²`, with equality exactly for simplices.
View manuscript on GitHub → Open PDF (raw) →M176
Symplectic Balls in Symmetric Polar Products
For every integer n ≥ 2 and every origin-symmetric convex body `K Rn`, we prove that the Gromov width of ` int K× int K^ ` is…
View manuscript on GitHub → Open PDF (raw) →The Mézard–Parisi formula for diluted spin glasses.
Proves the Mézard–Parisi hierarchical cavity formula for Poisson-diluted even-arity Ising models satisfying the Panchenko–Talagrand factorization and positivity assumptions, with only first-moment integrability. The limiting free energy equals the infimum over finite-depth hierarchical trial laws. This includes the Viana–Bray model, symmetric diluted even-spin models, and weighted soft even-K satisfiability. ([Lean](lean/docs/221.md))
PQC lens: Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
Primary manuscript on GitHub → Family index on GitHub →M411
The Mézard–Parisi formula for diluted spin glasses
We prove the Mézard–Parisi hierarchical cavity formula for diluted even-arity Ising models in the Panchenko–Talagrand class. This resolves the variational equality conjecture for that class: the limiting pressure equals the infimum of the trial functional over all finite hierarchy depths and trial laws. Only first moments of the interaction and external field are required.
View manuscript on GitHub → Open PDF (raw) →Global smoothness for relativistic Vlasov–Maxwell.
Proves large-data global existence and uniqueness for the three-dimensional, one-species relativistic Vlasov–Maxwell system. Smooth admissible initial data may be arbitrary provided the particle density is compactly supported and the electromagnetic fields have finite energy and bounded derivatives of every order; the solution remains smooth on every finite time interval. ([Lean](lean/docs/362.md))
PQC lens: Directly shapes quantum threat models, algorithm timelines, or query-complexity assumptions used in security proofs.
Primary manuscript on GitHub → Family index on GitHub →M693
Global classical solutions of the three-dimensional relativistic Vlasov–Maxwell system
We prove global existence and uniqueness for arbitrary smooth admissible initial data in the three-dimensional, one-species relativistic Vlasov–Maxwell system. The particle density is initially compactly supported, and the electromagnetic fields have finite energy and bounded derivatives of all orders. The solution remains smooth on every finite time interval. This resolves the large-data global classical regularity problem for this model, without size or symmetry restrictions on the data.
View manuscript on GitHub → Open PDF (raw) →Hilbert’s tenth problem over ℚ.
Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively.
PQC lens: Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
Primary manuscript on GitHub → Family index on GitHub →M008
Hilbert’s tenth problem over the rational numbers
We give a negative answer to Hilbert's tenth problem over the rational numbers: no algorithm decides whether a polynomial with integer coefficients has a rational zero. The number of variables is part of the input.
View manuscript on GitHub → Open PDF (raw) →M009
A pointwise 2-converse for elliptic curves with rational two-torsion
We prove a pointwise 2-converse for elliptic curves over ` Q` with nonzero rational two-torsion: if the `2^∞` -Selmer corank is zero or one, then the analytic rank and Mordell–Weil rank equal that corank, and the Shafarevich–Tate group is finite. The result allows arbitrary reduction at 2.
View manuscript on GitHub → Open PDF (raw) →The optimal quartic separation between randomized and quantum queries.
Shows that the universal bound `R(f)=O((1+Q(f))⁴)` for total Boolean functions is sharp in its exponent, ruling out every smaller power and disproving the conjectured cubic relation. Here R and Q are randomized and quantum worst-case bit-query complexities with error at most 1/3; computation between queries is unrestricted.
PQC lens: Directly shapes quantum threat models, algorithm timelines, or query-complexity assumptions used in security proofs.
Primary manuscript on GitHub → Family index on GitHub →M568
A Nearly Quartic Separation Between Randomized and Quantum Query Complexity
We construct total Boolean functions with a nearly quartic separation between bounded-error randomized and quantum query complexity. Writing these complexities as ` R(f)` and ` Q(f)`, the examples rule out every universal bound ` R(f)=O((1+ Q(f))^α)` with α < 4. Thus the known quartic upper bound has the optimal exponent, disproving the conjectured cubic bound. Both complexities count worst-case bit queries with error at most 1/3 on every input.
View manuscript on GitHub → Open PDF (raw) →Global quantum geometric Langlands at irrational level.
Proves the unramified de Rham quantum geometric Langlands equivalence for every connected simple complex algebraic group on every smooth projective connected complex curve, at every shifted level `c C Q`. It identifies the full derived categories of twisted D-modules for the group and its Langlands dual, retaining all global forms and connected components.
PQC lens: Directly shapes quantum threat models, algorithm timelines, or query-complexity assumptions used in security proofs.
Primary manuscript on GitHub → Family index on GitHub →M147
Global quantum geometric Langlands at irrational level
We prove the unramified quantum geometric Langlands equivalence for every connected simple complex algebraic group G and every level `c C Q`, including non-real levels. For every smooth projective connected complex curve X, it identifies the full twisted D-module categories on ` Bun _G(X)` and ` Bun (G^ )(X)` at the dual shifted levels c and `-1/(rc)`, where r is the lacing number. The equivalence uses the given global forms and includes all connected components.
View manuscript on GitHub → Open PDF (raw) →Browse families
Family 001 Milne’s rationality conjecture and algebraic specialization. 1 paper
Proves Milne's rationality conjecture for abelian varieties over `Q̄` with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology theory. Together with result 032, every specialized Hodge class is represented by a single rational algebraic cycle simultaneously in all prime-to-p and crystalline realizations, for every residue characteristic p.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M001
Milne's rationality conjecture for abelian varieties
We prove Milne's rationality conjecture for abelian varieties, including residue characteristic 2. After good reduction, the pairing of a rational Hodge class with any complementary product of divisor classes on the reduction is the same rational number in every prime-to-p realization and in crystalline cohomology. Using the Hodge theorem for CM abelian varieties, we also show that every such specialized Hodge class is represented by a single rational algebraic cycle in all these realizations.
View manuscript on GitHub → Open PDF (raw) →Family 005 Irrationality of Catalan’s constant. 1 paper
Proves that Catalan's constant `G=∑(j≥0)(-1)j/(2j+1)²`…
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M010
Catalan's constant is irrational
We prove that Catalan's constant `G=∑(j≥0)(-1)j/(2j+1)²`…
View manuscript on GitHub → Open PDF (raw) →Family 006 Goldfeld’s conjecture: densities and mean analytic rank. 2 papers
Proves Goldfeld's conjecture for quadratic twists of every elliptic curve over ℚ: analytic ranks zero and one each have density 1/2, and the mean analytic rank tends to 1/2. Both statements order signed squarefree twist parameters by absolute value.
PQC lens: Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
Primary manuscript on GitHub → Family index on GitHub →M011
Goldfeld's analytic density conjecture and the 2-converse for elliptic curves
We prove Goldfeld's analytic density conjecture: for every elliptic curve E over ℚ, the quadratic twists of E with analytic rank zero and one each have density 1/2 among signed squarefree twist parameters ordered by absolute value. We also prove the low-corank 2-converse: if the `2^∞` -Selmer corank of E is zero or one, then it equals the analytic and Mordell–Weil ranks, and the Tate–Shafarevich group is finite.
View manuscript on GitHub → Open PDF (raw) →M012
The mean analytic rank of quadratic twists of elliptic curves
For every elliptic curve over ℚ, we prove that the average analytic rank of its quadratic twists tends to 1/2 when signed squarefree twist parameters are ordered by absolute value. This resolves Goldfeld's mean analytic-rank conjecture in this counting convention.
View manuscript on GitHub → Open PDF (raw) →Family 007 Ordinary two-point correlations and the corrected Elliott conjecture. 1 paper
Proves the ordinary two-point Chowla conjecture, with a bound `O(X/(log X)c)` for Liouville correlation sums along fixed nonproportional affine forms, where c > 0 is absolute. More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times `nit` for `|t|≤ X`. ([Lean](lean/docs/007.md))
PQC lens: Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
Primary manuscript on GitHub → Family index on GitHub →M013
Ordinary two-point correlations of multiplicative functions
We prove the ordinary two-point Chowla conjecture. For every fixed pair of nonproportional affine forms, the Liouville correlation has a power-of-logarithm saving at every cutoff, with an absolute exponent. We also prove the binary corrected Elliott conjecture for ordinary averages of complex multiplicative functions of modulus at most one, under uniform nonpretentiousness of at least one original factor. This qualitative conclusion holds in fixed residue classes and for fixed nonproportional affine forms.
View manuscript on GitHub → Open PDF (raw) →Family 008 The Deligne–Drinfeld conjecture. 1 paper
Proves that the rational Grothendieck–Teichmüller Lie algebra, with the Ihara bracket, is freely generated by one element in each odd weight 3, 5, 7, …, resolving the Deligne–Drinfeld conjecture. ([Lean](lean/docs/008.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M014
The Deligne-Drinfeld conjecture
We prove the Deligne–Drinfeld conjecture: the rational Grothendieck–Teichmüller Lie algebra is freely generated by one element in every odd weight at least three, with the corresponding isomorphism after weight completion.
View manuscript on GitHub → Open PDF (raw) →Family 009 Function-field reconstruction from Milnor K-theory and Galois data. 3 papers
Reconstructs function fields of transcendence degree at least two over algebraically closed constants from `K M_1/ℓ`, `K M_2/ℓ`, and their product. These data recover the perfect closure and constants when ℓ differs from the characteristic, and the original field and its named base in equal characteristic. Also proves Bogomolov–Pop reconstruction from abelian-by-central pro-ℓ Galois data away from the characteristic. ([Lean](lean/docs/009.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M015
Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory
We prove a mod-ℓ Bogomolov–Pop reconstruction theorem for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from ℓ. The groups `K M_1/ℓ` and `K M_2/ℓ`, together with their full bilinear product, determine the perfect closure and its constant field. Every compatible isomorphism of these data is induced by a field isomorphism up to a single scalar in ` F_ℓ^×`, with only Frobenius ambiguity in the field isomorphism in positive characteristic.
View manuscript on GitHub → Open PDF (raw) →M016
Reconstruction from Milnor K-theory modulo the characteristic
Let K and L be finitely generated extensions of transcendence degree at least two over algebraically closed fields of characteristic p. We prove that every isomorphism of their first Milnor K-groups modulo p preserving the degree-two Steinberg relations is a nonzero scalar multiple of the map induced by a unique field isomorphism. The proof recovers projective lines over the subfields of pth powers by an elementary calculation with derivations.
View manuscript on GitHub → Open PDF (raw) →M017
The Bogomolov-Pop reconstruction theorem
We prove the Bogomolov–Pop reconstruction conjecture for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from ℓ. The pro-ℓ abelian-by-central datum determines the perfect closure and its constant field, with precisely the Frobenius and ℓ-adic unit ambiguities.
View manuscript on GitHub → Open PDF (raw) →Family 010 Unrestricted pro-modularity at the prime two. 3 papers
Every continuous odd absolutely irreducible two-dimensional 2-adic representation of `G( Q)` unramified outside finitely many primes occurs in a completed Hecke algebra at some odd tame level. Also proves classical modularity up to Tate twist for irreducible odd representations with these finiteness conditions that are de Rham at 2 with distinct Hodge–Tate weights, resolving the dyadic Fontaine–Mazur case without residual restrictions.
PQC lens: Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
Primary manuscript on GitHub → Family index on GitHub →M018
Unrestricted pro-modularity at the prime two
Every continuous, odd, absolutely irreducible two-dimensional 2-adic representation of `G( Q)` that is unramified outside finitely many finite primes occurs in the full completed Hecke algebra at some odd tame level. The level may contain auxiliary tame primes, and scalar and reducible residual representations are included. This is a completed-Hecke occurrence result, with no de Rham hypothesis; it does not assert classical modularity.
View manuscript on GitHub → Open PDF (raw) →M019
The Dimension of the Two-Adic Hecke Algebra at Odd Level
For every odd positive integer N, every irreducible component of the full two-adic Hecke algebra of level ` Gamma_1(N)` has Krull dimension four. This proves the p = 2 case of Emerton's dimension conjecture, including all residual components.
View manuscript on GitHub → Open PDF (raw) →M020
Fontaine–Mazur modularity at the prime 2
We prove that every continuous, irreducible, odd two-dimensional 2-adic representation of `G( Q)`, unramified outside finitely many primes and de Rham at 2 with distinct Hodge–Tate weights, is modular up to Tate twist. This resolves the odd, regular two-dimensional Fontaine–Mazur conjecture over ℚ at 2, including all residual representations.
View manuscript on GitHub → Open PDF (raw) →Family 011 Prime-factor statistics of `p-1`. 3 papers
Proves that the normalized ordered logarithms of the prime factors of `p-1`, counted with multiplicity, converge jointly to the Poisson–Dirichlet law `PD(1)` as p ranges uniformly over primes up to x and `x→∞`. This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers n have more than `n¹⁻ε` totient preimages, for every ε > 0.
PQC lens: Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
Primary manuscript on GitHub → Family index on GitHub →M021
Weighted dilation graphs, smooth shifted primes and totient fibers
We prove Erdős's conjecture on the largest fibers of Euler's totient function: for every ε > 0, infinitely many positive integers n have more than `n¹⁻ε ` preimages. We also show that, for every fixed δ > 0, there are at least `x¹⁻o(¹)` primes p in `2x p≤5x` whose predecessors have no prime factor exceeding xδ.
View manuscript on GitHub → Open PDF (raw) →M022
The Poisson-Dirichlet law for prime predecessors
For a prime p chosen uniformly from `3≤ p≤ x`, list the prime factors of `p-1` in decreasing order, with multiplicity. As `x→∞`, their logarithms, divided by `log(p-1)`, converge in every finite joint distribution to the Poisson–Dirichlet distribution with parameter one. This proves the conjecture of Ford, Konyagin and Luca.
View manuscript on GitHub → Open PDF (raw) →M023
Prime Predecessors with an Even Number of Prime Factors
We prove that there are infinitely many primes p for which `p-1` is squarefree and has an even number of prime factors. Equivalently, there are infinitely many primes p with `μ(p-1)=1`.
View manuscript on GitHub → Open PDF (raw) →Family 012 Independent largest prime factors of consecutive integers. 1 paper
Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of n and `n+1` are asymptotically independent in ordinary natural density. In particular, the integers satisfying `P⁺(n) P⁺(n⁺¹)` have density 1/2. ([Lean](lean/docs/012.md))
PQC lens: Affects polynomial or integer factorization—core to RSA/DSA analysis and to comparing classical vs quantum factoring.
Primary manuscript on GitHub → Family index on GitHub →M024
The joint Dickman law for consecutive integers
Let `P⁺(n)` denote the largest prime factor of n. We prove that `log P⁺(n)/log n` and `log P⁺(n⁺¹)/log n` are asymptotically independent in ordinary natural density, with Dickman marginals. This resolves the Erdős–Pomerance joint Dickman conjecture positively and implies that the ordering `P⁺(n) P⁺(n⁺¹)` has natural density 1/2.
View manuscript on GitHub → Open PDF (raw) →Family 013 Ostmann’s inverse Goldbach conjecture. 1 paper
Proves that no finite modification of the primes can be written as `A+B` with `A,B Z(≥0)` each containing at least two elements. This resolves Ostmann's inverse Goldbach conjecture on additive indecomposability. ([Lean](lean/docs/013.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M025
The additive indecomposability of the primes
We prove Ostmann's inverse Goldbach conjecture: no set differing from the primes by finitely many elements can be written as `A+B`, where A and B are sets of nonnegative integers with at least two elements each.
View manuscript on GitHub → Open PDF (raw) →Family 014 Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan. 8 papers
Proves the restricted geometric Langlands equivalence for connected reductive groups on smooth projective connected curves over ` F _q` under the four stated Lie-theoretic characteristic hypotheses. Over arbitrary algebraically closed fields of characteristic p > 0, the same conclusion holds assuming additionally that p is very good for the group and `p |W_G|`.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M026
Global Arthur Enhancements of Cuspidal Excursion Parameters
Under the finite-level Ramanujan–Arthur decomposition stated in Theorem 1.1, we construct a global Arthur enhancement for every occurring cuspidal excursion parameter at a specified full finite level for a split connected semisimple group over a global function field. A single algebraic SL2 and a commuting Weil centralizer map recover the given parameter by diagonal specialization on the entire Weil group, including inertia. The result does not assert ellipticity, Arthur-packet classification, or a multiplicity formula.
View manuscript on GitHub → Open PDF (raw) →M027
Rationality of the Canonical Unramified Arthur Filtration
Under the characteristic hypotheses of restricted geometric Langlands theory, we prove that the canonical Arthur filtration on finitely supported unramified automorphic functions is defined over ℚ for split connected semisimple groups. The result includes the noncuspidal part and every closed invariant nilpotent support, establishing the rationality conjecture of Gaitsgory–Lafforgue–Raskin in this setting.
View manuscript on GitHub → Open PDF (raw) →M028
Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level
We prove rational and `Q̄_ℓ` decompositions of cuspidal automorphic functions for split semisimple groups over global function fields into subspaces indexed by nilpotent orbits of the dual group. The decompositions hold at every full finite level, with arbitrary divisor multiplicities. Each summand is governed by the same orbit at every unramified place and every complex embedding. At trivial level this proves Conjectures 3.4.5 and 3.4.6 of Gaitsgory–Lafforgue–Raskin.
View manuscript on GitHub → Open PDF (raw) →M029
Temperedness at ramified places for globally generic exceptional groups
We prove the generalized Ramanujan conjecture for globally generic cuspidal automorphic representations of split connected adjoint exceptional groups over global function fields. Every local component is tempered, without restrictions on characteristic or ramification depth. The proof extends the unramified Ramanujan theorem of a companion paper to all ramified places.
View manuscript on GitHub → Open PDF (raw) →M030
The Restricted Geometric Langlands Equivalence in Positive Characteristic
We prove the `Q̄_ℓ` -linear restricted geometric Langlands equivalence for smooth projective connected curves and connected reductive groups in characteristic p > 0, with ℓ ≠ p, in two regimes. Over ` F _q`, we assume the four characteristic conditions of the restricted theory stated below. Over arbitrary algebraically closed fields, we assume these conditions and additionally that p is very good and does not divide the Weyl-group order.
View manuscript on GitHub → Open PDF (raw) →M031
Constructible tame Hecke eigensheaves in positive characteristic
We construct nonzero locally constructible perverse Hecke eigensheaves with Borel level at one marked point on a smooth projective curve of genus at least two over an algebraic closure of a finite field. The parameter is a Zariski-dense geometric ℓ-adic local system with tame regular-unipotent monodromy. The group is simple and simply connected and satisfies four explicit Lie-theoretic characteristic hypotheses.
View manuscript on GitHub → Open PDF (raw) →M032
Tame Hecke Eigensheaves with Several Marked Points
For SLn in characteristic p > n, we construct nonzero locally constructible perverse Hecke eigensheaves with Borel level at two or more marked points on a smooth projective curve of genus at least two over an algebraic closure of a finite field. The parameter is a Zariski-dense geometric ℓ-adic PGLn-local system with unipotent tame monodromy; its nilpotent logarithms may have any Jordan type, including zero.
View manuscript on GitHub → Open PDF (raw) →M033
Frobenius Structures on Tame Hecke Eigensheaves
We construct nonzero perverse Weil Hecke eigensheaves for SLn with Borel level at one marked point, after a finite extension of the field of constants. The parameter is a geometrically dense arithmetic PGLn-local system with tame regular-unipotent monodromy on a once-punctured curve of genus at least two over a finite field of characteristic p > n.
View manuscript on GitHub → Open PDF (raw) →Family 015 Torus-packet equidistribution in prime, quartic, and sextic degrees. 3 papers
Proves Haar equidistribution without escape of mass for complete volume-weighted torus packets from totally real fields: arbitrary lattices and prescribed local types in fixed prime degree at least five, arbitrary-order Picard packets in primitive quartic fields, and maximal-order ideal-class packets in primitive sextic fields. Here primitive means having no proper intermediate field; the relevant order or field discriminant tends to infinity. ([Lean](lean/docs/015.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M034
Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type
We prove the packet form of the higher-dimensional Duke equidistribution problem for totally real fields of any fixed prime degree at least five, allowing arbitrary local homothety types of full lattices. As the multiplier-order discriminant tends to infinity, the volume-weighted packet measures converge to Haar probability measure, with no escape of mass.
View manuscript on GitHub → Open PDF (raw) →M035
Equidistribution of primitive quartic torus packets for arbitrary orders
Let K range over totally real quartic fields with no proper intermediate field, and let ` O` be any order in K. We prove that the packets of periodic diagonal orbits attached to invertible ` O` -ideal classes, weighted by orbit volume, equidistribute with no escape of mass as `| Disc ( O)|` tends to infinity. The proof combines measure rigidity with a cubic-resolvent estimate that remains uniform at primes dividing the order index.
View manuscript on GitHub → Open PDF (raw) →M036
Equidistribution of Primitive Sextic Torus Packets
We prove that the complete ideal-class packets of maximal orders in totally real sextic fields with no proper intermediate fields become equidistributed, with no escape of mass, in the space of unimodular lattices as their field discriminants tend to infinity. The limit is Haar probability measure. Each packet includes all coordinate-sign translates and is weighted by diagonal-orbit volume.
View manuscript on GitHub → Open PDF (raw) →Family 016 Zilber–Pink in abelian varieties and the Siegel threefold. 4 papers
Proves the abelian Zilber–Pink conjecture over `Q̄`: every irreducible subvariety has finitely many maximal atypical subvarieties relative to its smallest containing torsion coset. It also proves the full curve case in the Siegel threefold ` A_2` for Hodge-generic curves defined over `Q̄`, without boundary or reduction assumptions.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M037
The abelian Zilber–Pink conjecture
We prove the abelian Zilber–Pink conjecture over `Q̄`. Every irreducible subvariety of an abelian variety has only finitely many maximal atypical subvarieties, where atypicality is measured inside its smallest containing torsion coset.
View manuscript on GitHub → Open PDF (raw) →M038
The E×CM component of Zilber–Pink for curves in A2
We prove the `E×CM` component of Zilber–Pink for Hodge-generic algebraic curves in ` A_2` over `Q̄`. Each such curve contains only finitely many points whose abelian surface is isogenous to a product of elliptic curves with at least one factor having complex multiplication.
View manuscript on GitHub → Open PDF (raw) →M039
Quaternionic division points on curves in the Siegel threefold
We prove the quaternionic-division component of Zilber–Pink for curves in ` A_2` over `Q̄`. A Hodge-generic algebraic curve contains only finitely many points whose full geometric rational endomorphism algebra is an indefinite quaternion division algebra over ℚ. No boundary or reduction hypothesis is required.
View manuscript on GitHub → Open PDF (raw) →M040
Elliptic squares and Zilber–Pink for curves in A2
We prove that every Hodge-generic algebraic curve in ` A_2` over `Q̄` contains only finitely many points whose abelian surface is isogenous to the square of an elliptic curve without complex multiplication (CM). Combining this result with the companion `E×CM` and quaternionic-division finiteness theorems, we prove the curve case of Zilber–Pink in ` A_2` over `Q̄`, without a boundary hypothesis.
View manuscript on GitHub → Open PDF (raw) →Family 017 The irrationality exponent of π is 2. 1 paper
Proves that the irrationality exponent of π is exactly 2: for every ε > 0 and all sufficiently large denominators q, every rational `p/q` satisfies `|π-p/q|≥ q⁻²⁻ε`. This also proves convergence of the Flint–Hills series `∑(n≥1)1/(n³sin² n)`, with angles in radians. ([Lean](lean/docs/017.md))
PQC lens: Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
Primary manuscript on GitHub → Family index on GitHub →M041
The irrationality exponent of pi is 2
We prove the conjecture that the irrationality exponent of π is 2. As a consequence, the classical Flint–Hills series `∑(n≥1)1/(n³sin² n)` converges, with angles in radians.
View manuscript on GitHub → Open PDF (raw) →Family 018 The Margulis–Platonov conjecture over global fields. 2 papers
Proves the Margulis–Platonov conjecture over every global field, including function fields of characteristic two. For an absolutely almost simple simply connected algebraic group G over k, every noncentral abstract normal subgroup of `G(k)` is the inverse image of an open normal subgroup in the finite product of its anisotropic nonarchimedean local groups.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M042
The Margulis–Platonov conjecture over global function fields
We prove the Margulis–Platonov conjecture over every global function field, including characteristic two. For an absolutely almost simple simply connected algebraic group, every noncentral abstract normal subgroup of its rational points is the inverse image of an open normal subgroup of the finite product of its anisotropic local groups.
View manuscript on GitHub → Open PDF (raw) →M043
The Margulis–Platonov conjecture over number fields
We prove the Margulis–Platonov conjecture over number fields. For an absolutely almost simple simply connected group, every noncentral abstract normal subgroup of its rational points is the inverse image of an open normal subgroup of the product of its anisotropic nonarchimedean local groups.
View manuscript on GitHub → Open PDF (raw) →Family 019 The local p-adic section conjecture and global consequences. 2 papers
Proves that rational points on every smooth proper geometrically connected curve of genus at least two over a finite extension of ℚp correspond bijectively to conjugacy classes of sections of its full arithmetic étale fundamental group. It also proves Grothendieck's section conjecture over ℚ for the modular curves `X_0(N)` and `X_1(N)` of genus at least two.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M044
Étale covers with a prescribed exterior sheet
Let X be a smooth proper hyperbolic curve over an algebraic closure of a p-adic local field. Given finitely many disjoint small open disks, we construct one connected finite étale cover with a sheet isomorphic to their entire exterior and with degree divisible by p on every connected component above each disk.
View manuscript on GitHub → Open PDF (raw) →M045
The p-adic section conjecture
We prove the local p-adic section conjecture for smooth proper geometrically connected curves of genus at least two, over every finite extension of ℚp. Combined with established finite-descent theorems, this also yields the global section conjecture for smooth proper geometrically connected curves of genus at least two over number fields when their finite-cover descent locus equals their rational points; this includes `X_0(N)` and `X_1(N)` of genus at least two over ℚ.
View manuscript on GitHub → Open PDF (raw) →Family 020 Squarefree quartics and power-free polynomial values. 1 paper
Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density. More generally, establishes the `(d-2)` -power-free density for irreducible integer polynomials of degrees four through eight under the necessary local condition; together with Browning's higher-degree theorem, this covers every d ≥ 4. ([Lean](lean/docs/020.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M046
Squarefree values of quartics and power-free values of polynomials
We prove that every irreducible integer quartic with no fixed prime-square divisor takes squarefree values with the predicted positive Euler-product density. More generally, we obtain the corresponding `(d-2)` -power-free density in degrees `4≤ d≤8`. The proof combines number-field factorization, determinant estimates with adaptive auxiliary primes, and explicit low-degree geometry. Together with Browning's theorem for higher degrees, this gives the `(d-2)` -power-free density for every d ≥ 4.
View manuscript on GitHub → Open PDF (raw) →Family 021 A quadratic bound for Jacobsthal’s function. 1 paper
Answers Jacobsthal's quadratic-bound question: every interval of `Ck²` consecutive integers contains an integer coprime to any prescribed positive integer with at most k distinct prime divisors, for an absolute constant C. The bound is uniform over prime sets and interval positions and removes the classical logarithmic loss. ([Lean](lean/docs/021.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M047
A quadratic bound for Jacobsthal's function
Let `h(k)` be the least integer such that every interval of `h(k)` consecutive integers contains an integer coprime to any prescribed positive integer having at most k distinct prime divisors. We prove `h(k) k²/(loglog(3k))²`, giving an affirmative answer to Jacobsthal's quadratic-bound question.
View manuscript on GitHub → Open PDF (raw) →Family 022 The weak inhomogeneous Duffin–Schaeffer conjecture. 1 paper
Proves that for every real shift γ and finite-valued `: N→[0,∞)`, divergence of `∑_qφ(q) (q)/q` implies ` |qx-γ | (q)` for infinitely many q, for almost every x. Here ϕ is Euler's totient and the norm is distance to the nearest integer. Numerators are unrestricted; no monotonicity or Diophantine condition on γ is needed.
PQC lens: Affects polynomial or integer factorization—core to RSA/DSA analysis and to comparing classical vs quantum factoring.
Primary manuscript on GitHub → Family index on GitHub →M048
The Weak Inhomogeneous Duffin–Schaeffer Conjecture
We prove the weak inhomogeneous Duffin–Schaeffer conjecture. For every fixed γ ∈ ℝ and finite-valued `: N→[0,∞)`, divergence of `∑(q≥1)φ(q) (q)/q` implies ` |qx-γ | (q)` for infinitely many q, for almost every x. Here ϕ is Euler's totient and ` |· |` is distance to the nearest integer. Numerators need not be reduced, and the shift satisfies no Diophantine restriction.
View manuscript on GitHub → Open PDF (raw) →Family 023 Patterson's first moment for cubic Gauss sums. 1 paper
Proves unconditionally the all-primary-prime form of Patterson's first-moment asymptotic: normalized cubic Gauss sums over primary Eisenstein primes of norm at most X, including both conjugates, have an explicit positive main term of order `X⁵/⁶/log X`. Every fixed nonzero prime-angle Fourier mode has smaller order. ([Lean](lean/docs/023.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M049
An unconditional first moment for cubic Gauss sums
We prove Patterson's first-moment conjecture for normalized cubic Gauss sums over all primary Eisenstein primes, unconditionally. The sharp-cutoff main term is `(6/5)c_*X⁵/⁶/log X`, where `c_*=(2π)²/³/(3 (2/3))`. For every fixed nonzero angular Fourier mode of the prime argument, we also prove cancellation at the first-moment scale.
View manuscript on GitHub → Open PDF (raw) →Family 024 An asymptotic formula for the number of totients. 1 paper
Gives an asymptotic equivalent for the number `V(x)` of distinct totient values up to x, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular, `V(cx)/V(x)→ c` for every fixed c > 0, answering Erdős and Hall's scaling question. ([Lean](lean/docs/024.md))
PQC lens: Affects polynomial or integer factorization—core to RSA/DSA analysis and to comparing classical vs quantum factoring.
Primary manuscript on GitHub → Family index on GitHub →M050
An asymptotic formula for the number of totients
Let `V(x)` count the distinct values of Euler's totient function up to x. We give an explicit asymptotic equivalent for `V(x)`. Its coefficient is a uniform limit of functions defined from finite arithmetic data. We also prove that `V(cx)/V(x)→ c` as `x→∞` for every fixed c > 0, answering a question of Erdős and Hall.
View manuscript on GitHub → Open PDF (raw) →Family 025 Short Egyptian fractions. 1 paper
Every rational `a/b` with `1≤ a b` is a sum of `O(loglog b)` distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős's conjecture on short Egyptian fractions. ([Lean](lean/docs/025.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M051
Short Egyptian fractions
We prove a conjecture of Erdős: for every sufficiently large integer b, every rational number `a/b` with `1≤ a b` is a sum of `O(loglog b)` distinct positive unit fractions, with an absolute implied constant. This order is best possible when the numerator varies. We also show that both the number of expansions of 1 with exactly k distinct terms and the least integer at least 2 that never occurs as a denominator in such an expansion grow doubly exponentially in k: their double logarithms have order k.
View manuscript on GitHub → Open PDF (raw) →Family 026 Positive lower density of large prime gaps. 1 paper
For every fixed C > 0, a positive proportion of consecutive prime gaps exceed `Clog p_n`, throughout every sufficiently large initial segment of the primes. The proportion may depend on C. Consequently, the indices where `p_n/n` increases have positive lower density, answering Erdős and Prachar. ([Lean](lean/docs/026.md))
PQC lens: Affects polynomial or integer factorization—core to RSA/DSA analysis and to comparing classical vs quantum factoring.
Primary manuscript on GitHub → Family index on GitHub →M052
Positive lower density of large prime gaps
For every fixed C > 0, we prove that a positive proportion of consecutive prime gaps exceed `Clog p`, where p is the smaller prime. The proportion is bounded below for every sufficiently large initial segment of the prime sequence, with a constant depending on C. It follows that the indices at which `p_n/n` increases have positive lower asymptotic density, answering a question of Erdős and Prachar.
View manuscript on GitHub → Open PDF (raw) →Family 027 Potential integral density on curve character varieties. 1 paper
Resolves the determinant-one curve case of Litt's integral-density question. For every smooth connected complex algebraic curve and every rank, integral points become Zariski dense in every component of its SLr character variety over the full ring of integers of one number field. Prescribed quasi-unipotent boundary conjugacy classes are allowed, including nonsemisimple classes.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M053
Integral points on character varieties of curves
We prove potential Zariski density of integral points on SLr-character varieties of smooth complex curves in every rank. The result allows prescribed quasi-unipotent boundary monodromy, including exact nonsemisimple conjugacy classes, and holds on every component over the full ring of integers of one finite extension. Here integrality is measured in the ambient character variety, while the exact boundary conditions are imposed on the complex representation.
View manuscript on GitHub → Open PDF (raw) →Family 028 Uniformly bounded components of Gaussian-prime graphs. 1 paper
Proves the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have uniformly bounded steps. More strongly, for every distance bound D, the graph joining Gaussian primes at distance at most D has uniformly bounded finite component sizes, depending only on D, including primes on the coordinate axes. ([Lean](lean/docs/028.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M054
Bounded-Step Walks on Gaussian Primes
We prove the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have bounded steps. More strongly, for each fixed finite step bound, the connected components of the Gaussian-prime graph have uniformly bounded size. This bound applies to every starting prime, including primes on the coordinate axes, and is nonexplicit. The proof constructs a finite periodic sieve obstruction using geometric sampling and information-theoretic estimates.
View manuscript on GitHub → Open PDF (raw) →Family 029 Primitive roots for every admissible integer base. 2 papers
Proves the infinitude assertion in Artin's primitive root conjecture for every integer a that is neither −1 nor a square. For each such base, at least `c_a x/(log x)²` primes in every sufficiently large interval `(x,2x)` have primitive root a, with `c_a 0`.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M055
Primitive roots for every admissible integer base
We prove the infinitude assertion in Artin's primitive root conjecture: for every integer a that is neither −1 nor a square, there are at least `c_a x/(log x)²` primes in `(x,2x)` with primitive root a, for some `c_a 0` and every sufficiently large x.
View manuscript on GitHub → Open PDF (raw) →M056
Simultaneous primitive roots: a conditional lower bound for prime bases
For every fixed finite set of distinct positive primes, we prove that at least `cx/(log x)²` primes in `(x,2x)` have every member of the set as a primitive root, for some c > 0 and all sufficiently large x. The result assumes four explicitly stated analytic and sieve inputs from the companion paper on primitive roots for admissible integer bases.
View manuscript on GitHub → Open PDF (raw) →Family 030 Modularity of elliptic curves over imaginary quadratic fields. 1 paper
Proves the modularity conjecture for elliptic curves over imaginary quadratic fields: every elliptic curve over every imaginary quadratic field is modular, with matching local parameters at every place.
PQC lens: Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
Primary manuscript on GitHub → Family index on GitHub →M057
Modularity of elliptic curves over imaginary quadratic fields
We prove the modularity conjecture for elliptic curves over imaginary quadratic fields: every such curve is modular, with matching local parameters at every place.
View manuscript on GitHub → Open PDF (raw) →Family 031 Uchida’s conjecture for open homomorphisms of Galois groups. 1 paper
Proves Uchida's conjecture: every continuous open homomorphism between Galois groups of possibly infinite solvably closed Galois extensions of number fields comes from a unique equivariant field embedding in the opposite direction. No restriction on the kernel or separate cyclotomic-compatibility assumption is needed.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M058
Open Homomorphisms of Global Solvably Closed Galois Groups
We prove Uchida's conjecture on open homomorphisms of Galois groups. Every continuous open homomorphism between Galois groups of solvably closed Galois extensions of number fields is induced by a unique equivariant field embedding in the opposite direction. No restriction on the kernel or additional compatibility hypothesis is required.
View manuscript on GitHub → Open PDF (raw) →Family 032 Hodge and Kuga–Satake results for all projective K3 surfaces. 8 papers
Proves the rational Hodge conjecture for every complex CM abelian variety, in every dimension and codimension. Through Milne's theorems, this also gives the Tate conjecture for all abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties in every characteristic. Companion results prove rational Hodge for arbitrary products of projective complex K3 surfaces and algebraicity of the Kuga–Satake correspondence for every such surface.
PQC lens: Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
Primary manuscript on GitHub → Family index on GitHub →M059
The rational Hodge conjecture for products of K3 surfaces
We prove the rational Hodge conjecture for every finite product of projective complex K3 surfaces: every rational Hodge class is algebraic. The factors may be distinct or repeated, with no restrictions on their Picard numbers, periods, or endomorphism fields.
View manuscript on GitHub → Open PDF (raw) →M060
Algebraicity of Kuga–Satake Correspondences for K3 Surfaces
We prove that the Kuga–Satake correspondence is algebraic for every smooth projective complex K3 surface. More precisely, the prescribed embedding of its transcendental cohomology into the cohomology of its Kuga–Satake abelian variety is induced by a rational algebraic cycle, with the fixed normalization and full even-Clifford target. The result also holds for isogenous Kuga–Satake models with the transported embedding.
View manuscript on GitHub → Open PDF (raw) →M061
The rational Hodge conjecture for CM abelian varieties
We prove the rational Hodge conjecture for complex abelian varieties with complex multiplication: every rational Hodge class on such a variety is a rational linear combination of algebraic cycle classes. As consequences, we obtain the generalized Hodge conjecture for CM abelian varieties, the Tate conjecture for abelian varieties over finite fields, and the Hodge standard conjecture for abelian varieties in arbitrary characteristic.
View manuscript on GitHub → Open PDF (raw) →M062
Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus
We prove the rational Hodge and generalized Hodge conjectures in every cohomological degree of every self-power of a projective complex K3 surface whose transcendental quadratic space, with its cup-product form, admits a rational isometric embedding in ` U( Q) oplus² -1 rangle⁴`.
View manuscript on GitHub → Open PDF (raw) →M063
Weil classes and Hodge classes on abelian powers
We prove the rational Hodge conjecture in every codimension on every self-power of a complex abelian sixfold with an imaginary-quadratic action and a compatible polarization whose rational homological Hermitian form is hyperbolic of signature `(3,3)`. We also prove it in every codimension on every self-power of a complex abelian variety of dimension at most five admitting an imaginary-quadratic action. Both results include nonsimple varieties and special periods with additional endomorphisms.
View manuscript on GitHub → Open PDF (raw) →M064
Abelian covers, Gale correspondences, and the Hodge conjecture for powers
We prove the rational Hodge conjecture on every self-power of the Jacobian at each tensor Hodge-generic point of the full marked variation of a connected abelian cover of curves. This holds in every base genus and for every compatible branching pattern. For any CM abelian variety, the conclusion also holds for every self-power of its product with the Jacobian, on the same Hodge-generic locus.
View manuscript on GitHub → Open PDF (raw) →M065
Algebraicity of Weil classes on split abelian eightfolds
We prove that every rational Weil class on a split abelian eightfold of Weil type is algebraic. The result holds for every imaginary quadratic field, every compatible polarization type, and every member of the split family, including those with additional endomorphisms. Thus the full two-dimensional rational Weil space in codimension four is generated by algebraic cycle classes.
View manuscript on GitHub → Open PDF (raw) →M066
A Conditional Reduction for Algebraic Kuga–Satake Correspondences
For a polarized K3 surface whose primitive cohomology has full orthogonal Hodge group, one algebraic correspondence inducing a nonzero map from that cohomology to the second cohomology of an abelian variety suffices to recover the prescribed full Kuga–Satake correspondence. We give an equivalent condition using holomorphic one-forms on a generically finite surface cover.
View manuscript on GitHub → Open PDF (raw) →Family 033 Iitaka subadditivity, variation, and logarithmic additivity. 5 papers
Proves Campana's orbifold Iitaka subadditivity conjecture for smooth Fujiki-class- ` C` manifolds with rational simple-normal-crossing boundaries. For projective fibrations `f:U→ V` of smooth complex quasi-projective varieties with connected fibers, general fiber F, and ` (V)≥0`, proves Popa's inequality ` (U)≥ (F)+max (V), Var (f) `, where variation measures the whole geometric generic fiber. ([Lean](lean/docs/033.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M067
Orbifold and logarithmic Iitaka subadditivity
We prove Campana's orbifold Iitaka subadditivity conjecture for rational simple normal crossing boundaries on compact manifolds in Fujiki class ` C`, including coefficient one. Ordinary and logarithmic subadditivity follow.
View manuscript on GitHub → Open PDF (raw) →M068
Logarithmic Kodaira dimension and whole-fiber variation
We prove the logarithmic Iitaka–Viehweg inequality for projective surjective morphisms with connected fibers between smooth complex quasi-projective varieties whose base has nonnegative logarithmic Kodaira dimension. The variation measures the birational field of definition of the whole geometric generic fiber. This resolves Popa's logarithmic variation conjecture positively.
View manuscript on GitHub → Open PDF (raw) →M069
The reverse logarithmic Kodaira inequality and additivity
We prove the reverse logarithmic Kodaira inequality for a surjective connected-fiber morphism `f:(X,E)→(Y,D)` of smooth projective reduced simple-normal-crossing pairs, with ` Supp (f*D) Supp E`, such that X and every boundary stratum are smooth over `Y Supp D`. Together with logarithmic subadditivity, the inequality gives additivity, including both negative-infinity cases. This resolves Popa's logarithmic additivity conjecture positively in the projective reduced-SNC, stratum-smooth setting.
View manuscript on GitHub → Open PDF (raw) →M070
Projective Hodge lines and ordinary Iitaka subadditivity
We prove the ordinary Iitaka subadditivity conjecture for surjective projective morphisms with connected fibers between smooth connected projective varieties over algebraically closed fields of characteristic zero. If F is the geometric generic fiber of `f:X→ Z`, then ` (X)≥ (F)+ (Z)`.
View manuscript on GitHub → Open PDF (raw) →M071
B-semiampleness for compact log-smooth Kähler fibrations
We prove the compact log-smooth Kähler case of b-semiampleness. Let `f:Y→ X` be a surjective holomorphic map with connected fibers between smooth compact connected Kähler manifolds, and let Δ be an effective rational divisor with simple normal crossing support and coefficients in `[0,1]`, with `K_Y+ ( Q)f*L` for `L Pic (X)( Q)`.
View manuscript on GitHub → Open PDF (raw) →Family 034 Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity. 14 papers
Using logarithmic Iitaka subadditivity, proves log abundance in every dimension for normal compact Kähler log canonical pairs with effective rational boundary: an analytically nef ℚ-Cartier adjoint is semiample. It also proves projective log abundance over every algebraically closed field of characteristic zero and the effective Iitaka fibration conjecture for smooth projective varieties of nonnegative Kodaira dimension in that setting.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M072
Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity
Assume logarithmic Iitaka subadditivity for surjective morphisms with connected fibers between smooth projective complex varieties with compatible reduced simple normal crossing boundaries. We prove log abundance for normal irreducible compact Kähler spaces in every dimension: for a log canonical pair `(X, )` with effective rational boundary and `K_X+ ` ℚ-Cartier, analytic nefness of `K_X+ ` implies semiampleness.
View manuscript on GitHub → Open PDF (raw) →M073
Uniform indices for semi-log-canonical log Calabi–Yau pairs
We prove a uniform index theorem for connected projective semi-log-canonical log Calabi–Yau pairs in every fixed dimension at least four over an algebraically closed field of characteristic zero. For boundary coefficients in a fixed finite rational set, a single multiple of the log canonical divisor is Cartier and linearly trivial. The multiple depends only on the dimension and coefficient set, not on the number of irreducible components.
View manuscript on GitHub → Open PDF (raw) →M074
Conditional good minimal models for compact Kähler fourfolds
Assuming orbifold Iitaka subadditivity, the specified pseudo-effective fourfold minimal model program, and abundance for nef fourfold adjoints of nonnegative Kodaira dimension, we prove the existence of good minimal models for globally strongly ℚ-factorial compact Kähler klt fourfold pairs with effective rational boundary and analytically pseudo-effective actual ℚ-Cartier adjoint. The additional step is nonvanishing.
View manuscript on GitHub → Open PDF (raw) →M075
Log abundance in characteristic zero
We prove the rational-boundary log abundance conjecture in every dimension over algebraically closed fields of characteristic zero: every nef ℚ-Cartier log canonical divisor on a projective log canonical pair with effective rational boundary is semiample. Over ℂ, the proof also establishes canonical nonvanishing for smooth projective varieties in every dimension.
View manuscript on GitHub → Open PDF (raw) →M076
Minimal metrics and interior injectivity for nef adjoints
Let `(H, )` be a projective complex klt pair with effective rational boundary and nef ℚ-Cartier adjoint. On any projective log resolution, minimal semipositive metrics on the pulled-back adjoint exist and have zero Lelong numbers everywhere. On smooth projective complex varieties, we also prove an H1-injectivity theorem for rational interior boundaries with simple-normal-crossing support when both endpoint bundles carry zero-Lelong semipositive metrics.
View manuscript on GitHub → Open PDF (raw) →M077
Fourfold nonvanishing by minimal metrics and moving jets
We prove canonical nonvanishing for smooth connected projective complex fourfolds: if KX is pseudo-effective, then `H⁰(X,mK_X)≠0` for some positive integer m.
View manuscript on GitHub → Open PDF (raw) →M078
Lifting sections from the reduced support of an adjoint
For a projective ℚ-factorial dlt pair with effective rational boundary over an algebraically closed field of characteristic zero, we prove that the adjoint has positive Iitaka dimension whenever a nonzero effective Cartier multiple is supported on the coefficient-one boundary and restricts to a semiample line bundle on its whole reduced support. This gives log abundance after nonvanishing in dimension at most four over ℂ.
View manuscript on GitHub → Open PDF (raw) →M079
Schnell fiber spaces and good canonical models
We prove the Campana–Peternell inequality ` (X)≥ (D)` for a smooth connected projective complex variety X and an effective Cartier divisor D whenever `m_0K_X-D` is pseudo-effective for some positive integer m0. For an algebraic fiber space `f X→ Y` between smooth connected projective complex varieties, the hypothesis that `m_0K_X-f*H` is pseudo-effective with H ample Cartier gives ` (X)= (F)+dim Y` for a very general smooth fiber F, as well as nonzero sections of `mK_X-f*H` for all sufficiently large divisibl…
View manuscript on GitHub → Open PDF (raw) →M080
Uniform log Iitaka fibrations and bounded moduli denominators
For normal projective log canonical pairs over algebraically closed fields of characteristic zero, of fixed dimension d ≥ 5 and with effective boundary coefficients in a fixed finite rational set, we prove that one complete rounded pluricanonical system generates the full Iitaka field whenever the ℚ-Cartier log canonical divisor has nonnegative Kodaira dimension. Its degree depends only on the dimension and coefficient set.
View manuscript on GitHub → Open PDF (raw) →M081
Uniform Pluricanonical Iitaka Fibrations
We prove the effective Iitaka fibration conjecture in characteristic zero. For each dimension, one pluricanonical degree defines the Iitaka fibration of every smooth integral projective variety of that dimension and nonnegative Kodaira dimension over an algebraically closed field. The associated sections generate the full Iitaka function field.
View manuscript on GitHub → Open PDF (raw) →M082
Relative denominators and effective systems for log Calabi-Yau fibrations
We prove uniform denominator and effective-system bounds for log Calabi-Yau fibrations from projective log canonical complex pairs of dimension at most four onto positive-dimensional bases, with boundary coefficients in a fixed finite rational set. The bounds give a uniform trivializing degree and a uniform b-Cartier multiple of the moduli b-divisor. When the base divisor is big, a uniform complete rounded adjoint system has section ratios generating the full base function field.
View manuscript on GitHub → Open PDF (raw) →M083
Arithmetic Stein-degree bounds for log Calabi–Yau pairs
Fix d ≥ 1 and t > 0. Let `(X,B)` be an ordinary projective log canonical ℚ-pair of dimension d over a characteristic-zero field k, with X normal and integral, `H⁰(X, O_X)=k`, B effective, and `K_X+B ( Q)0`. We prove that every prime component S of B with coefficient at least t satisfies `[k_S:k]≤ N(d,t)`, where kS is the relative algebraic closure of k in `k(S)`.
View manuscript on GitHub → Open PDF (raw) →M084
Uniform effective log Iitaka fibrations for fourfolds
We prove the finite-rational-coefficient case of the effective log Iitaka conjecture in dimension four. For normal projective log canonical complex fourfolds with boundary coefficients in a fixed finite rational set and pseudo-effective rational Cartier adjoint, one uniform degree makes the complete rounded reflexive system nonempty and its section ratios generate the full Iitaka field. We also prove a uniform canonical index bound for projective klt complex fourfolds with rationally trivial canonical divisor.
View manuscript on GitHub → Open PDF (raw) →M085
Abundance after nonvanishing for compact Kähler fourfolds
We prove semiampleness of the actual ℚ-Cartier adjoint `K_X+ ` of a normal connected compact Kähler klt fourfold whenever it is analytically nef and some positive Cartier multiple has a nonzero section. The boundary is effective and rational; the fourfold need not be projective or ℚ-factorial. In Iitaka dimension zero, a positive Cartier multiple is the trivial holomorphic line bundle.
View manuscript on GitHub → Open PDF (raw) →Family 035 Log-canonical threefold abundance in numerical dimension one. 2 papers
Proves log abundance for projective log canonical threefold pairs over algebraically closed fields of characteristic p > 3 when the effective boundary is rational and the ℚ-Cartier adjoint is nef of numerical dimension one. The adjoint is semiample, without requiring the original variety to be terminal or ℚ-factorial.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M086
Log abundance in numerical dimension one for threefolds in positive characteristic
We prove the numerical-dimension-one case of log abundance for threefolds over algebraically closed fields of characteristic p > 3. If `(X,B)` is a projective log canonical threefold pair with effective rational boundary, and `K_X+B` is ℚ-Cartier, nef, and of numerical dimension one, then `K_X+B` is semiample. Neither terminality nor ℚ-factoriality of X is required.
View manuscript on GitHub → Open PDF (raw) →M087
Abundance in numerical dimension one for terminal threefolds in positive characteristic
Let X be a projective ℚ-factorial terminal threefold over an algebraically closed field of characteristic p > 3. We prove the numerical-dimension-one case of abundance: if KX is nef with `ν(K_X)=1`, then KX is semiample and ` (X,K_X)=1`.
View manuscript on GitHub → Open PDF (raw) →Family 036 Numerical semiampleness and generalized minimal models. 4 papers
Proves numerical semiampleness for nef adjoints `K_X+B+M` with `K_X+B` pseudo-effective and M nef rational, for projective klt rational pairs over algebraically closed characteristic-zero fields and smooth compact Kähler rational klt simple-normal-crossing pairs, using Bott–Chern cohomology in the latter case. Separately, projective generalized log canonical rational pairs over such fields admit minimal models for pseudo-effective adjoints and Mori fiber spaces otherwise, with nef b-data fixed.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M088
Numerical semiampleness of nef adjoint classes on compact Kähler manifolds
Let X be a smooth connected compact Kähler manifold, let B be an effective rational simple normal crossing divisor with coefficients less than one, and let M be a nef rational holomorphic line bundle on X. If `K_X+B` is pseudo-effective and `K_X+B+M` is nef, we prove that its first Chern class in real Bott–Chern cohomology is represented by a semiample rational line bundle. This numerical statement allows a flat change of line bundle; it does not assert semiampleness of the original adjoint.
View manuscript on GitHub → Open PDF (raw) →M089
Numerical Semiampleness of Nef Adjoint Divisors
We prove the Generalised Abundance Conjecture: if `(X,B)` is a projective klt ℚ-pair over an algebraically closed field of characteristic zero, `K_X+B` is pseudo-effective, M is a nef ℚ-Cartier divisor on X, and `K_X+B+M` is nef, then `K_X+B+M` is numerically equivalent to a semiample ℚ-Cartier divisor on X.
View manuscript on GitHub → Open PDF (raw) →M090
Minimal models and Mori fibre spaces for generalized log canonical Q-pairs
We resolve the existence form of the minimal-model conjecture for projective generalized log canonical ℚ-pairs over algebraically closed fields of characteristic zero. Such a pair admits a minimal model when its adjoint divisor is pseudo-effective, and a Mori fibre space otherwise, while keeping the nef ℚ-Cartier b-divisor fixed. Taking the nef part to be zero gives the corresponding result for ordinary log canonical ℚ-pairs.
View manuscript on GitHub → Open PDF (raw) →M091
Minimal models in numerical dimension one
We resolve the numerical-dimension-one case of the minimal-model conjecture for smooth connected complex projective varieties of dimension at least three. If KX is pseudo-effective and ` kappa_σ(X,K_X)=1`, with κσ defined by section growth with a fixed ample twist, then X admits a projective ℚ-factorial terminal minimal model.
View manuscript on GitHub → Open PDF (raw) →Family 037 The ordinary-double-point volume gap. 2 papers
Proves the ordinary-double-point volume-gap conjecture: every singular complex algebraic klt germ of dimension n ≥ 2, with zero boundary, has normalized volume at most `2(n-1)n`. Equality holds precisely for an analytic ordinary double point.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M092
The ordinary-double-point gap in every dimension
We prove the ordinary-double-point gap conjecture for boundary-zero complex algebraic klt germs in every dimension: a singular n-dimensional germ has normalized volume at most `2(n-1)n`, with equality precisely at an analytic ordinary double point.
View manuscript on GitHub → Open PDF (raw) →M093
The normalized-volume gap in dimension four
We prove that every singular complex algebraic klt fourfold germ with zero boundary has normalized volume at most 162, with equality precisely for an analytic ordinary double point. This resolves the ordinary-double-point volume-gap conjecture in dimension four.
View manuscript on GitHub → Open PDF (raw) →Family 038 Fujita’s freeness conjecture. 1 paper
Proves Fujita's freeness conjecture at its sharp bound in every dimension: for a smooth projective complex variety X of dimension n and an ample line bundle L, the adjoint `K_X+mL` is globally generated for every integer `m≥ n+1`.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M094
Fujita's freeness conjecture
We prove Fujita's freeness conjecture. If X is a smooth complex projective variety of dimension n and L is an ample line bundle, then `K_X+mL` is globally generated for every integer `m≥ n+1`.
View manuscript on GitHub → Open PDF (raw) →Family 039 Nagata’s conjecture and maximal Seshadri constants. 4 papers
Proves Nagata's strict inequality `∑_i m_i d r` for every nonzero effective plane curve of degree d through r ≥ 10 very general complex points, with arbitrary multiplicities mi. It also proves maximal multipoint Seshadri constants `(Ln/r)¹/n` for every smooth polarized projective variety of dimension n ≥ 2 and all sufficiently large r: at very general points over ℂ, and at the geometric generic tuple over any algebraically closed field of positive characteristic. ([Lean](lean/docs/039.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M095
Nagata's conjecture for plane curves
We prove that a nonzero effective plane curve of degree d at r ≥ 10 very general complex points has total multiplicity strictly less than `d r`. The inequality holds simultaneously for all curves, including reducible and nonreduced curves, and establishes Nagata's conjecture in its strict, nonhomogeneous form.
View manuscript on GitHub → Open PDF (raw) →M096
Maximal Seshadri constants on arbitrary polarized surfaces
We prove that, for every smooth integral complex projective surface S and every ample line bundle L, the multipoint Seshadri constant at r very general points equals `√(L²/r)` for every sufficiently large integer r. This resolves positively the qualitative Nagata–Biran conjecture for surfaces.
View manuscript on GitHub → Open PDF (raw) →M097
Maximal Multipoint Seshadri Constants in Higher Dimensions
Let L be an ample line bundle on a smooth integral complex projective variety X of dimension n ≥ 3. We prove that there is a threshold `r_0=r_0(X,L)` such that for every integer `r≥ r_0`, the ordinary multipoint Seshadri constant at r very general points equals the volume bound `(Ln/r)¹/n`. This establishes the qualitative Nagata–Biran–Szemberg assertion in these dimensions.
View manuscript on GitHub → Open PDF (raw) →M098
Maximal multipoint Seshadri constants in positive characteristic
Let L be an ample line bundle on a smooth integral projective variety X over an algebraically closed field of positive characteristic, with `dim X=n≥3`. We prove that there is a threshold `r_0=r_0(X,L)` such that for every integer `r≥ r_0`, the ordinary multipoint Seshadri constant at the geometric generic tuple of r points equals the volume bound `(Ln/r)¹/n`. The same conclusion holds in dimension two. This establishes the positive-characteristic form of the qualitative Nagata–Biran–Szemberg assertion at geometric generic tuples.
View manuscript on GitHub → Open PDF (raw) →Family 040 Bloch’s conjecture for complex surfaces. 1 paper
Proves Bloch's conjecture: for every smooth connected projective complex surface S with `p_g(S)=0`, the Albanese map `CH_0(S)⁰→Alb(S)( C)` on integral degree-zero zero-cycles is an isomorphism. This combines the new `p_g=q=0` theorem with the classical theorem of Bloch, Kas, and Lieberman.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M099
Bloch’s conjecture for surfaces with p_g=0
We prove Bloch's conjecture for smooth connected projective complex surfaces with `p_g=0`: the Albanese homomorphism on integral degree-zero zero-cycles is an isomorphism.
View manuscript on GitHub → Open PDF (raw) →Family 041 Hyperkähler SYZ and projective-space bases. 2 papers
Proves the strong hyperkähler SYZ conjecture: every holomorphic line bundle with nonzero nef isotropic first Chern class on a compact irreducible holomorphic symplectic Kähler manifold is semiample. It also proves that every projective Lagrangian fibration with normal projective base has projective space as its base, in every dimension and deformation type.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M100
Projective-space bases of Lagrangian fibrations
We prove that the normal projective base of a projective Lagrangian fibration from a compact irreducible holomorphic symplectic Kähler manifold is projective space. This resolves the projective-space base conjecture for such fibrations in every dimension and deformation type.
View manuscript on GitHub → Open PDF (raw) →M101
The strong hyperkähler SYZ conjecture
We prove the strong hyperkähler SYZ conjecture: every holomorphic line bundle with nonzero nef isotropic first Chern class on a compact irreducible holomorphic symplectic Kähler manifold is semiample.
View manuscript on GitHub → Open PDF (raw) →Family 042 Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII. 1 paper
Proves that every complex K3 surface is Oka, including nonprojective surfaces. More generally, every connected minimal compact complex surface of Kodaira dimension zero is Oka; a connected minimal compact complex surface of class VII is Oka exactly when it is a Hopf or Enoki surface.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M102
Every complex K3 surface is Oka
We prove that every complex K3 surface X is Oka, resolving the K3 Oka conjecture. Equivalently, for every m ≥ 1, every holomorphic map to X from a neighborhood of a compact convex set in ℂm can be approximated uniformly on that set by entire maps ` Cm→ X`.
View manuscript on GitHub → Open PDF (raw) →Family 043 P = W for fixed-determinant SLn moduli spaces. 1 paper
Proves `P_k=W(2k)=W(2k+1)` on the full rational cohomology of smooth coprime fixed-determinant, trace-free Higgs moduli spaces and their character varieties for composite ranks over smooth projective complex curves of genus at least two. This includes variant cohomology and, together with the known prime-rank theorems, establishes the fixed-determinant P = W conjecture in every coprime rank.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M103
P=W in composite rank for fixed determinant
We prove the P = W conjecture on the full rational cohomology of fixed-determinant, trace-free Higgs moduli spaces in composite rank and coprime degree, for smooth projective complex curves of genus at least two. Together with the established prime-rank cases, this gives the equality in every coprime rank.
View manuscript on GitHub → Open PDF (raw) →Family 044 The equivariant cohomological Hikita conjecture. 1 paper
Proves the equivariant cohomological Hikita correspondence for every finite quiver, including loops and multiple arrows, with arbitrary dimension and framing vectors and commuting flavor torus. When every semistable point is stable and the gauge action is free, the equivariant cohomology of the Nakajima variety is canonically the coordinate ring of the scheme-theoretic cocharacter-fixed locus of its flavor-deformed Coulomb branch.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M104
The equivariant cohomological Hikita conjecture for arbitrary quivers
We prove the equivariant cohomological Hikita conjecture for arbitrary finite quivers, including loops and multiple arrows. For any dimension and framing vectors, any commuting flavor torus, and a stability character whose semistable locus is stable and has free gauge action, the equivariant cohomology of the Nakajima variety is canonically isomorphic to the coordinate ring of the scheme-theoretic fixed locus of the stability cocharacter on the flavor-deformed Coulomb branch.
View manuscript on GitHub → Open PDF (raw) →Family 046 Shafarevich counterexamples in dimension two and with large fundamental group. 2 papers
Constructs a smooth projective complex fourfold with large fundamental group whose universal cover contains no positive-dimensional compact analytic subvariety but is neither Stein nor holomorphically convex. A separate smooth projective complex surface already disproves unrestricted Shafarevich holomorphic convexity in dimension two.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M105
A projective fourfold with large fundamental group and non-Stein universal cover
We construct a smooth projective complex fourfold with large fundamental group whose universal cover is not Stein. The universal cover contains no positive-dimensional compact complex-analytic subvariety, so this disproves Shafarevich's holomorphic-convexity conjecture even under the large-fundamental-group hypothesis.
View manuscript on GitHub → Open PDF (raw) →M106
A surface counterexample to Shafarevich holomorphic convexity
We construct a smooth connected projective complex surface whose universal cover is not holomorphically convex. This disproves the Shafarevich conjecture on holomorphic convexity in complex dimension two.
View manuscript on GitHub → Open PDF (raw) →Family 047 Zariski cancellation and affine fibrations over the complex numbers. 1 paper
Constructs an integral complex affine fourfold `X A⁴` with `X× A¹ A⁵`, disproving affine-space cancellation over ℂ in dimension four. It also disproves the Dolgachev–Weisfeiler affine-fibration conjecture: smooth surjections `X→ A¹` and ` A⁵→ A²` have every residue-field fiber isomorphic to affine three-space but are not Zariski-locally trivial. ([Lean](lean/docs/047.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M107
An explicit failure of complex affine-space cancellation
We construct an explicit integral complex affine fourfold X with `X× A¹ A⁵` but `X A⁴`. This gives a negative answer to Zariski's affine-space cancellation problem over ℂ in dimension four. The same construction disproves the Stable Coordinate Conjecture in ambient dimension five and yields smooth 𝔸3-fibrations over 𝔸1 and 𝔸2 that are not Zariski-locally trivial. These fibrations disprove the Dolgachev–Weisfeiler affine-fibration conjecture over these bases.
View manuscript on GitHub → Open PDF (raw) →Family 048 A characteristic-zero counterexample to Lipman–Zariski. 1 paper
Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman–Zariski conjecture.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M108
A singular normal affine surface with free tangent sheaf
We construct a singular normal affine complex surface whose tangent sheaf is free of rank two. This disproves the Lipman–Zariski conjecture in characteristic zero.
View manuscript on GitHub → Open PDF (raw) →Family 049 A stable-coordinate counterexample in four variables. 2 papers
Constructs a polynomial in four complex variables that is not a coordinate but becomes one after adjoining a single variable, disproving the Stable Coordinate conjecture in four variables. Every fiber is affine three-space, yet none of its embeddings is rectifiable, also disproving the Abhyankar–Sathaye conjecture even when all fibers are affine spaces. ([Lean](lean/docs/049.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M109
A stable coordinate that is not a coordinate in four variables
We construct an explicit degree-five polynomial over ` C` that is not a coordinate in four variables but becomes one after adjoining a single variable. This gives a counterexample to the stable coordinate conjecture in four variables. Every fiber is isomorphic to affine three-space, yet its embedding in affine four-space is not rectifiable. Thus the example also disproves the Abhyankar–Sathaye embedding conjecture in ambient dimension four.
View manuscript on GitHub → Open PDF (raw) →M110
An explicit noncoordinate polynomial with affine three-space zero fibre
We construct an explicit counterexample to the Abhyankar–Sathaye conjecture: a noncoordinate polynomial in four complex variables whose zero fibre is affine three-space. Adjoining variables gives counterexamples in every ambient dimension at least four.
View manuscript on GitHub → Open PDF (raw) →Family 050 A counterexample to Griffiths’ positivity conjecture. 1 paper
Constructs ample rank-two bundles on ` P¹× P¹` with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface. ([Lean](lean/docs/050.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M111
Ample rank-two bundles on the quadric surface without Griffiths-positive metrics
We give counterexamples to Griffiths' conjecture in rank two on the quadric surface ` P¹× P¹`. We construct an explicit bundle G whose coordinatewise power pullbacks, tensored with ` O(1,1)`, are ample for every positive power, but admit no smooth strictly Griffiths-positive Hermitian metric for all sufficiently large powers.
View manuscript on GitHub → Open PDF (raw) →Family 051 Kobayashi’s canonical-ampleness conjecture. 1 paper
Every compact connected Kähler manifold of positive complex dimension with no nonconstant entire curve has ample canonical bundle and is therefore projective. This proves Kobayashi's canonical-ampleness conjecture in the smooth compact Kähler setting.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M112
Canonical ampleness of compact hyperbolic Kähler manifolds
We prove that every compact connected Kähler manifold of positive complex dimension containing no nonconstant entire curve has ample canonical bundle and is projective. This resolves positively Kobayashi's canonical-ampleness conjecture in the smooth compact Kähler category.
View manuscript on GitHub → Open PDF (raw) →Family 052 Tangent splittings and product decompositions. 2 papers
A splitting of the tangent bundle of a compact Kähler manifold into two integrable holomorphic subbundles induces a compatible product decomposition of its universal cover, proving the two-summand form of Beauville's splitting conjecture. On smooth rationally connected projective manifolds, both summands are automatically integrable, establishing Höring's conjecture and the corresponding product decomposition. ([Lean](lean/docs/052.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M113
Universal-cover splitting for compact Kähler manifolds
We prove the two-summand form of Beauville's compatible splitting conjecture. If the tangent bundle of a compact connected Kähler manifold decomposes into two integrable holomorphic subbundles of positive rank, then its ordinary universal cover admits a product decomposition whose factor tangent bundles are the lifted specified summands.
View manuscript on GitHub → Open PDF (raw) →M114
Integrability of split tangent bundles on rationally connected manifolds
We prove that both summands of every specified holomorphic splitting of the tangent bundle of a smooth rationally connected projective complex manifold into two positive-rank subbundles are integrable. This proves Höring's conjecture on rationally connected projective manifolds. Höring's product theorem then gives a product decomposition compatible with the specified splitting.
View manuscript on GitHub → Open PDF (raw) →Family 053 A counterexample to Pixton completeness in Chow. 1 paper
Constructs a tautological relation on a moduli space of stable pointed curves that vanishes in rational Chow, hence in rational cohomology, but lies outside Pixton's original relation span. This disproves the Chow and rational-cohomological forms of his original completeness conjecture.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M115
A high-arity counterexample to Pixton completeness in Chow
We disprove the Chow and rational-cohomological forms of Pixton's completeness conjecture for his original relation system. We construct a formal tautological class outside the original Pixton relation span whose image is zero in the Chow ring and in rational cohomology. The example has genus `10⁶⁰` and `3 10⁶⁰ 3` markings.
View manuscript on GitHub → Open PDF (raw) →Family 054 Irrational cubic fourfolds with Hodge-theoretic and categorical K3 associations. 1 paper
For every sufficiently large admissible Hassett discriminant, a very general smooth complex cubic fourfold is irrational despite having both an untwisted geometric K3 category and an integral Hodge-theoretic K3 association. This disproves Kuznetsov's rationality conjecture and the sufficiency of the associated-K3 criterion for rationality; the discriminant threshold is ineffective.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M116
Irrational cubic fourfolds with geometric K3 categories
For every sufficiently large admissible Hassett discriminant, we prove that a very general cubic fourfold of that discriminant is irrational, although its Kuznetsov component is equivalent to the ordinary derived category of a projective K3 surface. The discriminant threshold is ineffective. This disproves Kuznetsov's rationality conjecture. The same cubics have associated untwisted polarized K3 surfaces in the Hodge-theoretic sense, so they also disprove the sufficiency direction of the associated-K3 rationality prediction.
View manuscript on GitHub → Open PDF (raw) →Family 055 Gepner symmetry and large-volume stability on threefolds. 2 papers
Proves Toda's Gepner conjecture for every smooth complex quintic threefold, constructing a numerical Bridgeland stability condition with the prescribed phase shift 2/5. Also constructs numerical Bridgeland stability conditions at every sufficiently large volume on all smooth projective complex threefolds with trivial canonical bundle, with the exact ordinary and square-root-Todd central charges.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M117
A Gepner stability condition on every smooth quintic threefold
We prove Toda's normalized quintic Gepner conjecture. On every smooth complex quintic threefold, we construct a numerical Bridgeland stability condition for which tensoring by the hyperplane bundle, followed by the spherical twist at the structure sheaf, increases phase by 2/5.
View manuscript on GitHub → Open PDF (raw) →M118
Prescribed large-volume charges on threefolds with trivial canonical bundle
Let X be a smooth projective complex threefold with trivial canonical bundle. We construct numerical Bridgeland stability conditions with the exact ordinary and square-root-Todd central charges at every sufficiently large volume. One volume threshold works on an open set of real twists and ample directions. The resulting stability conditions have the support property on the full numerical Grothendieck group and stable point sheaves.
View manuscript on GitHub → Open PDF (raw) →Family 056 Termination of projective and Kähler fourfold minimal model programs. 5 papers
Proves termination of every existing generalized log canonical flip sequence on globally Weil ℚ-factorial compact Kähler fourfolds, with rational boundary, fixed rational analytically nef b-data, and projective small flip diagrams with the prescribed ample signs. Also proves termination of arbitrary permitted minimal model programs for projective log canonical fourfolds with rational boundary in characteristic zero.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M119
Termination of generalized log canonical flips on compact Kähler fourfolds
Every sequence of generalized log canonical flips on a normal irreducible globally Weil ℚ-factorial compact Kähler fourfold terminates, provided the flips are projective small diagrams with the stated opposite ample signs. The boundary is rational, and the nef b-divisor is fixed and represented by an analytically nef ℚ-Cartier divisor on a projective modification. No scaling rule or pseudo-effectivity assumption is required. The theorem concerns existing flip sequences; it does not assert the existence of all contractions or flips.
View manuscript on GitHub → Open PDF (raw) →M120
Termination of generalized-canonical flips on compact Kähler fourfolds
Every sequence of generalized-canonical flips on compact Kähler fourfolds with rational boundary and fixed rational analytically nef b-divisor terminates when the small morphisms are projective. We prove this for normal globally Weil ℚ-factorial models with boundary coefficients less than one, allowing exceptional log discrepancies equal to one. No scaling rule or pseudo-effectivity assumption is required. The theorem applies to existing projective small diagrams with the specified opposite ample signs.
View manuscript on GitHub → Open PDF (raw) →M121
Termination for projective log canonical fourfolds with rational boundary
We prove that every permitted minimal model program for a projective log canonical fourfold with rational boundary over an algebraically closed field of characteristic zero terminates. The result allows arbitrary negative extremal rays and mixed birational steps on the given models, without pseudo-effectivity or initial ℚ-factoriality.
View manuscript on GitHub → Open PDF (raw) →M122
Finite ordinary minimal model programs on compact Kähler fourfolds
We prove that every maximal ordinary negative-ray program starting from a compact Kähler klt fourfold pair with effective rational boundary in the global Weil-divisor ℚ-factorial category terminates. It ends at a nef model when the adjoint is pseudo-effective and at a projective Mori fibre space otherwise.
View manuscript on GitHub → Open PDF (raw) →M123
Finite ordinary minimal model programs on compact Kähler fourfolds
We prove that a globally Weil-ℚ-factorial compact Kähler klt fourfold pair with effective rational boundary and canonical rational line bundle admits a finite ordinary minimal model program starting on the given pair. It ends at a nef model when the adjoint class is pseudo-effective and at a Mori fibre space otherwise.
View manuscript on GitHub → Open PDF (raw) →Family 057 Fundamental groups of special complex varieties and root orbifolds. 3 papers
Proves Campana's abelianity conjecture: special compact Kähler manifolds have virtually abelian fundamental groups. Using this theorem, establishes the same conclusion for order-two root orbifolds of smooth projective complex fourfolds along one nonempty smooth connected divisor, when special in the stated differential-line sense. For smooth special complex quasi-projective varieties, proves that every finite-dimensional complex linear representation of the fundamental group has virtually nilpotent image of class at most two.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M124
The abelianity conjecture for special compact Kähler manifolds
We prove that the fundamental group of every special compact Kähler manifold is virtually abelian, resolving Campana's abelianity conjecture in all dimensions. In particular, every smooth compact connected Kähler manifold of Kodaira dimension zero has virtually abelian ordinary fundamental group.
View manuscript on GitHub → Open PDF (raw) →M125
Two-step monodromy of special quasi-projective varieties
We give an independent proof that every complex linear representation of the ordinary fundamental group of a connected smooth special complex quasi-projective variety has virtually nilpotent image of class at most two. This conclusion was previously announced by Cao–Deng–Hacon–Păun. We also construct special open surfaces whose general quasi-Albanese fibres are not special.
View manuscript on GitHub → Open PDF (raw) →M126
A conditional abelianity theorem for special fourfold pairs with a half-weight divisor
Using the abelianity theorem for special compact Kähler manifolds in every dimension, we prove virtual abelianity of the entire orbifold fundamental group of the special order-two root orbifold associated with a pair `(X, tfrac12D)`, where X is a smooth projective fourfold and D is a nonempty smooth connected divisor. We construct a special smooth projective eightfold whose fundamental group surjects onto the required group.
View manuscript on GitHub → Open PDF (raw) →Family 058 Semialgebraic universal covers and bounded domains. 2 papers
Proves the Kollár–Pardon conjecture: the semialgebraic universal covers of connected normal projective complex varieties are exactly products `D× Cm× F`, with D bounded symmetric and F simply connected, normal, and projective. A universal cover is quasi-projective exactly when the bounded symmetric factor is absent. In particular, a smooth projective variety covered by ℂn has a finite étale cover by an abelian variety. ([Lean](lean/docs/058.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M127
Semialgebraic universal covers of normal projective varieties
We prove the Kollár–Pardon conjecture: the universal cover of a connected normal projective complex variety is biholomorphic to a semialgebraic open subset of a projective variety if and only if it is a product of a bounded symmetric domain, a complex affine space, and a simply connected normal projective variety.
View manuscript on GitHub → Open PDF (raw) →M128
Symmetry of semialgebraic bounded domains with compact quotient
Every nonempty connected semialgebraic bounded open subset of a complex affine variety admitting a properly discontinuous cocompact group of biholomorphisms is smooth and biholomorphic to a bounded symmetric domain. This answers the bounded-domain question of Kollár and Pardon affirmatively.
View manuscript on GitHub → Open PDF (raw) →Family 059 Counterexamples to Zariski’s multiplicity conjecture. 2 papers
Disproves Zariski's multiplicity conjecture by constructing reduced holomorphic hypersurface germs that are ambiently homeomorphic but have different multiplicities. The examples include hypersurfaces in ℂ4 with isolated critical points and multiplicities four and five.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M129
Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three
We give a negative answer to the embedded Zariski multiplicity conjecture. We construct two reduced hypersurface germs that are ambiently homeomorphic but have multiplicities two and three. Both have isolated singularities and lie in a common complex affine space of dimension divisible by eight. Their defining function germs are also topologically right equivalent, and the same examples answer Arnold's corank problem negatively for ambient topological equivalence.
View manuscript on GitHub → Open PDF (raw) →M130
Ambiently homeomorphic isolated hypersurface germs in ℂ⁴ with multiplicities four and five
We give a negative answer to Zariski's multiplicity question in four complex variables. We construct two reduced convergent holomorphic function germs with isolated critical points and multiplicities four and five whose zero-set germs are ambiently homeomorphic.
View manuscript on GitHub → Open PDF (raw) →Family 060 The Global Spherical Shell conjecture. 1 paper
Every connected minimal compact complex surface of class VII with `b_2 0` contains a global spherical shell, proving the positive-b2 Global Spherical Shell conjecture. Such a shell is a holomorphically embedded neighborhood of the standard three-sphere in ` C² 0 ` whose complement is connected.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M131
Global Spherical Shells on Minimal Surfaces of Class VII
We prove the Global Spherical Shell conjecture: every connected minimal compact complex surface of class VII with positive second Betti number contains a global spherical shell. This is a holomorphically embedded neighborhood of the standard three-sphere in ` C² 0 ` whose complement is connected.
View manuscript on GitHub → Open PDF (raw) →Family 062 Projective contact classification and the LeBrun–Salamon conjecture. 1 paper
Proves the LeBrun–Salamon conjecture: every closed connected positive quaternionic-Kähler manifold of real dimension at least eight is homothetic to a compact symmetric Wolf space. It also proves contact-Fano homogeneity and classifies smooth connected complex projective contact manifolds of complex dimension at least three: those with `b_2=1` are adjoint varieties with their canonical contact structures, while those with `b_2≥2` have underlying manifold ` P(T*Z)` for a smooth projective variety Z.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M132
Contact Fano manifolds and the LeBrun–Salamon conjecture
We resolve the contact-Fano homogeneity conjecture and the Riemannian LeBrun–Salamon conjecture positively. Every smooth connected complex projective contact Fano manifold of complex dimension at least three, with its given contact distribution, is contact-isomorphic to the adjoint variety of a simple complex Lie algebra. Consequently, every closed connected smooth positive quaternionic-Kähler manifold of real dimension `4m≥8` is homothetic to a compact symmetric Wolf space.
View manuscript on GitHub → Open PDF (raw) →Family 063 The generalized Mukai conjecture. 1 paper
Proves the generalized Mukai conjecture.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M133
The generalized Mukai conjecture
We prove the generalized Mukai conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 064 Topological triviality of μ-constant surface singularities. 1 paper
Proves topological right-triviality for every holomorphic one-parameter family of isolated hypersurface singularities in ℂ3 with constant Milnor number, resolving the surface case of the μ-constant problem. After shrinking the parameter disk and representatives, ambient homeomorphisms vary jointly continuously, fix the origin section and preserve the defining functions.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M134
Topological triviality of mu-constant families of surface singularities
We prove that every holomorphic one-parameter family of isolated hypersurface singularities in ℂ3 with constant Milnor number is topologically right-trivial. This gives a positive answer to the surface case of the μ-constant problem. The trivialization fixes the parameter and the origin section, and preserves the defining functions.
View manuscript on GitHub → Open PDF (raw) →Family 065 Virasoro constraints for complete intersections and projective-bundle towers. 2 papers
Proves the full ordinary unreduced descendant Virasoro conjecture for smooth complete intersections in complex projective space, in every genus and curve class with arbitrary cohomology insertions. The constraints also pass from any smooth projective complex base satisfying them to the projectivization of every algebraic vector bundle of rank at least two, and hence to projective-bundle towers.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M135
Virasoro Constraints under Projectivization
We prove that full ordinary descendant Virasoro constraints pass from a smooth projective complex base to the projectivization of any algebraic vector bundle of rank at least two. The bundle need not split and satisfies no positivity requirement. Assuming the full constraints on the base, the conclusion includes every genus, each individual integral curve class, and all cohomology insertions, including primitive and odd classes. The result also applies successively to towers of projective bundles.
View manuscript on GitHub → Open PDF (raw) →M136
Virasoro Constraints for Projective Complete Intersections
We prove the Virasoro conjecture for the ordinary descendant Gromov–Witten theory of smooth complete intersections in projective space, in every genus and curve class and with arbitrary cohomology insertions. This includes primitive and odd cohomology classes, with no semisimplicity assumption.
View manuscript on GitHub → Open PDF (raw) →Family 066 Bounded klt complements for Fano contractions. 2 papers
Proves the finite-rational-coefficient form of Shokurov's bounded-klt-complement conjecture for ϵ-lc complex Fano-type pairs with nef anti-log-canonical divisor. For ϵ-lc Fano contractions over any algebraically closed characteristic-zero field, it gives klt complements near every base point, with index bounded only by dimension and positive rational ϵ.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M137
Bounded klt complements for Fano contractions
For fixed dimension d and positive rational ϵ, we prove that every ϵ-log-canonical Fano contraction over an algebraically closed field of characteristic zero admits, near each closed base point, a klt complement of index bounded only by d and ϵ. Over ℂ, we also obtain monotone klt complements for Fano type pairs with nef anti-log-canonical divisor and coefficients in a fixed finite rational set. This proves the finite-rational-coefficient form of Shokurov's bounded-klt-complement conjecture.
View manuscript on GitHub → Open PDF (raw) →M138
Uniform Cartier sections for Fano type contractions
We prove the Cartier-divisor conjecture of Birkar and Shokurov for rational boundaries in characteristic zero and for real boundaries over ℂ. For an ϵ-lc Fano type contraction with positive-dimensional base and nef negative log canonical class, a Cartier divisor through any prescribed base point can be chosen with pullback log canonical threshold bounded below in terms of the dimension and ϵ alone.
View manuscript on GitHub → Open PDF (raw) →Family 067 The Campana–Peternell conjecture in dimension six. 1 paper
Proves the Campana–Peternell conjecture in complex dimension six: every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M139
The Campana–Peternell conjecture in dimension six
We prove that every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous, resolving the Campana–Peternell conjecture in complex dimension six. As a consequence, a connected compact Kähler manifold X with nef holomorphic tangent bundle and `dim( C)X- q(X)≤6` has ordinary universal cover `F× C q(X)`, where F is a rational homogeneous manifold and ` q(X)` is the maximal irregularity of a connected finite étale cover.
View manuscript on GitHub → Open PDF (raw) →Family 068 Anticanonical nonvanishing in every dimension. 7 papers
If X is a smooth connected complex projective variety and `-K_X` admits a smooth Hermitian metric with nonnegative curvature, then `H⁰(X,-mK_X)≠0` for some m > 0. Thus smooth semipositivity forces a nonzero section of a positive tensor power of the anticanonical bundle in every dimension.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M140
Anticanonical nonvanishing from smooth semipositivity
Every smooth connected projective complex variety with smoothly semipositive anticanonical bundle has a nonzero section of some positive anticanonical power.
View manuscript on GitHub → Open PDF (raw) →M141
Invariant anticanonical indices and conversion of twisted differentials
For a holomorphic action of a compact torus on a compact complex manifold, we prove that the invariant Euler characteristic of naturally linearized anticanonical powers is polynomial on a divisible progression, with its actual value at exponent zero. Independently, on a smooth projective variety with smoothly semipositive anticanonical bundle, we remove a fixed pseudoeffective error from an unbounded sequence of effective twists.
View manuscript on GitHub → Open PDF (raw) →M142
Bounded anticanonical metrics on klt pairs and torus quotients
Let `(W,D)` be a projective ℚ-factorial klt pair with effective rational boundary. We prove that a positive Cartier multiple of `-(K_W+D)` has a nonzero section if this divisor is nef, its pullback to a resolution admits a semipositive metric with locally bounded weights, and the resolution has nonzero structure-sheaf Euler characteristic. Consequently a smooth rationally connected projective variety with smoothly semipositive anticanonical bundle has a nonzero naturally invariant anticanonical plurisection for every algebraic torus action.
View manuscript on GitHub → Open PDF (raw) →M143
Cohomological transfer and equivariant anticanonical sections
Let X be a smooth projective complex variety with smoothly semipositive anticanonical bundle. For any torus linearization of that bundle, we show that invariant sections in unbounded degrees with one fixed negative pseudoeffective error yield an invariant section in a positive untwisted degree. Combining the natural invariant index with compact-monodromy structure, we deduce that every such X has a nonzero section of some positive anticanonical power.
View manuscript on GitHub → Open PDF (raw) →M144
Metric descent and rank-preserving contractions
From a smooth semipositive anticanonical metric on a smooth projective complex variety, we construct an unweighted integrable semipositive metric on a corrected anticanonical line of a smooth base, using a normal equidimensional toroidal model and full generic adjoint rank one at exponent zero. The explicit rational boundary correction pulls back to an exceptional divisor.
View manuscript on GitHub → Open PDF (raw) →M145
Integrable metrics and effectivity with controlled boundary
Let Y be smooth projective and let C be an effective integral divisor. If `-K_Y+C` admits a metric with a global strict curvature lower bound for which the canonical section of C is locally square integrable, every pseudoeffective rational divisor becomes rationally effective after an effective correction supported on C.
View manuscript on GitHub → Open PDF (raw) →M146
Exact orders and invariant anticanonical linear systems
On a smooth connected projective complex variety with smoothly semipositive anticanonical bundle, we prove that asymptotically zero valuation on the curvature-null face is attained by an effective rational anticanonical divisor. The fixed auxiliary line bundle is arbitrary.
View manuscript on GitHub → Open PDF (raw) →Family 071 Koebe’s circle-domain conjecture. 2 papers
Resolves the existence part of Koebe's circle-domain conjecture: every domain in the Riemann sphere is conformally equivalent to a domain whose complementary components are round disks or points. It also proves that circle domains with conformally removable boundary are rigid, meaning every conformal equivalence to another circle domain is Möbius, establishing this direction of the He–Schramm conjecture. ([Lean](lean/docs/071.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M148
Removable Boundaries and Rigidity of Circle Domains
We prove that a circle domain with conformally removable boundary is conformally rigid, with no restriction on the number of complementary components. This establishes the removability-to-rigidity direction of the He–Schramm Conjecture.
View manuscript on GitHub → Open PDF (raw) →M149
Koebe's Circle-Domain Conjecture
We prove that every domain in the Riemann sphere is conformally equivalent to a circle domain, resolving Koebe's circle-domain conjecture positively.
View manuscript on GitHub → Open PDF (raw) →Family 072 Brennan's conjecture and the integral-means spectrum. 2 papers
Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, `|φ'|s` is area-integrable for `4/3 s 4`.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M150
Brennan's conjecture and sharp inverse-square integral means
We prove Brennan's conjecture: if a simply connected plane domain admits a conformal bijection φ onto the unit disk, then `| '|s` is area-integrable for every `4/3 s 4`.
View manuscript on GitHub → Open PDF (raw) →M151
A strict inverse-first-power bound for univalent functions
We prove a uniform upper bound for inverse-first-power integral means of normalized univalent disk maps with exponent strictly below 1/4.
View manuscript on GitHub → Open PDF (raw) →Family 073 The Falconer distance conjecture. 1 paper
Resolves the Falconer distance conjecture in every dimension d ≥ 2: every compact set `E Rd` with Hausdorff dimension greater than `d/2` determines a set of Euclidean distances of positive Lebesgue measure. ([Lean](lean/docs/073.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M152
The Falconer distance conjecture in all dimensions
We resolve the Falconer distance conjecture in every dimension. For every integer d ≥ 2, a compact subset of ℝd with Hausdorff dimension greater than `d/2` determines a set of Euclidean distances of positive Lebesgue measure.
View manuscript on GitHub → Open PDF (raw) →Family 074 Kakeya in three and four dimensions. 2 papers
Resolves the Kakeya maximal conjecture in three dimensions and the Hausdorff-dimension conjecture in four. In three dimensions, the radius-δ tube maximal operator maps `L³( R³)` to `L³(S²)` with norm `O_ε(δ⁻ε)` for every ε > 0. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M153
The Kakeya maximal conjecture in three dimensions
We prove the Kakeya maximal conjecture in three dimensions. For every ε > 0, the maximal average over unit tubes of radius δ maps `L³( R³)` to `L³(S²)` with norm at most `C_εδ⁻ε`.
View manuscript on GitHub → Open PDF (raw) →M154
Every four-dimensional Kakeya set has full Hausdorff dimension
We prove the four-dimensional Hausdorff-dimension Kakeya conjecture: every subset of ℝ4 containing a unit line segment in every direction has Hausdorff dimension four. No compactness or regularity assumption is imposed on the set or its witnessing line family.
View manuscript on GitHub → Open PDF (raw) →Family 075 The `Llog L` Fourier-convergence conjecture. 1 paper
Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in `Llog L( T)` converge almost everywhere along the full sequence. This resolves the classical sufficiency conjecture at the `Llog L` scale.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M155
Almost-everywhere Fourier convergence in L log L
We prove that the ordinary symmetric Fourier partial sums of every complex-valued function in `Llog L` on the circle converge almost everywhere to the function along the full sequence. This establishes the classical `Llog L` sufficiency conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 076 Real ultraflat Littlewood polynomials and unbounded binary merit factors. 3 papers
Constructs polynomials with N consecutive coefficients in ` -1,1 ` whose modulus is `(1+o(1)) N` uniformly on the entire unit circle, for every sufficiently large integer length N. Thus real Littlewood polynomials are ultraflat, including at the real endpoints. Their binary merit factors tend to infinity, disproving Turyn's bounded-merit-factor conjecture. ([Lean](lean/docs/076.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M156
Ultraflat real Littlewood polynomials
For every `ε (0,1)` and every sufficiently large integer N, there is a polynomial of length N with coefficients in ` -1,1 ` whose modulus lies between `(1-ε) N` and `(1+ε) N` everywhere on the unit circle. Thus real Littlewood polynomials can be ultraflat through every sufficiently large integer length. The signs may be chosen separately at each length.
View manuscript on GitHub → Open PDF (raw) →M157
Nearly minimal maxima and positive minima of Littlewood polynomials
For every η > 0 and every sufficiently large integer N, there is a polynomial with N consecutive coefficients in ` -1,1 ` whose modulus lies between ` N/16` and `(1+ ) N` everywhere on the unit circle.
View manuscript on GitHub → Open PDF (raw) →M158
Asymptotically minimal maxima of real Littlewood polynomials
We prove that the minimum possible maximum modulus on the unit circle of a polynomial with N consecutive real coefficients in ` -1,1 ` is `(1+o(1)) N`, as N tends to infinity through all integers. This disproves the real-sign analogue of Erdős's fixed relative-gap conjecture. As a consequence, the largest binary merit factor at length N tends to infinity through all integer lengths, disproving Turyn's conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 077 Fourier restriction for positively curved surfaces. 2 papers
Proves the diagonal Fourier extension conjecture for positively curved surfaces in three dimensions. For every compact smooth positively curved surface ` R³`, including surfaces with boundary, the extension operator is bounded from `Lp( )` to `Lp( R³)` for every p > 3.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M159
Elliptic capacity propagation and Fourier restriction to the sphere
We prove the bounded-data Fourier restriction conjecture for the sphere in three dimensions: the Fourier extension operator maps `L^∞(S²)` boundedly into `Lp( R³)` for every p > 3. This is the full conjectured open range, and the threshold p = 3 is sharp.
View manuscript on GitHub → Open PDF (raw) →M160
Diagonal Fourier extension for positively curved surfaces in three dimensions
We prove the diagonal Fourier extension conjecture for compact smooth positively curved surfaces ` R³`, including surfaces with smooth boundary. For every `3 p ∞`, the extension operator maps `Lp( )` boundedly into `Lp( R³)`. The compact paraboloid case also yields free Schrödinger local smoothing in two spatial dimensions for every `3 p ∞` and Sobolev order `s 2-6/p`.
View manuscript on GitHub → Open PDF (raw) →Family 078 The three-dimensional Bochner–Riesz conjecture. 1 paper
Resolves the three-dimensional Bochner–Riesz conjecture in its strict range: the Bochner–Riesz multipliers of order δ are bounded on `Lp( R³)` for every `1≤ p≤∞` whenever `δ max 3|1/p-1/2|-1/2,0 `.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M161
Bochner–Riesz multipliers in three dimensions
We prove the three-dimensional Bochner–Riesz conjecture in its strict-order formulation. The main result is boundedness of the Bochner–Riesz multipliers on `L³( R³)` for every positive order. Interpolation and duality then give the full conjectured strict range.
View manuscript on GitHub → Open PDF (raw) →Family 079 Local smoothing in three dimensions. 1 paper
Resolves Sogge's local smoothing conjecture for the Euclidean wave equation in three spatial dimensions. The estimate holds throughout the full strict range `2 p ∞`, with Sobolev regularity above `max 0,1-3/p `. In particular, the critical L3 estimate holds with every positive Sobolev loss.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M162
Critical local smoothing for the three-dimensional wave equation
We prove the critical L3 local smoothing estimate for the wave equation in three spatial dimensions, with every positive Sobolev loss. This resolves Sogge's Euclidean local smoothing conjecture in dimension three.
View manuscript on GitHub → Open PDF (raw) →Family 080 The exact Sobolev endpoint for Schrödinger convergence. 2 papers
Proves almost-everywhere convergence `eit → f` as `t downarrow0` for every `f Hn/(²(n⁺¹)( Rn)` and every dimension n ≥ 2. This attains the sharp Sobolev equality case of Carleson's Schrödinger convergence problem, including the planar endpoint H1/3.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M163
Endpoint convergence for the planar Schrodinger equation
We resolve the planar Sobolev endpoint of Carleson's convergence problem: for initial data in `H¹/³( R²)`, the Schrödinger evolution converges almost everywhere to the initial data as `t downarrow0`. The evolution is defined by taking the Gaussian regularization limit first; on one full-measure spatial set, this limit exists for every `0 t 1`, and the resulting evolution is continuous at t = 0.
View manuscript on GitHub → Open PDF (raw) →M164
Endpoint pointwise convergence for the Schrodinger equation in higher dimensions
We resolve the Sobolev endpoint of Carleson's pointwise convergence problem in every dimension n ≥ 3. For initial data in `Hn/(²(n⁺¹)( Rn)`, the free Schrödinger evolution converges almost everywhere to the initial data as `t downarrow0`. The evolution is defined by removing Gaussian regularization on one full-measure spatial set, uniformly over the interval `0 t 1`.
View manuscript on GitHub → Open PDF (raw) →Family 081 Riesz transforms and rectifiability in higher codimension. 1 paper
Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and `2≤ n≤ d-2`, an n-Ahlfors–David regular Radon measure on ℝd is uniformly n-rectifiable whenever its n-dimensional Riesz transform is uniformly L2-bounded over all positive hard truncations. The rectifiability bounds depend only on dimension, regularity and operator bounds. ([Lean](lean/docs/081.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M165
Riesz transforms and uniform rectifiability in higher codimension
We prove that an n-Ahlfors–David regular Radon measure on ℝd is uniformly n-rectifiable if its n-dimensional Riesz transform is uniformly bounded from scalar `L²(μ)` to vector-valued `L²(μ)` over all positive hard truncations, for integers d ≥ 4 and `2≤ n≤ d-2`. This gives a positive answer to the David–Semmes Riesz-transform question in its remaining higher-codimension range. The conclusion gives uniform big pieces of Lipschitz images of Euclidean balls.
View manuscript on GitHub → Open PDF (raw) →Family 082 Annular variation and dyadic absolute bounds for the triangular Hilbert transform. 3 papers
Proves maximal and annular r-variation bounds, for every r > 2, from complex `L³( R²)× L³( R²)` to `L³/²( R²)`. The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric `L³× L³× L³` point. ([Lean](lean/docs/082.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M166
Annular variation of the triangular Hilbert transform at the symmetric point
We prove the annular r-variation estimate for the triangular Hilbert transform from complex `L³× L³` to L3/2 for every r > 2. The partitions may depend on the output point and range over all positive scales. The estimate yields the two-endpoint maximal bound and joint almost-everywhere and norm principal values, and resolves the symmetric scalar triangular Hilbert transform problem.
View manuscript on GitHub → Open PDF (raw) →M167
An L³ bound for the dyadic triangular Hilbert form
We prove a uniform `L³× L³× L³` estimate for the dyadic triangular Hilbert form with unrestricted real inputs. The sum of the absolute local contributions over any finite set of scales is bounded by forty times the product of the input norms. In particular, the bound allows coefficients of modulus at most one to vary independently among admissible interval triples.
View manuscript on GitHub → Open PDF (raw) →M168
The maximal triangular Hilbert transform at the symmetric point
For arbitrary complex inputs in `L³( R²)`, we prove the pointwise maximal `L³× L³→ L³/²` estimate for the triangular Hilbert transform, with the supremum over both hard truncation endpoints. The estimate yields joint almost-everywhere and L3/2 convergence as the lower endpoint tends to zero and the upper endpoint tends to infinity. This also proves the conjectured scalar estimate at the symmetric point.
View manuscript on GitHub → Open PDF (raw) →Family 083 Hilbert transforms along Lipschitz directions. 1 paper
Proves a uniform strong L2 bound for the planar Hilbert transform along any Lipschitz unit vector field, at integration lengths bounded by an absolute multiple of its reciprocal Lipschitz constant. The estimate is uniform over inner truncations and yields an L2-bounded principal-value operator, establishing Stein's weak-type conjecture at this short scale. ([Lean](lean/docs/083.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M169
A uniform Hilbert transform estimate for Lipschitz directions
We prove a uniform L2 bound for the Hilbert transform along a Lipschitz unit vector field in the plane, with integration restricted to a fixed absolute multiple of the reciprocal Lipschitz seminorm. The bound is uniform in the inner truncation and holds for fields depending on both coordinates. This gives an affirmative answer to Stein's weak- `(2,2)` conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 084 The geometric case of the Erdős similarity conjecture. 2 papers
For every fixed `q (0,1)`, constructs compact subsets of `[0,1]` with measure arbitrarily close to one containing no translated and nontrivially dilated copy of ` qn:n≥1 `, with dilations of either sign. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio. ([Lean](lean/docs/084.md))
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Primary manuscript on GitHub → Family index on GitHub →M170
The geometric case of the Erdős similarity conjecture
We prove the geometric-progression case of the Erdős similarity conjecture. For every fixed `q (0,1)` and every ` (0,1)`, we construct a compact set `E(q, ) [0,1]` of measure greater than `1- ` containing no nontrivial affine copy of ` qn:n≥1 `, for any translation and either sign of nonzero dilation. The set may depend on q; the result makes no simultaneous assertion for different ratios.
View manuscript on GitHub → Open PDF (raw) →M171
The dyadic case of the Erdős similarity conjecture
We construct a compact subset of the unit interval, of measure arbitrarily close to one, that contains no affine copy of the dyadic sequence ` 2⁻n:n≥1 `. The conclusion holds for every translation and every nonzero real dilation, of either sign, proving the dyadic case of the Erdős similarity conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 085 Endpoint Sobolev regularity of centered disk averages. 1 paper
Resolves the planar centered-disk case of the Hajłasz–Onninen maximal-function regularity problem. For every real `f W¹,¹( R²)`, the centered disk maximal function satisfies ` | Mf |_1≤ C | f |_1` with an absolute constant. It belongs locally to `W¹,¹` and has a globally integrable weak gradient. ([Lean](lean/docs/085.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M172
An Endpoint Gradient Bound for the Centered Disk Maximal Operator
We prove the endpoint gradient bound ` | Mf |_1≤ C | f |_1` for every real-valued `f W¹,¹( R²)`, where `Mf(x)` is the supremum of the averages of `|f|` over disks centered at x, and C is an absolute constant. The maximal function belongs to `W¹,¹(loc)( R²)` and has a globally integrable weak gradient. This gives a positive resolution of the planar centered-disk case of the endpoint question of Hajłasz and Onninen.
View manuscript on GitHub → Open PDF (raw) →Family 086 An L3 bound for the trilinear Hilbert transform. 1 paper
Proves that the principal-value trilinear Hilbert transform with shifts `x-t`, `x-2t`, `x-3t` is bounded from `L³( R)³` to `L¹( R)`. This resolves the L3 exponent case of the standard conjecture for slopes 1, 2, 3.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M173
An L3 bound for the trilinear Hilbert transform
We prove that the trilinear Hilbert transform with fixed slopes 1, 2, 3 maps `L³( R)× L³( R)× L³( R)` to `L¹( R)`, resolving this case of the trilinear Hilbert transform conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 088 Sharp projection-body inequalities and a counterexample to simplex maximization. 2 papers
Proves Petty's projection-volume conjecture in the remaining dimensions n ≥ 4: ellipsoids uniquely minimize projection-body volume at fixed body volume. Also establishes the full Lutwak–Petty projection inequalities. In contrast, products of simplices exceed Brannen's proposed simplex maximum for normalized projection-body volume by an exponential factor in every sufficiently large dimension. ([Lean](lean/docs/088.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M177
Petty’s projection-volume conjecture in dimensions at least four
We prove that ellipsoids uniquely minimize the volume of the projection body among convex bodies of fixed volume in every dimension at least four. This proves Petty's projection-volume conjecture in these dimensions.
View manuscript on GitHub → Open PDF (raw) →M178
A product counterexample to the simplex maximum for projection-body volume
The product of two ten-dimensional simplices has larger normalized projection-body volume than a twenty-dimensional simplex. This gives a counterexample to Brannen's proposed simplex maximum.
View manuscript on GitHub → Open PDF (raw) →Family 089 Bounded-distortion L1 embeddings of planar and bounded-treewidth graphs. 2 papers
Resolves the planar and bounded-treewidth cases of the Gupta–Newman–Rabinovich–Sinclair conjecture. Shortest-path metrics of finite connected graphs with arbitrary positive edge lengths embed into real L1 with universal distortion for planar graphs, and distortion depending only on treewidth for bounded-treewidth graphs. The corresponding multicommodity flow–cut gaps are uniformly bounded. ([Lean](lean/docs/089.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M179
Planar Graph Metrics Embed into L1 with Constant Distortion
We prove that every finite connected planar graph with arbitrary positive real edge lengths embeds into real L1 with a universal distortion bound. This resolves the planar embedding conjecture positively.
View manuscript on GitHub → Open PDF (raw) →M180
L1 Embeddings of Graphs of Bounded Treewidth
For every fixed treewidth bound, the shortest-path metrics of finite connected graphs with arbitrary positive real edge lengths embed into real L1 with uniformly bounded distortion. This resolves the bounded-treewidth case of the Gupta–Newman–Rabinovich–Sinclair conjecture positively.
View manuscript on GitHub → Open PDF (raw) →Family 090 Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy. 4 papers
Proves that the triangular lattice minimizes the lower limit of energy per particle for every nonnegative completely monotone potential of squared distance among locally finite planar configurations of centered-disk density one. It also minimizes unit-background renormalized Riesz energies for `0 s 2` and Coulomb energy, resolving Sandier–Serfaty and the two-dimensional Brauchart–Hardin–Saff conjecture on the linear term of optimal spherical logarithmic energy. ([Lean](lean/docs/090.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M181
An atomic certificate for triangular-lattice universal optimality
We prove that the density-one triangular lattice minimizes the lower energy per particle for every nonnegative completely monotone function of squared distance, among all locally finite planar configurations of centered disk density one. The comparison includes infinite energies. The proof constructs sharp Gaussian Fourier minorants using an atomic interpolation certificate.
View manuscript on GitHub → Open PDF (raw) →M182
Universal optimality of the triangular lattice
We prove universal energy minimality for the triangular lattice in the plane. Among locally finite configurations of centered density one, it minimizes the lower limit of centered-ball energy averages for every nonnegative completely monotone function of squared distance, including when the energy is infinite. We also prove triangular minimality for planar logarithmic and Riesz renormalized energies, with `0 s 2` in the Riesz case, and the corresponding jellium minima.
View manuscript on GitHub → Open PDF (raw) →M183
A sharp Fourier certificate for planar circle packing
We resolve the planar Cohn–Elkies sharpness conjecture: the two-point Fourier bound attains the optimal circle-packing density `π/(2 sqrt3)`. We construct a radial Schwartz certificate and prove its global sign conditions using rigorous interval arithmetic and analytic estimates. The same certificate recovers the classical uniqueness of the triangular packing among periodic equality cases.
View manuscript on GitHub → Open PDF (raw) →M184
Triangular minimality for planar Coulomb renormalized energy
We prove the Sandier–Serfaty conjecture: the triangular lattice of covolume one minimizes planar Coulomb renormalized energy over all admissible curl-free fields with a unit uniform background. Combined with Bétermin and Sandier's asymptotic formula, this also proves the Brauchart–Hardin–Saff conjecture for the linear term of optimal ordered-pair logarithmic energy on the unit two-sphere. The proof uses a direct Voronoi-cell comparison with rigorous interval arithmetic.
View manuscript on GitHub → Open PDF (raw) →Family 091 Logarithmic and Lp Brunn–Minkowski inequalities and the B-conjecture. 1 paper
Proves the logarithmic Brunn–Minkowski inequality for origin-symmetric convex bodies in every dimension, and the scalar-dilation B-conjecture for all even log-concave Radon measures. For Lebesgue volume it also proves the additive Lp Brunn–Minkowski inequality for full-dimensional origin-symmetric convex bodies throughout `0 p 1`. ([Lean](lean/docs/091.md))
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Primary manuscript on GitHub → Family index on GitHub →M185
The logarithmic Brunn–Minkowski conjecture
We prove the logarithmic Brunn–Minkowski conjecture for arbitrary origin-symmetric convex bodies in every dimension. The theorem also gives the symmetric Lp Brunn–Minkowski inequality for every `0 p 1`. Combined with Saroglou's transfer theorem and a support-subspace reduction, it yields the logarithmic inequality for every even log-concave Radon measure and the scalar-dilation `(B)` -conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 092 The optimal order of convex-body covering density. 2 papers
Determines the optimal worst-case covering density as ` (nlog n)`, for both lattice and unrestricted translative coverings. Every convex body in ℝn, n ≥ 2, admits a lattice covering of density at most `Cnlog n`; centrally symmetric examples in every sufficiently large dimension require at least `cnlog n` even without the lattice restriction, for absolute `c,C 0`. ([Lean](lean/docs/092.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M186
A single-lattice covering bound of order n log n
Every convex body in ℝn, n ≥ 2, admits a covering by translates along one full-rank lattice with density at most `Cnlog n`, for an absolute constant C. No symmetry or boundary regularity is assumed.
View manuscript on GitHub → Open PDF (raw) →M187
Translative covering densities of order n log n
For every sufficiently large dimension n, we construct a centrally symmetric convex body whose translative covering density exceeds `c nlog n`, where c > 0 is absolute. This disproves the existence of a universal linear upper bound and matches the order of Rogers' upper bound.
View manuscript on GitHub → Open PDF (raw) →Family 093 Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures. 1 paper
Proves a dimension-free logarithmic Sobolev inequality for centered log-concave densities with uniformly subgaussian linear marginals, with constant bounded by a universal multiple of the squared linear subgaussian parameter.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M188
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures
We prove that every centered log-concave probability measure with a Lebesgue density on ℝn and linear subgaussian parameter a satisfies ` Ent _μ(f²)≤ Ca²∫|Df|² dμ` for compactly supported smooth f, with one universal constant C. This resolves positively the dimension-free logarithmic Sobolev conjecture for subgaussian log-concave measures.
View manuscript on GitHub → Open PDF (raw) →Family 094 Subpolynomial dimension reduction in Lp. 1 paper
For every fixed `1 p ∞` and distortion D > 1, every n-point subset of real Lp embeds into `ℓ_pd` with distortion at most D and dimension `d=no(¹)`, answering Naor's sublinear-dimension question for p ≠ 2. In contrast, exact embeddings require worst-case dimension ` (n²)` when p ≠ 2. ([Lean](lean/docs/094.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M189
Subpolynomial dimension reduction in Lp
For every fixed `1 p ∞` and D > 1, every n-point subset of a real Lp space embeds into `ℓ_pd` with distortion at most D and subpolynomial dimension `d=no(¹)`. The target has the same exponent p, and the embedding need not be linear.
View manuscript on GitHub → Open PDF (raw) →Family 095 Hyperbolicity cones without semidefinite lifts. 3 papers
Disproves the Projected Lax conjecture: some hyperbolicity cones are not spectrahedral shadows. The examples admit no exact finite affine semidefinite lift, regardless of the number of auxiliary variables or the real coefficients used. This also disproves the generalized Lax conjecture that every hyperbolicity cone is spectrahedral. ([Lean](lean/docs/095.md))
PQC lens: Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
Primary manuscript on GitHub → Family index on GitHub →M190
Hyperbolicity Cones Without Semidefinite Lifts
We prove that not every hyperbolicity cone is a spectrahedral shadow: some closed hyperbolicity cones admit no finite affine semidefinite lift, even with arbitrary real coefficients and any finite number of auxiliary variables. This disproves the Projected Lax Conjecture and hence the generalized Lax conjecture.
View manuscript on GitHub → Open PDF (raw) →M191
A nonspectrahedral hyperbolicity cone
We construct a homogeneous polynomial of degree 16 in 23 real variables whose hyperbolicity cone has no representation by a finite homogeneous real symmetric linear matrix inequality. This disproves the geometric Generalized Lax conjecture.
View manuscript on GitHub → Open PDF (raw) →M192
An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone
We construct an exact semidefinite lift of the explicit nonspectrahedral hyperbolicity cone in twenty-three variables defined in the companion paper. The lift is a homogeneous real symmetric pencil of size 100 with 307 auxiliary variables and represents the entire closed cone, including every point with singular X. Thus, although this cone has no semidefinite representation in its original coordinates, it admits one when auxiliary variables are allowed. The stated sizes are not claimed to be minimal.
View manuscript on GitHub → Open PDF (raw) →Family 096 The Gaussian propeller conjecture in every dimension. 1 paper
Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most `9/(8π)`. In dimension at least two, three planar sectors of angle `2π/3`, extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor `(8π/9)(1-1/k)` for identity-target kernel clustering with fixed k ≥ 3 on rational centered positive semidefinite inputs. ([Lean](lean/docs/096.md))
PQC lens: Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
Primary manuscript on GitHub → Family index on GitHub →M193
The Gaussian propeller bound in every dimension
We prove the Gaussian propeller conjecture: for every finite measurable partition of a Euclidean space, the sum of the squared lengths of its Gaussian first moments is at most `9/(8π)`. For dimension at least two and at least three cells, three planar sectors of angle `2π/3`, extended by an orthogonal Euclidean factor, attain the bound.
View manuscript on GitHub → Open PDF (raw) →Family 097 The Euclidean Steinitz–Bergström bound. 1 paper
Proves that any finite sequence in the Euclidean unit ball of ℝd admits signs keeping every partial sum within `C d`, independently of length. Consequently every zero-sum family can be reordered with the same bound on unsigned partial sums. A matching lower bound gives the optimal order `S_2(d)= ( d)`. ([Lean](lean/docs/097.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M194
The Euclidean Steinitz–Bergström theorem
Every prescribed-order finite sequence of vectors in the Euclidean unit ball of ℝd has one signing for which every signed prefix has norm at most `C d`, with C absolute and independent of the sequence length. Consequently, every indexed zero-sum family of unit-ball vectors admits an ordering with the same bound for its unsigned partial sums. This determines the Euclidean Steinitz constant up to absolute factors, `S_2(d)= ( d)`, and resolves the Euclidean Steinitz–Bergström conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 098 Compact counterexamples to bi-Lipschitz dimension reduction. 1 paper
Every infinite-dimensional real Banach space contains a compact doubling set that admits no bi-Lipschitz embedding into any finite-dimensional normed space. The doubling constant is universal. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space. ([Lean](lean/docs/098.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M195
A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding
Every infinite-dimensional real Banach space contains a compact doubling subset that admits no bi-Lipschitz embedding into any finite-dimensional real normed space. The doubling constant has a universal bound, independent of the ambient Banach space. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space.
View manuscript on GitHub → Open PDF (raw) →Family 099 The sharp exponential scale of edit-distance distortion. 3 papers
Determines the least distortion of embedding edit distance on words of length at most d into real ℓ1: it is `exp( (√(log d loglog d)))`. Insertions, deletions and substitutions have unit cost. The constants are uniform over all finite alphabets with at least two symbols, even when the alphabet grows with d; binary words already force the lower bound. ([Lean](lean/docs/099.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M196
Edit Distance in l1: Matching Bounds up to Constants in the Exponent
We determine the exponential scale of the least ℓ1 distortion of unit-cost edit distance on all strings of length at most d. For every sufficiently large d, uniformly over finite alphabets of size at least two, the distortion lies between `exp(c√(log d loglog d))` and `exp(C√(log d loglog d))` for absolute constants `c,C 0`. The lower bound already holds on binary strings of one common length. Thus the order of logarithmic distortion is sharp up to absolute constants.
View manuscript on GitHub → Open PDF (raw) →M197
Finite-Circle Obstructions, Binary Codes, and Histogram Embeddings for Edit Distance
We give two finite-circle constructions of binary strings whose least ℓ1 distortion is `exp( (√(log d loglog d)))`, where d bounds their length. Both constructions supply words of one common length for every sufficiently large cap. Two direct binary coding arguments transfer the constructions with absolute distortion and logarithmic block width.
View manuscript on GitHub → Open PDF (raw) →M198
Tree Constructions for the l1 Distortion of Binary Edit Distance
We give two independent constructions of binary words of one length at most d whose ordinary edit-distance metrics require ℓ1 distortion `exp( (√(log d loglog d)))` for every sufficiently large d. We also prove a constant-distortion binary conversion for one prescribed input length. Together with the companion upper embedding theorem, these lower bounds determine the order of logarithmic distortion uniformly over finite alphabets with at least two symbols.
View manuscript on GitHub → Open PDF (raw) →Family 100 Cylinder coverings below the half-area bound. 4 papers
Covers the entire closed regular tetrahedron by finitely many cylinders with compact triangular perpendicular bases whose total area is less than half its smallest orthogonal projection area. This disproves Bang's half-area cylinder-covering bound and the stronger directionwise normalized conjecture in dimension three. ([Lean](lean/docs/100.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M199
Finite angular cylinder covers below the half-area bound
A regular tetrahedron admits a finite cylinder covering with compact triangular perpendicular bases whose total area is less than half its minimum orthogonal projection area. This disproves the half-area cylinder-covering conjecture. By affine invariance, the same construction gives a counterexample to the directionwise normalized half-bound for every nondegenerate tetrahedron.
View manuscript on GitHub → Open PDF (raw) →M200
Finite cylinder approximation of ruled sets
We approximate compact ruled families of segments by finitely many cylinders with square intercept tiles and perpendicular-base area at most their integral projection cost plus any positive error. The velocity field is C1, and its differential has opposite real eigenvalues whose magnitudes are strictly below the inverse segment half-length; both eigenvalues may vanish. The result includes square-zero differentials with unrestricted shear and fields that pass between the two regimes. It holds for arbitrary compact label sets.
View manuscript on GitHub → Open PDF (raw) →M201
Slope-field perturbations of the two-cylinder covering
The two-cylinder covering of a regular tetrahedron can be perturbed to give finite covers with total perpendicular base area strictly below half its minimum projection area. These covers give negative answers to both the half-area question and the directionwise normalized half-bound conjecture. By affine invariance, the directionwise conclusion holds for every nondegenerate tetrahedron.
View manuscript on GitHub → Open PDF (raw) →M202
Finite triangular approximation of radial sweeps
We prove that radially aligned segment sweeps admit finite cylinder covers with one triangular base for each interval of any tagged partition. As the mesh tends to zero, the total perpendicular base area converges to a weighted parameter area. An explicit application covers every regular tetrahedron with total base area below half its minimum projection area, giving negative answers to the half-area question and the directionwise normalized 1-Codimensional Cylinder Covering Conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 101 The sharp simplex conjecture for isotropic constants. 1 paper
Proves that simplices uniquely maximize the isotropic constant among convex bodies in every dimension, resolving the strong isotropic constant conjecture. Also establishes the sharp entropy lower bound for log-concave probability densities, with equality precisely for invertible affine images of products of one-sided exponential laws.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M203
A sharp entropy bound and the simplex inequality for isotropic constants
We prove the strong isotropic constant conjecture: in each dimension, simplices are the unique maximizers of the isotropic constant among convex bodies. We also prove the sharp entropy bound `h(f)≥ m+ tfrac12logdet Cov (f)` for every log-concave probability density on ℝm, with equality precisely for invertible affine images of products of one-sided exponential laws.
View manuscript on GitHub → Open PDF (raw) →Family 103 Exact derandomization of logarithmic space: ` L= RL = BPL `. 1 paper
Proves ` L= RL = BPL `, resolving derandomization for bounded-error logarithmic-space computation. An effective compiler converts each randomized polynomial-time logarithmic-space machine deciding a language with one-sided or two-sided error into a deterministic logarithmic-space decider with explicit polynomial running-time bounds.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M209
Exact derandomization of logarithmic space: L = RL = BPL
We prove ` L= RL = BPL `, resolving the derandomization problem for polynomial-time randomized logarithmic space.
View manuscript on GitHub → Open PDF (raw) →Family 104 Quasipolynomial algorithms for mean-payoff, stochastic and parity games. 4 papers
Gives deterministic algorithms.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M210
Turn-Based Stochastic Mean-Payoff Games in Deterministic Quasipolynomial Time
We give a uniform deterministic quasipolynomial-time algorithm for finite turn-based stochastic mean-payoff games with signed integer rewards and rational chance-transition probabilities encoded in binary.
View manuscript on GitHub → Open PDF (raw) →M211
Mean-payoff parity games in quasipolynomial time
We give a uniform deterministic quasipolynomial-time algorithm for mean-payoff parity games.
View manuscript on GitHub → Open PDF (raw) →M212
Deterministic quasipolynomial-time mean-payoff games
We give a deterministic algorithm that computes the complete zero-threshold winning set of a finite mean-payoff game with arbitrary signed integer edge weights encoded in binary.
View manuscript on GitHub → Open PDF (raw) →M213
Randomized quasipolynomial-time mean-payoff games
We give a randomized algorithm that computes the complete zero-threshold winning set of a finite mean-payoff game with arbitrary signed integer edge weights encoded in binary.
View manuscript on GitHub → Open PDF (raw) →Family 105 Perfect completeness for 2-to-1 games. 1 paper
Proves Khot's 2-to-1 Games Conjecture with perfect completeness: for every fixed rational `δ (0,1)`, it is NP-hard to distinguish satisfiable games from games whose optimum is at most δ, on explicit unweighted instances. The alphabet depends only on δ, and every right-hand label has exactly two preimages under each constraint map. ([Lean](lean/docs/105.md))
PQC lens: Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
Primary manuscript on GitHub → Family index on GitHub →M214
Perfect completeness for 2-to-1 games
We prove the 2-to-1 Games Conjecture with perfect completeness. For every fixed rational `δ (0,1)`, it is NP-hard to distinguish satisfiable 2-to-1 games from games of value at most δ, with a fixed alphabet and an explicitly listed unweighted multiset of constraints.
View manuscript on GitHub → Open PDF (raw) →Family 106 Hardness of coloring three-colorable graphs. 1 paper
It is NP-hard to color a three-colorable graph using any fixed number c ≥ 3 of colors. More strongly, for every fixed `0 δ 1/3`, a deterministic polynomial-time reduction from 3SAT produces simple unweighted graphs that are three-colorable in the satisfiable case and have no independent set of size `δ n` otherwise, where n is the number of vertices. ([Lean](lean/docs/106.md))
PQC lens: Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
Primary manuscript on GitHub → Family index on GitHub →M215
Hardness of finding large independent sets in three-colorable graphs
We prove that, for every fixed `0 δ 1/3`, it is NP-hard to distinguish three-colorable graphs from graphs in which every independent set has fewer than δ times the number of vertices. Consequently, for every fixed integer c ≥ 3, finding a proper c-coloring of a three-colorable graph is NP-hard.
View manuscript on GitHub → Open PDF (raw) →Family 107 Matrix multiplication with exponent at most 9/4. 3 papers
Proves `ω≤9/4` over ℂ, giving `O_ε(n⁹/⁴⁺ε)` arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension na with a > 0.465 permits `n²⁺o(¹)` rectangular multiplication. Further square bounds give ω < 2.258 outside finitely many positive characteristics and ω < 2.371054886006746 over every fixed field. ([Lean](lean/docs/107.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M216
An Upper Bound of 9/4 for the Matrix Multiplication Exponent
We prove that the exponent of matrix multiplication over the complex numbers is at most 9/4.
View manuscript on GitHub → Open PDF (raw) →M217
Complex Matrix Multiplication Below 2.258 and Rectangular Bounds
— secondary writeup Over every field of characteristic zero, we prove that the square matrix-multiplication exponent satisfies ω < 2.258, the dual exponent satisfies α > 0.465, and `ω(1,0.709,1) 2.092`. The strict square and k = 0.709 rectangular bounds also hold over every field except possibly in one finite set of positive characteristics, in the arithmetic-operation model.
View manuscript on GitHub → Open PDF (raw) →M218
Staggered extraction for exact matrix multiplication over every field
We prove that the arithmetic exponent of square matrix multiplication over every fixed field satisfies ω < 2.371054886006746. This includes every positive characteristic.
View manuscript on GitHub → Open PDF (raw) →Family 108 A cubic permanent–determinant lower bound. 1 paper
Proves an ` (n³)` lower bound for the border determinantal complexity of the `n× n` permanent over ℂ. Even coefficientwise limits of determinants of affine-linear matrices require matrix size at least `cn³`, for an absolute c > 0 and all sufficiently large n; the same bound therefore holds for exact representations. ([Lean](lean/docs/108.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M219
A cubic lower bound for border determinantal complexity of the permanent
We prove that the complex border determinantal complexity of the `m× m` permanent is ` (m³)`, allowing arbitrary affine-linear determinant representations and coefficientwise limits. It also gives cubic lower bounds for exact determinantal complexity and for the numbers of vertices and edges in affine-linear algebraic branching programs, including coefficientwise limits with a fixed vertex or edge budget.
View manuscript on GitHub → Open PDF (raw) →Family 109 Integer multiplication below `nlog n`. 1 paper
Multiplies two n-bit integers exactly at every input length in deterministic worst-case time `O(n(log n)¹⁻ )`, with ` =2⁻¹⁸²`, on one fixed finite-alphabet Turing machine with finitely many one-dimensional tapes. This disproves the Schönhage–Strassen `nlog n` optimality conjecture in the ordinary multitape bit model.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M220
Integer multiplication below n log n
We give a deterministic algorithm that multiplies two n-bit integers in `O(n( n)¹⁻ )` worst-case time, with ` =2⁻¹⁸²`, on one fixed finite-alphabet Turing machine with a fixed finite number of one-dimensional tapes. The algorithm is exact for every input length and disproves the `nlog n` optimality conjecture of Schönhage and Strassen in this model.
View manuscript on GitHub → Open PDF (raw) →Family 110 Optimal-order randomized k-server on arbitrary metrics. 2 papers
Establishes a randomized competitive ratio `O(log²(k⁺¹)` for k-server on every metric space, matching the worst-case lower-bound order. One policy serves every finite oblivious request sequence, including on infinite unbounded metrics. On finite rational metrics, a uniform implementation has polynomial preprocessing and per-request bit cost in the input length and `log(t+1)` at request t, with a finite instance-dependent additive movement constant. ([Lean](lean/docs/110.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M221
Squared-logarithmic randomized k-server on arbitrary metrics
We prove that randomized k-server has competitive ratio `O((log(k+1))²)` on every metric space against oblivious request sequences, matching the known worst-case lower bound. For each metric and initial configuration, one policy works for all finite request sequences, including on infinite and unbounded spaces. When the initial server positions are distinct, no additive term is needed.
View manuscript on GitHub → Open PDF (raw) →M222
Uniform computation of the squared-logarithmic k-server bound
We construct a uniform randomized k-server algorithm on finite rational metrics with competitive ratio `O(log²(k⁺¹)` against oblivious request sequences. Preprocessing is polynomial in the input length, and per-request bit complexity is polynomial in that length and the binary request-counter length. The additive movement constant is finite and instance-dependent, but may be enormous. The construction uses the companion squared-logarithmic existence theorem.
View manuscript on GitHub → Open PDF (raw) →Family 111 One-sample matroid prophet inequalities against an almighty adversary. 1 paper
For every finite matroid known in advance, gives a distribution-independent online rule using one independent sample per element and earning a universal constant fraction of the expected offline optimum. Values are independent and nonnegative, with finite expected optimum. The guarantee holds even when the arrival-order adversary sees all samples, values, and the rule's entire random seed; no polynomial-time implementation is asserted. ([Lean](lean/docs/111.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M223
One Sample Suffices for Matroid Prophet Inequalities against an Almighty Adversary
We prove that one independent sample per element suffices for a constant-competitive prophet inequality on every finite matroid. The guarantee holds even when the arrival-order adversary observes all samples, all online values, and the algorithm's entire random seed. The rule needs no description of the value distributions and achieves the absolute competitive ratio `2⁻³¹⁰`.
View manuscript on GitHub → Open PDF (raw) →Family 112 Beyond the square-root exponent for depth-three circuits. 1 paper
Constructs a single language in deterministic polynomial time whose n-bit membership function requires `2ω( n)` total gates in unbounded-fan-in OR–AND–OR circuits, at every sufficiently large input length. This crosses the square-root-exponent threshold for explicit depth-three Boolean circuit lower bounds. ([Lean](lean/docs/112.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M224
Beyond the Square-Root Exponent for Depth-Three Boolean Circuits
We construct a language in deterministic polynomial time whose n-bit membership function requires `2ω( n)` gates in an unbounded-fan-in OR–AND–OR circuit. The bound holds at every sufficiently large input length and counts all gates, including the bottom layer.
View manuscript on GitHub → Open PDF (raw) →Family 113 Approximate counting and entropy of perfect matchings. 2 papers
Gives a fully polynomial randomized approximation scheme for counting perfect matchings in arbitrary finite simple graphs, with exact detection of zero counts. Also proves the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant, bounding the maximum entropy of a matching law at every feasible edge-marginal vector in a loopless labelled multigraph, including boundary points. ([Lean](lean/docs/113.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M225
A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs
We give a fully polynomial randomized approximation scheme (FPRAS) for counting perfect matchings in arbitrary finite simple undirected graphs, resolving the general-graph perfect-matching approximation problem. The algorithm returns zero with certainty when no perfect matching exists. Otherwise, it achieves relative error ε with failure probability at most δ in worst-case bit time polynomial in the input length, `ε ⁻¹`, and `logδ⁻¹`.
View manuscript on GitHub → Open PDF (raw) →M226
Entropy and Face Dimension of the Perfect-Matching Polytope
We prove the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant. For every feasible vector x of perfect-matching edge marginals in a loopless labelled multigraph on `2m≥2` vertices, the maximum entropy `H(x)` of a matching law with marginals x satisfies ` F(x)-(2-2/m)B(x)≤ H(x)≤ F(x),` where `F(x)=-∑_e x_elog x_e` and `B(x)=-∑_e(1-x_e)log(1-x_e)`. This bound holds throughout the polytope, including its boundary.
View manuscript on GitHub → Open PDF (raw) →Family 114 Approximate counting of common integer polymatroid bases. 2 papers
Gives a fully polynomial randomized approximation scheme for counting common integer bases of two integral polymatroids of equal total rank, supplied by exact rank-value oracles. Capacities are binary-encoded, each integer vector counts once, and oracle calls and bit operations outside the oracles are polynomial on every execution. For matroids presented by independence oracles, the results also cover common independent sets of prescribed, unrestricted, or maximum cardinality, even when the ranks differ. ([Lean](lean/docs/114.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M227
An FPRAS for Common Integer Polymatroid Bases with Binary Capacities
We give a fully polynomial randomized approximation scheme for counting common integer bases of two polymatroids with the same total rank, supplied by exact rank-value oracles. The total rank and capacities are encoded in binary, and each integer vector is counted once. On every execution, the number of oracle calls and the bit work outside the oracles are bounded by a fixed polynomial in the ground-set size, the binary input length, the inverse relative-error tolerance, and the logarithm of the inverse failure probability.
View manuscript on GitHub → Open PDF (raw) →M228
Approximate counting of common bases of two matroids
We give a fully polynomial randomized approximation scheme for counting the common bases of two arbitrary matroids of the same rank, supplied by independence oracles. The algorithm requires no explicit representation of either matroid and has polynomial bounds on both oracle calls and bit operations on every execution.
View manuscript on GitHub → Open PDF (raw) →Family 115 Sampling and counting contingency tables with arbitrary margins. 2 papers
For nonnegative integer matrices with prescribed row and column sums, gives exact uniform sampling in expected polynomial bit time and almost-uniform sampling in worst-case polynomial bit time. The dimensions and binary-encoded margins are unrestricted. Also gives a fully polynomial randomized approximation scheme for counting such tables with arbitrary individual cell bounds, including structural zeros, with polynomial cost on every execution. ([Lean](lean/docs/115.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M229
Exact Uniform Sampling of Contingency Tables with Arbitrary Margins
We give an exact uniform sampler for nonnegative integer contingency tables with arbitrary prescribed margins. It terminates almost surely and has expected bit complexity polynomial in both dimensions and the binary length of the margins. No positivity, balance, sparsity, or fixed-dimension assumption is required.
View manuscript on GitHub → Open PDF (raw) →M230
An FPRAS for Cell-Bounded Contingency Tables
We give a fully polynomial randomized approximation scheme for counting nonnegative integer matrices with prescribed row sums, column sums, and individual entry bounds. Both dimensions vary, all numerical data are encoded in binary, and zero bounds are allowed. The algorithm uses only unbiased random bits and has a polynomial bound on its bit operations on every execution.
View manuscript on GitHub → Open PDF (raw) →Family 116 Uniform black-box noncommutative identity testing across characteristics. 3 papers
For each characteristic, constructs in deterministic polynomial bit time a polynomial-dimensional matrix tuple detecting every nonzero division-free noncommutative formula of bounded size over any field of that characteristic. Rational formulas over ℚ also admit polynomial-size hitting lists whenever they have a defined rational-matrix evaluation. ([Lean](lean/docs/116.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M231
Uniform Matrix Hitting Points in Every Positive Characteristic
We construct a single matrix substitution that detects every nonzero size-s division-free noncommutative formula in n variables over every field of a given positive characteristic. One deterministic machine, given a promised prime p in binary and n, s in unary, outputs matrices over 𝔽p of dimension `O(n³s⁶)` in polynomial bit time. The same tuple works with arbitrary extension-field coefficients, including in characteristic two.
View manuscript on GitHub → Open PDF (raw) →M232
One Rational Matrix Hitting Point for Noncommutative Formulas
We construct, in deterministic polynomial bit time, one tuple of rational matrices that detects every nonzero polynomial computed by a noncommutative division-free formula of a prescribed size. The matrices have dimension `O(ns²)` for n variables and formula size s, and the same tuple works over every field of characteristic zero.
View manuscript on GitHub → Open PDF (raw) →M233
Polynomial Hitting Lists for Noncommutative Rational Formulas
We construct, in deterministic polynomial bit time, a polynomial-size list of rational matrix tuples for noncommutative rational formulas over ℚ of bounded tree size. Every nonzero admissible formula has a defined, invertible value at one tuple, with no separate bounds on inverse nesting or rational constant heights. Matrix dimensions, entry bit lengths, and total output length are polynomially bounded.
View manuscript on GitHub → Open PDF (raw) →Family 117 Uniform sparsest cut: hardness and semidefinite gaps. 2 papers
Proves that approximating Uniform Sparsest Cut within any fixed constant factor is NP-hard, even with nonnegative rational capacities and unit demands. The Goemans–Linial semidefinite relaxation also has integrality gaps of order at least `√(log n)/(loglog n)³`, approaching the square-root-logarithmic upper bound. ([Lean](lean/docs/117.md))
PQC lens: Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
Primary manuscript on GitHub → Family index on GitHub →M234
Constant-factor hardness of uniform sparsest cut
We prove that, for every fixed C > 1, approximating Uniform Sparsest Cut within factor C is NP-hard. The output graphs have nonnegative rational capacities and unit demand between every pair of distinct vertices.
View manuscript on GitHub → Open PDF (raw) →M235
Near-square-root logarithmic integrality gaps for uniform sparsest cut
We construct uniform sparsest-cut instances whose Goemans–Linial semidefinite integrality gap is at least `c√(log n)/(loglog n)³` along a sequence `n→∞`. The demand is one between every pair of distinct vertices, and the capacities are nonnegative real numbers. This matches the Arora–Rao–Vazirani upper bound up to a power of `loglog n`.
View manuscript on GitHub → Open PDF (raw) →Family 118 Bin packing and unbounded configuration-LP gaps. 1 paper
Disproves the modified integer round-up conjecture of Scheithauer and Terno: the integral bin-packing optimum can exceed its configuration linear-programming value by an arbitrarily large additive constant. Approximating the optimum within any fixed additive constant is also NP-hard, even when every item exceeds 1/6 and each bin holds at most five items. ([Lean](lean/docs/118.md))
PQC lens: Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
Primary manuscript on GitHub → Family index on GitHub →M236
Additive hardness and unbounded configuration gaps in bin packing
We disprove the Modified Integer Round-Up Conjecture for bin packing by constructing instances with arbitrarily large additive gaps between the configuration-LP value and the integral optimum. We also prove that, for every fixed nonnegative integer c, distinguishing instances that fit in B bins from those requiring more than `B+c` bins is NP-hard. Both results hold with rational item sizes greater than 1/6, so each bin contains at most five items.
View manuscript on GitHub → Open PDF (raw) →Family 119 The Courtade–Kumar and Hellinger conjectures. 2 papers
Proves the Courtade–Kumar conjecture: among Boolean functions of independent uniform bits, a single coordinate retains the most mutual information after independent bit-flip noise. A stronger theorem treats randomized binary summaries at fixed initial information. The Hellinger conjecture is also proved for every Boolean output bias and noise correlation. ([Lean](lean/docs/119.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M237
Sharp binary-information contraction on the discrete cube
We prove sharp contraction of the information carried by a binary channel under independent symmetric noise on a uniform discrete cube. At fixed initial information, a noisy coordinate retains the most information. The Boolean specialization resolves the Courtade–Kumar conjecture and gives an output-entropy refinement. We also establish a stronger mean-dependent entropy-production bound.
View manuscript on GitHub → Open PDF (raw) →M238
Hellinger contraction with arbitrary Boolean output bias
We prove the Hellinger conjecture for Boolean functions on the uniform discrete cube, with arbitrary output bias. For a Boolean function of mean m and every `ρ [-1,1]`, the loss `√(1-m²)⁻ E√(1-(T_ρ f)²)` is at most `1-√(1-ρ²)`, with equality for signed coordinates. The proof combines asymmetric dimension induction, a calibrated noise-semigroup energy estimate, and finite exact arithmetic certificates. The Hellinger inequality also yields the Courtade–Kumar information bound.
View manuscript on GitHub → Open PDF (raw) →Family 120 Almost-linear-time exact matching and prescribed-degree factors in general graphs. 1 paper
Gives a randomized algorithm finding an exact maximum-cardinality matching in any simple undirected graph in `(n+m)¹⁺o(¹)` word time, with success probability at least 2/3. The time bound holds on every computation path. The same guarantees apply to finding a spanning subgraph with prescribed admissible vertex degrees, or deciding that none exists.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M239
Almost-Linear-Time Maximum-Cardinality Matching in General Graphs
We prove that maximum-cardinality matching in a simple undirected graph with n vertices and m edges can be found by one uniform randomized algorithm in `(n+m)¹⁺o(¹)` time. The bound holds on every computation path in a logarithmic-word model, and the algorithm returns an explicit maximum matching with probability at least 2/3. An explicit reduction gives the same time and probability guarantees for deciding whether a simple host graph has a spanning subgraph with prescribed valid vertex degrees, and for finding one when it exists.
View manuscript on GitHub → Open PDF (raw) →Family 121 Almost-linear approximation of edit distance. 1 paper
For every fixed rational `ε (0,1)`, gives a randomized `(1+ε)` approximation to unit-cost edit distance in worst-case expected time `N¹⁺o(¹)`, with success probability at least 2/3. The strings have total length N and polynomially bounded integer symbols. This is an asymptotic guarantee at fixed accuracy. ([Lean](lean/docs/121.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M240
An Almost-Linear Approximation Scheme for Edit Distance
We give a uniform randomized approximation scheme for unit-cost edit distance. For every fixed rational `ε (0,1)`, it estimates the distance between arbitrary explicitly stored strings of total length N within a factor `1+ε` with probability at least 2/3, in worst-case expected time `N¹⁺o(¹)` on a logarithmic-word RAM. The algorithm supports polynomially bounded integer alphabets and returns zero deterministically on equal strings.
View manuscript on GitHub → Open PDF (raw) →Family 122 Quantitative trace-reconstruction bounds with a uniform decoder. 3 papers
At every fixed deletion probability in `(0,1)`, reconstructing an arbitrary length-n binary string requires `n (loglog n)` independent traces, ruling out polynomial-sample reconstruction. A uniform decoder achieves quasipolynomial sample and running-time bounds for known fixed rational retention probabilities. When the deletion probability is at most `n⁻ε` for fixed ε > 0, both bounds become polynomial in the input and parameter encoding. ([Lean](lean/docs/122.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M241
Uniform quasipolynomial-time trace reconstruction
We give a uniform algorithm that reconstructs every binary string from independent deletion traces when its length and rational retention probability are known. For each fixed retention probability, both the number of traces and the bit complexity are quasipolynomial in the string length. More generally, we give an explicit sample bound uniform over all rational retention probabilities, with running time polynomial in the sample budget and the binary input length. If the deletion probability is at most `n⁻ε` for fixed ε >
View manuscript on GitHub → Open PDF (raw) →M242
A latest-anchor induction with spectrally compact masks for worst-case trace reconstruction
We give an improved worst-case sample bound for reconstructing a string from independent deletion traces, with its length and retention probability known. For each fixed retention probability, the number of traces is quasipolynomial: the logarithm of the sample budget is `O((log n)³(¹⁺loglog(²n)⁶)`. If the deletion probability is at most `n⁻ε` for fixed ε > 0, polynomially many traces suffice. These bounds apply to binary strings and to strings of general symbols observed exactly.
View manuscript on GitHub → Open PDF (raw) →M243
Quantitative lower bounds for trace reconstruction
Exact worst-case reconstruction of a binary word from independent deletion traces requires `n (loglog n)` samples for every fixed deletion probability `q (0,1)`, even with unrestricted computation and any fixed positive success probability. This gives a negative answer to the polynomial-sample question for binary trace reconstruction. More generally, when `q³log n→∞`, we prove a lower bound of `nclog(q^³log n)` samples for every fixed `0 c 1/(4log2)`. Here q is the known deletion probability, and all logarithms are natural.
View manuscript on GitHub → Open PDF (raw) →Family 124 Polynomial-time scheduling on three identical machines. 1 paper
Resolves the three-processor unit-job scheduling problem of Garey and Johnson: a deterministic polynomial-time algorithm minimizes makespan for nonpreemptive unit-length jobs with arbitrary precedence constraints on three identical parallel machines. For an explicitly given precedence graph, it decides deadline feasibility exactly and constructs a feasible schedule. ([Lean](lean/docs/124.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M244
A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling
We give a uniform deterministic polynomial-time algorithm for scheduling unit-length jobs with arbitrary precedence constraints on three identical parallel machines. The algorithm constructs a schedule of minimum makespan and decides exactly whether all jobs can finish by a specified deadline. The proof reorganizes feasible schedules into intervals whose job sets have descriptions of bounded size. A dynamic program searches a family containing polynomially many such descriptions.
View manuscript on GitHub → Open PDF (raw) →Family 125 The metric k-median approximation threshold and recovery. 2 papers
Gives a deterministic polynomial-time `(1+2/e+ε)` -approximation for finite rational metric k-median with specified candidate facilities, for every fixed ε > 0. Assuming `P≠ NP`, the optimal infimum approximation factor is `1+2/e`. ([Lean](lean/docs/125.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M245
Single-exponential recovery and bounded-price strictness for metric k-median
We give an exact-budget recovery algorithm for metric k-median with single-exponential dependence on the number of comparison clusters without accurate, distinct proxies in a supplied anchor solution. On positive integral metrics of polynomially bounded diameter, a sufficiently small total proxy error and logarithmically many such clusters yield a `(1+2/e+ε)` approximation in polynomial time with arbitrarily high success probability. We also prove bounded-price strictness for one compatible execution of the logarithmic-surplus construction.
View manuscript on GitHub → Open PDF (raw) →M246
The approximation threshold for metric k-median
For every fixed ε > 0, we give a deterministic polynomial-time `(1+2/e+ε)` -approximation for finite rational metric k-median with specified candidate facilities, opening at most k facilities. Under `P≠ NP`, the infimum approximation factor in this model is therefore `1+2/e`.
View manuscript on GitHub → Open PDF (raw) →Family 126 Exponential semidefinite complexity of perfect matching. 1 paper
Proves that every exact semidefinite lift of the perfect matching polytope has exponential size, answering Rothvoss's polynomial-size lift question negatively. The bound holds even for the positive semidefinite rank of its odd-cut slack matrix after any fixed shift `0 ρ 1`, allowing arbitrary real positive semidefinite factors. ([Lean](lean/docs/126.md))
PQC lens: Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
Primary manuscript on GitHub → Family index on GitHub →M247
Exponential PSD rank of positively shifted matching matrices
For every fixed `0 ρ 1`, the matrix indexed by odd vertex sets U and perfect matchings M of Kn, with entries `|M capδ(U)|-1+ρ`, has real positive semidefinite rank `2 (n)` as even n tends to infinity. Here `δ(U)` is the edge cut of U. Consequently, every exact semidefinite lift of the perfect matching polytope has exponential size.
View manuscript on GitHub → Open PDF (raw) →Family 127 Average sensitivity of polynomial threshold functions. 1 paper
Proves that a degree-at-most-d polynomial threshold function on the uniform n-dimensional Boolean cube has average sensitivity at most `8d n`, uniformly for `1≤ d≤ n`. Average sensitivity counts expected output changes under single-bit flips. This establishes the asymptotic Gotsman–Linial conjecture, allowing polynomial zeros with ` sign (0)=1`. ([Lean](lean/docs/127.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M248
Average sensitivity of polynomial threshold functions
For every n ≥ 1 and `1≤ d≤ n`, we prove that a polynomial threshold function of degree at most d on the uniform Boolean cube has average sensitivity at most `8d n`. This proves the asymptotic form of the Gotsman–Linial conjecture. The bound is uniform in both parameters and uses the convention ` sgn (0)=1`.
View manuscript on GitHub → Open PDF (raw) →Family 128 A factor-two approximation for shortest common superstring. 1 paper
Gives a deterministic polynomial-time algorithm constructing a common superstring of length at most twice the optimum for every finite family of explicitly represented strings. The running time is polynomial in the full encoded input length, including symbol labels. ([Lean](lean/docs/128.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M249
A Polynomial-Time 2-Approximation for Shortest Common Superstring
We give a deterministic algorithm that, for every finite family of explicitly represented ordinary strings, outputs a common superstring of length at most twice the optimum in time polynomial in the total encoded input size, including symbol labels. The guarantee applies to the algorithm constructed here, not the classical maximum-overlap Greedy procedure.
View manuscript on GitHub → Open PDF (raw) →Family 129 Exponential state costs for two-way automata. 2 papers
Proves exponential lower bounds both for complementing two-way nondeterministic finite automata and for simulating one-way nondeterministic automata by two-way deterministic ones. The latter resolves the Sakoda–Sipser state-succinctness conjecture over growing finite alphabets; both results rule out polynomial state bounds independent of alphabet size. ([Lean](lean/docs/129.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M250
An exponential state lower bound for two-way nondeterministic complementation
We prove that two-way nondeterministic finite automata cannot be complemented with a polynomial number of states independent of the alphabet. For each n ≥ 4 we construct an n-state automaton over a finite alphabet whose complement requires at least ` tfrac12 2 (n⁻⁴)/¹²⁷ -1` states.
View manuscript on GitHub → Open PDF (raw) →M251
An exponential two-way deterministic state lower bound for one-way liveness
One-way liveness on h points accepts a word of binary relations when their ordered product is nonempty. For every h ≥ 2, it has a nondeterministic automaton with `h+3` states and no left moves, whereas every equivalent s-state two-way deterministic automaton satisfies `4(s+2)²≥2 (h⁻²)/³¹ `. Partial transition rules, stay moves, and nonaccepting infinite computations are allowed.
View manuscript on GitHub → Open PDF (raw) →Family 130 Exact Fourier transforms below `nlog n`. 2 papers
Gives a deterministic length-n discrete Fourier transform algorithm using `O(n(log n)¹⁻δ)` operations for every n, with explicit `δ=10⁻¹³`. The model uses exact complex arithmetic, unrestricted coefficients and a supplied root of unity, and counts scalar preparation and logarithmic-word indexing. ([Lean](lean/docs/130.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M252
An explicit power saving for the exact discrete Fourier transform
We give a deterministic algorithm that computes the discrete Fourier transform at every length n in `O(n(log n)¹⁻¹⁰^ ⁻¹³ )` operations. The model uses exact complex arithmetic, unrestricted coefficients, specified Fourier roots, and unit-cost logarithmic-size indexing; scalar preparation and array organization are included.
View manuscript on GitHub → Open PDF (raw) →M253
Finite tensor savings and exact Fourier circuits
We construct exact nonuniform Fourier circuits of size `o(nlog n)` along an unbounded sequence of lengths, counting every addition, subtraction, and scalar multiplication. This refutes the ` (nlog n)` lower bound in the unrestricted complex linear-circuit model. The construction uses a finite tensor saving: a tensor power of some invertible nonmonomial complex matrix can be computed with fewer matrix calls than the standard tensor-axis algorithm on the same coordinates, when invertible monomial maps are allowed freely between calls.
View manuscript on GitHub → Open PDF (raw) →Family 131 Rapid mixing of graph switches for every degree sequence. 1 paper
Resolves the simple-undirected Kannan–Tetali–Vempala conjecture: the lazy edge-switch chain mixes in `O(n⁸)` time for every graphical labeled degree sequence. The same degree-constrained graphs can also be sampled exactly uniformly by an almost-surely terminating algorithm with expected polynomial bit running time. ([Lean](lean/docs/131.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M254
Polynomial mixing of the switch chain for every graphical degree sequence
We prove the simple-undirected form of the Kannan–Tetali–Vempala conjecture: the switch chain on simple undirected graphs mixes in polynomial time for every graphical degree sequence. For a lazy chain that proposes switches uniformly on four vertices, the total-variation mixing time at distance 1/4 is at most `2n⁸`. We also give an exactly uniform sampler for every graphical labeled degree vector. It uses unbiased random bits, terminates almost surely, and has expected polynomial bit running time.
View manuscript on GitHub → Open PDF (raw) →Family 132 A superquadratic separation of sensitivity and block sensitivity. 1 paper
Constructs total Boolean functions with block sensitivity ` bs (f)≥ s(f)^α` for a fixed α > 2, disproving the quadratic strengthening of the Sensitivity Conjecture. Here `s(f)` counts influential individual-bit flips, while block sensitivity allows disjoint groups of bits to change together. ([Lean](lean/docs/132.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M255
A superquadratic separation between sensitivity and block sensitivity
We disprove the quadratic strengthening of the Sensitivity Conjecture by constructing nonconstant total Boolean functions whose block sensitivity grows faster than any constant multiple of sensitivity squared. In fact, for some fixed α > 2, our examples have unbounded block sensitivity and satisfy ` bs (f)≥ s(f)^α`.
View manuscript on GitHub → Open PDF (raw) →Family 133 The computational complexity of Weisfeiler–Leman refinement. 4 papers
Proves unconditional `n (k)` deterministic time lower bounds for joint and separate k-dimensional Weisfeiler–Leman equivalence, for sufficiently large fixed k in the specified sequential adjacency-matrix models. With dimension as input, joint equivalence is EXPTIME-complete even on subcubic graphs; deciding whether refinement identifies a graph is also EXPTIME-complete. ([Lean](lean/docs/133.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M256
Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence
For k ≥ 4, we construct two uncolored graphs that are k-dimensional Weisfeiler–Leman equivalent exactly when a prescribed finite-domain choice system has no compatible choice. The system has `k+1` domains for joint refinement and k for separate-coordinate refinement. A successful choice is detected after two joint rounds or one separate round. Applied to sparse satisfiability, the reduction gives fixed-dimension `n (k)` time exclusions under positive-rate ETH, even for deciding equality of these early histograms.
View manuscript on GitHub → Open PDF (raw) →M257
The complexity of identifying a graph by Weisfeiler–Leman refinement
We prove that deciding whether Weisfeiler–Leman refinement of an input dimension identifies a given graph is EXPTIME-complete. The input is a nonempty finite simple uncolored graph in adjacency-matrix form and a positive binary-encoded dimension. Identification quantifies over every comparison graph.
View manuscript on GitHub → Open PDF (raw) →M258
Unconditional time lower bounds for Weisfeiler–Leman equivalence
For every sufficiently large fixed k, deciding whether two n-vertex graphs are k-Weisfeiler–Leman equivalent requires `n (k)` deterministic sequential time in the worst case. The bound holds at every sufficiently large graph order, even for simple connected uncolored graphs of diameter at most two, without a complexity assumption. Inputs are explicit adjacency matrices; the models are multitape Turing machines and sequential logarithmic-word RAMs with fixed polynomial-bit-time instructions.
View manuscript on GitHub → Open PDF (raw) →M259
Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs
Deciding joint-update k-dimensional Weisfeiler–Leman equivalence is ` EXPTIME ` -complete when the two graphs are given by explicit adjacency matrices and k ≥ 2 is encoded in binary. The result holds even for connected simple uncolored graphs of equal positive order and maximum degree at most three.
View manuscript on GitHub → Open PDF (raw) →Family 134 Generalized star height at most three. 3 papers
Every regular language over a finite alphabet has a generalized regular expression with at most three nested Kleene stars, allowing union, concatenation and complement over the same alphabet. This establishes an absolute bound independent of automaton size, resolving the uniform-boundedness version of the generalized star-height problem. ([Lean](lean/docs/134.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M260
Finite Monoid Computations and a Uniform Generalized Star-Height Bound
Every regular language over a finite alphabet has a generalized regular expression of star height at most thirteen over that same alphabet. We prove this uniform bound by representing finite monoid computations as affine updates and recovering them through twelve successive split constructions.
View manuscript on GitHub → Open PDF (raw) →M261
Generalized Star Height at Most Four
Every regular language over a finite alphabet has generalized star height at most four over that same alphabet. We give a complete construction using an affine correction that hides one interval product, a finite clock, and several scales for moving boundaries through periodic words.
View manuscript on GitHub → Open PDF (raw) →M262
Generalized Star Height at Most Three
Every regular language over a finite alphabet has generalized star height at most three, with complement taken in the same free monoid. We express finite-monoid computations using a prefix code of word pieces.
View manuscript on GitHub → Open PDF (raw) →Family 135 Homogeneous depth-five lower bounds for iterated matrix multiplication. 1 paper
Over every characteristic-zero field, the `(1,1)` entry of a product of n independent `n× n` variable matrices requires `n ( n)` gates in homogeneous depth-five sum–product circuits. This sharp bound allows shared gates and bottom linear forms involving all variables. ([Lean](lean/docs/135.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M263
Homogeneous depth-five lower bounds for iterated matrix multiplication
Let ` IMM (n,n)` be the `(1,1)` entry of a product of n independent `n× n` matrices of variables. Over every field of characteristic zero, every syntactically homogeneous ` ` circuit computing ` IMM (n,n)` has at least `n n/⁴⁰⁰` gates for all sufficiently large n, with an absolute threshold independent of the field. Bottom linear forms may have arbitrary support, and arbitrary finite fan-in, fan-out, and gate sharing are allowed. Over every field, a block expansion gives such circuits with at most `n n⁺⁴` gates for n ≥ 2.
View manuscript on GitHub → Open PDF (raw) →Family 136 A quasilinear PCP theorem for PPAD. 1 paper
Resolves the quasilinear PCP-for-PPAD conjecture. An End-of-Line instance of length N reduces to numerical circuit constraints of total length `N(log N)O(¹)` such that any polynomially encoded rational assignment satisfying all but a fixed fraction to fixed accuracy yields an endpoint solution. Such assignments always exist, giving robust local verification with only quasilinear size overhead.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M264
The PCP-for-PPAD conjecture: a quasilinear reduction
We prove the quasilinear-size PCP-for-PPAD conjecture of Babichenko, Papadimitriou, and Rubinstein. There are fixed positive rational constants ε and δ and a deterministic polynomial-time reduction that transforms an End-of-Line instance of binary length N into a generalized circuit of total binary length `N(log N)O(¹)`.
View manuscript on GitHub → Open PDF (raw) →Family 137 One-tape time simulation in two-fifths-power space. 1 paper
Determines the halting and finite-control outcome of a fixed deterministic one-writable-tape machine up to time T using `O(T²/⁵logC(T⁺²)` space, improving the square-root exponent. Heads move at most one cell per step; finitely many read-only input heads are allowed. Initial contents are independent of T, and contents and input symbols have polylogarithmic-space access. Simulation time is unrestricted.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M265
Simulating One-Tape Time in Two-Fifths-Power Space
We show that a fixed deterministic Turing machine with one writable tape and head can be simulated in `O(T²/⁵ polylog (T+2))` work-space bits when a binary time cap T ≥ 2 is supplied. The simulator computes the finite-control and halting outcome by time T; its running time is unrestricted. The result allows a fixed number of read-only input heads and requires a fixed accessor that supplies every initial writable and read-only symbol within distance T of the relevant head origin in polylogarithmic space.
View manuscript on GitHub → Open PDF (raw) →Family 138 Subset Sum in `O(2⁰.⁴⁹n)` time. 2 papers
Gives a uniform randomized classical algorithm for worst-case Subset Sum in ordinary `O(2⁰.⁴⁹n)` word-RAM time on polynomial-bit inputs, where n counts the integers. The time bound holds on every execution and success probability is at least 2/3 on every input. Inputs may repeat positive integers; words have `O(n+b)` bits for maximum input bit length b.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M266
Subset Sum in Time `O(2⁰.⁴⁹n)`
We give a uniform randomized classical algorithm for Subset Sum with bounded error and worst-case running time `O(2⁰.⁴⁹n)` on polynomial-bit inputs in a word-RAM model, where n is the number of input integers. The time bound holds on every random execution.
View manuscript on GitHub → Open PDF (raw) →M267
A Low-Space Algorithm for Worst-Case Subset Sum
We give a uniform classical randomized decision algorithm for worst-case Subset Sum. Under every fixed polynomial bound on input-integer bit length, it uses ` poly (n)2n/²` time and ordinary `O(2n/⁵)` writable words of `O(n+b)` bits, where b is the largest input bit length. Both resource bounds hold on every execution. The error is one-sided: the algorithm always rejects unsolvable instances and accepts each solvable instance with probability at least 2/3.
View manuscript on GitHub → Open PDF (raw) →Family 139 Subpolynomial query complexity for log-concave sampling. 1 paper
For C2 potentials with a supplied minimizer and `I nabla²V preceq2I`, proves that sampling within total variation 1/10 requires only `C_ε d^ε` exact value-and-gradient queries for every fixed ε > 0. The bound holds on every run, with unrestricted computation between queries. A logarithmic lower bound also holds, so the optimal power-law exponent in this oracle model is zero. ([Lean](lean/docs/139.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M268
Subpolynomial query complexity for well-conditioned log-concave sampling
For every fixed ε > 0, we give a sampling algorithm using at most `C_ε d^ε` exact first-order queries on every execution for C2 potentials on ℝd with a known minimizer and Hessian between Id and `2I_d`. The output has total-variation distance at most 1/10 from the target Gibbs law. Computation between queries is unrestricted. We also prove an ` (log d)` query lower bound for arbitrary randomized adaptive algorithms, determining the optimal dimension exponent to be zero.
View manuscript on GitHub → Open PDF (raw) →Family 140 Memory–sample lower bounds for noiseless Gaussian regression. 6 papers
For fixed A > 0, a one-pass learner with `Ad²` persistent bits needs ` Omega_A(dlog(1/ε))` noiseless Gaussian samples to recover a unit vector to angular error `0 ε≤1/10` with probability 2/3, uniformly in accuracy for large d. Computation and randomized updates are unrestricted, but output uses only the terminal state, stopping index and fresh randomness. ([Lean](lean/docs/140.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M269
Memory and precision in noiseless Gaussian regression
For every fixed A > 0, a learner that retains at most `Ad²` bits between fresh exact Gaussian linear measurements needs ` Omega_A(dlog(1/ε))` measurements to estimate a uniformly random unit vector to angular error at most ϵ, for any `0 ε≤ 1/10`, with probability at least 2/3. The constant is absolute for `o(d²)` memory.
View manuscript on GitHub → Open PDF (raw) →M270
Posterior replicas and conditional information in Gaussian regression
For a signal with density bounded by L relative to uniform probability on `Sd⁻¹`, we bound the information in a finite message W formed from exact Gaussian measurements, conditional on an independent projection revealed only to the analyst. For explicit row counts proportional to d, the bound is `O(H(W)/d+d+log(2+log L))`.
View manuscript on GitHub → Open PDF (raw) →M271
Localization costs and information growth for exact Gaussian observations
For the image of a uniform cube under a spherical coordinate map, we prove that finite messages from t blocks of ` (d)` exact Gaussian measurements reveal only `O_A(dt)` information when each message has at most `exp(Ad²)` values, for fixed A. The same bound holds when each message is supplemented with a nested cell that restores the required geometric spread.
View manuscript on GitHub → Open PDF (raw) →M272
Projection moments, positive cap domination, and Riesz estimates on the sphere
We prove moment estimates for exact random projections of finite measures whose mass is controlled on Euclidean balls, and derive positive domination by countable sums of spherical cap measures. For learners with `M=o(d²)` bits of memory, these estimates give three proofs that uniform-sphere average success at least 2/3 at angular accuracy `0 ε≤1/10` requires ` (dlog(1/ε))` noiseless Gaussian observations. The three proofs keep their different stopping and accuracy costs explicit.
View manuscript on GitHub → Open PDF (raw) →M273
Replacing Gaussian observations in memory-constrained inference
Replacing the Gaussian rows used to select a finite message by independent rows increases the remaining conditional information by at most `Cd`, for a uniform spherical signal, message entropy at most d2, and the specified row dimensions proportional to d. As an application, we prove that learners with `M=o(d²)` persistent bits need `T= (dlog(1/ε))` exact observations to attain uniform-sphere angular success at least 3/5, for `0 ε≤1/10` and a deterministic finite horizon.
View manuscript on GitHub → Open PDF (raw) →M274
Subsphere methods for memory-sample lower bounds in noiseless Gaussian regression
Let a finite-state streaming learner estimate a uniformly random unit vector from independent exact Gaussian linear measurements. We prove that `o(d²)` bits of persistent memory and angular success probability at least 2/3 require at least `2⁻¹⁶dlog_2(1/ε)` samples for all sufficiently large d, uniformly for `0 ε≤1/10`. The proof conditions each batch on its observed projection and controls the resulting random residual subsphere.
View manuscript on GitHub → Open PDF (raw) →Family 141 Existential–universal real sentences in the counting hierarchy. 1 paper
Proves that the existential theory of the reals lies in the counting hierarchy. More generally, truth of existential–universal real sentences can be decided at one fixed level of that hierarchy, even when their integer polynomials are specified by arithmetic circuits.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M275
Existential–universal real sentences in the counting hierarchy
We prove that the existential theory of the reals lies in the counting hierarchy. More generally, we show that the truth of existential–universal sentences over the reals can be decided in a fixed level of the counting hierarchy, even when the integer polynomials are given by arithmetic circuits.
View manuscript on GitHub → Open PDF (raw) →Family 142 Deterministic polynomial factorization over prime fields. 1 paper
Gives a uniform deterministic algorithm that completely factors every nonzero dense degree-n polynomial over a prime field 𝔽p, including multiplicities, in bit complexity polynomial in `(n+1)log p`. The prime is supplied in binary. No randomness, integer-factorization or primitive-root oracle, or GRH assumption is required.
PQC lens: Affects polynomial or integer factorization—core to RSA/DSA analysis and to comparing classical vs quantum factoring.
Primary manuscript on GitHub → Family index on GitHub →M276
Deterministic Polynomial Factorization over Prime Fields
We give a uniform deterministic polynomial-time algorithm for complete factorization over prime fields. For a prime p in binary and a nonzero polynomial `f F_p[x]` given by its dense coefficient list, the algorithm computes the irreducible factors and their multiplicities using a number of bit operations polynomial in `(deg f+1)log p`. The proof uses the uniform Hecke zero-free theorem from the companion paper *Primitive roots for every admissible integer base*.
View manuscript on GitHub → Open PDF (raw) →Family 143 Hilbert's sixteenth problem: uniform bounds for limit cycles. 2 papers
Resolves the uniform boundedness assertion in Hilbert's sixteenth problem: the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on its degree. For classical quintic Liénard systems, the exact maximum is two limit cycles. ([Lean](lean/docs/143.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M277
Uniform bounds for planar polynomial limit cycles
For every degree, we prove that the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on that degree. This establishes the uniform boundedness assertion in the second part of Hilbert's sixteenth problem. The proof uses separation of asymptotic expansions on nested complex domains and a finite-dimensional counting argument.
View manuscript on GitHub → Open PDF (raw) →M278
Two limit cycles for quintic Liénard systems
Every classical Liénard system ` x=y-F(x)`, ` y=-x`, with F an arbitrary real polynomial of degree at most five, has at most two geometrically distinct isolated periodic orbits, and the bound is attained. This proves the degree-five case of the Lins Neto–de Melo–Pugh conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 144 Banach’s simple Lebesgue-spectrum problem. 1 paper
Resolves the probability-preserving form of Banach's simple Lebesgue-spectrum problem within smooth dynamics. A smooth volume-preserving diffeomorphism of the standard-volume three-torus has simple Lebesgue spectrum on its entire complex mean-zero L2 space: the bilateral iterates of one real observable form an orthonormal basis of that space. ([Lean](lean/docs/144.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M279
A smooth three-torus diffeomorphism with simple Lebesgue spectrum
We solve the probability-preserving form of Banach's simple Lebesgue-spectrum problem in smooth dynamics. We construct a C∞ diffeomorphism of the three-torus that preserves standard volume and whose Koopman operator has simple Lebesgue spectrum on the entire mean-zero L2 space.
View manuscript on GitHub → Open PDF (raw) →Family 145 Rokhlin’s multiple-mixing problem. 1 paper
Proves that every invertible mixing probability-preserving transformation is mixing of all finite orders, resolving Rokhlin's multiple-mixing problem for a single transformation. Correlations among any finite collection of measurable sets converge to the product of their measures whenever all pairwise time separations diverge. ([Lean](lean/docs/145.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M280
Rokhlin's multiple-mixing problem for one transformation
Every invertible mixing probability-preserving transformation is mixing of every finite order. Thus Rokhlin's multiple-mixing problem for a single transformation has an affirmative answer.
View manuscript on GitHub → Open PDF (raw) →Family 146 Positive metric entropy for the standard map. 1 paper
Proves that the standard sine map on the two-dimensional torus has positive metric entropy with respect to area for every sufficiently large positive parameter. This establishes Sinai's positive-parameter-measure conjecture for the original family, with the stronger conclusion of a full parameter tail. ([Lean](lean/docs/146.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M281
Positive Metric Entropy for the Standard Map at Large Parameters
We prove that the standard sine map of the two-dimensional torus has positive metric entropy with respect to normalized area for every sufficiently large positive parameter. This gives a full parameter tail, and hence answers Sinai's positive-parameter-measure conjecture affirmatively.
View manuscript on GitHub → Open PDF (raw) →Family 147 The near-boundary Birkhoff conjecture. 2 papers
Resolves the near-boundary Birkhoff conjecture for smooth strictly convex planar billiards of positive curvature. Such a billiard is an ellipse whenever a full grazing annulus is continuously foliated by individually invariant essential curves. A continuous physical collar of smooth closed convex caustics also suffices.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M282
Continuous Phase Foliations Create Analytic Caustic Collars
For a smooth strictly convex planar billiard of positive curvature, a continuous foliation of a full grazing annulus by individually invariant essential curves forces an analytic boundary and a jointly analytic collar of smooth strictly convex caustics. The physical-collar rigidity theorem proved in the companion article then implies that the table is an ellipse.
View manuscript on GitHub → Open PDF (raw) →M283
Rigidity of Smooth Billiards with a Continuous Caustic Collar
We prove that a smooth strictly convex planar billiard with positive curvature is an ellipse whenever a full neighborhood of its boundary is continuously foliated by smooth closed convex caustics. No differentiability across the leaves is assumed. This resolves the near-boundary Birkhoff conjecture in the stated smooth class.
View manuscript on GitHub → Open PDF (raw) →Family 148 The entropy-rate dimension formula for self-similar measures. 1 paper
For every self-similar measure on the line generated by finitely many contracting similarities, proves `dim( H)μ=min 1,h(RW)/ `, where `h(RW)` is the entropy rate of random composed maps and χ the average logarithmic contraction. This resolves the entropy-rate dimension conjecture without a separation assumption, allowing exact overlaps and unequal contraction ratios. ([Lean](lean/docs/148.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M284
The entropy-rate dimension formula for self-similar measures on the line
We prove that the Hausdorff dimension of every finite real self-similar measure equals the minimum of one and its random-walk entropy rate divided by its Lyapunov exponent. Exact overlaps are allowed, and the contraction ratios may be unequal and negative. This resolves the entropy-rate dimension conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 149 Classwise permanence for weakly reversible mass-action systems. 2 papers
Proves the permanence conjecture for every finite weakly reversible mass-action system with fixed positive rate constants. Every positive stoichiometric compatibility class, even an unbounded one, has a common compact convex forward-invariant absorbing set. All positive trajectories in that class therefore eventually share positive lower and finite upper concentration bounds. ([Lean](lean/docs/149.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M285
Uniform Permanence in Weakly Reversible Mass-Action Systems
We prove the permanence conjecture for finite weakly reversible mass-action systems with fixed positive reaction rates. Every positive stoichiometric compatibility class admits one compact convex forward-invariant set that every positive trajectory in that class enters in finite time, even when the class is unbounded. The set depends only on the network, the rates, and the class; the entry time may depend on the initial state. Thus, after its entry time, every concentration satisfies the same positive lower and finite upper bounds throughout that class.
View manuscript on GitHub → Open PDF (raw) →M286
Boundedness and persistence of weakly reversible mass-action systems
We prove the boundedness and persistence conjectures for finite weakly reversible mass-action systems with positive constant reaction rates. For every positive initial condition, the solution exists for all forward time, and every concentration remains bounded above and bounded away from zero. The bounds may depend on the initial condition, and no boundedness assumption is imposed on its stoichiometric compatibility class.
View manuscript on GitHub → Open PDF (raw) →Family 150 Weak mixing of triangular billiards with an irrational angle. 2 papers
Proves that the billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized area times uniform direction. This strengthens ergodicity on the entire irrational-angle class, with no genericity or Diophantine restrictions. ([Lean](lean/docs/150.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M287
Weak mixing of triangular billiards with an irrational angle
We prove that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized Liouville measure. Equivalently, the product of the flow with itself is ergodic.
View manuscript on GitHub → Open PDF (raw) →M288
Ergodicity of triangular billiards with an irrational angle
We prove that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is ergodic for normalized area times uniform angular measure. No genericity or Diophantine condition is required. The flow is considered outside the null set of trajectories that hit a vertex.
View manuscript on GitHub → Open PDF (raw) →Family 151 A C1 counterexample to the entropy conjecture. 1 paper
Constructs a noninvertible C1 self-map of a compact smooth manifold with zero topological entropy but eigenvalue 2 on second homology. This disproves the homological entropy lower bound for general C1 self-maps: homological growth need not force positive orbit complexity. ([Lean](lean/docs/151.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M289
A C^1 Counterexample to the Entropy Conjecture
We disprove the general C1 self-map formulation of Shub's entropy conjecture. We construct a noninvertible C1 self-map of a compact smooth manifold without boundary whose topological entropy is zero, while its action on second real homology has eigenvalue 2. Thus the topological entropy is strictly smaller than the logarithm of the homological spectral radius.
View manuscript on GitHub → Open PDF (raw) →Family 152 Zero entropy does not guarantee a smooth positive-volume model. 1 paper
Constructs a zero-entropy ergodic invertible transformation of a standard nonatomic probability space that is not measurably conjugate to any C∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. One example rules out every finite dimension. ([Lean](lean/docs/152.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M290
A zero-entropy system without a smooth positive-volume model
We construct an ergodic invertible transformation of a standard nonatomic probability space with zero Kolmogorov–Sinai entropy that has no smooth positive-volume model. More precisely, it is not measurably conjugate to any C∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. A single example excludes every finite dimension, including models on nonorientable manifolds and manifolds with smooth boundary.
View manuscript on GitHub → Open PDF (raw) →Family 153 Arithmetic classification and non-Pisot singularity for Bernoulli convolutions. 1 paper
Classifies singular and absolutely continuous unbiased Bernoulli convolutions for every `λ (0,1)` by an infinite, one-sided approximation condition using explicit finite sets of algebraic units. It also proves singularity at reciprocals of every quartic Salem number in `(1,2)`, giving examples beyond reciprocal Pisot parameters.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M291
Arithmetic classification and non-Pisot singularity for Bernoulli convolutions
We give an arithmetic classification of singular and absolutely continuous unbiased Bernoulli convolutions for every parameter `λ (0,1)`. The criterion is expressed through one-sided approximation by explicitly defined finite sets of algebraic units; it is an infinite approximation condition, not a finite membership algorithm.
View manuscript on GitHub → Open PDF (raw) →Family 154 Pointwise multiple ergodic averages for mixing transformations. 4 papers
Proves almost-everywhere convergence of consecutive multiple ergodic averages of every finite length for invertible mixing probability-preserving transformations. For each fixed tuple of bounded functions, the limit is the product of their integrals, along all positive averaging lengths. No mixing rate or standardness assumption on the probability space is required.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M292
Pointwise Multiple Ergodic Averages for Mixing Transformations
Let T be an invertible mixing probability-preserving transformation. For every integer n ≥ 2 and every fixed tuple of bounded measurable functions, we prove that the consecutive multiple ergodic averages of length n converge almost everywhere to the product of the integrals, as the averaging length tends to infinity through all positive integers. The probability space need not be standard, and no rate of mixing is required.
View manuscript on GitHub → Open PDF (raw) →M293
Pointwise convergence of fourfold ergodic averages for mixing transformations
We prove that fourfold ergodic averages along the times `n,2n,3n,4n` converge almost everywhere to the product of the integrals for every invertible, bimeasurable, mixing probability-preserving transformation and every fixed choice of bounded measurable inputs. Convergence holds along all positive integer averaging lengths. No quantitative mixing rate is assumed, and the probability space need not be standard.
View manuscript on GitHub → Open PDF (raw) →M294
Triple ergodic averages with distinct integer slopes
For every invertible mixing probability-preserving transformation, triple ergodic averages of bounded measurable functions along any three pairwise distinct nonzero integer slopes converge almost everywhere to the product of their integrals. The slopes may be positive or negative.
View manuscript on GitHub → Open PDF (raw) →M295
Pointwise convergence of triple ergodic averages for mixing transformations
We prove pointwise convergence of triple ergodic averages for every invertible mixing probability preserving transformation T of an arbitrary probability space `(X, F,μ)`, with measurable inverse. For every triple of bounded measurable functions `f_1,f_2,f_3:X→ C`, ` frac1N∑(n=1)N f_1(Tnx)f_2(T²nx)f_3(T³nx) ∏(j=1)³∫_X f_j dμ` for μ-almost every x as `N→∞` through all positive integers.
View manuscript on GitHub → Open PDF (raw) →Family 155 A counterexample to periodic tiling in dimension three. 1 paper
Constructs a finite translational tile in ℤ3 that tiles space but admits no fully periodic tiling, disproving the periodic tiling conjecture in the smallest possible lattice dimension. Its unit-cube thickening gives the same counterexample in ℝ3, even with arbitrary real translation vectors. ([Lean](lean/docs/155.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M296
A translational tile with no fully periodic tiling in dimension three
We construct a finite translational tile in ℤ3 that admits tilings but no fully periodic tiling. Its unit-cube thickening has the same property in ℝ3, even when arbitrary real translations are allowed. This gives a negative resolution of the periodic tiling conjecture in dimension three.
View manuscript on GitHub → Open PDF (raw) →Family 156 Borsuk's conjecture fails in dimension nine. 1 paper
Constructs a compact subset of ℝ9 that cannot be covered by ten sets of strictly smaller diameter, disproving Borsuk's covering assertion already in dimension nine. The example consists of rank-one orthogonal projectors onto lines in ℝ4, with the Frobenius metric. ([Lean](lean/docs/156.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M297
A nine-dimensional counterexample to Borsuk's covering assertion
The compact set of rank-one orthogonal projectors on ℝ4, with the Frobenius metric, cannot be covered by ten sets of strictly smaller diameter. It therefore gives a counterexample to Borsuk's conjecture in dimension nine.
View manuscript on GitHub → Open PDF (raw) →Family 157 Graph coloring, clique minors, and Colin de Verdière invariants. 3 papers
Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy ` chi_f(G) h(G)`, where `h(G)` is the largest clique-minor order. Also disproves the fractional Colin de Verdière chromatic bound ` chi_f(G)≤μ(G)+1`. In the positive direction, every finite nonempty graph satisfies ` (list)(G)≤ C h(G)` for a universal constant C. ([Lean](lean/docs/157.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M298
A counterexample to Hadwiger's conjecture
We disprove Hadwiger's conjecture by constructing arbitrarily large graphs whose chromatic number exceeds their Hadwiger number. The examples have independence number at most two, and even their ordinary fractional chromatic number exceeds their Hadwiger number. Thus they also disprove the fractional-coloring weakening discussed by Reed and Seymour.
View manuscript on GitHub → Open PDF (raw) →M299
A counterexample to the Colin de Verdière chromatic conjecture
We disprove the Colin de Verdière chromatic conjecture by constructing graphs whose chromatic number exceeds their Colin de Verdière invariant by more than one. The examples have independence number at most two. In fact, their ordinary fractional chromatic number also exceeds their Colin de Verdière invariant by more than one.
View manuscript on GitHub → Open PDF (raw) →M300
A linear list-coloring bound in terms of the Hadwiger number
We prove that every finite nonempty graph G satisfies ` (list)(G)≤ C h(G)` for an absolute integer C, where `h(G)` is the largest order of a clique minor. This resolves the Linear List Hadwiger conjecture affirmatively.
View manuscript on GitHub → Open PDF (raw) →Family 158 The Euclidean plane cannot be colored with five colors. 1 paper
Proves that every five-coloring of the Euclidean plane has a monochromatic pair at distance one, with no restriction on the color classes. This advances the Hadwiger–Nelson problem: together with the classical seven-coloring, only six and seven remain possible chromatic numbers of the plane. ([Lean](lean/docs/158.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M301
The Euclidean plane is not five-colorable
We prove that every coloring of the Euclidean plane with five colors has a monochromatic unit-distance pair, with no regularity assumption on the color classes. Consequently, the chromatic number of the plane is either six or seven.
View manuscript on GitHub → Open PDF (raw) →Family 159 Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds. 1 paper
Proves Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. Quantitatively, for each fixed k ≥ 3, every subset of ` 1,N ` with no nonconstant k-term progression has size at most `C_kNexp[-c_k(log N)ε_k]`, with positive constants depending only on k. ([Lean](lean/docs/159.md))
PQC lens: Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
Primary manuscript on GitHub → Family index on GitHub →M302
Quasipolynomial Bounds for Arithmetic Progressions
We prove Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length.
View manuscript on GitHub → Open PDF (raw) →Family 160 Superexponential van der Waerden numbers. 1 paper
Resolves Erdős's superexponential-growth question for van der Waerden numbers. If `W_r(k)` is the least interval length forcing a monochromatic k-term progression in every r-coloring, then `W_r(k) kck lfloorlog_² r ` for an absolute c > 0, all r ≥ 2 and sufficiently large k, uniformly in r. In particular, `W_r(k)¹/k→∞` for each fixed r. ([Lean](lean/docs/160.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M303
Quantitative Superexponential Bounds for van der Waerden Numbers
We prove that there are absolute constants c > 0 and K0 such that `W_r(k) kck lfloorlog_² r ` for every `k≥ K_0` and r ≥ 2. Consequently `W_r(k)¹/k→∞` for each fixed r ≥ 2, giving a quantitative positive resolution of Erdős's superexponential-growth question, including the two-color case.
View manuscript on GitHub → Open PDF (raw) →Family 161 Counterexamples to Sidorenko’s conjecture and the forcing conjecture. 1 paper
Disproves Sidorenko's conjecture with a connected bipartite pattern on 35 vertices and 66 edges that occurs less frequently than in a random graph of the same edge density. The same pattern disproves the forcing conjecture of Skokan and Thoma: matching its density and the edge density of a constant graphon need not force quasirandomness. ([Lean](lean/docs/161.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M304
A counterexample to Sidorenko's conjecture
We disprove Sidorenko's conjecture with a bipartite graph on 35 vertices and 66 edges: its homomorphism density in some finite simple graph is smaller than the conjectured lower bound. The same connected graph also disproves the forcing conjecture: at one fixed density, asymptotically matching the edge and pattern densities does not imply quasirandomness.
View manuscript on GitHub → Open PDF (raw) →Family 162 Counterexamples to Ryser’s covering conjecture. 2 papers
Disproves Ryser's covering conjecture by constructing intersecting `(q+1)` -partite, `(q+1)` -uniform hypergraphs with covering number `q+1`, rather than the predicted bound q, for every sufficiently large prime q. A separate construction over extension fields also disproves Gyárfás's monochromatic tree-cover conjecture. ([Lean](lean/docs/162.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M305
Balanced counterexamples to Ryser's conjecture at prime orders
For every sufficiently large prime q, we construct a finite intersecting `(q+1)` -partite `(q+1)` -uniform hypergraph with covering number `q+1` and exactly `q+1` nonisolated vertices in each part. This disproves Ryser's covering conjecture, even for intersecting hypergraphs with equally sized parts.
View manuscript on GitHub → Open PDF (raw) →M306
A counterexample to Ryser's covering conjecture
For every sufficiently large prime `s≡2 pmod3` and every sufficiently large odd integer n, with the threshold depending on s, we construct an intersecting `(sn⁺¹)` -partite `(sn⁺¹)` -uniform hypergraph with covering number `sn⁺¹`. This disproves Ryser's covering conjecture in its intersecting case.
View manuscript on GitHub → Open PDF (raw) →Family 164 Hindman’s finite sums and products conjecture. 1 paper
Proves Hindman's finite sums and products conjecture: every finite coloring of the positive integers contains sets of any prescribed finite size whose nonempty subset sums and nonempty subset products all have one common color.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M307
Monochromatic finite sums and products in the positive integers
We prove Hindman's finite sums and products conjecture: for every finite coloring of the positive integers and every positive integer k, there is a k-element set whose nonempty subset sums and nonempty subset products all have the same color.
View manuscript on GitHub → Open PDF (raw) →Family 165 The Harary–Hill and Zarankiewicz crossing-number formulas. 2 papers
Resolves the Harary–Hill conjecture and Turán's brickyard problem in the Zarankiewicz formulation, determining the crossing numbers of every complete and complete bipartite graph. The result proves the optimality of the classical drawings among all plane drawings with continuous edge arcs. ([Lean](lean/docs/165.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M308
The crossing number of complete graphs
We prove the Harary–Hill conjecture: for every positive integer n, the ordinary crossing number of the complete graph Kn is ` frac14≤ft n2 ≤ft n-1 2 ≤ft n-2 2 ≤ft n-3 2.`
View manuscript on GitHub → Open PDF (raw) →M309
The crossing number of complete bipartite graphs
We prove the Zarankiewicz crossing-number conjecture, resolving Turán's brickyard problem. For all positive integers m, n, the ordinary crossing number of the complete bipartite graph `K(m,n)` is ` ≤ft m2 ≤ft m-1 2 ≤ft n2 ≤ft n-1 2.`
View manuscript on GitHub → Open PDF (raw) →Family 166 The higher-dimensional Erdős distinct-distances conjecture. 1 paper
For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝd determine at least `c_dn²/d` distinct distances, with `c_d 0` depending only on dimension. This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M310
The higher-dimensional Erdős distinct-distances conjecture
For every fixed integer d ≥ 3, we prove that every set of n ≥ 2 distinct points in ℝd determines at least `c_d n²/d` distinct distances, where `c_d 0` depends only on d. This resolves the higher-dimensional Erdős distinct-distances conjecture positively.
View manuscript on GitHub → Open PDF (raw) →Family 167 Planar distinct distances and unit-distance bounds. 2 papers
Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but `o(n)` points of any n-point planar set determine at least `n¹⁻ε` distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by `O(n⁴/³⁻δ)` for an absolute δ > 0. ([Lean](lean/docs/167.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M311
The weak pinned planar distance theorem
We prove the weak pinned Erdős distinct-distance conjecture. For every fixed ε > 0, all but `o(n)` points of any n-point planar set determine at least `n¹⁻ε` distinct nonzero distances.
View manuscript on GitHub → Open PDF (raw) →M312
A power saving for planar unit distances
We prove a power saving for the planar unit-distance problem: for some absolute `β 4/3`, every set of n points in the Euclidean plane determines `O(n^β)` unordered pairs at unit distance.
View manuscript on GitHub → Open PDF (raw) →Family 168 Combinatorial invariance of Kazhdan–Lusztig polynomials. 1 paper
Resolves the full combinatorial invariance conjecture: isomorphic Bruhat intervals in arbitrary Coxeter systems have identical equal-parameter Kazhdan–Lusztig polynomials. Thus the abstract order of the interval determines the polynomial, even across different Coxeter systems. ([Lean](lean/docs/168.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M313
Combinatorial invariance of Kazhdan–Lusztig polynomials
We prove that an isomorphism of Bruhat intervals in arbitrary Coxeter systems preserves their equal-parameter Kazhdan–Lusztig polynomials. This resolves the full combinatorial invariance conjecture positively.
View manuscript on GitHub → Open PDF (raw) →Family 169 Shareshian–Wachs elementary positivity. 1 paper
Resolves the elementary-positivity part of the Shareshian–Wachs conjecture: the chromatic quasisymmetric function of every natural unit interval graph has elementary-basis coefficients in ` N[q]`. The coefficients count explicitly described permutations, giving a combinatorial explanation of positivity. ([Lean](lean/docs/169.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M314
Elementary positivity of chromatic quasisymmetric functions
We prove that the chromatic quasisymmetric function of every natural unit interval graph is elementary-positive over ` N[q]`. This resolves the elementary-positivity part of the Shareshian–Wachs conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 170 Sharp logarithmic exponents for off-diagonal Ramsey numbers. 2 papers
For every fixed integer s ≥ 5, proves `r(s,t)=ts⁻¹/(log t)s⁻²⁺o(¹)` as `t→∞`, determining the logarithmic exponent and matching the classical upper bound at that scale. Here `r(s,t)` is the least number of vertices forcing an s-clique or a t-vertex independent set. ([Lean](lean/docs/170.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M315
The sharp logarithmic exponent of r(5,t)
We determine the sharp logarithmic exponent of the off-diagonal Ramsey number `r(5,t)`: ` r(5,t)= t⁴ (log t)³⁺o(¹) (t ∞).`
View manuscript on GitHub → Open PDF (raw) →M316
Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers
For every fixed integer s ≥ 6, we determine the sharp logarithmic exponent of the off-diagonal Ramsey number: ` r(s,t)= ts⁻¹ (log t)s⁻²⁺o(¹) (t ∞).`
View manuscript on GitHub → Open PDF (raw) →Family 171 The hypercube Ramsey conjecture. 1 paper
Resolves the Burr–Erdős hypercube Ramsey conjecture: the two-color Ramsey number of the n-dimensional cube is ` (2n)`. Thus every red-blue coloring of a complete graph on a universal constant times the cube's number of vertices contains a monochromatic copy of the cube.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M317
The hypercube Ramsey number has linear order
We prove that the two-color Ramsey number of the n-dimensional binary cube is at most `C2n`, where C is an absolute constant. This resolves positively the hypercube Ramsey conjecture of Burr and Erdős.
View manuscript on GitHub → Open PDF (raw) →Family 172 Classification of finite Euclidean Ramsey configurations. 1 paper
Classifies finite point configurations that occur monochromatically, at their original scale, in every finite coloring of sufficiently high-dimensional Euclidean space. The characterization is an algebraic condition over the coordinate field. It also disproves the Leader–Russell–Walters conjecture that every such configuration is a subset of a finite transitive set. ([Lean](lean/docs/172.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M318
A classification of finite Euclidean Ramsey configurations
We classify finite Euclidean Ramsey configurations by a necessary and sufficient tensor condition over their coordinate fields. The Ramsey property here concerns monochromatic congruent copies at the original scale under arbitrary finite colorings. The criterion shows that every nonempty subtransitive set and every nonempty set of at most five points on a circle is Ramsey. In particular, some Ramsey cyclic quadrilaterals are not subtransitive, disproving the necessity direction of the Leader–Russell–Walters conjectured characterization.
View manuscript on GitHub → Open PDF (raw) →Family 173 Seymour’s second-neighborhood conjecture. 1 paper
Proves Seymour's second-neighborhood conjecture: every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance exactly two as at directed distance one. Oriented graphs may be arbitrary apart from the exclusion of loops and oppositely directed edge pairs. ([Lean](lean/docs/173.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M319
A proof of Seymour’s second-neighborhood conjecture
We prove that every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance two as at directed distance one. This resolves Seymour's second neighborhood conjecture positively.
View manuscript on GitHub → Open PDF (raw) →Family 174 Deterministic construction of strong thin spanning trees. 2 papers
Resolves the strong thin-tree conjecture constructively. Every finite loopless k-edge-connected multigraph on at least two vertices has a spanning tree containing at most a universal `C/k` fraction of the edges of every cut. Such a tree can be found deterministically in polynomial time, even with binary-encoded parallel-edge multiplicities. ([Lean](lean/docs/174.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M320
The strong thin tree conjecture
We prove that every finite loopless k-edge-connected multigraph on at least two vertices, with k ≥ 1, has a spanning tree meeting each cut in at most `C/k` times the size of the cut, where C is a universal constant. This resolves the strong thin tree conjecture.
View manuscript on GitHub → Open PDF (raw) →M321
A polynomial-time construction of strong thin trees
We give a deterministic polynomial-time construction of strong thin trees. Given a finite k-edge-connected loopless multigraph on at least one vertex, the algorithm constructs a spanning tree meeting every cut in at most a `C/k` fraction of its edges, for a universal constant C. The running time is polynomial in the binary input length, including when parallel-edge multiplicities are encoded in binary.
View manuscript on GitHub → Open PDF (raw) →Family 175 Talagrand’s expectation thresholds, discrete convexity, and graph decompositions. 3 papers
Proves that integral and fractional expectation thresholds differ by at most a universal factor, and resolves Talagrand's discrete-convexity conjecture. An application proves the Ascoli–He–Park–Talagrand graph-decomposition conjecture: every graph's edges split into a universally bounded number of fixed pieces, each with containment threshold at most a universal constant times the original graph's integral expectation threshold. The pieces' embeddings need not agree on shared vertices. ([Lean](lean/docs/175.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M322
Graph Decompositions at the Integral Expectation Threshold
We prove the graph-decomposition conjecture of Ascoli, He, Park, and Talagrand. Every graph admits a partition into a universally bounded number of fixed edge pieces, each having ordinary containment threshold at most a universal constant times the original graph's integral expectation threshold. The partition is chosen before sampling the random host, and the separate embeddings of the pieces need not agree on shared vertices.
View manuscript on GitHub → Open PDF (raw) →M323
Integral and fractional expectation thresholds are equivalent
We prove Talagrand's conjecture that integral and fractional expectation thresholds are within a universal constant factor, with the same covering budget.
View manuscript on GitHub → Open PDF (raw) →M324
Talagrand’s discrete-convexity conjecture
We prove Talagrand's discrete-convexity conjecture. There is a universal integer k such that, whenever an arbitrary family has Bernoulli product measure at least `1-1/k`, the sets not contained in a union of k members admit a cover of total cost at most 1/2 at the same density.
View manuscript on GitHub → Open PDF (raw) →Family 176 The second Kahn–Kalai conjecture with an edge-count bound. 1 paper
Proves the second Kahn–Kalai conjecture: for every finite simple graph H with h ≥ 1 edges and at most n vertices, its appearance threshold in `G(n,p)` is at most `C p( E)(n,H)(1+log_2 h)`, with universal C. Here `p( E)` is the least density at which every subgraph of H has expected copy count at least 1/2. ([Lean](lean/docs/176.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M325
The second Kahn–Kalai conjecture
We prove the second Kahn–Kalai conjecture. For every finite simple graph H with h ≥ 1 edges and at most n vertices, the threshold for `G(n,p)` to contain an ordinary copy of H is at most `C p( E)(n,H)(1+log_2 h)`, where C is universal. Here `p( E)(n,H)` is the least density at which every subgraph of H has expected copy count at least one half.
View manuscript on GitHub → Open PDF (raw) →Family 177 Bounded-degree coboundary expanders. 1 paper
Constructs arbitrarily large finite d-dimensional simplicial complexes, for every d ≥ 3, with uniformly bounded vertex degrees and uniform 𝔽2 coboundary expansion in every degree below d. Together with the known graph and two-dimensional cases, this establishes the existence of such expanders in every positive dimension. ([Lean](lean/docs/177.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M326
Bounded-degree coboundary expanders in every dimension
For every integer d ≥ 3, we construct arbitrarily large finite d-dimensional simplicial complexes with uniformly bounded vertex degrees and uniform 𝔽2 coboundary expansion in every degree below d. Together with the known graph and two-dimensional cases, this establishes the existence of bounded-degree 𝔽2 coboundary expanders in every positive dimension.
View manuscript on GitHub → Open PDF (raw) →Family 178 Deterministic nonbipartite Ramanujan graphs in every fixed degree. 1 paper
For every fixed d ≥ 3, constructs a simple d-regular nonbipartite Ramanujan graph on every sufficiently large even number n of vertices, with every nonconstant adjacency eigenvalue strictly between `-2√(d-1)` and `2√(d-1)`. A deterministic algorithm outputs the full adjacency list in polynomial bit time, with exponent depending on d.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M327
Deterministic nonbipartite Ramanujan graphs in every fixed degree
For every fixed integer d ≥ 3, we give a deterministic algorithm that constructs a simple nonbipartite d-regular Ramanujan graph on every sufficiently large even number n of vertices. It outputs the full adjacency list in polynomially many bit operations, with an exponent that may depend on d. Every nonconstant adjacency eigenvalue lies strictly between `-2√(d-1)` and `2√(d-1)`.
View manuscript on GitHub → Open PDF (raw) →Family 179 The circulant Hadamard and Barker-sequence conjectures. 1 paper
Proves that real circulant Hadamard matrices exist exactly in orders 1 and 4, resolving the circulant Hadamard conjecture. Together with classical Barker-sequence results, this shows that binary sequences whose nontrivial aperiodic autocorrelations have magnitude at most 1 exist at lengths n > 1 exactly when `n 2,3,4,5,7,11,13 `. ([Lean](lean/docs/179.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M328
The circulant Hadamard conjecture
We prove the circulant Hadamard conjecture: a real circulant Hadamard matrix has order 1 or 4. As a consequence, Barker sequences of length greater than one exist exactly at lengths 2, 3, 4, 5, 7, 11, 13, proving the Barker-sequence conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 180 Barnette’s Hamiltonian-cycle conjecture. 1 paper
Proves that every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle, resolving Barnette's conjecture. Equivalently, every three-edge path in a finite simple cubic 3-vertex-connected bipartite Pfaffian graph lies in a Hamiltonian cycle. ([Lean](lean/docs/180.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M329
Paired states and Hamiltonian cycles in cubic bipartite planar graphs
We prove Barnette's conjecture: every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle.
View manuscript on GitHub → Open PDF (raw) →Family 181 The Erdős–Gallai cycle-decomposition conjecture. 1 paper
Proves that the edges of every finite simple undirected graph on n vertices can be partitioned into at most `Cn` simple cycles and single edges, for an absolute constant C. This resolves the Erdős–Gallai cycle-decomposition conjecture, bounding the number of pieces linearly even for dense graphs. ([Lean](lean/docs/181.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M330
A linear cycle-and-edge decomposition of every graph
We prove that every finite simple undirected graph on n vertices has an edge partition into at most `Cn` simple cycles and single edges, for an absolute constant C. This resolves the Erdős–Gallai cycle decomposition conjecture positively.
View manuscript on GitHub → Open PDF (raw) →Family 182 Power savings for intersective polynomial differences and prime arguments. 3 papers
For every fixed intersective integer polynomial h of degree k ≥ 2 with positive leading coefficient, proves that a subset of ` 1,N ` avoiding nonzero values `h(1),h(2), ` as differences has size `O_h(N¹⁻c_k)`, with `c_k 0` depending only on degree. Here intersective means having a root modulo every modulus. For prime arguments, a power saving also holds when h has a unit root modulo every modulus, with exponent allowed to depend on h. ([Lean](lean/docs/182.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M331
A power saving for intersective polynomial differences with an exponent depending only on the degree
An integer polynomial is intersective if it has a root modulo every positive integer. For each degree k ≥ 2, we prove that there is an exponent `c_k 0` such that every set `A 1,N ` whose differences avoid all nonzero values `h(1),h(2), `, where h is an intersective polynomial of degree k with positive leading coefficient, satisfies `|A|=O_h(N¹⁻c_k)`. The implied constant may depend on h, but the power-saving exponent depends only on its degree.
View manuscript on GitHub → Open PDF (raw) →M332
A Power Saving for Polynomial Differences at Prime Arguments
Let h be a fixed integer polynomial of degree at least two with positive leading coefficient, having a unit root modulo every positive integer. We prove that any set `A 1,N ` whose differences avoid all nonzero values `h(p)` at primes satisfies `|A|≤ C_hN¹⁻c_h`, where `c_h 0` and `C_h≥1` depend only on h. Thus the local unit-root condition gives a fixed power saving even when polynomial arguments are restricted to primes.
View manuscript on GitHub → Open PDF (raw) →M333
A power saving for square-difference-free sets
We prove that there are absolute constants c > 0 and C < ∞ such that every set `A 1,N ` with no nonzero square difference satisfies `|A|≤ C N¹⁻c`. This answers the fixed-power question posed by Green and Sawhney.
View manuscript on GitHub → Open PDF (raw) →Family 183 Power savings for planar halving lines and k-sets. 1 paper
Improves the planar halving-line bound to `O(n⁴/³⁻ε)` for sets with no three collinear and an absolute ε > 0. More generally, an n-point set with no three collinear has `O(n(k+1)¹/³⁻ε_⁰)` strictly separable k-subsets for `1≤ k≤ n/2`, with an absolute `ε_0 0`. The constants and positive exponents are nonquantitative. ([Lean](lean/docs/183.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M334
A power saving for planar halving lines
There are absolute constants ε > 0 and C such that every sufficiently large even n-point set in the plane with no three collinear has at most `Cn⁴/³⁻ε` unordered halving pairs. This gives a power saving over the classical `O(n⁴/³)` bound for planar halving lines. The proof is nonquantitative and does not supply explicit constants.
View manuscript on GitHub → Open PDF (raw) →Family 184 Correspondence coloring with a fixed forbidden subgraph. 2 papers
Proves the Alon–Krivelevich–Sudakov coloring conjecture in correspondence-coloring form: graphs avoiding any fixed subgraph F need `O_F( /log )` colors when their maximum degree Δ is sufficiently large. Also proves the Ajtai–Erdős–Komlós–Szemerédi independence conjecture: for fixed r ≥ 4, every n-vertex Kr-free graph of average degree d ≥ 2 has an independent set of size ` Omega_r(nlog d/d)`. ([Lean](lean/docs/184.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M335
Correspondence coloring graphs with a forbidden clique
For every fixed integer r ≥ 4, we prove that every Kr-free graph of sufficiently large maximum degree Δ has correspondence chromatic number `O_r( /log )`. This resolves the Alon–Krivelevich–Sudakov coloring conjecture in the stronger correspondence-coloring form. The same bound, with a constant depending on F, holds when any fixed graph F is excluded as an ordinary subgraph. Ordinary and list coloring satisfy the same bounds.
View manuscript on GitHub → Open PDF (raw) →M336
A logarithmic independence bound for clique-free graphs
For every fixed integer r ≥ 4, every Kr-free graph on n vertices with average degree d ≥ 2 has an independent set of size at least `c_r nlog d/d`, where `c_r 0` depends only on r. This proves the fixed-clique-size independence conjecture of Ajtai, Erdős, Komlós and Szemerédi.
View manuscript on GitHub → Open PDF (raw) →Family 185 Counterexamples to infinite matroid intersection and packing/covering. 1 paper
Disproves the unrestricted infinite matroid intersection and packing/covering conjectures in ZFC, using two self-dual partitional matroids on a countably infinite ground set. The same examples answer Joó's partitional-matroid question negatively. They are neither finitary nor cofinitary, so Nash-Williams' original finitary conjecture remains outside the result. ([Lean](lean/docs/185.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M337
A Counterexample to the Infinite Matroid Packing/Covering Conjecture
We construct in ZFC two self-dual partitional matroids on a countably infinite common ground set that admit neither a packing/covering partition nor an intersection witness. This disproves the unrestricted infinite matroid packing/covering and intersection conjectures and answers Joó's question for two partitional matroids negatively. The examples are neither finitary nor cofinitary.
View manuscript on GitHub → Open PDF (raw) →Family 186 Uniform influence and sharp thresholds for graph and hypergraph properties. 2 papers
Proves the Friedgut–Kalai threshold-width conjectures for graphs and fixed-uniformity hypergraphs. For fixed `0 ε 1/2`, every nontrivial increasing relabeling-invariant property crosses from probability ε to `1-ε` within width `O((log n)⁻²)` for graphs and `O_r((log n)⁻r/(r⁻¹))` for r-uniform hypergraphs, r ≥ 3. The hypergraph influence bound also applies to nonmonotone properties. ([Lean](lean/docs/186.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M338
A uniform influence bound for hypergraph properties
For every fixed integer r ≥ 3, we prove that every relabeling-invariant Boolean property of simple r-uniform hypergraphs on n vertices satisfies ` Var _p(f)≤ C_r I_p(f)/(log n)r/(r⁻¹)`. The constant depends only on r, and the bound holds uniformly for all `0 p 1` without a monotonicity assumption. For increasing properties, it gives the corresponding threshold-width bound with exponent `r/(r-1)`, proving the hypergraph threshold-width conjecture of Friedgut and Kalai.
View manuscript on GitHub → Open PDF (raw) →M339
A Sharp Threshold Bound for Monotone Graph Properties
We prove the Friedgut–Kalai sharp-threshold conjecture. For every integer n ≥ 2, every nontrivial increasing family of graphs on n vertices invariant under all vertex permutations, and every `0 ε 1/2`, the edge probabilities at which its probability equals ε and `1-ε` differ by at most `Clog(1/(2ε))/(log n)²`, for a universal constant C.
View manuscript on GitHub → Open PDF (raw) →Family 187 Snaky in 21 Maker moves. 1 paper
Settles the Snaky achievement problem: Maker can force the six-cell Snaky shape within 21 of its own moves on the initially empty infinite square board. Maker moves first, each player claims one free cell per turn, and translations, rotations and reflections count as wins. ([Lean](lean/docs/187.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M340
Snaky in 21 Maker moves
We prove that Maker can achieve the Snaky hexomino within 21 actual Maker moves against arbitrary legal Breaker play on the initially empty infinite square board. The same bound holds on a `17×17` square; in fact, Maker can confine its claims to a fixed 251-cell board.
View manuscript on GitHub → Open PDF (raw) →Family 188 The sharp terminal leave in random triangle removal. 1 paper
Starting from the complete graph on n vertices, repeatedly delete a uniformly chosen remaining triangle. The terminal edge count is asymptotic to `n³/²/(2 sqrt2)`, with mean-square convergence after normalization by n3/2. This proves the triangle case of the Joos–Kühn sharp-constant conjecture. ([Lean](lean/docs/188.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M341
The sharp terminal leave in random triangle removal
Starting from the complete graph on n vertices, repeatedly remove the three edges of a uniformly chosen remaining triangle. We prove that the number of edges left at termination, divided by n3/2, converges in L2 to `1/(2 sqrt2)`. This proves the triangle case of the sharp-constant conjecture of Joos and Kühn. In particular, the same limit holds in probability and for the normalized expectation.
View manuscript on GitHub → Open PDF (raw) →Family 189 Cycle–clique Ramsey numbers. 1 paper
Proves the Erdős–Faudree–Rousseau–Schelp conjecture: `R(C_m,K_n)=(m-1)(n-1)+1` for every `m≥ n≥3`, except `R(C_3,K_3)=6`. This is the exact threshold forcing a red m-cycle or a blue n-clique in every red–blue coloring of a complete graph. ([Lean](lean/docs/189.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M342
Cycle–clique Ramsey numbers
We prove that `R(C_m,K_n)=(m-1)(n-1)+1` for every pair of integers `m≥ n≥3` other than `(m,n)=(3,3)`, for which `R(C_3,K_3)=6`. This establishes the cycle–clique conjecture of Erdős, Faudree, Rousseau and Schelp. The proof combines expansion in a minimal counterexample with a large-clique lemma and an optimization of paths joining clique vertices. These arguments reduce the remaining cases to `3, 099` finite parameter-pattern instances, which are excluded by two exact implementations of proved inference rules.
View manuscript on GitHub → Open PDF (raw) →Family 190 Polynomial removal fails for ordered binary matrices. 1 paper
Disproves polynomial ordered binary matrix removal with one fixed `66×66` zero–one pattern. Matrices can require many binary-entry changes to become pattern-free while their copy density is smaller than every proposed polynomial bound in that distance. Copies preserve row and column orders and match both zeros and ones. ([Lean](lean/docs/190.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M343
Polynomial removal fails for ordered binary matrices
We construct a fixed `66×66` binary matrix for which ordered matrix removal has no polynomial bound. This disproves the polynomial ordered binary matrix-removal conjecture. Ordered copies preserve the separate row and column orders and match both zeros and ones; removal permits changing entries in either direction.
View manuscript on GitHub → Open PDF (raw) →Family 191 A power improvement in the Heilbronn triangle lower bound. 1 paper
For every sufficiently large n, constructs n points in the unit square such that every triangle has area at least `n⁻²⁺c` for one absolute c > 0. This disproves the conjectured almost-n−2 upper bound in Heilbronn's triangle problem, which asks how large the smallest determined triangle can be. ([Lean](lean/docs/191.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M344
A power improvement in the Heilbronn triangle lower bound
There are absolute constants `,c_1 0` such that, for every sufficiently large integer n, one can choose n points in the unit square so that every triangle they determine has area at least `c_1n⁻²⁺ `. Thus the almost n−2 upper-bound formulation of Heilbronn's triangle problem is false. The exponent η is fixed but extremely small.
View manuscript on GitHub → Open PDF (raw) →Family 192 Boolean functions violate the square-root degree bound by arbitrary factors. 1 paper
Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every C > 0, there is a sign-valued Boolean function f with `∑_i f( i ) C√(deg(f))`. Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor. ([Lean](lean/docs/192.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M345
Unbounded Violations of the Square-Root Degree Bound
We disprove the Gopalan–Servedio square-root conjecture, even up to an arbitrary constant factor. For every real C > 0, there is a nonconstant Boolean function `f: -1,1 n→ -1,1 ` on a finite sign cube such that ` ∑(i=1)n f( i ) C√(deg(f)).` Here ` f( i )` is the linear Fourier coefficient associated with the ith input, and `deg(f)` is the degree of the real multilinear polynomial representing f.
View manuscript on GitHub → Open PDF (raw) →Family 193 Serre’s intersection-multiplicity conjecture. 1 paper
Proves strict positivity of Serre's intersection multiplicity ` (M,N)` for nonzero finitely generated modules over any regular local ring, provided `M otimes_R N` has finite length and `dim M+dim N=dim R`. This resolves the positivity conjecture, including ramified mixed characteristic.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M346
Positivity of Serre's Intersection Multiplicity
We prove Serre's positivity conjecture, including ramified mixed characteristic. If two nonzero finitely generated modules over a regular local ring have tensor product of finite length and complementary dimensions, then their intersection multiplicity is strictly positive.
View manuscript on GitHub → Open PDF (raw) →Family 194 Lech’s multiplicity conjecture. 1 paper
Proves `e(R)≤ e(S)` for every flat local homomorphism of nonzero Noetherian local rings, where e is Hilbert–Samuel multiplicity. This resolves Lech's conjecture in every dimension and characteristic. ([Lean](lean/docs/194.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M347
Lech's multiplicity conjecture
We prove Lech's multiplicity conjecture: Hilbert–Samuel multiplicity cannot decrease under a flat local homomorphism of nonzero Noetherian local rings. The result holds in arbitrary dimension, with no restrictions on the residue fields or characteristics.
View manuscript on GitHub → Open PDF (raw) →Family 195 A counterexample to the small Cohen–Macaulay module conjecture. 1 paper
Constructs a three-dimensional complete Noetherian normal local domain over ℂ with no nonzero finitely generated maximal Cohen–Macaulay module. A three-dimensional local domain essentially of finite type over ℂ has the same property, disproving the domain form of the small Cohen–Macaulay module conjecture.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M348
A Complete Local Domain without a Small Cohen–Macaulay Module
We construct a three-dimensional complete Noetherian normal local domain containing ℂ, with residue field ℂ, that has no nonzero finitely generated maximal Cohen–Macaulay module. This disproves the domain form of the small Cohen–Macaulay module conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 196 A counterexample to Kaplansky’s zero-divisor conjecture. 1 paper
Constructs a finitely presented torsion-free group G whose group algebra ` F_2[G]` has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space. ([Lean](lean/docs/196.md))
PQC lens: Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
Primary manuscript on GitHub → Family index on GitHub →M349
A Torsion-Free Group Algebra with Zero Divisors
We disprove Kaplansky's zero-divisor conjecture by constructing a finitely presented torsion-free group G for which ` F_2[G]` has nonzero zero divisors. The group admits a finite two-dimensional classifying space.
View manuscript on GitHub → Open PDF (raw) →Family 198 A counterexample to finitistic-dimension finiteness. 1 paper
Constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture. ([Lean](lean/docs/198.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M354
An algebra of infinite little finitistic dimension
We construct a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 199 Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures. 2 papers
Constructs finite-dimensional algebras over a characteristic-two rational-function field that disprove the Auslander–Reiten and Gorenstein-projective conjectures, and Tachikawa's second conjecture. An associated endomorphism algebra also disproves the classical, generalized and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. The counterexamples persist under every extension of the base field. ([Lean](lean/docs/199.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M355
An explicit counterexample to the Auslander-Reiten conjecture
We disprove the Auslander–Reiten conjecture for Artin algebras. We construct a finite-dimensional algebra Λ over `k= F_2(q,H_1,H_2)` and a finite-dimensional nonprojective left module Z such that ` Ext i_ (Z,Z)= Ext i_ (Z, )=0` for every i > 0. The module is Gorenstein-projective, so the same example also disproves the Gorenstein-projective conjecture. Both counterexamples persist after every extension of k.
View manuscript on GitHub → Open PDF (raw) →M356
A counterexample to Tachikawa's second conjecture
We disprove Tachikawa's second conjecture by constructing a finite-dimensional symmetric algebra over `k= F_2(q,H_1,H_2)` with a finite-dimensional nonprojective module whose self-extension groups vanish in every positive degree. The associated endomorphism algebra also gives counterexamples to the classical, generalized, and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. These conclusions persist after every extension of k.
View manuscript on GitHub → Open PDF (raw) →Family 200 Eisenbud–Green–Harris and lex-plus-powers. 2 papers
Proves the Eisenbud–Green–Harris and lex-plus-powers conjectures over every characteristic-zero field. Any homogeneous ideal containing a regular sequence, of arbitrary length and degrees at least two, admits a lex-plus-powers ideal with the same Hilbert function and no smaller graded Betti numbers.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M357
The Artinian Lex-Plus-Powers Betti Theorem
We prove the Eisenbud–Green–Harris and lex-plus-powers conjectures over every characteristic-zero field for homogeneous regular sequences of any positive length, with degrees at least two. For every homogeneous ideal containing such a sequence, the corresponding lex-plus-powers ideal has the same Hilbert function and at least as large a graded Betti number in every homological and internal degree.
View manuscript on GitHub → Open PDF (raw) →M358
Commuting Division-Coefficient Forms and the Artinian Eisenbud–Green–Harris Conjecture
We prove the Eisenbud–Green–Harris conjecture over every characteristic-zero field for homogeneous regular sequences of any positive length, with degrees at least two. Thus every homogeneous ideal containing such a sequence has the Hilbert function of a monomial ideal containing the corresponding pure powers.
View manuscript on GitHub → Open PDF (raw) →Family 201 A counterexample to Kurosh’s division-ring problem. 1 paper
Constructs a countable characteristic-zero division ring that is algebraic over its center and generated by two elements over that center, but has infinite dimension over it. This answers Kurosh's division-ring problem on local finiteness negatively.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M359
A Counterexample to Kurosh’s Division-Ring Problem
We construct a countable division ring of characteristic zero that is algebraic over its center, generated by two elements as an algebra over that center, and infinite-dimensional over it. This gives a negative answer to the Kurosh problem for division rings.
View manuscript on GitHub → Open PDF (raw) →Family 202 The blockwise Alperin weight conjecture. 1 paper
Proves the numerical blockwise Alperin weight conjecture for every prime and every finite group: the number of irreducible Brauer characters in a block equals the number of conjugacy classes of its weights.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M360
The Blockwise Alperin Weight Conjecture
We prove the numerical blockwise Alperin weight conjecture for every finite group and every prime p. For each p-block B, the number of irreducible Brauer characters in B equals the number of conjugacy classes of B-weights.
View manuscript on GitHub → Open PDF (raw) →Family 203 Donovan's conjecture over fields and complete mixed-characteristic DVRs. 2 papers
Proves Donovan's conjecture: over each fixed algebraically closed field of characteristic p, blocks of finite groups with bounded defect-group order have only finitely many Morita-equivalence classes, for every prime p. Also proves the integral form over each fixed complete mixed-characteristic discrete valuation ring with algebraically closed residue field.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M361
Donovan's Conjecture over Algebraically Closed Fields
We prove Donovan's conjecture over every algebraically closed field K of characteristic p > 0. For each fixed K and bound on defect-group order, blocks of finite groups represent only finitely many K-linear Morita equivalence classes. The defect groups need not be abelian, and the result includes p = 2.
View manuscript on GitHub → Open PDF (raw) →M362
Integral Donovan Finiteness over Witt Vectors
We prove integral Donovan finiteness: for every prime p and positive integer M, the blocks of all finite groups with defect groups of order at most M have only finitely many Morita equivalence classes over `W( F _p)`. The defect groups need not be abelian, and the result includes p = 2. The same bounded-defect finiteness holds over each fixed complete discrete valuation ring of characteristic zero with algebraically closed residue field of characteristic p, including ramified rings.
View manuscript on GitHub → Open PDF (raw) →Family 204 Tensor saturation for even spin groups. 1 paper
Proves saturation factor one for ` Spin (2n)`, n ≥ 2: for three dominant integral weights whose sum lies in the root lattice, an invariant at any common positive integral dilation already gives an invariant at the original weights. This resolves the type-D part of the simply-laced saturation conjecture.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M363
Tensor saturation for even spin groups
We prove the saturation conjecture for ` Spin (2n)`, n ≥ 2. If three dominant integral weights sum to an element of the root lattice, then the existence of a nonzero tensor invariant after a positive integral dilation implies the existence of one at the original weights.
View manuscript on GitHub → Open PDF (raw) →Family 205 Saxl’s conjecture and universal tensor squares. 2 papers
Proves Saxl's conjecture: the tensor square of every staircase representation contains every irreducible complex representation of the corresponding symmetric group. More generally, every Sn with `n 2,4,9 ` has an irreducible representation whose tensor square contains all irreducibles. ([Lean](lean/docs/205.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M364
Universal Tensor Squares for Symmetric Groups
For every positive integer n other than 2, 4, and 9, we prove that some irreducible complex representation of Sn has a tensor square containing every irreducible representation. This resolves the tensor square conjecture for symmetric groups affirmatively.
View manuscript on GitHub → Open PDF (raw) →M365
A Cyclic Polytabloid Proof of Saxl's Conjecture
For every staircase partition, we prove that the tensor square of the corresponding irreducible complex representation of the symmetric group contains every irreducible representation of that group. This proves Saxl's conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 206 Finite lattice representation and undecidability. 2 papers
Some finite lattices are not congruence lattices of any finite algebra, answering the finite lattice representation problem negatively. Moreover, no algorithm decides whether a finite lattice has such a representation, or whether it is a full subgroup interval of a finite group. ([Lean](lean/docs/206.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M366
Finite congruence lattices: characterization and undecidability
We give an explicit colored-graph characterization of the finite nonempty lattices that occur as full congruence lattices of finite algebras, and prove that deciding this representation property is undecidable. In particular, the finite lattice representation problem has a negative answer. We also prove that recognition of full subgroup intervals in finite groups is undecidable.
View manuscript on GitHub → Open PDF (raw) →M367
A negative solution to the finite lattice representation problem
We give a negative solution to the finite lattice representation problem. We prove that there is a finite nonempty lattice that is not the full congruence lattice of any finite nonempty algebra of any finite signature.
View manuscript on GitHub → Open PDF (raw) →Family 207 The ℓ¹-Bass conjecture for all discrete groups. 2 papers
Proves the ℓ1-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over `ℓ¹(G)` are supported on finitely many finite-order conjugacy classes. The algebraic companion proves the integral Bass trace conjecture and Kaplansky's idempotent conjecture for torsion-free groups over every commutative unital characteristic-zero domain. ([Lean](lean/docs/207.md))
PQC lens: Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
Primary manuscript on GitHub → Family index on GitHub →M368
The ℓ¹-Bass Conjecture for Discrete Groups
We prove the ℓ1-Bass conjecture for every discrete group. The Hattori–Stallings trace of every idempotent matrix over the complex ℓ1 group algebra is supported on finitely many conjugacy classes of finite-order elements.
View manuscript on GitHub → Open PDF (raw) →M369
The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture
We prove the complex group-ring Bass trace conjecture for every discrete group: the Hattori–Stallings trace of a finitely generated projective module over its complex group ring is supported on conjugacy classes of finite-order elements. As a consequence, for every torsion-free group G and every commutative unital domain R of characteristic zero, the only idempotents in `RG` are 0 and 1. This proves Kaplansky's idempotent conjecture in characteristic zero.
View manuscript on GitHub → Open PDF (raw) →Family 208 Finite symmetric tensor categories and the Verlinde tower. 1 paper
Proves that every finite symmetric tensor category over an algebraically closed field k of characteristic p > 0 admits a k-linear exact faithful strong symmetric monoidal fiber functor to a higher Verlinde category `Ver(pn)`. The level may depend on the category, and the theorem includes characteristic two, resolving the finite case of the Benson–Etingof–Ostrik conjecture.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M370
Fiber functors for finite symmetric tensor categories in positive characteristic
We prove the finite case of the Benson–Etingof–Ostrik conjecture: every finite symmetric tensor category over an algebraically closed field k of characteristic p > 0 admits a k-linear exact faithful strong symmetric monoidal functor to a finite higher Verlinde category ` Ver (pn)(k)`. The result includes characteristic two, and the level n may depend on the category.
View manuscript on GitHub → Open PDF (raw) →Family 209 Integral counterexamples to Gersten’s conjecture. 2 papers
Disproves unrestricted integral Gersten injectivity in degrees 3 and 5. Two explicit two-dimensional ramified regular local rings of mixed characteristic `(0,5)` have nonzero integral K-theory classes that vanish over their fraction fields.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M371
An integral counterexample to Gersten's conjecture
We construct a two-dimensional ramified regular local ring A of mixed characteristic `(0,5)` for which `K_5(A)→ K_5( Frac A)` has a nonzero kernel. This disproves Gersten's conjecture in its unrestricted integral form.
View manuscript on GitHub → Open PDF (raw) →M372
An integral degree-three Gersten counterexample
We construct a two-dimensional ramified regular local ring A in mixed characteristic `(0,5)` for which the integral map `K_3(A)→ K_3( Frac A)` has nonzero kernel. This gives a negative answer to the unrestricted integral Gersten conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 210 Foulkes' conjecture for sixth powers and quadratic stabilization. 2 papers
Proves the sixth case of Foulkes' conjecture: ` Sym ⁶( Sym bV)` embeds equivariantly in ` Sym b( Sym ⁶V)` for every b ≥ 6 and finite-dimensional complex V. More generally, the canonical multiplication map ` Sym b( Sym aV)→ Sym a( Sym bV)` is surjective for a ≥ 2 and `b≥ a(a-1)`, giving dimension-independent quadratic stabilization. ([Lean](lean/docs/210.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M373
Foulkes' conjecture for the sixth symmetric power
We prove the sixth-symmetric-power case of Foulkes' conjecture. For every integer b ≥ 6 and every finite-dimensional complex vector space V, there is a ` GL (V)` -equivariant injection ` Sym ⁶( Sym b V) Sym b( Sym ⁶ V)`.
View manuscript on GitHub → Open PDF (raw) →M374
Quadratic stabilization of the canonical Foulkes–Howe map
For every finite-dimensional complex vector space V, we prove that the canonical Foulkes–Howe map ` Sym b( Sym a V) Sym a( Sym b V)` is surjective whenever a ≥ 2 and `b≥ a(a-1)`. This gives a quadratic stabilization bound independent of `dim V`.
View manuscript on GitHub → Open PDF (raw) →Family 211 The geometric phase diagram, diffusion, and spectra of random planar maps. 7 papers
Critical Fortuin–Kasteleyn planar maps converge to Liouville quantum gravity spheres for `0 q≤4` and to the Brownian continuum random tree for q > 4, establishing the surface-to-tree geometric transition. For FK–Ising and spanning-tree-weighted maps, stationary random walks converge to Liouville Brownian motion on the limiting sphere. The FK–Ising spectral result also gives convergence of eigenvalues and heat traces, using the stated Brownian/LQG inputs. ([Lean](lean/docs/211.md))
PQC lens: Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
Primary manuscript on GitHub → Family index on GitHub →M375
Random Walks on Critical FK–Ising Maps and Liouville Brownian Motion
We prove that stationary random walks on critical spherical FK–Ising planar maps converge to Liouville Brownian motion on the ordinary unit-area ` sqrt3` -quantum sphere, using the geometric and electrical results of the spectral companion. The walk chooses uniformly among all incident half-edges, retaining loops and multiple edges. With the corner measure as the stationary law, the deterministic time acceleration is exactly the number of map edges.
View manuscript on GitHub → Open PDF (raw) →M376
Spectral convergence for critical FK–Ising planar maps
Using the conformal and metric-measure companion results and the stated Brownian/Liouville quantum gravity inputs, we prove spectral convergence for critical spherical FK–Ising maps to Liouville Brownian motion on the ordinary unit-area ` sqrt3` -quantum sphere. The discrete walk has total attempt rate one, uses every map edge including loops and multiplicities, and has the corner measure as its stationary law.
View manuscript on GitHub → Open PDF (raw) →M377
A Linear Clock for Random Walk on Tree-Weighted Planar Maps
We prove that stationary random walk on a planar map sampled with weight equal to its number of spanning trees converges to Liouville Brownian motion on the unit-area ` sqrt2` -Liouville quantum sphere. The convergence retains the conditional path law jointly with the measured metric space. The walk chooses uniformly among all incident half-edges and starts from the stationary degree measure. For total attempt rate one and the continuum Dirichlet form with factor 1/2, the time acceleration is exactly the number of map edges.
View manuscript on GitHub → Open PDF (raw) →M378
Canonical conformal limits of subcritical FK planar maps
For every fixed `0 q 4`, we prove joint convergence of spherical Fortuin–Kasteleyn planar maps in their flag-triangle uniformization to the corresponding unit-area Liouville quantum gravity sphere decorated by an independent conformal loop ensemble. The convergence includes the area measure, deterministically rescaled graph distances between all vertex pairs, and the full nested interface collection, with interfaces converging uniformly up to reparameterization.
View manuscript on GitHub → Open PDF (raw) →M379
The critical Liouville quantum sphere and geometric limits of FK maps at q=4
We construct the field and area law of the unit-area critical Liouville quantum sphere as a limit of ordinary subcritical quantum spheres, and equip it with its critical intrinsic metric. We then prove that spherical Fortuin–Kasteleyn planar maps at q = 4, embedded by their equilateral flag uniformizations, converge jointly to this sphere decorated by an independent nested conformal loop ensemble CLE4.
View manuscript on GitHub → Open PDF (raw) →M380
Metric-measure limits of subcritical FK and spanning-tree planar maps
We resolve the finite spherical cases of Gwynne and Miller's graph-metric conjecture for critical Fortuin–Kasteleyn maps at each fixed `q (0,4)` and for uniform spanning-tree-decorated maps. After deterministic rescaling of graph distances, these maps, equipped with the vertex probability measure proportional to degree, converge in Gromov–Hausdorff–Prokhorov law to their ordinary unit-area Liouville quantum gravity spheres. Distances use every primal edge, and convergence holds through all positive integer edge counts.
View manuscript on GitHub → Open PDF (raw) →M381
Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four
We prove the finite-volume continuum-random-tree prediction for critical Fortuin–Kasteleyn planar maps at every fixed q > 4. After rescaling graph distances by a constant times n−1/2, an n-edge map with normalized degree measure converges to the Brownian continuum random tree in the Gromov–Hausdorff–Prokhorov topology. The convergence holds through all positive integer sizes.
View manuscript on GitHub → Open PDF (raw) →Family 212 Planar first-passage geometry and the absence of bigeodesics. 2 papers
Proves that planar first-passage percolation has no doubly infinite geodesic for iid nonnegative nonatomic edge weights when the minimum of four weights has finite second moment. For exponential weights, the limit shape is strictly convex with C1 boundary. Differentiability also holds for every Gamma law with positive shape and rate. ([Lean](lean/docs/212.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M382
No bigeodesics in planar first-passage percolation
We prove that planar first-passage percolation with independent identically distributed nonnegative nonatomic edge weights has almost surely no doubly infinite geodesic, provided the minimum of four independent weights has finite second moment. This resolves the planar no-bigeodesics conjecture under that moment assumption. The conclusion rules out all bigeodesics simultaneously, without any regularity assumption on the limit shape.
View manuscript on GitHub → Open PDF (raw) →M383
Strict convexity and differentiability of the planar exponential first-passage limit shape
We prove that the limit shape of undirected nearest-neighbor first-passage percolation on ℤ2 with independent exponential edge weights is strictly convex and has a C1 boundary. This resolves the strict convexity and differentiability conjectures for the planar exponential model. More generally, we prove differentiability of the time-constant norm for every Gamma edge-weight law with positive shape and rate.
View manuscript on GitHub → Open PDF (raw) →Family 213 Critical percolation on every quasi-transitive graph. 2 papers
Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with `p_c 1`: at the critical probability, there is almost surely no infinite cluster. The family also establishes this conclusion for both nearest-neighbor bond and site percolation on ℤ3. ([Lean](lean/docs/213.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M384
Critical bond and site percolation on the cubic lattice
We prove that nearest-neighbor Bernoulli bond and site percolation on ℤ3 have no infinite cluster at their respective critical parameters. The proof combines a finite connection inequality for independent hyperedges with a finite-scale extension estimate and an adaptive exploration.
View manuscript on GitHub → Open PDF (raw) →M385
No percolation at criticality on quasi-transitive graphs
We prove that critical Bernoulli bond percolation has no infinite cluster on any infinite connected locally finite quasi-transitive graph with critical probability less than one. This resolves the bond form of the criticality conjecture of Benjamini and Schramm.
View manuscript on GitHub → Open PDF (raw) →Family 214 The Benjamini–Schramm nonuniqueness conjecture. 1 paper
Proves `p_c p_u` for Bernoulli bond percolation on every infinite connected locally finite nonamenable quasi-transitive graph, resolving the Benjamini–Schramm nonuniqueness conjecture. Thus there is a nonempty range of probabilities with infinitely many infinite clusters. A stronger operator bound also establishes the critical triangle condition. ([Lean](lean/docs/214.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M386
Nonuniqueness of percolation on nonamenable quasi-transitive graphs
We prove the bond-percolation nonuniqueness conjecture of Benjamini and Schramm: every infinite connected, locally finite, nonamenable quasi-transitive graph has a nonempty interval of parameters for which Bernoulli bond percolation almost surely has infinitely many infinite clusters. We also prove Hutchcroft's stronger operator-threshold conjecture, establishing `p_c p(2→2)≤ p_u`.
View manuscript on GitHub → Open PDF (raw) →Family 215 Canonical `O(3)` continuum limit and exact `O(4)` mass asymptotics. 5 papers
Constructs the canonical continuum limit of the two-dimensional nearest-neighbor `O(3)` model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice `O(4)` model, determines the exact leading asymptotic of the full transfer gap, `m(lat)(β) sim32eπ/⁴⁻¹/² sqrtβ e⁻πβ`. The family also proves exponential spin-correlation decay for two-dimensional nearest-neighbor `O(n)` models with n ≥ 3 at every positive temperature. ([Lean](lean/docs/215.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M387
The canonical massive continuum limit of the two-dimensional O(3) model
We construct a canonical interacting massive continuum limit of the two-dimensional nearest-neighbor `O(3)` model with unit-length spins, no external field, and no topological term. Normalized by susceptibility and second-moment correlation length, the limit exists as the bare coupling tends to infinity through all positive real values, without selecting subsequences. The limiting fields satisfy the Osterwalder–Schrader axioms and have a nonzero connected four-point correlation on separated time supports.
View manuscript on GitHub → Open PDF (raw) →M388
An Isolated Particle Pole for the Two-Dimensional O(3) Spin Field
We consider the continuum spin field constructed from the nearest-neighbor two-dimensional `O(3)` model by the fixed prescription of the companion paper. We prove that its vector two-point spectral measure has a positive atom corresponding to the lowest mass, separated by a positive gap from all remaining mass support.
View manuscript on GitHub → Open PDF (raw) →M389
Exact mass asymptotics for the two-dimensional O(4) lattice model
For the nearest-neighbor `O(4)` model on the square lattice at inverse temperature β, we prove the exact low-temperature asymptotic ` m(lat)(β) 32exp(π/4-1/2) sqrtβ exp(-πβ) (β→∞).` Here the mass is the gap of the…
View manuscript on GitHub → Open PDF (raw) →M390
Sharp mass bounds for the two-dimensional O(4) model
We prove sharp mass bounds for the two-dimensional nearest-neighbor `O(4)` model. For all sufficiently large inverse couplings β, the full transfer gap, including rotation-invariant local-observable sectors, is bounded above and below by positive multiples of ` sqrtβ e⁻πβ`. We also prove that the model has a unique periodic local limit and a positive full gap at every finite β > 0, resolving the all-temperature lattice mass-generation conjecture for this periodic state.
View manuscript on GitHub → Open PDF (raw) →M391
Exponential decay in two-dimensional classical O(n) models
We prove exponential decay of two-point correlations for the classical nearest-neighbor `O(n)` model on the square lattice, for every n ≥ 3 and every finite positive inverse temperature. The estimate is uniform over finite free-boundary subgraphs and bounded nonnegative edge strengths. This resolves positively the all-temperature exponential spin-decay conjecture for these models.
View manuscript on GitHub → Open PDF (raw) →Family 216 Critical and near-critical XY scaling and BKT universality. 6 papers
For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations `C(β_c)(r) Ar⁻¹/⁴(log r)¹/⁸` and the Berezinskii–Kosterlitz–Thouless essential singularity `√(β_c-β)log (β)→ B`, with `A,B 0` after the free-box thermodynamic limit. For finite square-symmetric interactions containing nearest neighbors, discrete Gaussian heights converge to Gaussian fields throughout the rough phase, including its threshold, along geometric torus sizes.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M392
The critical logarithmic correction for the planar XY model
We prove the critical logarithmic correction for the nearest-neighbor cosine XY model on the square lattice. Let `C(b_c)(r)` be the two-point correlation at critical inverse temperature, obtained by taking the free-box thermodynamic limit while the two sites remain r lattice steps apart along a coordinate axis. Then ` C(b_c)(r)=B(XY)r⁻¹/⁴(log r)¹/⁸(1+o(1)), B(XY) (0,∞),` as `r→∞`.
View manuscript on GitHub → Open PDF (raw) →M393
Critical Center Magnetization in the Planar XY Model
Assuming the stated critical-height, local-renormalization, and spin-field inputs from the companion papers, we determine the center magnetization of the planar XY model in a square with aligned boundary spins at its mass-defined critical threshold. As `n→∞`, the magnetization is `A(XY)n⁻¹/⁸(log n)¹/¹⁶(1+o(1))`, where `A(XY)` is a finite, strictly positive model-specific constant.
View manuscript on GitHub → Open PDF (raw) →M394
The Critical Spin Field of the Planar XY Model
Using the companion critical Bessel-height and pin-limit theorems, we prove that at the mass-defined critical inverse temperature of the nearest-neighbor XY model on the square lattice, the spin field on a square with boundary angles fixed to zero converges along the full sequence to the full-variance imaginary exponential of a zero-Dirichlet Gaussian free field with stiffness `2/π`.
View manuscript on GitHub → Open PDF (raw) →M395
Essential Singularity of the Correlation Length in the Planar XY Model
We prove the Berezinskii–Kosterlitz–Thouless essential singularity for the correlation length of the nearest-neighbor cosine XY model on the square lattice. Let `m(b)` be the mass obtained by taking first the free-box thermodynamic limit and then the separation limit along a coordinate axis. As the inverse temperature b approaches bc from the massive side, ` √(b_c-b)log 1 m(b) A(XY),` where `A(XY)` is a finite, strictly positive model-specific constant.
View manuscript on GitHub → Open PDF (raw) →M396
The critical correlation exponent of the planar XY model
For the ordinary nearest-neighbor XY model on the square lattice, we prove the Berezinskii–Kosterlitz–Thouless prediction that the critical spin-correlation exponent is 1/4. More precisely, at the mass-defined critical inverse temperature, the infinite-volume correlation is `n⁻¹/⁴⁺o(¹)`. The infinite-volume limit through free square boxes is taken before the separation limit.
View manuscript on GitHub → Open PDF (raw) →M397
BKT universality for height and planar spin fields
We prove Gaussian scaling limits for the two-dimensional discrete Gaussian height model throughout its rough phase, including the physical roughening threshold, for every finite square-symmetric interaction set containing the nearest neighbors. After the natural lattice normalization, the critical effective temperature has the universal value `8π`.
View manuscript on GitHub → Open PDF (raw) →Family 217 The low-temperature Sherrington–Kirkpatrick fluctuation law. 2 papers
For every fixed inverse temperature β > 1, determines the fluctuation scale and limiting law of the zero-field Gaussian Sherrington–Kirkpatrick log partition function. Its variance is asymptotic to `c_β n¹/³`, with `c_β 0`, confirming the predicted n1/6 standard-deviation scale. Exact centering and standardization give full-sequence convergence to a uniquely characterized nondegenerate law.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M398
The low-temperature Sherrington–Kirkpatrick free-energy limiting law
For every fixed inverse temperature β > 1, we prove that the zero-field Gaussian Ising Sherrington–Kirkpatrick free energy, centered by its expectation and divided by its standard deviation, converges in distribution to a nondegenerate law as the system size tends to infinity through all integers. We also prove that its variance divided by n1/3 converges to a finite positive constant.
View manuscript on GitHub → Open PDF (raw) →M399
The low-temperature Sherrington–Kirkpatrick fluctuation scale
For the zero-field Gaussian Sherrington–Kirkpatrick model at every fixed inverse temperature β > 1, we prove that the standard deviation of the log partition function is `n¹/⁶⁺o(¹)`. The same exponent describes its typical centered absolute fluctuations, establishing the predicted one-sixth exponent in this regime.
View manuscript on GitHub → Open PDF (raw) →Family 218 Conformal universality for weakly interacting and random-bond Ising models. 6 papers
Weak finite-range square-symmetric even multispin perturbations of the square-lattice Ising model preserve critical bulk spin and energy limits; weak square-symmetric contour interactions also yield chordal SLE3 interface limits. With sufficiently weak iid bond disorder of any fixed bounded nondegenerate mean-zero law, critical spin interfaces converge to the same law in probability over environments. ([Lean](lean/docs/218.md))
PQC lens: Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
Primary manuscript on GitHub → Family index on GitHub →M400
Quenched SLE₃ limits for general weak random-bond Ising models
For the planar Ising model with bonds `J_e=1+ε xi_e`, where the ξe have any fixed bounded, nondegenerate, mean-zero iid law, we prove quenched chordal SLE3 convergence for sufficiently small ε > 0. The temperature is the spontaneous-magnetization threshold. Convergence holds in probability over environments for the full oriented curve law in deterministic Jordan-domain approximations.
View manuscript on GitHub → Open PDF (raw) →M401
Quenched SLE₃ Universality for the Weak Random-Bond Ising Model
We prove quenched chordal SLE3 convergence for the planar Ising model with sufficiently weak independent symmetric two-valued ferromagnetic bonds. The temperature is the critical point defined by spontaneous magnetization. Convergence holds in probability over environments for the full oriented curve law in deterministic Jordan-domain approximations.
View manuscript on GitHub → Open PDF (raw) →M402
Logarithmic Relative Fluctuations in the Weakly Disordered Planar Ising Model
Assuming the stated deterministic critical-reference estimates, we prove that the relative second moment of the quenched critical spin correlation in the square-lattice Ising model with independent fair bonds `1 pmε` grows as `(log r)¹/⁴⁺o(¹)` for each sufficiently small fixed ε > 0. The correlation is evaluated at the physical critical temperature, with the thermodynamic limit taken before the disorder moments and the large-distance limit.
View manuscript on GitHub → Open PDF (raw) →M403
Conformal universality of bulk Ising correlations under weak interactions
We prove conformal universality of mixed bulk spin and energy correlations for the square-lattice Ising model with sufficiently small, square-symmetric, finite-range even multispin perturbations of either sign. One critical-temperature branch and two field normalizations apply in every bounded simply connected C2 Jordan domain with free, plus, or minus boundary conditions, and in the periodic thermodynamic plane state. Energies are centered in their actual states.
View manuscript on GitHub → Open PDF (raw) →M404
SLE3 universality for weak finite-range Ising interactions
We prove that every sufficiently small, square-symmetric finite-range perturbation of the planar Ising contour energy has a domain-independent inverse temperature at which its spin interface converges to chordal SLE3. The result allows interactions of either sign, arbitrary admissible exterior contours, and uniformly approximated Jordan domains. Convergence holds for the full oriented curve law in uniform distance modulo increasing reparametrization.
View manuscript on GitHub → Open PDF (raw) →M405
Buffered comparison and stopping-band resolution in critical Ising
We prove pointwise comparison and finite stopping-band approximation theorems for the critical nearest-neighbor Ising model on the square lattice. Arbitrary common pinned spins are allowed in the comparison: the likelihood ratio is controlled by a zero-field FK connection probability on the graph with those vertices deleted.
View manuscript on GitHub → Open PDF (raw) →Family 219 GOE bulk universality for regular graphs with weak Anderson disorder. 2 papers
For every fixed degree d ≥ 3, the bulk adjacency-eigenvalue point process of a uniform simple random d-regular graph converges to the Gaussian orthogonal ensemble law, including for cubic graphs. The same fixed-energy universality persists under sufficiently weak fixed iid uniform diagonal disorder, throughout compact bands strictly inside the clean spectral edges.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M406
Fixed-energy universality for weak Anderson disorder on random regular graphs
For every fixed degree d ≥ 3, we prove fixed-energy GOE universality for the adjacency matrix of a uniformly random simple labelled d-regular graph with independent uniform diagonal disorder. The disorder strength is positive, sufficiently small, and fixed as the graph grows. At each fixed energy in a compact subinterval of the clean spectral band, the full microscopic eigenvalue point process converges to the GOE bulk process after rescaling by the positive density of states of the corresponding infinite-tree operator.
View manuscript on GitHub → Open PDF (raw) →M407
GOE bulk universality for fixed-degree random regular graphs
We prove the fixed-degree bulk-universality conjecture for random regular graphs. For every fixed integer d ≥ 3 and every fixed energy in the open Kesten–McKay bulk, the unfolded eigenvalue point process of the adjacency matrix of a uniform simple labelled d-regular graph converges to the GOE bulk process. The convergence holds along all admissible graph sizes, without additional conditioning.
View manuscript on GitHub → Open PDF (raw) →Family 220 Directional zero–one laws beyond iid environments and iid ballisticity. 3 papers
On ℤd, d ≥ 3, directional escape has probability zero or one for iid strictly elliptic nearest-neighbor environments, and for stationary ergodic finite-range-dependent environments under uniform ellipticity. In iid uniformly elliptic environments with d ≥ 2, almost-sure directional transience implies a deterministic limiting velocity with positive projection in that direction, resolving the ballisticity conjecture. ([Lean](lean/docs/220.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M408
A directional zero–one law for finite-range-dependent random environments
We prove a directional zero–one law for uniformly elliptic nearest-neighbor random walks in stationary, ergodic, finite-range-dependent environments on ℤd, d ≥ 3. For every fixed nonzero real direction, the probability of escape in that direction, averaged over the environment, is either zero or one. Finite-range dependence is imposed on the full transition rows: collections of rows at distance greater than a fixed range are independent, including collections indexed by infinite deterministic sets.
View manuscript on GitHub → Open PDF (raw) →M409
A directional zero–one law under strict ellipticity
We prove the directional zero–one conjecture for nearest-neighbor random walks in independent and identically distributed strictly elliptic environments on ℤd, d ≥ 3: the probability of escape in each fixed nonzero real direction is zero or one. Only strict positivity of the transition probabilities is required; no uniform lower bound or moment assumption is imposed.
View manuscript on GitHub → Open PDF (raw) →M410
Directional transience implies ballisticity
We prove that almost-sure transience in a fixed direction implies a deterministic limiting velocity with positive projection in that direction for nearest-neighbor random walks in independent and identically distributed uniformly elliptic environments on ℤd, d ≥ 2. This resolves the ballisticity conjecture positively.
View manuscript on GitHub → Open PDF (raw) →Family 222 Perceptron free energies and microscopic jamming exponents. 4 papers
Determines finite-temperature variational free energies for Gaussian Ising perceptrons with bounded Borel log-potentials and Gaussian spherical perceptrons with bounded continuous potentials, at every positive pattern density. A spherical extension treats bi-orthogonally invariant disorder with compact limiting singular-value distributions and no outliers. At margin −1, the quadratic-penalty spherical model has a sharp feasibility threshold and limiting gap and force laws, with system size, zero temperature, and critical density taken in that order.
PQC lens: Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
Primary manuscript on GitHub → Family index on GitHub →M412
The free energy of the Ising random perceptron
We determine the limiting free energy of the Ising perceptron with independent Gaussian patterns for every bounded Borel log-potential, at every fixed positive temperature and pattern density. We give an explicit variational formula for the limit and prove convergence in expectation and probability.
View manuscript on GitHub → Open PDF (raw) →M413
Microscopic jamming in the negative spherical perceptron
We prove a sharp feasibility threshold and limiting gap and force laws for the spherical perceptron with margin −1 and quadratic penalty. The limits are taken successively in system size, inverse temperature, and density approaching the threshold from above. They agree for Gaussian coordinates and for the equal mixture of centered Gaussian coordinates with variances `1-ε` and `1+ε`, for every sufficiently small fixed ε.
View manuscript on GitHub → Open PDF (raw) →M414
The spherical perceptron with bi-orthogonally invariant disorder
We determine the limiting free energy of a spherical perceptron with a bounded continuous activation and a bi-orthogonally invariant disorder matrix. The singular values may have any compact limiting distribution, provided there are no outliers. We give an explicit variational formula for this limit.
View manuscript on GitHub → Open PDF (raw) →M415
The free energy of the spherical random perceptron
We prove an exact variational formula for the limiting pressure of the spherical random perceptron with an arbitrary bounded continuous single-pattern potential. The formula holds at every fixed positive density and inverse temperature, with convergence in expectation and in probability.
View manuscript on GitHub → Open PDF (raw) →Family 223 Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling. 6 papers
Proves chordal SLEκ limits for critical square-lattice random-cluster interfaces for `0 q≤4`, with ` =4π/ (- q/2)`.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M416
Square-lattice FK interfaces and nested loops for 1 <= q < 4
For every fixed `1≤ q 4`, critical square-lattice random-cluster Dobrushin interfaces converge as ordered curves to chordal ` SLE ( (q))`, where ` (q)=4π/ (- q/2)`, confirming the Rohde–Schramm prediction in this parameter…
View manuscript on GitHub → Open PDF (raw) →M417
Self-dual random-cluster interfaces below one
For every fixed `0 q 1`, we prove that the Dobrushin interface of the square-lattice random-cluster model at its self-dual parameter converges to chordal SLEκ, where ` =4π/ (- q/2)…
View manuscript on GitHub → Open PDF (raw) →M418
Quenched SLE Universality for Weakly Disordered FK–Ising Interfaces
We prove that critical FK–Ising interfaces with sufficiently weak, symmetric, independent two-valued bond disorder converge to chordal SLE16/3. The disorder strength is fixed as the mesh tends to zero, and convergence of the conditional curve laws holds in probability over the environment.
View manuscript on GitHub → Open PDF (raw) →M419
Thermal FK–Ising interfaces and massive SLE
We prove convergence of thermal FK–Ising interfaces, for every fixed nonzero mass of either sign, on uniformly angle-bounded isoradial lattices in bounded simply connected domains. The limit is independent of the lattice and the admissible domain approximation. For positive mass it is the unique massive SLE `(16/3)` law with locally finite-energy drift prescribed by a massive boundary value problem; negative mass follows by duality and reversal. The theorem also allows Carathéodory approximations whose diameters diverge.
View manuscript on GitHub → Open PDF (raw) →M420
Natural Occupation Measures for Critical Square-Lattice FK Interfaces
We prove that the rescaled counting measure of a critical square-lattice Fortuin–Kasteleyn Dobrushin interface in the unit square converges to the Minkowski-content measure of its Schramm–Loewner limit, for every fixed cluster weight `1≤ q 4`. A single deterministic constant times the predicted power of the mesh gives the normalization. Convergence is joint with the ordered curve and includes the total mass, giving the scaling limit of the interface's total number of steps.
View manuscript on GitHub → Open PDF (raw) →M421
Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces
For every fixed `1≤ q≤4`, critical square-lattice random-cluster Dobrushin interfaces converge to chordal ` SLE ( (q))`, where ` (q)=4π/ (- q/2)`, and the complete nested plane loop collections converge to whole-plane ` CLE (…
View manuscript on GitHub → Open PDF (raw) →Family 224 Critical and quenched near-critical universality for Poisson–Voronoi percolation. 3 papers
Proves Cardy's formula for annealed critical Poisson–Voronoi crossing probabilities in every bounded Jordan quadrilateral. With each model normalized by its own expected unit-square pivotal count, the conditional joint near-critical crossing-threshold laws for rational polygonal quads converge in environment probability to the triangular-lattice reference law. This establishes quenched near-critical universality for crossing thresholds.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M422
From critical crossings to quenched near-critical universality in Voronoi percolation
Taking Cardy's crossing formula for critical Poisson–Voronoi percolation as an input, we prove a universal joint limit for the near-critical crossing thresholds of rational polygonal quadrilaterals under the monotone coupling. Conditional on the Poisson tessellation, the threshold law converges in probability over tessellations to the same law as on the triangular lattice. Each model is normalized by its own expected number of color-pivotal sites for a unit-square crossing.
View manuscript on GitHub → Open PDF (raw) →M423
A Pivotal Amplitude for Voronoi Percolation from Cardy's Formula
Taking Cardy's conformal crossing formula for critical planar Poisson–Voronoi percolation in every bounded Jordan quadrilateral as an input, we prove that the expected number of color-pivotal cells for a unit-square crossing is asymptotic to a positive constant times `ε ⁻³/⁴`, where the point intensity is `ε ⁻²`. No rate of convergence in Cardy's formula is required.
View manuscript on GitHub → Open PDF (raw) →M424
Cardy’s formula for critical Poisson–Voronoi percolation
We prove Cardy's formula for annealed crossing probabilities in critical planar Poisson–Voronoi percolation in every bounded Jordan quadrilateral. This proves the annealed crossing-probability form of the conformal-invariance conjecture for this model.
View manuscript on GitHub → Open PDF (raw) →Family 225 Gaussian free field limits throughout the balanced six-vertex regime. 1 paper
The balanced square-lattice six-vertex height field with `a=b=1` and `0 c≤2` converges to a Gaussian free field, including at the endpoint c = 2. The plane state is defined by balanced-torus limits. For unit height increments and Green kernel `-(2π)⁻¹log|x-y|`, the exact variance multiplier is `1/ (c/2)`.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M425
The Gaussian free field limit of the balanced six-vertex model with variance multiplier 1/arcsin(c/2)
We prove that the height function of the square-lattice six-vertex model with weights `a=b=1` and `0 c≤2`, in the plane state obtained from balanced tori, converges to a multiple of the Gaussian free field. For unit height jumps and Green kernel `-(2π)⁻¹log|x-y|`, the squared multiplier is `1/ (c/2)`.
View manuscript on GitHub → Open PDF (raw) →Family 226 The double-dimer loop ensemble converges to CLE4. 1 paper
Resolves the half-plane Temperleyan form of the double-dimer scaling-limit conjecture: the complete loop ensemble formed by two independent dimer coverings of the Temperleyan square lattice converges to nested CLE4. Convergence matches every macroscopic loop as an unparametrized curve, upgrading convergence of loop observables to convergence of the loops themselves.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M426
The curve scaling limit of half-plane double dimers
We prove that the complete double-dimer loop ensemble for the Temperleyan square lattice in the upper half-plane converges to nested CLE4. The convergence holds along the full mesh limit and matches every macroscopic loop as an unparametrized curve. This resolves the half-plane Temperleyan form of the double-dimer CLE4 scaling-limit conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 227 Critical SK autocorrelation processes and dynamics across the temperature transition. 8 papers
For zero-field Gaussian SK heat-bath dynamics with rate-one updates per spin, proves worst-start cutoff on the `log n` scale for fixed `0≤β 1`, mixing time `n²/³⁺o(¹)` at β = 1, and stretched-exponential mixing from a Gibbs-sampled fixed starting configuration for β > 1, in probability over disorder. At criticality, rescaled stationary and quench autocorrelation processes have universal random limits for Gaussian and Rademacher disorder; the quench limit relaxes to the stationary limit. ([Lean](lean/docs/227.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M427
Universality of critical quench autocorrelations in the Sherrington–Kirkpatrick model
We prove joint functional convergence of the stationary and quench autocorrelations of zero-field Sherrington–Kirkpatrick heat-bath dynamics at inverse temperature β = 1, with the same random limit for Gaussian and Rademacher couplings. Each site has a rate-one clock, mean spin autocorrelations are multiplied by n1/3, and waiting times and lags are measured in units n2/3. The quench starts from independent fair spins, and convergence is uniform on compact sets of positive waiting times and lags.
View manuscript on GitHub → Open PDF (raw) →M428
Functional universality of critical SK autocorrelations
We prove that the stationary spin autocorrelation of the zero-field Sherrington–Kirkpatrick model at inverse temperature β = 1 has a common functional scaling limit for Gaussian and Rademacher couplings. With rate-one heat-bath clocks at every site, time is scaled by n2/3 and the mean spin autocorrelation is multiplied by n1/3. The functions converge in law uniformly on compact positive-time intervals.
View manuscript on GitHub → Open PDF (raw) →M429
A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase
For every fixed inverse temperature `0 β 1`, we prove that the unscaled spectral gap of single-site heat-bath dynamics for the zero-field Gaussian Sherrington–Kirkpatrick model is bounded away from zero with probability tending to one over the disorder. Equivalently, the Gibbs law satisfies a dimension-free Poincaré inequality for all functions.
View manuscript on GitHub → Open PDF (raw) →M430
Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase
We prove worst-case total-variation cutoff for the zero-field Gaussian Sherrington–Kirkpatrick heat-bath dynamics at every fixed inverse temperature `0≤β 1`. With rate-one refresh at each spin, the cutoff location is `log n/(2λ(β))` for a positive deterministic rate `λ(β)`. The location for uniformly chosen single-site update attempts is n times as large. Convergence is in probability over the disorder.
View manuscript on GitHub → Open PDF (raw) →M431
Critical slowing down in the Sherrington–Kirkpatrick model
At the critical inverse temperature β = 1, we prove slow mixing for zero-field Gaussian Sherrington–Kirkpatrick heat-bath dynamics from typical equilibrium configurations held fixed as initial states. For every deterministic sequence `t_n=o(n²/³)` in rate-one-per-site time, the Gibbs mass of initial states whose time-tn total-variation distance from equilibrium exceeds 1/4 tends to one in probability over the disorder.
View manuscript on GitHub → Open PDF (raw) →M432
Stretched-exponential barriers for typical SK initial states
At every fixed inverse temperature β > 1, we prove a stretched-exponential obstruction to mixing for the zero-field Sherrington–Kirkpatrick model from typical equilibrium configurations held fixed as initial states. The Gibbs mass of states whose total-variation distance from equilibrium at time `exp(n¹/¹⁰⁰⁰⁰)` exceeds 1/4 tends to one in probability over the disorder. This holds for rate-one-per-site heat-bath dynamics and after the same stated number of discrete update attempts.
View manuscript on GitHub → Open PDF (raw) →M433
A typical-start upper bound for low-temperature SK Glauber dynamics
For every fixed inverse temperature β > 1, we prove subexponential mixing for rate-one-per-site heat-bath dynamics in the zero-field Gaussian Sherrington–Kirkpatrick model from a typical equilibrium configuration. If one Gibbs-sampled configuration is held fixed as the initial state, its total-variation distance from equilibrium is at most 1/4 by time `exp(n¹⁻¹/⁴⁰⁰⁰⁰⁰⁰⁰)`, with joint probability tending to one over the disorder and the sampled state.
View manuscript on GitHub → Open PDF (raw) →M434
Critical mixing in the Sherrington–Kirkpatrick model
At the critical inverse temperature β = 1, we determine the worst-start total-variation mixing exponent for single-site heat-bath dynamics in the zero-field Gaussian Sherrington–Kirkpatrick model. The mixing time is `n²/³⁺o(¹)` when every site has rate one, or `n⁵/³⁺o(¹)` attempted uniform-site updates, in probability over the disorder.
View manuscript on GitHub → Open PDF (raw) →Family 228 Continuum phase transitions for radial pair potentials. 2 papers
Constructs stable distance-dependent pair interactions for three-dimensional classical particles with a first-order phase transition: the canonical free energy has a derivative jump at one inverse temperature throughout an open density interval. One potential has a divergent repulsive core; another is bounded and continuous with an integrable power-law tail, realizing the type of transition sought in Simon's continuum problem. ([Lean](lean/docs/228.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M435
A continuum temperature singularity for a radial pair potential
We construct a stable radial pair potential in three dimensions whose canonical free energy has a strict downward derivative jump at one common finite positive inverse temperature throughout an open interval of positive densities. The potential has a divergent repulsive core, a nontrivial attractive interval, and an integrable tail satisfying `φ(r)=o(r⁻³)`. Its free-cube canonical free energy is finite at every positive inverse temperature and density.
View manuscript on GitHub → Open PDF (raw) →M436
A radial continuum phase transition with algebraic decay
We construct a bounded, continuous, stable radial pair potential in three dimensions with `|φ(r)|≤ Cr⁻³⁻¹/³²` for r ≥ 1. Throughout an open interval of positive densities, its canonical thermodynamic free energy is finite at every positive inverse temperature and has a strict downward derivative jump at one common finite positive inverse temperature.
View manuscript on GitHub → Open PDF (raw) →Family 229 Exact three- and four-state reconstruction thresholds and four-state tree capacity. 3 papers
Proves the exact reconstruction threshold `dλ² 1`, with nonreconstruction at equality, for three-state symmetric and four-state ferromagnetic broadcasting on regular trees (d ≥ 2) and observed Poisson trees (mean d > 1 and d > 0, respectively), with Poisson advantage averaged without conditioning on survival. The three-state theorem allows both signs of λ and gives the exact weak-recovery threshold for the symmetric three-community stochastic block model. ([Lean](lean/docs/229.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M437
The Reconstruction Threshold for the Ferromagnetic Four-State Potts Model
We establish the exact Kesten–Stigum reconstruction threshold for the ferromagnetic four-state Potts broadcast model on every regular d-ary tree with d ≥ 2 and every Poisson Galton–Watson tree of mean d > 0. Reconstruction occurs exactly when `dλ² 1`; we prove nonreconstruction at and below the threshold, including equality. In the Poisson model the whole tree is observed and the reconstruction advantage is averaged without conditioning on survival.
View manuscript on GitHub → Open PDF (raw) →M438
A Capacity Criterion for Four-State Potts Reconstruction on Trees
For the ferromagnetic four-state broadcast model with `0 λ 1`, we prove that reconstruction on a bounded-degree deterministic rooted tree occurs exactly when its L3 capacity with edge resistances `λ⁻²|e|` is positive. This gives an exact criterion without regularity or growth-rate assumptions on the tree, including at the exponential critical boundary.
View manuscript on GitHub → Open PDF (raw) →M439
The exact reconstruction threshold for the three-state symmetric channel
We determine the exact reconstruction threshold for the symmetric three-state broadcast process on every regular b-ary tree, b ≥ 2, and every observed Poisson Galton–Watson tree of mean d > 1. Reconstruction occurs exactly when `dλ² 1`, with d = b in the regular model; there is non-reconstruction at equality for either sign of the channel parameter. The Poisson advantage is averaged over trees and spins without conditioning on survival. This resolves the all-degree three-state regular-tree prediction.
View manuscript on GitHub → Open PDF (raw) →Family 230 Exact Hausdorff gauges for SLE. 2 papers
Resolves Schramm's Hausdorff-measure question for chordal SLEκ, `0 8`. The explicit gauge `rd(loglog(¹/r))(²⁻d)/²`, `d=1+ /8`, gives almost surely positive finite measure to every trace segment `γ([s,t])` with `0 s t ∞`, and finite expected measure to the trace in every bounded disk. ([Lean](lean/docs/230.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M440
An exact Hausdorff gauge for SLE
For each `0 8`, we construct a deterministic Hausdorff gauge that almost surely assigns positive finite measure to every nontrivial compact positive-time segment of chordal Schramm–Loewner evolution. This answers Schramm's Hausdorff-measure existence problem in this parameter range. The entire trace has finite expected gauge measure in each bounded box.
View manuscript on GitHub → Open PDF (raw) →M441
An explicit exact Hausdorff gauge for SLE
For each fixed `0 8`, let `d=1+ /8`. The gauge `h(r)=rd(loglog(¹/r))(²⁻d)/²` at sufficiently small radii almost surely gives positive finite Hausdorff measure to every nontrivial positive-time compact segment of chordal SLEκ. This gives an explicit solution to Schramm's Hausdorff-measure problem in this parameter range. The entire trace has finite expected measure in every bounded disk. The exponent-one iterated-logarithm gauge suggested by Schramm is not sigma-finite on any such segment.
View manuscript on GitHub → Open PDF (raw) →Family 231 The free uniform spanning forest is a factor of IID. 1 paper
On every infinite connected locally finite simple unweighted graph, the free uniform spanning forest is a factor of independent vertex labels, by one isomorphism-equivariant rule using no root. Translation-invariant strongly Rayleigh binary processes on every countable group, including invariant determinantal processes with Hermitian positive-contraction kernels, are also factors of IID. ([Lean](lean/docs/231.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M442
The free uniform spanning forest is a factor of IID
We prove that the free uniform spanning forest is a factor of IID on every infinite connected locally finite simple unweighted graph. One Borel rule works for all such graphs and uses no root, answering affirmatively the general factor question for unimodular random graphs. We also show that every translation-invariant strongly Rayleigh process indexed by a countable group is a factor of IID, including invariant determinantal processes with Hermitian positive-contraction kernels. This group-action conclusion requires no amenability.
View manuscript on GitHub → Open PDF (raw) →Family 232 Gaussian fields and interfaces for triangular-lattice Lipschitz heights. 3 papers
Proves Gaussian free field limits on bounded smooth simply connected domains for triangular-lattice height models: uniform odd heights with increments `0, pm2` and two-arc boundary values ` pm1`, and zero-boundary integer Lipschitz heights weighted by fixed `x [1/ sqrt2,1]`. Uniform real Lipschitz heights also converge to a Gaussian field; at a tuned opposite-boundary amplitude, their interface converges to chordal SLE4, establishing Schramm's real-field/interface predictions.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M443
Gaussian free-field limits of weighted integer Lipschitz heights
We prove a Gaussian free field scaling limit for weighted integer Lipschitz heights on the triangular lattice. With zero boundary values and a factor x for each edge on which the height changes, the field converges after division by a positive constant depending only on x to the zero-Dirichlet Gaussian free field, for every fixed `x [1/ sqrt2,1]`. The convergence holds as a random distribution on every bounded C2 Jordan domain under inside lattice approximations with uniformly convergent boundary parametrizations.
View manuscript on GitHub → Open PDF (raw) →M444
The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data
We prove that the centered uniform odd integer height function on triangular-lattice approximations of a smooth simply connected domain, with neighboring differences zero or two and boundary values +1 and −1 on two arcs, converges to a universal multiple of the Dirichlet Gaussian free field. This resolves the field part of Schramm's Problem 2.2. We give an absolutely convergent finite-volume formula for the normalization. The proof combines reflection positivity, a spectral sum rule, and boundary comparison with Gaussian moment identities.
View manuscript on GitHub → Open PDF (raw) →M445
Uniform real Lipschitz surfaces on the triangular lattice
We prove the Gaussian free field and SLE `_4` scaling limits for uniformly sampled real nearest-neighbor Lipschitz heights on the triangular lattice, resolving Schramm's Problem 2.3. On approximations of smooth simply connected domains, the centered height field converges as a random distribution to a multiple of the Dirichlet Gaussian free field. At one tuned two-arc boundary amplitude, the zero-height interface converges in uniform curve distance to chordal SLE `_4`.
View manuscript on GitHub → Open PDF (raw) →Family 233 The joint critical Ashkin–Teller current limit. 1 paper
Identifies the joint scaling limit of Ashkin–Teller heights and both complete current-cluster collections throughout the critical line, including the four-state Potts endpoint. In bounded Jordan domains with admissible lattice approximations and wired primal/free dual boundaries, the height converges to the predicted Gaussian free field, and the clusters to canonical recursive sets of that same field, retaining every nesting depth.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M446
The joint scaling limit of critical Ashkin-Teller currents
For each fixed point on the critical Ashkin–Teller line, including the four-state Potts endpoint, we prove the conjectured joint scaling limit of the height and both current-cluster collections in every bounded Jordan domain, for every admissible polygonal approximation. The height converges to a Gaussian free field with the predicted coupling constant, and the clusters converge to canonical recursive two-valued local sets of that same field. The joint limit includes all nesting depths and the distinguished wired boundary cluster.
View manuscript on GitHub → Open PDF (raw) →Family 234 All-temperature pressure of orthogonally invariant Ising spin glasses. 1 paper
Gives an exact variational formula for the limiting pressure of orthogonally invariant Ising spin glasses at every fixed temperature, both almost surely and in expectation. The coupling matrix is a Haar-random rotation of a deterministic spectrum converging to a compactly supported law, with extreme eigenvalues converging to its support edges. The zero-field ground-state energy follows as temperature tends to zero. ([Lean](lean/docs/234.md))
PQC lens: Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
Primary manuscript on GitHub → Family index on GitHub →M447
All-temperature pressure for orthogonally invariant Ising spin glasses
We determine the limiting pressure of an Ising spin glass with Haar orthogonal eigenvectors, a compact limiting spectral law, and no asymptotic outliers, at every fixed temperature. The pressure converges in expectation and almost surely to a variational formula. We also give the limiting pressure with deterministic external fields whose empirical laws converge in first-moment transport distance, and derive a formula for the zero-field ground-state energy by taking temperature to zero.
View manuscript on GitHub → Open PDF (raw) →Family 235 Limiting random SAT thresholds, sharp variance and computability. 4 papers
For random k-SAT with independent uniformly signed proper clauses sampled with replacement, proves finite positive limiting thresholds and hitting-time variance ` Theta_k(n)` for every fixed k ≥ 3, and computability of the 3-SAT threshold. We credit Gaia Carenini with priority for resolving the threshold-existence conjecture in her concurrent [ECCC TR26-229](https://eccc.weizmann.ac.il/report/2026/229/), made public October 5, 2026; this family supplies another proof and the sharper variance and computability results.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M448
A Limiting Satisfiability Threshold for Every Fixed Clause Size
For every fixed integer k ≥ 3, random k-SAT with independent uniformly signed clauses on distinct variables, sampled with replacement, has a finite positive limiting satisfiability threshold. We credit Gaia Carenini [[5]](https://eccc.weizmann.ac.il/report/2026/229/) with priority for resolving the satisfiability conjecture. This paper gives an alternative proof, using concentration of a capped last satisfiable index and a comparison between different system sizes.
View manuscript on GitHub → Open PDF (raw) →M449
Linear Variance of the Random 3-SAT Hitting Time
For random 3-SAT on n Boolean variables, with independent uniformly signed clauses on three distinct variables sampled with replacement, we prove that the first unsatisfiable prefix has variance ` (n)`. The upper bound removes the logarithmic loss in the earlier variance estimate; the matching lower bound follows from Wilson's transition-width theorem.
View manuscript on GitHub → Open PDF (raw) →M450
Variance of the Random k-SAT Hitting Time
Let Hn be the index of the first unsatisfiable prefix in random k-SAT on n variables, with independent uniformly signed clauses using k distinct variables and sampled with replacement. For every fixed k ≥ 4, we prove ` Var (H_n)= Theta_k(n)`. For k = 3, the variance is bounded below by a positive multiple of n and above by a constant multiple of `nlog n`; the companion paper on random 3-SAT sharpens this to ` (n)`.
View manuscript on GitHub → Open PDF (raw) →M451
Computing the Random 3-SAT Threshold
The limiting satisfiability threshold of uniform random 3-SAT is a computable real. We credit Gaia Carenini [[4]](https://eccc.weizmann.ac.il/report/2026/229/) with priority for resolving the satisfiability conjecture, which establishes the threshold's existence. We prove that one finite deterministic machine can approximate it to any prescribed accuracy. A deletion estimate gives explicit lower certificates, while a finite hierarchical approximation of the soft pressure gives upper certificates at every larger rational density.
View manuscript on GitHub → Open PDF (raw) →Family 236 The exact factor-of-IID threshold for free Ising spins on trees. 1 paper
Determines when the free zero-field ferromagnetic Ising state on the infinite d-regular tree is a factor of independent vertex labels: exactly when `tanhβ≤(d-1)⁻¹/²`, including equality, for d ≥ 3 and β ≥ 0. The construction uses no root and is almost surely equivariant for each fixed tree automorphism, resolving the ferromagnetic case of Lyons's question. ([Lean](lean/docs/236.md))
PQC lens: Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
Primary manuscript on GitHub → Family index on GitHub →M452
The sharp factor-of-IID threshold for the free Ising model on regular trees
For every integer d ≥ 3 and inverse temperature β ≥ 0, the free zero-field ferromagnetic Ising measure on the infinite d-regular tree is a factor of IID exactly when `tanhβ≤(d-1)⁻¹/²`. We prove the positive implication, including equality, resolving the conjecture of Nam, Sly and Zhang; the strict converse is known. The spin factor can be chosen to commute with every tree automorphism on every label input.
View manuscript on GitHub → Open PDF (raw) →Family 237 The three-quarter exponent for honeycomb self-avoiding walk. 13 papers
Proves the diameter form of Nienhuis's predicted three-quarter exponent: a uniformly chosen n-step self-avoiding walk on the honeycomb lattice has diameter `n³/⁴⁺o(¹)`. Its local mass and covering numbers have exponent 4/3. These estimates hold at every sufficiently large fixed length, simultaneously across scales, with arbitrarily high polynomial probability. ([Lean](lean/docs/237.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M453
Radial transfer estimates and polygon length laws for honeycomb walks
We prove critical diameter-tail exponents −2 for unrooted honeycomb polygons and −2/3 for length-weighted polygons, together with a truncated second-length-moment bound of exponent 2/3. Consequently, polygons conditioned to have length at least n have diameter `n³/⁴⁺o(¹)` in probability under either weight.
View manuscript on GitHub → Open PDF (raw) →M454
Critical honeycomb chords with prescribed boundary endpoints
Critical self-avoiding walks between prescribed, macroscopically separated boundary ports of a regular honeycomb hexagon have length `R⁴/³⁺o(¹)` in probability, where R is the scale of the hexagon. We prove the corresponding statements for half-plane arches, parallel cuts and nonparallel pure cuts. The half-plane law also has mean length `R⁴/³⁺o(¹)`. A separate strip argument gives endpoint mean laws on one density-one set of heights and in an aligned, critically weighted mixture of all even heights in a macroscopic interval.
View manuscript on GitHub → Open PDF (raw) →M455
Cylinder loop weights and planar nesting
We determine the growth exponent, at every fixed positive loop fugacity, of the critical honeycomb partition function for disjoint polygons separating two prescribed markers on a balanced cylinder. At fugacity two the cylinder exponent is 1/6; the corresponding planar nesting exponent and middle-strip nesting exponent are 1/12.
View manuscript on GitHub → Open PDF (raw) →M456
Mass and covering exponents for fixed-length honeycomb walks
For uniform self-avoiding walks of every sufficiently large integer length on the honeycomb lattice, we prove diameter exponent 3/4 and simultaneous local-mass and covering exponent 4/3, with arbitrary positive exponent slack and arbitrary polynomial failure probability.
View manuscript on GitHub → Open PDF (raw) →M457
Signed cylinder propagation and marked polygons on the honeycomb lattice
At the critical activity and loop fugacity two, we prove a 1/6 partition exponent for disjoint honeycomb polygons separating opposite marks on a balanced infinite cylinder, and a 1/12 planar nesting exponent. For ordered first-exit chords in a regular hexagon of side R, with both boundary ports summed and diameter at least `R/10`, the finite critical partition is `R³/⁴⁺o(¹)` and the normalized mean length is `R⁴/³⁺o(¹)`.
View manuscript on GitHub → Open PDF (raw) →M458
Cylinder amplitudes and logarithmic bridge-length windows on the honeycomb lattice
We prove a logarithmic window of critical honeycomb bridge lengths: summed through height `2h(log h)¹/⁶⁴`, bridges of lengths between `h⁴/³(log h)⁻¹/⁸` and `2h⁴/³(log h)¹/²` have mass at least `h³/⁴(log h)⁻C`.
View manuscript on GitHub → Open PDF (raw) →M459
Marked polygon correlations and one-arc bounds
At the critical honeycomb vertex activity, the squared-length mass of simple polygons of diameter at most H, counted modulo translations, is at most `H²/³⁺o(¹)`. We also prove a quantitative two-mark cylinder estimate and a polynomial one-arc bound uniform even for arbitrarily unequal marked intervals.
View manuscript on GitHub → Open PDF (raw) →M460
Disk transfer representations and confined bridge mass
We prove that critical honeycomb bridges crossing a strip of width R have mean length `R⁴/³⁺o(¹)`, with the same exponent after confinement to a fixed multiple of the strip width. The initial boundary port is fixed, the terminal port is summed, and all finite lengths receive their critical weights. The bridge mass is comparable to R−1/4; the half-plane arch kernel at endpoint separation m is comparable to m−5/4.
View manuscript on GitHub → Open PDF (raw) →M461
Polynomial vacuum representations and bridge mass for honeycomb walks
We prove that, under the all-length critical measure, a honeycomb bridge crossing a strip of h layers, with its initial port fixed and its terminal port free, has mean length `h⁴/³⁺o(¹)`. We also obtain central visit probability `R⁻²/³⁺o(¹)` and mean length `R⁴/³⁺o(¹)` for macroscopic free-boundary chords in a regular hexagon.
View manuscript on GitHub → Open PDF (raw) →M462
Renewal and changes of law for critical honeycomb walks
For critical honeycomb self-avoiding walk, we prove spatial exponent 3/4 for the infinite irreducible-bridge law and for bridges of every sufficiently large compatible even length. We establish the corresponding thermal laws at every large discount scale, including all positive length and spatial moments. For unrestricted uniform walks, the endpoint lower law holds on a common set of lengths of natural density one.
View manuscript on GitHub → Open PDF (raw) →M463
Critical strip-crossing mass on the honeycomb lattice
At the critical weight of the regular honeycomb lattice, the total weight of self-avoiding paths crossing a strip of height N is comparable to N−1/4. The first horizontal-displacement moment of return paths is comparable to N3/4.
View manuscript on GitHub → Open PDF (raw) →M464
Uniform marked-polygon estimates and sharp finite bridge moments
We prove a uniform bound for critical honeycomb polygons through two axial marks on a periodic staircase. The bound retains an explicit power of the ratio between the period and the marked separation. We also prove sharp finite strip estimates: bridge mass of order h−1/4, first-length mass at most `Ch¹³/¹²`, and mass at least `ch⁻¹/⁴` on bridges with length at least `ch⁴/³`.
View manuscript on GitHub → Open PDF (raw) →M465
Cap-selected amplitudes and triangle chords for honeycomb walks
At the critical honeycomb fugacity, self-avoiding port-to-port chords in an equilateral lattice triangle of side R, summed over both boundary endpoints and restricted to diameter at least `R/100`, have partition sum comparable to R3/4 and mean length `R⁴/³⁺o(¹)`. We also determine the amplitude selected by two vacuum caps on a cylinder of circumference N and prove that it grows as `N¹/⁶⁺o(¹)`.
View manuscript on GitHub → Open PDF (raw) →Family 238 Optimal logarithmic mixing of the Thorp shuffle. 12 papers
Proves that the Thorp shuffle randomizes `N=2d` labeled cards in ` (log N)` physical shuffles, settling its optimal mixing order for power-of-two deck sizes. Convergence is in total variation from the worst initial ordering and concerns the entire permutation, not just individual card positions. ([Lean](lean/docs/238.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M466
Optimal-order mixing of the Thorp shuffle
We prove that the Thorp shuffle on `2d` cards mixes in ` (d)` complete shuffles. The full permutation law after `1600d` shuffles converges to uniform in total variation as `d→∞`, uniformly over the initial deck, while a support count gives a lower bound of `2d-O(1)`.
View manuscript on GitHub → Open PDF (raw) →M467
Random coordinate frames and partial permutation laws
For the Thorp shuffle on `2d` cards, we prove that the worst-start total-variation distance after `32800d` physical shuffles tends to zero as `d→∞`. Combining our fixed-list estimate with the companion Fourier transfer improves this bound to `512d` shuffles. These results follow from bounds on partially observed permutation laws in random coordinate frames.
View manuscript on GitHub → Open PDF (raw) →M468
From partial permutation information to Fourier bounds
We show how information about partial permutations controls full permutation laws. Let n tend to infinity through multiples of eight. If the images of a uniformly chosen `7n/8` labels approach the uniform injection law in average total variation, and the sign mean tends to zero, then the product of two independent permutations with the given law converges to uniform on Sn.
View manuscript on GitHub → Open PDF (raw) →M469
Conditional information under deterministic coordinate sweeps
We bound the information remaining after paths of specified cards in the Thorp shuffle on `n=2d` cards have been observed. After a fixed number of deterministic coordinate sweeps, the joint endpoint law of further cards is close to uniform on the available positions, on average over the observed paths, provided a fixed positive fraction of labels lies outside both lists. Combined with a Fourier transfer, these bounds give full-deck mixing after `2048d` physical shuffles.
View manuscript on GitHub → Open PDF (raw) →M470
Conditional permutations in a revealed switching environment
Fix half the labels in a Thorp shuffle on `2d` positions and reveal their complete trajectories. We prove that, after an explicit absolute number of coordinate sweeps, the conditional permutation of the remaining labels approaches uniform in expected total variation as `d→∞`, uniformly in the initial layout. The resulting constructions give full-deck mixing in a constant multiple of d physical shuffles.
View manuscript on GitHub → Open PDF (raw) →M471
Routing densities and representation contraction for Thorp sweeps
For `N=2d` cards, we prove that an absolute number of coordinate sweeps of the Thorp shuffle brings the full permutation law to total-variation distance tending to zero from uniform. The mixing time therefore has optimal order ` (log N)` in physical shuffles.
View manuscript on GitHub → Open PDF (raw) →M472
Row–column symmetry and contraction of coordinate sweeps
We prove that the Thorp shuffle on `N=2d` labeled cards has full-permutation total-variation mixing time ` (log N)` in physical shuffles. An absolute number of coordinate sweeps suffices for the upper bound.
View manuscript on GitHub → Open PDF (raw) →M473
Random-subspace tests and trace smoothing for coordinate sweeps
We prove that an absolute number of coordinate sweeps of the Thorp shuffle on `n=2d` positions brings the full permutation to total-variation distance at most ` frac12n⁻⁵` from uniform, uniformly over the initial deck. Thus `O(d)` physical shuffles suffice, which is optimal in order.
View manuscript on GitHub → Open PDF (raw) →M474
Compatibility entropy and the spectrum of a Thorp sweep
We prove that the Thorp shuffle on `n=2d` cards mixes in ` (d)` physical shuffles. After a sufficiently large fixed number of coordinate sweeps, the total-variation distance of the full permutation from uniform tends to zero as `d→∞`.
View manuscript on GitHub → Open PDF (raw) →M475
Signed tensor densities and diagram budgets for the Thorp shuffle
We prove that a fixed number of coordinate sweeps of the Thorp shuffle on `2d` cards makes the squared L2 distance of the full permutation density from uniform tend to zero as `d→∞`. In particular, the total-variation mixing time is ` (d)` physical shuffles.
View manuscript on GitHub → Open PDF (raw) →M476
Conditional coordinate sweeps and analytic transfer
For coordinate sweeps with near-uniform line laws and line sizes among powers of two in a suitable fixed interval, we prove a Schatten-moment bound after conditioning on any feasible collection of prescribed card trajectories. An analytic transfer to binary sweeps then shows that a fixed number of sweeps mixes the Thorp shuffle on `2d` cards in `O(d)` physical steps, matching the order of the support lower bound.
View manuscript on GitHub → Open PDF (raw) →M477
A strict four-row permanent inequality and permutation moments
We prove a permanent inequality with exponent strictly below two for laws on S4 sufficiently close to uniform and with exactly uniform coordinate marginals. Applied to a four-row recursion, it shows that an absolute number of coordinate sweeps brings the full permutation of the Thorp shuffle on `N=2d` cards to total-variation distance tending to zero from uniform as `N→∞`. Thus the mixing time has the optimal order `O(log N)` in physical shuffles.
View manuscript on GitHub → Open PDF (raw) →Family 239 Sharp singularity rates for symmetric random sign matrices. 2 papers
Determines the sharp exponential singularity rate of symmetric random sign matrices with independent entries on and above the diagonal. Uniform signs give ` (det A_n=0)=(1/2+o(1))n`; for fixed bias `p (0,1) 1/2 `, the rate is `(p²⁺(¹⁻p)²⁺o(¹)n`. In the biased case, agreeing rows attain this rate.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M478
The sharp exponential rate of singularity for symmetric Bernoulli matrices
Let An be a symmetric `n× n` matrix whose entries on and above the diagonal are independent uniform signs. We prove ` (det A_n=0)=≤ft( frac12+o(1) )n.`
View manuscript on GitHub → Open PDF (raw) →M479
The sharp singularity rate for biased symmetric sign matrices
Let An be a symmetric random matrix whose entries on and above the diagonal are independent signs, equal to 1 with fixed probability `p (0,1) 1/2 `. We prove that ` P(det A_n=0)=(p²⁺(¹⁻p)²⁺o(¹)n`. The rate is attained by the event that two rows agree.
View manuscript on GitHub → Open PDF (raw) →Family 240 Shelah's eventual categoricity and the prescribed-threshold obstruction. 2 papers
Proves Shelah's eventual categoricity conjecture in ZFC: for each bound on the Löwenheim–Skolem number, a uniform threshold makes categoricity of an abstract elementary class in one cardinal above that threshold imply categoricity throughout the same tail. Categoricity means uniqueness up to isomorphism at a given cardinality. Under the continuum hypothesis, a proposed specific Hanf threshold need not suffice. ([Lean](lean/docs/240.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M480
A CH obstruction to a prescribed categoricity threshold
Assuming the continuum hypothesis, we construct an abstract elementary class with Löwenheim–Skolem number ℵ0 that is categorical in every sufficiently large cardinal but has at least two nonisomorphic models of cardinality ` (ω_2)`. Thus categoricity does not transfer down to the proposed bound ` ((2 aleph_⁰)^⁺`, which equals ` (ω_2)` under CH. Consequently, if ZFC is consistent, the prescribed-threshold form of Shelah's categoricity conjecture is not provable in ZFC.
View manuscript on GitHub → Open PDF (raw) →M481
Eventual categoricity for abstract elementary classes
We prove Shelah's eventual categoricity conjecture for abstract elementary classes in ZFC. For each infinite bound on the Löwenheim–Skolem number there is a uniform threshold such that categoricity in any one cardinal at or above that threshold implies categoricity in every cardinal at or above the same threshold.
View manuscript on GitHub → Open PDF (raw) →Family 241 Rigidity of the Turing degrees. 1 paper
Every order automorphism of the Turing degrees is the identity, resolving their rigidity problem. Thus no nontrivial relabeling of degrees preserves the ordering by relative computability. ([Lean](lean/docs/241.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M482
Rigidity of the Turing degrees
We prove that every order automorphism of the full partial order of Turing degrees is the identity, resolving the rigidity conjecture for the Turing degrees positively.
View manuscript on GitHub → Open PDF (raw) →Family 242 Single-fold Diophantine representations and undecidability under an at-most-one-solution promise. 1 paper
Every recursively enumerable set of tuples of natural numbers has a Diophantine representation with exactly one auxiliary solution for each member and none for nonmembers. This proves the single-fold conjecture and hence the finite-fold conjecture. Diophantine solvability over the nonnegative integers remains undecidable even with an at-most-one-solution promise. ([Lean](lean/docs/242.md))
PQC lens: Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
Primary manuscript on GitHub → Family index on GitHub →M483
Single-fold Diophantine representations
Every recursively enumerable set of natural-number tuples has a polynomial Diophantine representation with exactly one complete auxiliary tuple for each member. This proves the single-fold conjecture and, consequently, the finite-fold conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 243 Separating choiceless counting from polynomial time and witnessed choice. 2 papers
Confirms the Blass–Gurevich–Shelah noncapture conjecture: consistency of a linear system over 𝔽3 defines a polynomial-time query on unordered finite structures that choiceless polynomial time with counting cannot express. A separate result shows that adding witnessed symmetric choice strictly increases expressive power. Both separations hold for the full counting formalism, allowing hereditarily finite sets of arbitrary finite rank. ([Lean](lean/docs/243.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M484
Choiceless polynomial time with counting does not capture polynomial time
We prove that choiceless polynomial time with counting does not capture polynomial time on unordered finite structures, confirming the noncapture conjecture of Blass, Gurevich and Shelah. A linear-consistency query over 𝔽3 in a fixed binary vocabulary is decidable in polynomial time but not in the full counting formalism.
View manuscript on GitHub → Open PDF (raw) →M485
Witnessed symmetric choice is strictly stronger than choiceless polynomial time with counting
We prove that witnessed symmetric choice strictly increases the expressive power of choiceless polynomial time with counting. A fixed sentence with one witnessed-choice occurrence defines a Boolean query on every finite input that is not definable in the original counting formalism.
View manuscript on GitHub → Open PDF (raw) →Family 244 The Partition Principle does not imply Choice. 1 paper
Assuming ZF is consistent, constructs a model in which every surjective image of a set injects into that set, yet the axiom of choice fails. Choice for ordinal-indexed families still holds. From any countable transitive model of ZFC, a separate construction gives a transitive symmetric extension with these properties and no new countable sequences of ground-model elements. ([Lean](lean/docs/244.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M486
The Partition Principle does not imply Choice
We prove that the Partition Principle does not imply the Axiom of Choice: if ZF is consistent, then so is ZF with the Partition Principle, Choice for ordinal-indexed families, and the negation of the Axiom of Choice. Separately, over every countable transitive model of ZFC, we construct a transitive symmetric model of this theory with the same ordinals and no new countable sequences of ground elements.
View manuscript on GitHub → Open PDF (raw) →Family 245 Weak normalization implies strong normalization in pure type systems. 1 paper
Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a β-normal form, every β-reduction sequence terminates. This resolves the β-Barendregt–Geuvers–Klop conjecture, including nonfunctional rules and open contexts. ([Lean](lean/docs/245.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M487
Weak and strong normalization in pure type systems
We prove that every weakly β-normalizing pure type system is strongly β-normalizing. Both properties quantify over all legal expressions in all valid contexts, and reduction acts inside type annotations. No functionality hypothesis is required. This resolves the β-Barendregt–Geuvers–Klop conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 246 Cannon's conjecture. 1 paper
Every word-hyperbolic group with boundary homeomorphic to S2 admits a proper cocompact isometric action on hyperbolic three-space with finite kernel, proving Cannon's conjecture. Every torsion-free such group is therefore the fundamental group of a closed hyperbolic three-manifold. ([Lean](lean/docs/246.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M488
A Modulus Proof of Cannon’s Conjecture
We prove that every hyperbolic group whose boundary is homeomorphic to the two-sphere admits a proper cocompact isometric action on hyperbolic three-space with finite kernel. This resolves Cannon's conjecture positively.
View manuscript on GitHub → Open PDF (raw) →Family 247 An infinite finitely presented residually finite 2-group and a finitely presented nil algebra. 2 papers
Constructs an infinite finitely presented residually finite group whose elements all have finite 2-power order, answering the finitely presented Burnside problem negatively even in this class. The construction also yields an infinite-dimensional finitely presented nil associative 𝔽2-algebra and a finitely presented infinite-dimensional algebraic unitization, giving negative answers to the corresponding nilpotence and Kurosh finiteness questions. ([Lean](lean/docs/247.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M489
An infinite finitely presented residually finite 2-group
We prove that the infinite, ordinarily finitely presented periodic Steinberg group ` = St (12)(R)` of a companion paper is residually finite. Its finite-index subgroup `G= ( → St (12)( F_2))` is infinite, ordinarily finitely presented, and residually finite, and every element of G has finite 2-power order. The orders of its elements are unbounded. Phases of long words in the companion's graded algebra allow finite degree truncations to detect all elements of Γ, including the central kernel.
View manuscript on GitHub → Open PDF (raw) →M490
An infinite finitely presented periodic group
We construct an infinite group with an ordinary finite presentation in which every element has finite order, answering the finitely presented Burnside question negatively. We also construct an infinite-dimensional finitely presented nonunital nil associative algebra over 𝔽2 that is Jacobson radical but not nilpotent. Its unitization is finitely presented, algebraic, and infinite-dimensional.
View manuscript on GitHub → Open PDF (raw) →Family 248 Thompson's group F is nonamenable. 1 paper
Proves that Thompson's group F, the group of dyadic piecewise linear homeomorphisms of the interval, is nonamenable, resolving its longstanding amenability problem. ([Lean](lean/docs/248.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M491
Thompson's group F is nonamenable
We prove that Thompson's group F is nonamenable. This confirms Geoghegan's conjecture and resolves the amenability problem for F.
View manuscript on GitHub → Open PDF (raw) →Family 249 A finitely generated Eilenberg–Ganea counterexample. 1 paper
Constructs a finitely generated residually finite group with integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture. It has no two-dimensional classifying space, even with infinitely many cells. ([Lean](lean/docs/249.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M492
A finitely generated counterexample to the Eilenberg–Ganea conjecture
We construct a finitely generated residually finite group of integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 250 Boone–Higman embeddings with higher finiteness. 3 papers
A finitely generated group has decidable word problem exactly when it embeds in a finitely presented simple group, proving the Boone–Higman conjecture. The target can have type F∞: a classifying space with finitely many cells in each dimension. A single group of type F∞ can also contain every finitely presented group. ([Lean](lean/docs/250.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M493
Finite algebraic envelopes and the Boone–Higman conjecture
We prove the Boone–Higman conjecture. A finitely generated group has decidable word problem if and only if it embeds in a finitely presented simple group.
View manuscript on GitHub → Open PDF (raw) →M494
Simple F∞ overgroups of groups with decidable word problem
Every finitely generated group with decidable word problem embeds in a simple group of type F∞. This resolves the higher-finiteness strengthening of the Boone–Higman conjecture.
View manuscript on GitHub → Open PDF (raw) →M495
A universal group of type F∞
We construct a single group of type F∞ containing every finitely presented group. Its finitely generated subgroups, up to isomorphism, are exactly the finitely generated recursively presented groups. This answers the F∞ form of the higher-dimensional Higman embedding question.
View manuscript on GitHub → Open PDF (raw) →Family 251 Amenability, unitarizability, and strong Ulam stability. 2 papers
Resolves Dixmier's problem for all discrete groups: amenability is equivalent to every uniformly bounded Hilbert-space representation being similar to a unitary representation. For countable discrete groups, amenability is also equivalent to strong Ulam stability: sufficiently accurate unitary approximate representations are uniformly close in operator norm to genuine representations on the same, possibly infinite-dimensional, Hilbert space. ([Lean](lean/docs/251.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M496
Unitarizability implies amenability for discrete groups
We prove that a discrete group is amenable if and only if every uniformly bounded representation on a complex Hilbert space is similar to a unitary representation. This resolves Dixmier's unitarizability problem affirmatively for discrete groups.
View manuscript on GitHub → Open PDF (raw) →M497
Strong Ulam Stability Characterizes Amenability
A countable discrete group is amenable if and only if it is strongly Ulam stable: every sufficiently accurate unitary almost representation, on any complex Hilbert space, is uniformly close in operator norm to a genuine representation on the same space. We prove the converse to Kazhdan's amenable stability theorem, answering the question of Burger, Ozawa, and Thom. The inclusion of infinite-dimensional Hilbert spaces is essential.
View manuscript on GitHub → Open PDF (raw) →Family 252 A torsion-free hyperbolic group that is neither residually finite nor linear over any field. 1 paper
Constructs a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question negatively. One fixed nonidentity element is killed by every finite-dimensional linear representation over every commutative field, so the group is not linear over any such field. ([Lean](lean/docs/252.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M498
A torsion-free hyperbolic group that is not residually finite
We construct a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question for hyperbolic groups negatively.
View manuscript on GitHub → Open PDF (raw) →Family 253 An infinite finitely presented simple amenable group. 1 paper
Constructs an infinite finitely presented simple amenable group, answering the longstanding question of whether these properties can occur simultaneously. ([Lean](lean/docs/253.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M499
An infinite finitely presented simple amenable group
We construct an infinite finitely presented simple amenable group, giving a positive answer to the finite-presentation existence question for simple amenable groups.
View manuscript on GitHub → Open PDF (raw) →Family 254 Classifying spaces and geometric obstructions for Artin groups. 3 papers
The Salvetti complex of every finite-rank Artin group is aspherical, proving the Artin `K(π,1)` conjecture. Arbitrary intersections of its parabolic subgroups are parabolic, proving the Parabolic Intersection Conjecture. An explicit Artin group admits no proper cocompact isometric action on any nonempty proper CAT `(0)` space. ([Lean](lean/docs/254.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M500
Harmonic heights and the Artin K(pi,1) conjecture
We prove that the standard Salvetti complex of every Artin group with finitely many standard generators is aspherical. This resolves the Artin `K(π,1)` conjecture in finite rank.
View manuscript on GitHub → Open PDF (raw) →M501
An Artin group with no geometric CAT(0) action
We construct an Artin group on 116 generators that admits no proper, cocompact isometric action on a nonempty proper CAT(0) space. This refutes the CAT(0) conjecture for Artin groups.
View manuscript on GitHub → Open PDF (raw) →M502
Parabolic intersections in Artin groups
We prove that every intersection of parabolic subgroups of any finite-rank Artin group, with arbitrary finite or infinite Coxeter labels, is parabolic. This resolves the Parabolic Intersection Conjecture affirmatively.
View manuscript on GitHub → Open PDF (raw) →Family 255 Quasi-isometric recognition of virtually polycyclic groups. 1 paper
Proves that every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic, resolving the Eskin–Fisher–Whyte lattice-recognition conjecture. Equivalently, a group quasi-isometric to a lattice in a connected simply connected solvable Lie group is virtually a uniform lattice in some such Lie group, possibly a different one. ([Lean](lean/docs/255.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M503
Quasi-isometric recognition of virtually polycyclic groups
We prove that every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic. This resolves the lattice-recognition conjecture of Eskin, Fisher and Whyte, which allows the ambient solvable Lie group in the conclusion to differ from the one in the hypothesis.
View manuscript on GitHub → Open PDF (raw) →Family 256 Nonsingular systems of equations over arbitrary groups. 2 papers
Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over ℚ has a simultaneous solution in an overgroup, resolving Howie's conjecture. The coefficient group embeds in the presented quotient. A companion proves Kervaire's conjecture: adjoining one generator and one relation cannot trivialize a nontrivial group. ([Lean](lean/docs/256.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M504
Nonsingular systems of equations over arbitrary groups
Every finite nonsingular system of equations over an arbitrary group has a simultaneous solution in an overgroup. This proves Howie's conjecture on nonsingular systems.
View manuscript on GitHub → Open PDF (raw) →M505
The Kervaire theorem for groups
We prove that no free product of a nontrivial group with an infinite cyclic group is normally generated by one element. This resolves the Kervaire conjecture positively.
View manuscript on GitHub → Open PDF (raw) →Family 257 A hyperbolic group without a geometric CAT(0) action. 1 paper
Answers negatively whether every word-hyperbolic group is a CAT(0) group. Constructs one with a finite classifying space but no proper cocompact isometric action on any proper complete ` CAT (0)` space, in any dimension. ([Lean](lean/docs/257.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M506
A hyperbolic group with no geometric CAT(0) action
We construct a hyperbolic group with a finite classifying space that admits no geometric action on a proper complete CAT(0) space. Consequently, a finite aspherical simplicial complex with a linear combinatorial disk-filling inequality need not have a finite locally CAT(0) homotopy model, and hence need not have a finite locally CAT(−1) model. Here metric models carry geodesic length metrics inducing the given complex topology.
View manuscript on GitHub → Open PDF (raw) →Family 258 Gersten’s conjecture and virtual compact specialness of one-relator groups. 2 papers
Proves Gersten's conjecture: every finitely generated one-relator group containing no Baumslag–Solitar subgroup `BS(m,n)`, with `m,n≠0`, is word-hyperbolic. It also proves that every word-hyperbolic one-relator group is virtually compact special.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M507
Baumslag-Solitar-free one-relator groups are hyperbolic
We prove Gersten's conjecture: every finitely generated one-relator group containing no subgroup isomorphic to a Baumslag–Solitar group ` BS (m,n)`, for nonzero integers m, n, is word-hyperbolic.
View manuscript on GitHub → Open PDF (raw) →M508
Virtual compact specialness of hyperbolic one-relator groups
We prove that every word-hyperbolic one-relator group is virtually compact special. Combined with the work of Kielak–Linton, this resolves Wise's virtual free-by-cyclic conjecture for hyperbolic one-relator groups: every such group is virtually free-by-cyclic, with the free kernel allowed to have infinite rank.
View manuscript on GitHub → Open PDF (raw) →Family 259 A group without fixed price. 1 paper
Constructs a finitely generated group with two essentially free probability-measure-preserving actions of different costs, answering the general fixed-price problem negatively. Its Bernoulli action has cost bounded away from one, while a sequence of finite height extensions has costs tending to one.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M509
A group without fixed price
We construct a finitely generated group with two essentially free probability-measure-preserving actions of different costs. This gives a negative answer to Gaboriau's general fixed-price problem. The group is an amalgam of a free group of rank 100 with a direct product. Its Bernoulli action has cost bounded away from one, whereas finite height extensions have costs tending to one.
View manuscript on GitHub → Open PDF (raw) →Family 260 Spacetime Penrose inequalities: enclosing area, charge, rotation, and anti-de Sitter extensions. 13 papers
Proves the sharp enclosing-area spacetime Penrose inequality for smooth one-ended asymptotically flat initial data in every spatial dimension n ≥ 3, under dominant energy, weak future trapping, positive enclosing area, and the stated decay assumptions. It bounds invariant ADM mass below using minimum enclosing area, with equality rigidity under additional horizon hypotheses. Charged upper-area bounds treat dyonic three-dimensional data; the higher-dimensional purely electric extension uses the matched neutral theorem. ([Lean](lean/docs/260.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M510
The spacetime Penrose inequality with charge and original-data rigidity
Under the stated energy, decay, and trapping hypotheses, we prove the sharp charged spacetime Penrose upper-area inequality for one-ended three-dimensional initial data with source-free electric and magnetic fields. The theorem allows arbitrary second fundamental form, nonzero ADM momentum, and disconnected boundary. Writing m for invariant ADM mass, Q for total charge magnitude, and rA for the minimum-enclosing-area radius, the bound is m ≥ Q and `r_A≤ m+√(m²⁻Q²)`.
View manuscript on GitHub → Open PDF (raw) →M511
A Charged Reduction of the Spacetime Penrose Inequality in Spatial Dimensions at Least Four
Using the companion neutral spacetime Penrose theorem exactly at its stated strong-decay, future-trapped, positive-area, and future-timelike scope, we prove the sharp purely electric upper-area inequality in every spatial dimension n ≥ 4 for one-ended charged data satisfying the charged dominant energy condition and ` div _g E=0`, with the specified decay, integrability, and finite-flux assumptions. The data may have arbitrary interior topology and ADM momentum, with a future or past trapping sign chosen independently on each boundary component.
View manuscript on GitHub → Open PDF (raw) →M512
Spacetime Penrose inequalities: enclosing area, charge, and rigidity
We prove sharp spacetime Penrose inequalities for invariant ADM mass and minimum enclosing area in three and four spatial dimensions. Under their respective designated-end conventions, the neutral numerical results allow arbitrary second fundamental form, nonzero momentum, disconnected weakly future trapped boundary, and finitely many ends.
View manuscript on GitHub → Open PDF (raw) →M513
The Kerr–Newman Penrose Inequality for Axisymmetric Electrovacuum Exteriors
We prove the sharp Kerr–Newman Penrose inequality ` m²≥ A 16π + Q² 2 + π(Q⁴⁺⁴J²) A.` Here `Q²⁼Q_e²⁺Q_b²`, and J is the conserved total angular momentum, including its electromagnetic contribution. The result applies to smooth axisymmetric electrovacuum exteriors in the stated one-ended topological and decay class, with a connected outermost, outer-area-minimizing future marginally outer trapped boundary, Coulomb electromagnetic asymptotics, zero ADM momentum, and the explicitly assumed physical-area condition `A≥4π√(Q⁴⁺⁴J²)`.
View manuscript on GitHub → Open PDF (raw) →M514
Electromagnetic tails and the Kerr–Newman Penrose inequality
We construct smooth axisymmetric electrovacuum exteriors that violate the Kerr–Newman Penrose inequality when J is the bare gravitational ADM angular momentum and the electromagnetic fields have only `O(r⁻²)` decay, allowing angularly varying leading tails. The examples have zero total electric and magnetic charge even though both electromagnetic fields are nonzero. They have a connected outermost, outer-area-minimizing future marginally outer trapped boundary and lie strictly on the physical area branch.
View manuscript on GitHub → Open PDF (raw) →M515
The nonmaximal anti-de Sitter Penrose Inequality and original-data rigidity
We prove the spacetime Penrose Inequality for smooth three-dimensional initial-data exteriors with one spherical asymptotically anti-de Sitter end, satisfying the stated decay and integrability assumptions, the dominant energy condition, and a compact weakly future outer trapped boundary. We assume that the hyperbolic metric four-flux is future timelike; its Lorentz norm is the mass, and the area is the infimum over full enclosing cuts.
View manuscript on GitHub → Open PDF (raw) →M516
The Penrose inequality for maximal asymptotically hyperbolic initial data
We prove the sharp Penrose inequality for three-dimensional maximal asymptotically hyperbolic initial data with spherical conformal infinity. Under the dominant energy condition, the stated decay and integrability assumptions, and a future-timelike mass covector, the invariant mass is bounded below by the Schwarzschild–anti-de Sitter mass associated with the minimum enclosing area of a weakly future outer-trapped boundary. The boundary may be disconnected, and no restriction is imposed on the compact topology.
View manuscript on GitHub → Open PDF (raw) →M517
A local Penrose inequality for conformal perturbations of Schwarzschild–anti-de Sitter data
We prove the asymptotically hyperbolic Penrose inequality for sufficiently small maximal vacuum conformal perturbations of a positive-mass Schwarzschild–anti-de Sitter exterior, for every fixed decaying transverse-traceless seed and every solution branch satisfying the stated decay and mass assumptions. The bound uses the area of the marginally outer trapped boundary itself; it requires neither outermostness nor outer area-minimization, and no bulk-versus-boundary domination condition.
View manuscript on GitHub → Open PDF (raw) →M518
The spacetime Penrose inequality and enclosing area
We prove the sharp spacetime Penrose inequality for smooth one-ended initial-data exteriors in every spatial dimension n ≥ 3, under the dominant energy condition, weak future trapping, the stated differentiated decay and positive enclosing area. These hypotheses force the ADM energy-momentum to be future timelike. Area is the infimum over full enclosing cuts in the original metric.
View manuscript on GitHub → Open PDF (raw) →M519
Conformal flow and the Riemannian Penrose inequality with minimizing frontiers
We give a detailed conformal-flow proof of the numerical Riemannian Penrose inequality in every dimension n ≥ 3 for complete asymptotically flat exteriors with nonnegative scalar curvature and full compact frontiers that are outer minimizing and locally perimeter minimizing. The metric extends smoothly through the possibly singular frontier; neither spin nor frontier connectedness is assumed. The proof follows the conformal-flow approach of Bray, Bray–Lee, and Bi–Zhu.
View manuscript on GitHub → Open PDF (raw) →M520
Equality and rigidity in the spacetime Penrose inequality
Equality in the spacetime Penrose inequality identifies the original initial data as a spacelike slice of a Schwarzschild–Tangherlini exterior, smoothly attached to its horizon, under the stated outermostness and decay hypotheses. We prove this in every spatial dimension n ≥ 3 for a strong-decay exterior class and for weaker decay in dimensions three and four. In dimension three the finite-component theorem forces a single spherical horizon and gives qualitative strictness for disconnected boundaries.
View manuscript on GitHub → Open PDF (raw) →M521
Boundary graph deformations for the spacetime Penrose inequality
We construct graph and conformal deformations of trapped initial-data exteriors in three and four spatial dimensions. The resulting metrics have nonnegative scalar curvature, strictly negative inner mean curvature, a lower bound protecting every enclosing cut, and an arbitrarily small upper error in ADM energy. We also give a direct four-dimensional maximal vacuum construction that retains a noncompact decaying second fundamental form. These constructions give boundary routes to the corresponding numerical Penrose inequalities.
View manuscript on GitHub → Open PDF (raw) →M522
Area-controlled end replacement and the Bondi Penrose inequality in the CKS class
We replace a three-dimensional Cha–Khuri–Sakovich hyperboloidal end by an asymptotically flat end while preserving the dominant energy condition and each fixed compact interior. For strictly future-timelike CKS initial-data charge, the replacements lose asymptotically no enclosing area and their ADM masses tend to the invariant Bondi mass. Combining this construction with the companion numerical spacetime Penrose theorem yields the sharp Bondi bound for possibly disconnected weakly future trapped boundaries in this class.
View manuscript on GitHub → Open PDF (raw) →Family 261 Localization and delocalization in the Anderson model. 2 papers
Resolves the predicted spectral contrast for the lattice Anderson model with independent uniform site potentials. In dimension two, every positive disorder strength gives almost surely pure-point spectrum. In every fixed dimension d ≥ 3, sufficiently weak positive disorder gives purely absolutely continuous spectrum on a fixed open interval with nonzero spectral weight. ([Lean](lean/docs/261.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M523
Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three
For each fixed dimension d ≥ 3 and each fixed sufficiently small positive disorder strength, we prove that the Anderson operator on ℤd with independent uniform site potentials almost surely has purely absolutely continuous spectrum with nonzero weight on an open energy interval. The interval may depend on d but is independent of the disorder strength. This settles the purely absolutely continuous energy-range question in Simon's Problem 1 for this model.
View manuscript on GitHub → Open PDF (raw) →M524
Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder
For each fixed positive disorder strength, we prove that the nearest-neighbor Anderson operator on the square lattice with independent uniform site potentials almost surely has pure-point spectral type throughout its spectrum. This resolves the pure-point assertion of the two-dimensional Anderson localization conjecture for the uniform single-site law.
View manuscript on GitHub → Open PDF (raw) →Family 262 Sharp finite-matrix Lieb–Thirring inequalities and all equality cases. 3 papers
Proves the sharp one-dimensional Lieb–Thirring inequality for `1/2 γ 3/2` and arbitrary finite-matrix potentials W ≥ 0 with `∫ tr (Wγ⁺¹/²) ∞`: the optimal constant is the scalar one-bound-state value, independent of matrix size. All equality cases are direct sums, in one constant unitary basis, of scalar sech2 solitons with independent scales and centers, and zero channels. ([Lean](lean/docs/262.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M525
Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality
We classify all equality cases in the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and `1/2 γ 3/2`. For measurable Hermitian positive semidefinite potentials W with `∫_ R tr (Wγ⁺¹/²) ∞`, equality holds precisely for direct sums, in one constant unitary basis, of scalar one-bound-state solitons and zero channels. The nonzero solitons may have independent scales and centers.
View manuscript on GitHub → Open PDF (raw) →M526
Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials
We prove the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and every exponent `1/2 γ 3/2`. The optimal constant is the scalar one-bound-state constant, independently of the matrix size. The inequality bounds the full sum of negative eigenvalue moments for every measurable Hermitian positive semidefinite potential W satisfying `∫_ R tr (Wγ⁺¹/²) ∞`. The matrices may have arbitrary rank and need not commute at different points.
View manuscript on GitHub → Open PDF (raw) →M527
Sharp one-dimensional Lieb–Thirring constants
We resolve affirmatively the remaining cases of the scalar one-dimensional Lieb–Thirring conjecture: for every ` frac12 γ frac32`, the optimal constant is the one-bound-state constant. The estimate holds for every nonnegative potential in `Lγ⁺¹/²( R)`, with all negative eigenvalues included.
View manuscript on GitHub → Open PDF (raw) →Family 263 The ionization and generalized ionization conjectures. 3 papers
For the full nonrelativistic Coulomb model with two electron spin states, proves that a molecule with M fixed nuclei of charges at least one and total charge Z strictly binds at most `Z+CM` electrons. Neutral-atom first ionization energies and radii containing all but an expected half-electron have universal positive upper and lower bounds. The energy cost of removing m electrons has Thomas–Fermi asymptotics as `m→∞` and `Z/m→∞`; neutral-atom outer radii have the corresponding iterated-limit asymptotics, taking `Z→∞` first.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M528
Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms
We prove that a molecule with M fixed nuclei of real charges at least one and total nuclear charge Z strictly binds at most `Z+CM` electrons, where C is universal. The result holds for the full nonrelativistic Coulomb Hamiltonian with two spin states and arbitrary distinct nuclear positions. For neutral atoms with integer nuclear charge Z ≥ 1, the first ionization energy and, for every ground state, the radius outside which one half of an electron remains are each bounded above and below by positive universal constants.
View manuscript on GitHub → Open PDF (raw) →M529
Generalized ionization energies for full Coulomb atoms
We prove the energy part of the generalized ionization conjecture for full nonrelativistic Coulomb atoms with two electron spin states. The energy needed to remove m electrons is asymptotic to `a(TF)m⁷/³` whenever `m→∞` and `Z/m→∞`, with the Thomas–Fermi constant in atomic units. We also obtain both conjectured iterated limits.
View manuscript on GitHub → Open PDF (raw) →M530
Generalized outer-electron radii of neutral Coulomb atoms
We prove the radius part of the generalized ionization conjecture for neutral nonrelativistic Coulomb atoms with two spin states. For every choice of ground states, the upper and lower large-nuclear-charge limits of the radius defined by an expected exterior electron mass m are both asymptotic to `(81π²/2)¹/³m⁻¹/³` as m tends to infinity.
View manuscript on GitHub → Open PDF (raw) →Family 264 Strong cosmic censorship near two-ended Kerr data. 3 papers
Proves local strong cosmic censorship near each fixed rotating subextremal Kerr bridge. A dense Gδ subset of a weighted smooth neighborhood of smooth complete two-ended asymptotically flat vacuum data has full maximal globally hyperbolic developments with no future continuous nondegenerate extension whose weak connection is locally square-integrable. No symmetry is imposed; extensions need not satisfy the vacuum equations.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M531
Generic Future Inextendibility with Square-Integrable Connection Near a Fixed Kerr Spacetime
For each fixed rotating subextremal Kerr background with mass M > 0 and rotation `0 | a| M`, we prove that the smooth vacuum data near its complete two-ended bridge whose full maximal globally hyperbolic development admits a future `C⁰ W¹,²(loc)` extension form a meagre set in the weighted smooth topology. The extension metric is continuous and nondegenerate, and its weak connection is locally square-integrable. The neighborhood requires smallness of only the tenth weighted seminorm, while the topology tests every finite order.
View manuscript on GitHub → Open PDF (raw) →M532
Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes
For each fixed Kerr spacetime with mass M > 0 and rotation `0 a M`, we prove that a dense Gδ set of nearby smooth, complete two-ended vacuum data has a maximal globally hyperbolic development with no future C1 extension. The neighborhood and genericity are defined in a weighted smooth topology, and ambient extensions may be nonvacuum.
View manuscript on GitHub → Open PDF (raw) →M533
Quantitative Near-Kerr Evolution and Generic C2 Future Inextendibility
For every fixed rotating subextremal Kerr bridge, we prove that smooth vacuum data admitting a C2 future extension of their full maximal globally hyperbolic development form a meagre set in a neighborhood defined by one finite-order seminorm. The topology allows arbitrary symbol-bounded asymptotically flat tails and imposes no symmetry.
View manuscript on GitHub → Open PDF (raw) →Family 265 Area laws and tensor networks for two-dimensional gapped systems. 2 papers
Proves an entropy area law for unique ground states of finite-range Hamiltonians on arbitrary finite induced square-lattice domains, using only a uniform full-system spectral gap and bounds on the local interactions. On open `L× L` squares, uniformly gapped nearest-neighbor ground states also admit projected entangled-pair state approximations with polynomial bond dimension and global vector error at most L−1.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M534
A two-dimensional area law from a global spectral gap
We prove an entropy area law for the unique ground state of a finite-range Hamiltonian on any finite induced subgraph of the square lattice. A lower bound on the spectral gap of the full Hamiltonian and fixed bounds on the local dimension, interaction range, and interaction strength suffice. For every set of sites, its entanglement entropy is bounded by a constant times the number of edges crossing its boundary, independently of the size and shape of the domain.
View manuscript on GitHub → Open PDF (raw) →M535
Polynomial PEPS approximation of gapped square-grid ground states
We prove that the unique ground state of a uniformly gapped nearest-neighbor Hamiltonian on an `L× L` square lattice admits a projected entangled-pair state approximation with bond dimension polynomial in L and global vector error at most L−1 after normalization. Only the gap of the full Hamiltonian is assumed. The result is an existence theorem, with constants uniform over Hamiltonians of fixed local dimension, interaction strength, and gap.
View manuscript on GitHub → Open PDF (raw) →Family 266 Exactly three mutually unbiased bases in dimension six. 2 papers
Proves `N(6)=3`, resolving Zauner's dimension-six mutually unbiased bases conjecture: three such bases exist in ℂ6, but four cannot. The exclusion is a complete certified computation under the stated binary64 arithmetic and compiler conditions. An independent companion proves the Matolcsi–Ruzsa–Weiner Fourier-vanishing conjecture for order-six complex Hadamard matrices outside Tao's cubic equivalence class. ([Lean](lean/docs/266.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M536
The maximum number of mutually unbiased bases in dimension six
We prove that the maximum number of mutually unbiased orthonormal bases in ℂ6 is three, resolving Zauner's dimension-six MUB conjecture. The upper bound is computer-assisted: under the stated binary64 arithmetic and compiler conditions, a complete execution of the documented verification pipeline excludes four arbitrary complex bases.
View manuscript on GitHub → Open PDF (raw) →M537
Exact Fourier certificates for complex Hadamard matrices of order six
We prove the Fourier-vanishing conjecture of Matolcsi, Ruzsa, and Weiner: every complex Hadamard matrix of order six outside Tao's cubic equivalence class has a vanishing character sum at every permutation of `(1,1,1,-1,-1,-1)`. We also give an exact certificate excluding seven mutually unbiased bases in ℂ6; by Weiner's completion theorem, this yields an upper bound of five. Both certificates use only integer and rational arithmetic, and the complete verifier is included.
View manuscript on GitHub → Open PDF (raw) →Family 267 Positive-temperature Bose–Einstein condensation and exact quantum depletion. 5 papers
Proves Bose–Einstein condensation for the exact canonical Gibbs state of the three-dimensional hard-sphere gas: each fixed exclusion distance and sufficiently small fixed density admit a strictly positive temperature, independent of volume, with positive condensate fraction in the thermodynamic limit. At zero temperature, proves the Bogoliubov leading quantum-depletion law for hard spheres and fixed bounded nonnegative radial finite-range potentials of positive scattering length, taking the thermodynamic limit before the dilute limit.
PQC lens: Directly shapes quantum threat models, algorithm timelines, or query-complexity assumptions used in security proofs.
Primary manuscript on GitHub → Family index on GitHub →M538
Bose–Einstein condensation at positive temperature in the dilute hard-sphere gas
We prove Bose–Einstein condensation at positive temperature in the three-dimensional dilute hard-sphere gas. For each fixed exclusion distance and every sufficiently small fixed density, there is a strictly positive temperature, independent of the volume, at which the exact canonical Gibbs state has a positive condensate fraction in the thermodynamic limit. The condensate occupies the constant orbital.
View manuscript on GitHub → Open PDF (raw) →M539
Quantum Depletion and Momentum Distribution in the Dilute Hard-Sphere Bose Gas
We prove the full scaled Bogoliubov momentum distribution for the depleted particles in the three-dimensional hard-sphere Bose gas at zero temperature. The thermodynamic limit is taken at each fixed density before the dilute limit, uniformly over all pure and mixed ground states. All thermodynamic accumulation values have the same dilute asymptotic: the limiting nonzero-momentum occupation measure has total mass `8/(3 sqrtπ)`, giving the depletion fraction ` 8 3 sqrtπ √(ρ a³)⁺o(√(ρ a³)` for density ρ and hard-sphere exclusion distance a.
View manuscript on GitHub → Open PDF (raw) →M540
Quantum Depletion for Fixed Bounded Repulsive Potentials
We prove the Bogoliubov quantum-depletion asymptotic for the ground state of a three-dimensional Bose gas with a fixed bounded, nonnegative, radial interaction of finite range and positive scattering length a. The thermodynamic limit is taken at each fixed density ρ before the dilute limit. Every thermodynamic accumulation value of the fraction outside the constant mode is ` 8 3 sqrtπ √(ρ a³)⁺o(√(ρ a³)` as `ρ downarrow0`.
View manuscript on GitHub → Open PDF (raw) →M541
A density-uniform condensate bound for dilute Bose gases
For each fixed bounded measurable, nonnegative, radial interaction of finite range in three dimensions that is not zero almost everywhere, we prove a positive lower bound on the constant-orbital condensate fraction that is uniform over all sufficiently small densities and all temperatures between zero and the square of the density. The interaction and density remain fixed in the thermodynamic limit.
View manuscript on GitHub → Open PDF (raw) →M542
Ground-state condensation in the dilute hard-sphere gas
We prove Bose–Einstein condensation in every ground state of a dilute three-dimensional hard-sphere Bose gas. At every sufficiently small fixed gas parameter, the constant orbital contains a positive fraction of the particles in the thermodynamic limit. The fraction can be chosen independently of the gas parameter, and the conclusion holds for arbitrary complex ground states.
View manuscript on GitHub → Open PDF (raw) →Family 268 The spin-one Haldane gap. 2 papers
Proves the spin-one Haldane gap conjecture for the pure antiferromagnetic Heisenberg chain on even periodic rings: the spectral gap stays uniformly positive as the chain grows. A companion establishes a gap for odd open chains with endpoint field `h=3/5` and gives boundary-selected infinite-volume states with topological index −1.
PQC lens: Directly shapes quantum threat models, algorithm timelines, or query-complexity assumptions used in security proofs.
Primary manuscript on GitHub → Family index on GitHub →M543
The periodic spin-one Haldane gap
We prove a uniform positive spectral gap for the pure antiferromagnetic spin-one Heisenberg chain on even periodic rings, establishing the positive even-periodic formulation of the spin-one Haldane conjecture.
View manuscript on GitHub → Open PDF (raw) →M544
A boundary-field gap for the spin-one Heisenberg chain
We prove a uniform spectral gap for odd open spin-one antiferromagnetic Heisenberg chains with the same fixed magnetic field at both endpoints. The field `h=3/5` gives a gap greater than `log(10)/392` on every chain of `2L+1` sites with L ≥ 960. Tasaki's index theorem then gives index −1 for every subsequential local limit of these boundary-selected ground states.
View manuscript on GitHub → Open PDF (raw) →Family 269 Uniform Laughlin gap and stability under bounded scalar disorder. 2 papers
Proves the fermionic Laughlin spectral-gap conjecture for the full V1 interaction at filling 1/3 on the round sphere. The unique ground state remains uniformly gapped under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. Both the gap and disorder threshold are uniform over all sufficiently large particle numbers and all normalized potential profiles. ([Lean](lean/docs/269.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M545
Uniform Stability of the Spherical Laughlin Gap
We prove that the fermionic Laughlin V1 Hamiltonian at filling 1/3 on expanding round spheres retains a unique ground state and a uniform spectral gap under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. With coefficient one for each pair projector and Laughlin flux `q=3(N-1)`, the gap and perturbation threshold are uniform for all sufficiently large particle numbers and all potential profiles of supremum norm at most one.
View manuscript on GitHub → Open PDF (raw) →M546
A Fock-space inequality and the Laughlin spectral gap
We prove the spherical fermionic Laughlin spectral-gap conjecture for the full V1 interaction. With coefficient one for each pair projector, the gap above the Laughlin state at filling 1/3 on the round sphere is at least 1/25 for all sufficiently large systems. More generally, `H_Q²≥γ H_Q` for some fixed `γ 1/25` on the entire lowest-Landau-level Fock space for all sufficiently large flux Q, independently of particle number.
View manuscript on GitHub → Open PDF (raw) →Family 270 Threshold and positive-energy bound states of the BFSS matrix model. 2 papers
Proves that the undeformed relative `SU(N)` BFSS model has exactly one normalizable zero-energy state for every finite N ≥ 2, resolving the threshold-bound-state conjecture. For `SU(2)`, a companion proves infinitely many normalizable positive-energy eigenstates with unbounded energies, contradicting the original BFSS paper's exclusion of additional bound states at N = 2.
PQC lens: Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
Primary manuscript on GitHub → Family index on GitHub →M547
The unique threshold bound state of the SU(N) BFSS model
For every finite N ≥ 2, we prove that undeformed ` SU (N)` BFSS matrix quantum mechanics has exactly one normalizable zero-energy state after removing the center of mass. This establishes the threshold-bound-state conjecture.
View manuscript on GitHub → Open PDF (raw) →M548
Positive eigenvalues of the relative SU(2) BFSS Hamiltonian
We prove that the relative ` SU (2)` BFSS Hamiltonian has infinitely many positive eigenvalues tending to infinity, with square-integrable eigenvectors. The operator is defined by closing the gauge-invariant supercharge form. This refutes, at N = 2, the exclusion of normalizable positive-energy states stated in the original BFSS paper.
View manuscript on GitHub → Open PDF (raw) →Family 272 Entanglement without distillable secret key. 1 paper
Constructs an entangled state on ` C¹⁰ C¹⁰` with zero distillable secret key for the specified local-instrument protocols that complete almost surely. These allow joint local processing and authenticated two-way public communication, with no other shared private resource and an eavesdropper holding the input purification and public record. A trace-preserving PPT channel on `M(21)( C)` whose square is not entanglement breaking disproves Christandl's PPT-square conjecture. ([Lean](lean/docs/272.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M553
Entanglement with zero distillable secret key in local dimension ten
We construct an entangled state on ` C¹⁰ C¹⁰` with zero distillable secret key for the local-instrument protocols specified here. They allow joint processing of all copies and unlimited two-way public communication, with the input as the only shared private resource and an eavesdropper holding a purification and the complete public record. The construction also disproves the unrestricted two-map PPT-composition conjecture. A separate trace-preserving PPT channel on `M(21)( C)` has a square that is not entanglement breaking.
View manuscript on GitHub → Open PDF (raw) →Family 273 The entropy photon-number inequality. 1 paper
Proves the entropy photon-number inequality for beam-splitter mixing of two independent finite-energy bosonic inputs in any finite number of modes: the output's entropy photon number is at least the transmissivity-weighted average of the inputs'. Arbitrary entanglement within each input is allowed, and product thermal inputs attain equality even when their entropies differ. ([Lean](lean/docs/273.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M554
The entropy photon-number inequality
We prove the entropy photon-number inequality for two independent bosonic inputs with finite mean energy, for every finite number of modes. This resolves the entropy photon-number conjecture in this finite-energy setting while allowing arbitrary entanglement among the modes within either input. As a consequence, we determine the exact minimum output entropy at fixed input entropy for every finite tensor power of an identical thermal attenuator, over finite-energy inputs that may be entangled across modes.
View manuscript on GitHub → Open PDF (raw) →Family 274 Parity is not in QAC0. 2 papers
Resolves Moore's parity conjecture in the measured-output model: constant-depth quantum circuits with arbitrary one-qubit gates, unbounded-arity Toffoli gates and polynomially many total qubits cannot compute parity with any fixed positive worst-case advantage. Ancillas start in zero, one output qubit is measured, and all other registers may be discarded. Xu–Li's reductions give the same bounded-error obstruction for strict majority. ([Lean](lean/docs/274.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M555
Product-projection localization and the QAC0 parity lower bound
We prove that constant-depth quantum circuits with arbitrary one-qubit and unbounded-arity Toffoli gates cannot compute parity with any fixed positive worst-case advantage using polynomially many qubits. Ancillas start in zero, only one output qubit is measured, and all final garbage is unrestricted. This resolves Moore's parity conjecture in the measured-output model.
View manuscript on GitHub → Open PDF (raw) →M556
Regular trajectories, pruning and quantum parity
We prove that constant-depth quantum circuits with arbitrary one-qubit gates and unbounded-arity Toffoli gates cannot compute parity with any fixed positive worst-case advantage using polynomially many total qubits. Ancillary qubits are initialized to `|0 `, one output qubit is measured, and all other final registers may be discarded without restriction. This resolves Moore's parity conjecture in the measured-output model.
View manuscript on GitHub → Open PDF (raw) →Family 275 QMA-hardness of continuum Coulomb energy. 2 papers
Proves QMA-hardness of approximating the electronic Coulomb energy infimum in three dimensions, minimizing over the full spinful fermionic continuum space. Deterministic polynomial-time reductions work even with only unit-charge nuclei at distinct rational positions, polynomially many electrons and an energy-threshold separation of at least one. ([Lean](lean/docs/275.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M557
Continuum Coulomb hardness with binary nuclear charges
We prove that approximating the electronic Coulomb spectral infimum in three-dimensional space is QMA-hard when positive integer nuclear charges are encoded in binary. The nuclei have distinct rational positions, the electron number is unary, and the energy is minimized over all antisymmetric continuum states and spin sectors. A deterministic classical polynomial-time reduction produces instances with threshold separation at least one. The nuclear charges may be exponentially large, but every output has polynomial bit length.
View manuscript on GitHub → Open PDF (raw) →M558
QMA-hardness of continuum Coulomb energy with unit nuclear charges
We prove that approximating the electronic ground-energy infimum for clamped unit-charge nuclei is QMA-hard on the full spinful fermionic continuum space. A deterministic classical polynomial-time reduction produces polynomially many nuclei at distinct rational positions and polynomially many electrons, with polynomial rational bit lengths and threshold separation at least one. No orbital basis, magnetic field, or additional external potential is supplied, and no binding assumption is imposed.
View manuscript on GitHub → Open PDF (raw) →Family 276 Classical capacity of generalized amplitude damping. 1 paper
Determines the unassisted classical capacity of every qubit generalized amplitude-damping channel, including all damping and thermal parameters. An explicit one-variable optimization gives the capacity, attained by independent two-state signal ensembles with collective decoding. Holevo capacity, minimum output entropy and regularized classical capacity are additive when tensoring with any finite-dimensional quantum channel. ([Lean](lean/docs/276.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M559
Classical capacity and entropy inequalities for generalized amplitude damping
We determine the unassisted classical capacity of every qubit generalized amplitude-damping channel, for all damping strengths and thermal occupations. It equals the one-shot Holevo capacity and is given by a one-variable maximum. A binary pure-state ensemble attains the one-use optimum, and its independent products attain the optimum at every block length.
View manuscript on GitHub → Open PDF (raw) →Family 277 Threshold repetition for entangled games. 1 paper
Proves exponential threshold repetition for every finite two-player one-round game: if its entangled value is v < 1, the probability of winning at least a fraction `v+δ` of k independent repetitions decays exponentially in k, for `0 δ 1-v`. Arbitrary joint finite-dimensional entangled strategies and correlated question distributions are allowed. ([Lean](lean/docs/277.md))
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Primary manuscript on GitHub → Family index on GitHub →M560
Threshold parallel repetition for finite-dimensional entangled games
For every finite two-player game with entangled value v < 1, we prove exponential decay for the probability of winning at least a `v+δ` fraction of k independent repetitions, uniformly over all finite-dimensional joint strategies. The result allows arbitrary correlated question distributions and holds for every `0 δ 1-v` and k ≥ 1. Its universal rate is proportional to `δ⁵/(1+log(| A|| B|))`, where ` A, B` are the answer alphabets. For each fixed question distribution, we also obtain an explicit cubic rate in δ.
View manuscript on GitHub → Open PDF (raw) →Family 278 Failure of Kohn–Sham ensemble representation. 1 paper
Constructs a three-electron Coulomb molecule with two equal positive-integer-charge nuclei whose absolute ground-state density has no noninteracting ground-state ensemble representation by a single real spin-independent local potential in `L³/²( R³)⁺L^∞( R³)`. This disproves Kohn–Sham ensemble representability for that potential class; the required nuclear charge is specified nonnumerically.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M561
A Coulomb ground-state density without Kohn-Sham ensemble representation
We construct a finite three-electron Coulomb molecule whose spin-summed ground-state density cannot be reproduced by any ground-state ensemble of noninteracting electrons in a single real, spin-independent local potential in `L³/²( R³)⁺L^∞( R³)`. The molecule has two equal positive integer nuclear charges, specified by an exact finite nonnumerical formula.
View manuscript on GitHub → Open PDF (raw) →Family 280 Unitary vertex operator algebras and conformal nets. 1 paper
Proves the strongly rational case of the strong-locality conjecture: every simple unitary strongly rational complex vertex operator algebra generates a completely rational conformal net. Its simple modules are unitarizable, and its representation category agrees with the net's finite-index sectors as a braided unitary tensor category. ([Lean](lean/docs/280.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M563
Strongly rational unitary vertex operator algebras and conformal nets
Every simple unitary strongly rational vertex operator algebra generates a completely rational conformal net. All its simple grading-restricted modules are unitarizable, its canonical fusion forms are positive, and the Carpi–Weiner–Xu functor gives a braided unitary tensor equivalence from its grading-restricted finite-length module category onto the finite-index sectors of the net.
View manuscript on GitHub → Open PDF (raw) →Family 281 QAOA attains the SK optimum in the thermodynamic-first limit. 2 papers
Proves that QAOA approaches the ground-state energy of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity before circuit depth. For every accuracy, finite depth and deterministic angles independent of size and disorder achieve the required limiting expected energy per spin. This also yields leading-order optimal expected MaxCut values on large-degree random regular graphs, with size tending to infinity before degree. ([Lean](lean/docs/281.md))
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Primary manuscript on GitHub → Family index on GitHub →M564
QAOA attains the SK ground-state energy in the thermodynamic-first limit
We prove that the Quantum Approximate Optimization Algorithm (QAOA) approaches the ground-state energy per spin of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity first and circuit depth then increases. For every accuracy, some finite depth and deterministic angles, independent of system size and disorder, achieve that accuracy in the limiting expected energy per spin using the standard cost Hamiltonian and transverse-field mixer.
View manuscript on GitHub → Open PDF (raw) →M565
Full support of the zero-temperature Sherrington-Kirkpatrick order parameter
We prove that every admissible integrable minimizer of the zero-temperature Parisi functional for the pure, zero-field Sherrington–Kirkpatrick model has full relative Stieltjes support on `[0,1)`. Thus its support has no gaps at any overlap scale below one. We use the covariance normalization ` (t)=t²/2`.
View manuscript on GitHub → Open PDF (raw) →Family 282 From scale symmetry to local conformal symmetry in four-dimensional QFT. 1 paper
Under the stated bounded-local-net and field-reconstruction hypotheses, proves that scale symmetry implies local conformal symmetry for four-dimensional unitary positive-energy theories with a discrete bounded-below scaling spectrum of finite multiplicity, finite scaling support and a physical local scale current. The stress tensor has a traceless improvement with unchanged spacetime charges. The conclusion concerns local Ward identities, not a global conformal action on the whole net.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M566
Scale and conformal symmetry in four-dimensional operational quantum field theory
We prove that scale symmetry implies local conformal symmetry for a class of four-dimensional unitary, positive-energy quantum field theories with a Poincaré- and scale-invariant vacuum. The class has a discrete, bounded-below spectrum of scaling dimensions with finite multiplicities, and each original field has finite scaling support.
View manuscript on GitHub → Open PDF (raw) →Family 283 Polynomial-time unitary synthesis from a Boolean oracle. 1 paper
Solves the constant-error Aaronson–Kuperberg unitary synthesis problem: a uniform polynomial-size quantum oracle circuit approximates every n-qubit unitary channel within diamond-norm error 1/2, after a suitable Boolean oracle is chosen. Gates, qubits, oracle calls and query length are polynomially bounded. The target-dependent oracle may have an unrestricted truth table; its efficient classical construction is not asserted.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M567
Polynomial-Time Unitary Synthesis from a Boolean Oracle
We give a positive answer to the constant-error formulation of the Aaronson–Kuperberg unitary synthesis problem. For every n, a quantum oracle circuit generated in polynomial time from n alone can approximate the channel of every n-qubit unitary to full diamond-norm error at most 1/2, after a suitable Boolean oracle is chosen. Using the fixed gates `H,T,T^,CNOT`, the circuit has polynomially many qubits, elementary gates, and oracle calls, and its oracle queries have polynomial length. The oracle may depend on the target unitary;
View manuscript on GitHub → Open PDF (raw) →Family 285 Counterexamples to Baum–Connes and Kadison–Kaplansky. 3 papers
Disproves the coefficient-free reduced Baum–Connes conjecture through a failure of rational injectivity and a separate failure of surjectivity of assembly. A finitely generated torsion-free example witnesses the injectivity failure. Separately, a torsion-free group has a nontrivial projection in its reduced group C∗-algebra, disproving the Kadison–Kaplansky conjecture.
PQC lens: Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
Primary manuscript on GitHub → Family index on GitHub →M569
A torsion-free counterexample to reduced Baum–Connes injectivity
We disprove rational injectivity of the coefficient-free reduced Baum–Connes assembly map for torsion-free groups. Specifically, we construct a finitely generated torsion-free discrete group whose degree-zero assembly map has a kernel class of infinite order.
View manuscript on GitHub → Open PDF (raw) →M570
A torsion-free counterexample to the Kadison–Kaplansky projection conjecture
We construct a finitely generated torsion-free discrete group whose reduced group C∗-algebra contains a projection other than zero and the identity. This disproves the Kadison–Kaplansky projection conjecture.
View manuscript on GitHub → Open PDF (raw) →M571
An irrational-trace counterexample to reduced Baum–Connes
We construct a finitely generated discrete group whose reduced group C∗-algebra contains a projection of irrational canonical trace. Its K0-class lies outside the image of the coefficient-free reduced Baum–Connes assembly map. This disproves the coefficient-free reduced Baum–Connes conjecture for countable discrete groups.
View manuscript on GitHub → Open PDF (raw) →Family 286 Rigidity and arithmetic of lattice von Neumann algebras. 2 papers
Classifies finite-index bimodules between scalar-twisted group factors of ICC groups commensurable with property- `(T)` lattices over characteristic-zero local fields and arbitrary ICC group factors. Every such bimodule is a summand of finite sums of models arising from finite-index subgroup isomorphisms and finite-dimensional projective representations. The classification also recovers the group, scalar cocycle and amplification scale up to the stated stable equivalence.
PQC lens: Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
Primary manuscript on GitHub → Family index on GitHub →M572
Arithmeticity of twisted finite correspondences for lattices over local fields
We prove arithmetic exhaustion for bifinite correspondences between arbitrarily scalar-twisted group factors of ICC groups commensurable with property- `(T)` lattices over characteristic-zero local fields and twisted group factors of arbitrary countable ICC groups. Every such correspondence is a closed summand of a finite direct sum of models obtained from actual finite-index subgroup isomorphisms and finite-dimensional projective representations of the matched cocycle ratio.
View manuscript on GitHub → Open PDF (raw) →M573
The arithmetic category and stable recovery of lattice factors
We prove stable canonical recovery for scalar-twisted group factors of ICC groups commensurable with property- `(T)` lattices over characteristic-zero local fields. Every specified stable isomorphism to a scalar-twisted factor of any countably infinite ICC group recovers the group, the cocycle up to a normalized cochain, and equal amplification scales, with an implementing partial isometry in the given matrix corners.
View manuscript on GitHub → Open PDF (raw) →Family 287 Isomorphism of the free group factors. 1 paper
Resolves the free group factor isomorphism problem: `L( F_2) L( F_3)`, and hence all interpolated free group factors, including `L( F_∞)`, are isomorphic. Their common factor has fundamental group ` R( 0)`. ([Lean](lean/docs/287.md))
PQC lens: Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
Primary manuscript on GitHub → Family index on GitHub →M574
An isomorphism of the free group factors
We solve the free group factor isomorphism problem affirmatively by proving that `L( F_2)` and `L( F_3)` are isomorphic as tracial von Neumann algebras. The classical free group factor alternative then implies that all interpolated free group factors, including `L( F_∞)`, are isomorphic and have fundamental group ` R( 0)`.
View manuscript on GitHub → Open PDF (raw) →Family 288 Kadison's similarity conjecture. 1 paper
Proves Kadison's similarity conjecture: every bounded complex-linear unital algebra homomorphism from a unital complex C∗-algebra to operators on a Hilbert space becomes a `*` -homomorphism after conjugation by a bounded invertible operator. ([Lean](lean/docs/288.md))
PQC lens: Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
Primary manuscript on GitHub → Family index on GitHub →M575
Kadison's similarity theorem through uniform derivation estimates
We prove that every bounded complex-linear unital algebra homomorphism from a unital complex C∗-algebra into the bounded operators on an arbitrary Hilbert space is similar to a `*` -homomorphism. This resolves Kadison's similarity conjecture positively. We also obtain one universal hyperreflexivity constant for all unital von Neumann algebras on arbitrary complex Hilbert spaces.
View manuscript on GitHub → Open PDF (raw) →Family 289 Strong Kadison–Kastler stability and its spatial boundaries. 3 papers
Proves that sufficiently close unital von Neumann algebras on the same Hilbert space are conjugate by a unitary arbitrarily close to the identity, with a universal tolerance in the operator-norm distance between unit balls. Counterexamples show that near-identity conjugacy fails for one-sided near inclusions, and that arbitrarily close norm-separable C∗-algebras need not be ambiently unitarily conjugate. ([Lean](lean/docs/289.md))
PQC lens: Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
Primary manuscript on GitHub → Family index on GitHub →M576
Universal strong Kadison–Kastler stability
We prove that sufficiently close unital von Neumann algebras are conjugate by a unitary arbitrarily close to the identity. The tolerance depends only on the prescribed distance of that unitary from the identity, uniformly over all algebras, representations, and Hilbert spaces. This resolves the strong Kadison–Kastler conjecture.
View manuscript on GitHub → Open PDF (raw) →M577
Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings
We give a negative answer to the unrestricted small-spatial-embedding problem for one-sided near inclusions of von Neumann algebras. On separable complex Hilbert spaces, we construct pairs of unital von Neumann algebras with common identity whose one-sided gaps tend to zero. Spatial embeddings of the source into the target exist, but every implementing unitary stays a fixed positive distance from the identity. The obstruction therefore concerns small implementing unitaries, rather than the existence of spatial embeddings.
View manuscript on GitHub → Open PDF (raw) →M578
Close Separable C*-Algebras Without Spatial Conjugacy
We disprove the separable C∗-algebraic spatial form of the Kadison–Kastler conjecture. For every ε > 0, we construct unital, norm-separable C∗-algebras on a common separable complex Hilbert space, with the same identity and Kadison–Kastler distance less than ε, that are not conjugate by any unitary. The two algebras have the same von Neumann closure.
View manuscript on GitHub → Open PDF (raw) →Family 290 Relative bicentralizers and modular spectral recovery. 2 papers
Proves Connes' bicentralizer conjecture for every type III1 factor with separable predual and every faithful normal state. More generally, for every inclusion `N M` of von Neumann algebras with separable preduals admitting a faithful normal conditional expectation, constructs an amenable expected subalgebra `P N` with `P' c(M)=N' c(M)`, resolving the relative bicentralizer conjecture. ([Lean](lean/docs/290.md))
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Primary manuscript on GitHub → Family index on GitHub →M579
Expected amenable subalgebras preserving core commutants
We prove that every inclusion `N M` of von Neumann algebras with separable preduals and a faithful normal conditional expectation contains an expected amenable subalgebra `P N` such that `P' c(M)=N' c(M)`, where `c(M)` denotes the continuous core of M. This resolves the relative bicentralizer conjecture in this setting.
View manuscript on GitHub → Open PDF (raw) →M580
Bounded recovery for modular spectral averages
We prove a bounded spectral recovery theorem for a von Neumann algebra with a faithful normal state whose centralizer consists only of scalars. A positive averaged squared norm for an arbitrary bounded Hilbert-space operator on shrinking modular spectral bands can be recovered on uniformly bounded algebra elements with shrinking spectral support. We apply this theorem to spectral intertwining rigidity and obtain an alternative proof of bicentralizer triviality for type III1 factors with separable predual.
View manuscript on GitHub → Open PDF (raw) →Family 291 Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability. 4 papers
Proves equivariant Jiang–Su stability for every countable discrete amenable group action on a simple separable unital infinite-dimensional nuclear stably finite Jiang–Su-stable C∗-algebra, resolving this case of Szabó's conjecture without restrictions on trace dynamics. The family also proves the unital Toms–Winter conjecture, equating strict comparison, finite nuclear dimension and Jiang–Su stability in the simple separable unital infinite-dimensional nuclear setting. ([Lean](lean/docs/291.md))
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Primary manuscript on GitHub → Family index on GitHub →M581
Equivariant Jiang–Su Stability for Amenable Actions in the Unital Stably Finite Case
Every action of a countable discrete amenable group on a simple, separable, unital, infinite-dimensional, nuclear, stably finite complex C∗-algebra that is already Jiang–Su stable absorbs the trivial action on the Jiang–Su algebra up to cocycle conjugacy. No restriction is imposed on the action on the tracial-state simplex. This proves the unital, stably finite case of Szabó's Conjecture A on automatic equivariant Jiang–Su stability.
View manuscript on GitHub → Open PDF (raw) →M582
Cuntz comparison and Jiang–Su absorption
We prove that strict comparison in the extended-functional sense implies Jiang–Su absorption for separable simple nuclear non-elementary C∗-algebras. This resolves the corresponding implication of the Toms–Winter regularity problem, including nonunital algebras and allowing unbounded traces. More generally, every separable nuclear C∗-algebra whose Cuntz semigroup is almost unperforated and fully almost divisible absorbs the Jiang–Su algebra.
View manuscript on GitHub → Open PDF (raw) →M583
Nuclear dimension and Jiang–Su stability without elementary subquotients
For separable nuclear C∗-algebras with no nonzero elementary ideal subquotients, finite nuclear dimension is equivalent to Jiang–Su stability. More generally, `dim(nuc)(A_0 Z)≤1` for every separable nuclear A0. This proves Robert and Tikuisis's Conjecture (C1) and the nuclear-dimension equivalence in the nonsimple Toms–Winter regularity question.
View manuscript on GitHub → Open PDF (raw) →M584
Tracial projection methods and uniform property Gamma
For a simple, separable, unital, infinite-dimensional, nuclear, stably finite C∗-algebra with traces, real rank zero of the uniform tracial ultrapower of its uniform tracial completion implies uniform property Γ. This answers Problem XXI of Schafhauser, Tikuisis and White affirmatively. Independently, strict comparison implies Jiang–Su absorption for simple, separable, unital, infinite-dimensional nuclear algebras, resolving the unital Toms–Winter conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 292 Kirchberg's ` O_2` norm-ultrapower embedding problem. 1 paper
Constructs an explicit separable unital full group C∗-algebra that cannot embed unitally into the norm ultrapower of any fixed nonzero unital nuclear C∗-algebra, for any free ultrafilter on the natural numbers. Taking the target to be ` O_2` answers Kirchberg's norm-ultrapower embedding problem negatively. ([Lean](lean/docs/292.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M585
An explicit obstruction to nuclear norm-ultrapower embeddings
We give a negative answer to Kirchberg's norm-ultrapower embedding problem. We construct an explicit separable unital full group C∗-algebra that admits no unital embedding into Bω, for any nonzero unital nuclear C∗-algebra B and any free ultrafilter ω on ℕ. In particular, it does not embed unitally into ` O_2^ω`.
View manuscript on GitHub → Open PDF (raw) →Family 293 Invariant projections, hyperinvariant subspaces, and transitive algebras. 2 papers
Constructs a nonzero norm-quasinilpotent operator on every infinite-dimensional separable complex Hilbert space with no nonzero proper closed subspace invariant under every commuting operator. The construction also gives operators with no nontrivial invariant projection in the hyperfinite type II1 factor. ([Lean](lean/docs/293.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M586
Invariant-projection counterexamples for every irrational rotation
For every irrational angle, we construct a continuous nonnegative circle weight with exactly one zero and logarithmic integral `-∞` whose weighted rotation has no nontrivial invariant projection in the associated hyperfinite type II1 factor. This answers negatively the question of Zhu, Fang, and Shi, with the angle prescribed in advance. The resulting operator is nonzero and norm-quasinilpotent.
View manuscript on GitHub → Open PDF (raw) →M587
Backward intertwiners and a transitive commutant
We give a negative answer to the hyperinvariant-subspace problem by constructing, on every infinite-dimensional separable complex Hilbert space, a nonzero bounded norm-quasinilpotent operator with no nonzero proper closed hyperinvariant subspace. Its commutant is a proper strongly closed unital transitive complex operator algebra.
View manuscript on GitHub → Open PDF (raw) →Family 294 Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness. 1 paper
Disproves Kaplansky's quasitrace conjecture by constructing a separable unital complex C∗-algebra admitting normalized 2-quasitraces, all of which are nonadditive. As a consequence, two unital simple stably finite C∗-algebras can have a properly infinite minimal tensor product, with one factor `C_r*( F_2)`. ([Lean](lean/docs/294.md))
PQC lens: Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
Primary manuscript on GitHub → Family index on GitHub →M588
A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness
We refute Kaplansky's quasitrace conjecture by constructing a separable unital complex C∗-algebra that admits normalized 2-quasitraces but has no tracial state. As a consequence, we show that the minimal tensor product of two unital simple stably finite complex C∗-algebras can be properly infinite, even when one factor is the reduced free-group algebra `C_r*( F_2)`.
View manuscript on GitHub → Open PDF (raw) →Family 295 The Kadison–Ringrose cohomology conjecture. 1 paper
Proves that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with coefficients in the algebra itself, has a bounded primitive. Equivalently, all higher bounded Hochschild cohomology groups vanish, resolving the Kadison–Ringrose conjecture. ([Lean](lean/docs/295.md))
PQC lens: Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
Primary manuscript on GitHub → Family index on GitHub →M589
Vanishing of higher bounded Hochschild cohomology
We prove that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with values in the algebra itself, has a bounded primitive. Together with the established degree-one inner-derivation theorem, this resolves the Kadison–Ringrose cohomology conjecture positively.
View manuscript on GitHub → Open PDF (raw) →Family 296 The generator problem for finite factors. 1 paper
Proves that every type II1 factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion `P M` of such factors, the unitaries u with `M=W*(P,u)` form a dense Gδ subset in the trace 2-norm topology. ([Lean](lean/docs/296.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M590
Relative generation and the generator problem for finite factors
We prove that every type II1 factor with separable predual is generated by one operator, equivalently by two self-adjoint operators. Combined with the established direct-integral reduction to this case, this gives an affirmative solution of the generator problem for von Neumann algebras with separable predual.
View manuscript on GitHub → Open PDF (raw) →Family 297 A ZFC counterexample to Naimark's problem. 1 paper
Gives an alternative to [Tanaka's ZFC construction](https://arxiv.org/abs/2609.26930v1) of a unital infinite-dimensional simple complex C∗-algebra with a faithful tracial state and exactly one nonzero irreducible representation up to unitary equivalence. Thus the unrestricted compact-operator characterization fails without additional set-theoretic assumptions; the counterexample is nonseparable. ([Lean](lean/docs/297.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M591
A counterexample to Naimark's problem in ZFC
We construct in ZFC a unital infinite-dimensional simple complex C∗-algebra with a faithful tracial state whose nonzero irreducible representations are all unitarily equivalent. This gives a negative answer to Naimark's problem without additional set-theoretic assumptions.
View manuscript on GitHub → Open PDF (raw) →Family 298 Two notions of free entropy differ even when both are finite. 1 paper
Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy `-∞ * ∞`. This answers Voiculescu's finite-entropy equality question negatively: the matrix-approximation and free-Fisher-information definitions differ even when both are finite. ([Lean](lean/docs/298.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M592
A finite-entropy separation of microstates and nonmicrostates free entropy
We answer the finite-entropy equality question for microstates and nonmicrostates free entropy negatively. We construct a bounded self-adjoint tuple X in a von Neumann algebra with faithful normal tracial state such that `-∞ (X)≤ *(X)- tfrac12 ∞`. Here χ is the original microstates entropy with an operator-norm cutoff and a limsup over matrix sizes. The counterexample uses a large but fixed number of variables.
View manuscript on GitHub → Open PDF (raw) →Family 299 The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability. 1 paper
A nonzero unital separable complex C∗-algebra is Jiang–Su stable exactly when its norm central-sequence algebra has no characters, for every free ultrafilter. This answers the Kirchberg–Rørdam character question. Also, the infinite minimal tensor power of every such algebra without characters is Jiang–Su stable, answering the Dadarlat–Toms question. ([Lean](lean/docs/299.md))
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Primary manuscript on GitHub → Family index on GitHub →M593
The Kirchberg–Rørdam character criterion
We prove the Kirchberg–Rørdam character criterion: for every free ultrafilter on ℕ, a nonzero unital separable complex C∗-algebra absorbs the Jiang–Su algebra if and only if its norm central-sequence algebra has no characters. We also prove that the infinite minimal tensor power of every nonzero unital separable complex C∗-algebra without characters is ` Z` -stable and contains a unital copy of ` Z`.
View manuscript on GitHub → Open PDF (raw) →Family 300 Approximation and quadratic strong-operator paving. 2 papers
Proves that every self-adjoint element of a complex von Neumann algebra admits strong-operator paving relative to any maximal abelian subalgebra with `O(ε⁻²)` blocks. The norm bound holds after compression by a projection arbitrarily close to the identity in the strong topology, resolving the Popa–Vaes quadratic paving conjecture.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M594
Approximation Paving over Arbitrary Maximal Abelian Subalgebras
We prove the approximation-paving conjecture of Popa and Vaes for every maximal abelian subalgebra of a complex von Neumann algebra. For each `0 ε 1`, every self-adjoint operator is a strong limit of self-adjoint operators of norm at most three times its norm, each admitting a norm paving with error at most ε times the approximant's own norm. The number of projections is at most `Cε⁻⁶` for a universal constant C. No separability or conditional-expectation hypothesis is required.
View manuscript on GitHub → Open PDF (raw) →M595
Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras
We prove the quadratic strong-operator paving conjecture of Popa and Vaes. For every `0 ε 1`, every self-adjoint element of a von Neumann algebra admits strong-operator paving over each maximal abelian subalgebra with at most `5×10⁸ε ⁻²` projections. The bound is uniform over representations and requires no separability or conditional-expectation assumption.
View manuscript on GitHub → Open PDF (raw) →Family 301 Trace cones and Razak–Jacelon stabilization. 1 paper
Classifies separable nuclear complex C∗-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their full topological cones of extended lower-semicontinuous tracial weights. This answers Robert's trace-cone question, including algebras with arbitrary ideal structure and both finite and infinite subquotients.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M596
The trace cone classifies Razak–Jacelon stabilizations
We prove that the canonical topological cone of all extended lower-semicontinuous tracial weights determines a separable nuclear C∗-algebra after tensoring with the Razak–Jacelon algebra and the compact operators, answering Robert's trace-cone classification question positively. The isomorphism realizes the prescribed cone map, with arbitrary ideal structure and without a density assumption on the finite domains of the weights.
View manuscript on GitHub → Open PDF (raw) →Family 302 Radius of comparison equals half the mean dimension. 2 papers
For every minimal homeomorphism h of an infinite compact metrizable space X, the radius of comparison of `C(X) rtimes_h Z` equals ` tfrac12mdim(X,h)`, including infinite values. Zero mean dimension is equivalent to the small boundary property, Jiang–Su stability and finite nuclear dimension; in this case nuclear dimension is at most one.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M597
Filtered products and boundary-preserving compression in complex cobordism
We prove that, in sufficiently large smash powers, an `MU` -null restriction to a finite pointed subcomplex becomes stably null on the entire union of products with a fixed positive proportion of restricted factors. This coherent vanishing theorem yields boundary-preserving compression of cube-valued maps on every compact metrizable input space.
View manuscript on GitHub → Open PDF (raw) →M598
Radius of comparison equals half the mean dimension
We prove the integer-action case of the Phillips–Toms conjecture: for every minimal homeomorphism of an infinite compact metrizable space, the radius of comparison of its crossed product equals one half of its mean dimension, including equality at infinity. For these systems, zero mean dimension and the small boundary property are equivalent to Jiang–Su stability and to finite nuclear dimension of the crossed product; in this case its nuclear dimension is at most one.
View manuscript on GitHub → Open PDF (raw) →Family 303 Weak pure infiniteness and Cuntz-algebra absorption. 1 paper
Resolves the ordinary-to-strong pure-infiniteness question of Kirchberg and Rørdam for complex C∗-algebras. For exact algebras, proper infiniteness of one fixed finite amplification of every positive element also suffices. Consequently, every separable nuclear algebra with this property absorbs ` O_∞`, without unitality or simplicity assumptions. ([Lean](lean/docs/303.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M599
Weak pure infiniteness and O-infinity absorption
We prove that every complex C∗-algebra in which each positive element is properly infinite is strongly purely infinite. This answers the ordinary-to-strong part of Kirchberg–Rørdam's comparison question. For exact algebras, we also prove that proper infiniteness of one fixed finite amplification of every positive element implies proper infiniteness of each positive element. Consequently, every separable nuclear algebra with this fixed-amplification property absorbs ` O_∞`, without assumptions of unitality or simplicity.
View manuscript on GitHub → Open PDF (raw) →Family 304 The Hilbert–Smith conjecture in every dimension. 1 paper
Every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected finite-dimensional topological manifold is a Lie group. This proves the Hilbert–Smith conjecture in all finite dimensions, for Hausdorff second-countable manifolds without boundary.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M600
The Hilbert–Smith conjecture in every finite dimension
We prove the Hilbert–Smith conjecture in every finite dimension: every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected Hausdorff second-countable finite-dimensional topological manifold without boundary is a Lie group.
View manuscript on GitHub → Open PDF (raw) →Family 305 Four-dimensional disk embedding and Wall's conjecture. 3 papers
The unrestricted four-dimensional disk-embedding conjecture fails: framed algebraic dual spheres do not suffice to obtain disjoint locally flat spanning disks. In particular, the free group F2 is not good in the sense of Freedman–Quinn. Also constructs a finitely presented integral Poincaré duality group of dimension four with a finite classifying space but no realization as the fundamental group of a closed aspherical topological four-manifold, disproving Wall's conjecture.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M601
A boundary-only obstruction to four-dimensional disk embedding
We disprove the four-dimensional disc embedding conjecture without a fundamental-group hypothesis, even when no homotopy classes or output framings are prescribed. We construct a compact oriented smooth four-manifold containing finitely many disc maps with framed algebraic dual spheres whose boundary circles bound no disjoint locally flat discs. Consequently, the free group on two generators is not good in the sense of Freedman–Quinn, and neither is any group containing it as a subgroup.
View manuscript on GitHub → Open PDF (raw) →M602
A marked tensor obstruction to four-dimensional disk embedding
We disprove the unrestricted four-dimensional disk-embedding conjecture. We construct immersed disks in a compact oriented smooth four-manifold with framed algebraic dual spheres satisfying the usual equivariant intersection and reduced self-intersection conditions, but with no pairwise disjoint locally flat replacements that preserve the boundary maps and induced normal framings. The obstruction holds even when the replacement disks' relative homotopy classes are not prescribed.
View manuscript on GitHub → Open PDF (raw) →M603
A PD4 group without an aspherical manifold model
We construct a finitely presented integral Poincaré duality group of dimension four that has a finite classifying space but is not the fundamental group of any closed aspherical topological four-manifold. This gives a negative answer to Wall's manifold-realization question in dimension four.
View manuscript on GitHub → Open PDF (raw) →Family 306 The purely cosmetic surgery conjecture. 1 paper
Distinct Dehn surgery slopes on a nontrivial smooth knot in S3 never produce orientation-preservingly homeomorphic manifolds, proving the purely cosmetic surgery conjecture. The statement includes the meridional slope.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M604
Purely cosmetic surgery on knots in the three-sphere
We prove the purely cosmetic surgery conjecture for knots in the three-sphere. Distinct Dehn surgery slopes on a nontrivial smooth knot never give orientation-preservingly homeomorphic manifolds.
View manuscript on GitHub → Open PDF (raw) →Family 307 Failure of rational injectivity for maximal coarse assembly. 2 papers
Constructs a uniformly discrete bounded-geometry space whose maximal coarse assembly map is not rationally injective. The example is a coarse disjoint union of finite connected graphs of uniformly bounded degree, with an infinite-order kernel class. A companion gives the analogous failure for reduced coarse assembly, disproving the rational coarse Novikov conjecture. ([Lean](lean/docs/307.md))
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Primary manuscript on GitHub → Family index on GitHub →M605
Failure of rational injectivity for maximal coarse assembly
We construct a uniformly discrete bounded-geometry space whose maximal coarse assembly map has an infinite-order element in its kernel. The space is a coarse disjoint union of finite connected graphs of uniformly bounded degree, so maximal coarse assembly need not be rationally injective even for such graph unions.
View manuscript on GitHub → Open PDF (raw) →M606
A counterexample to the coarse Novikov conjecture
We disprove the coarse Novikov conjecture: ordinary coarse assembly need not be rationally injective for uniformly discrete spaces of bounded geometry. We construct a coarse disjoint union of finite graphs of uniformly bounded degree and an infinite-order class in its degree-one coarse K-homology whose image under ordinary coarse assembly in the K-theory of the reduced, locally compact Roe algebra vanishes.
View manuscript on GitHub → Open PDF (raw) →Family 308 Finite Smith–Toda complexes at every height. 2 papers
For every n ≥ 0, constructs a finite Smith–Toda spectrum at a prime p depending on n, with Brown–Peterson homology `BP_*/(p,v_1,v_n)` and the canonical comodule structure. Thus Smith–Toda complexes exist at every height when the prime may vary; an explicit example realizes `V(4)` at p = 1009.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M607
Finite Smith–Toda Complexes at Varying Primes
For every nonnegative integer n, we construct a Smith–Toda complex `V(n)` at some prime p depending on n. It is a finite p-local spectrum whose Brown–Peterson homology is `BP_*/(p,v_1,v_n)` with its canonical comodule structure and generator in degree zero. Each listed generator is killed to its first power.
View manuscript on GitHub → Open PDF (raw) →M608
A finite Smith–Toda complex V(4) at the prime 1009
We construct a Smith–Toda complex `V(4)` at the prime 1009. It is an ordinary finite 1009-local spectrum whose Brown–Peterson homology is `BP_*/(1009,v_1,v_2,v_3,v_4)`, with its canonical comodule structure and generator in degree zero.
View manuscript on GitHub → Open PDF (raw) →Family 309 The Kervaire invariant problem at the prime three. 1 paper
Resolves the odd-primary Kervaire invariant problem at the prime three: exactly the standard classes with indices 0, 2, and 3 survive in the mod-three Adams spectral sequence, in stems 10, 106, and 322. Each surviving detection coset contains an element of exact additive order three.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M609
The Kervaire invariant problem at the prime three
We solve the Kervaire invariant problem at the prime three for the standard Kervaire classes in the mod-three Adams spectral sequence. These classes survive precisely at indices 0, 2, and 3, and each surviving detection coset contains an element of additive order three. In particular, there is an order-three Kervaire element in stem 322.
View manuscript on GitHub → Open PDF (raw) →Family 310 Quillen's conjecture in rational homology. 1 paper
Proves the rational-homology form of Quillen's conjecture for every finite group and every prime. If the largest normal p-subgroup of G is trivial, the poset of nontrivial elementary abelian p-subgroups has nonzero augmented reduced rational homology and is therefore not contractible.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M610
Rational homology and Quillen's conjecture
We prove Quillen's conjecture for all finite groups and all primes. More precisely, if a finite group G has trivial largest normal p-subgroup `O_p(G)`, then the poset of nontrivial elementary abelian p-subgroups of G has nonzero augmented reduced rational homology. This establishes the stronger rational-homology form of the conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 311 The Hovey–Strickland and Chai conjectures. 1 paper
Proves Chai's invariant-ideal conjecture for Lubin–Tate deformation rings over finite residue fields, at every prime and positive height n. Through the implication of Barthel–Heard–Naumann, this proves the Hovey–Strickland conjecture: dualizable `K(n)` -local spectra have exactly `n+2` thick tensor ideals, and their Balmer spectrum is a chain of `n+1` points.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M611
Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra
We classify the prime and radical ideals of the Lubin–Tate deformation ring invariant under an open Morava stabilizer subgroup, at every prime and positive height. This proves Chai's invariant-ideal conjecture in the standard finite-residue-field formulation. It also proves the Hovey–Strickland conjecture: the dualizable `K(n)` -local category has exactly `n+2` thick tensor ideals, and its Balmer spectrum is a chain of `n+1` points.
View manuscript on GitHub → Open PDF (raw) →Family 312 The Grothendieck homotopy hypothesis. 1 paper
Proves the Grothendieck homotopy hypothesis for ∞-groupoids associated with every Grothendieck coherator in the Ara–Henry convention: these algebraic objects recover the homotopy theory of spaces. ([Lean](lean/docs/312.md))
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Primary manuscript on GitHub → Family index on GitHub →M612
The Grothendieck homotopy hypothesis via elementary expansions
We prove the Grothendieck homotopy hypothesis for every Grothendieck coherator in the Ara–Henry convention: its weak globular infinity-groupoids recover the homotopy theory of spaces. We also resolve Henry's pushout conjecture, showing that elementary expansions preserve components and all homotopy groups of cellular infinity-groupoids.
View manuscript on GitHub → Open PDF (raw) →Family 313 Finite generation for the `K(n)` -local sphere. 1 paper
Answers the degreewise finiteness question of Hovey and Hovey–Strickland: every homotopy group of the `K(n)` -local sphere is a finitely generated ℤp-module, for every prime, positive height and integer degree. The same conclusion holds after `K(n)` -localizing any finite p-local spectrum.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M613
Finite generation for the K(n)-local sphere
We prove that the homotopy groups of the `K(n)` -local sphere are finitely generated ℤp-modules in every integer degree, for every prime p and positive height n. Equivalently, the same conclusion holds after `K(n)` -localizing any finite p-local spectrum. This answers the degreewise finiteness question of Hovey and Hovey–Strickland.
View manuscript on GitHub → Open PDF (raw) →Family 314 Cyclic length and chromatic fixed-point loss. 1 paper
Determines the optimal chromatic loss from geometric H-fixed points to geometric G-fixed points for every subgroup H of a finite p-group G. At every nonnegative height, the loss equals the shortest subnormal-chain length from H to G with cyclic quotients. Each quotient counts once regardless of order, and finite spectra witness sharpness.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M614
Cyclic length and chromatic fixed-point loss
For a finite p-group G and a subgroup H, we prove that the optimal chromatic fixed-point loss equals the shortest length of a subnormal chain from H to G with cyclic quotients. The equality holds at every prime and every nonnegative height, resolving positively the equality proposed by Kuhn and Lloyd.
View manuscript on GitHub → Open PDF (raw) →Family 315 The four-dimensional Singer conjecture. 1 paper
Proves that the L2-Betti numbers of the universal cover of every closed connected aspherical topological four-manifold vanish outside degree two. More generally, the same conclusion holds for every finite connected aspherical integral Poincaré complex of formal dimension four, proving the four-dimensional Singer conjecture in this wider class.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M615
The Singer conjecture in dimension four
We prove the four-dimensional Singer conjecture: the L2-Betti numbers of the universal cover of a closed connected aspherical topological four-manifold vanish outside degree two. More generally, the same vanishing holds for finite connected aspherical integral Poincaré complexes of formal dimension four, including nonorientable ones.
View manuscript on GitHub → Open PDF (raw) →Family 316 Curtis’s conjecture. 1 paper
Proves Curtis's conjecture: the positive-degree mod-two stable Hurewicz image of the sphere is spanned by the images of the Hopf-invariant-one classes η, ν, σ and the Kervaire-invariant-one classes that exist.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M616
The Stable Hurewicz Image of the Sphere at Two
We prove Curtis's conjecture: in every positive degree, the mod-two stable Hurewicz image of the sphere is spanned by the images of η, ν, σ and the Kervaire-invariant-one classes that exist. Consequently, Eccles's conjecture holds for every sphere Sn with n > 0.
View manuscript on GitHub → Open PDF (raw) →Family 317 Thomason model structures in all strict higher dimensions. 1 paper
Resolves the Ara–Maltsiniotis conjecture: for every n ≥ 1 and n = ω, small strict globular n-categories admit proper combinatorial Thomason model structures Quillen equivalent to simplicial sets. Thus strict higher categories model the homotopy theory of spaces in every stated dimension. ([Lean](lean/docs/317.md))
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Primary manuscript on GitHub → Family index on GitHub →M617
Thomason Model Structures in Every Strict Higher Dimension
We prove the higher-dimensional Thomason model-structure conjecture of Ara and Maltsiniotis. For every `1≤ n≤∞`, the category of small strict globular n-categories admits a proper combinatorial model structure that is Quillen equivalent to simplicial sets. Its weak equivalences and fibrations are detected by the twice-extended Street nerve `Ex²N_n`, and the Quillen equivalence is given by `c_nSd² dashvEx²N_n`. Thus strict higher categories model the homotopy theory of spaces in every positive finite dimension and in dimension ω.
View manuscript on GitHub → Open PDF (raw) →Family 318 Chromatic splitting: filtrations and counterexamples. 5 papers
Disproves strong chromatic splitting at height three for primes p ≥ 5, and weak splitting for the derived p-completed sphere at heights p (p ≥ 5) and `p+1` (p ≥ 7). Nevertheless, for n ≥ 1 and `p n+1`, the overlap `L(n-1)L(K(n))S_p^ ` admits a `2n` -stage filtration by the predicted localized-sphere pieces. At height three and prime three, even finite assembly from such pieces fails in the category of `E(2)` -local modules over the derived completed sphere.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M618
Filtered chromatic splitting at generic primes
For every n ≥ 1 and prime `p n+1`, we construct a `2n` -stage ordered filtration of `L(n-1)L(K(n))S_p^ ` with the classical chromatic-splitting cofibers. The map from the first stage to the target is the canonical localization unit.
View manuscript on GitHub → Open PDF (raw) →M619
The height-three chromatic overlap: an explicit filtration and its attachments
For every prime p ≥ 5, we construct an explicit eight-stage filtration of `L_2L(K(3)) S_p^ ` by the local-sphere layers in the height-three chromatic-splitting pattern. The map from its first stage to the overlap is the canonical unit, and two signed fracture formulas identify all attachments for the chosen local maps and compatibility homotopy. A companion canonical-map theorem further implies that the first height-one attachment is nonzero.
View manuscript on GitHub → Open PDF (raw) →M620
A rational obstruction to strong chromatic splitting at height three
For every prime p ≥ 5, the canonical map `L_0L(K(3))S→ L_0L(K(2))L(K(3))S` for the sphere spectrum S is nonzero on π−3. Consequently, the height-three strong chromatic splitting formula is false in this range, even as an equivalence of underlying `E(2)` -local spectra without specified summand maps.
View manuscript on GitHub → Open PDF (raw) →M621
Failure of finite assembly for a chromatic overlap at the prime three
At the prime three and height three, the chromatic overlap cannot be constructed from the rational, height-one, and height-two local spheres by finitely many sums, shifts, cofibers, and retracts in the category of `E(2)` -local modules over the derived 3-complete sphere. This gives a negative answer to the ordinary finite-assembly question for chromatic overlaps.
View manuscript on GitHub → Open PDF (raw) →M622
Counterexamples to weak chromatic splitting: sphere kernels and descent exponents
For the derived p-completion of the sphere, the canonical weak chromatic splitting map has no homotopy retraction at height p for every prime p ≥ 5, and at height `p+1` for every prime p ≥ 7. The classical sphere product `β_1(p⁻¹)^²` gives a nonzero kernel class in both ranges.
View manuscript on GitHub → Open PDF (raw) →Family 319 Counterexamples to finite generation at chromatic height two. 1 paper
Refutes the Hahn–Wilson conjecture at chromatic height two. For every sufficiently large prime p, constructs a connective p-complete spectrum of exact fp-type two that cannot be built from completed `BP langle2 ` by finitely many sums, shifts, cones and retracts. The examples nevertheless satisfy the finite and telescopic localization comparisons.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M623
Counterexamples to the Hahn-Wilson conjecture at height two
We disprove the Hahn–Wilson conjecture at height two. For every sufficiently large prime p, we construct a connective p-complete spectrum X of exact fp-type two outside the ordinary thick subcategory generated by the specified standard form of `BP langle2 rangle_p^ `. The same spectrum satisfies both localization comparisons `L_2fX L_2X` and `L(T(2))X L(K(2))X`.
View manuscript on GitHub → Open PDF (raw) →Family 320 Nonhomeomorphic closed aspherical four-manifolds. 1 paper
Constructs closed connected aspherical topological four-manifolds that are homotopy equivalent but not homeomorphic, with a common word-hyperbolic fundamental group. This disproves even the homeomorphism-existence formulation of the Borel conjecture in dimension four.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M624
Nonhomeomorphic closed aspherical four-manifolds with the same homotopy type
We construct closed connected aspherical topological four-manifolds that are homotopy equivalent but not homeomorphic, disproving the homeomorphism-existence formulation of the Borel conjecture in dimension four. Their common fundamental group is word-hyperbolic. One of the manifolds also has a self-homotopy equivalence not homotopic to a homeomorphism.
View manuscript on GitHub → Open PDF (raw) →Family 321 A counterexample to Wall's finite D(2) problem. 1 paper
Constructs a finite connected three-dimensional CW complex whose universal cover has no integral homology above degree two and whose third cohomology vanishes for every local coefficient module, but which has no finite two-dimensional homotopy model. This disproves Wall's finite D(2) conjecture; the example has infinite fundamental group.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M625
A Counterexample to Wall's D(2) Problem
We give a negative answer to Wall's finite `D(2)` problem. We construct a finite connected three-dimensional CW complex satisfying the `D(2)` finiteness condition but not homotopy equivalent to any finite CW complex of dimension at most two. The example has infinite fundamental group.
View manuscript on GitHub → Open PDF (raw) →Family 322 Tingley’s sphere-isometry problem. 1 paper
Resolves Tingley's problem: every surjective isometry between the unit spheres of nonzero real Banach spaces extends uniquely to a surjective real-linear isometry of the whole spaces. No dimension restriction is imposed, so the metric geometry of the unit sphere determines the Banach space up to linear isometry. ([Lean](lean/docs/322.md))
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Primary manuscript on GitHub → Family index on GitHub →M626
A positive solution to Tingley’s problem
Every surjective isometry between the unit spheres of real Banach spaces extends uniquely to a surjective real-linear isometry, giving an affirmative solution to Tingley's problem. If distances between different radii are not preserved, we realize the positive maximal defect in a possibly enlarged pair of Banach spaces. We then align extremal chords using common supports and Darbo's fixed-point theorem and obtain a contradiction from support and convexity estimates.
View manuscript on GitHub → Open PDF (raw) →Family 323 Independence of the separable quotient problem. 1 paper
Establishes, relative to the consistency of a measurable cardinal, that the separable quotient problem is independent of ZFC. The assertion that every infinite-dimensional Banach space has a separable infinite-dimensional quotient can hold for all real and complex Banach spaces, whereas the continuum hypothesis yields counterexamples over both fields. ([Lean](lean/docs/323.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M627
Relative independence of the separable quotient problem
The separable quotient problem asks whether every infinite-dimensional Banach space has a separable infinite-dimensional quotient. We prove that this statement is independent of ZFC, relative to the consistency of ZFC with a measurable cardinal. This holds over both the real and complex fields. For spaces of norm density ℵ1, we obtain independence relative to the consistency of ZFC alone.
View manuscript on GitHub → Open PDF (raw) →Family 324 Lipschitz equivalent Banach spaces need not be linearly isomorphic. 2 papers
Constructs separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic, resolving the separable Lipschitz-isomorphism problem negatively. Thus even the complete metric structure up to bi-Lipschitz equivalence does not determine a separable Banach space's linear isomorphism class. ([Lean](lean/docs/324.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M628
Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic
There are separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic. This gives a negative answer to the separable Banach-space Lipschitz-isomorphism problem.
View manuscript on GitHub → Open PDF (raw) →M629
Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0
We construct a separable real Banach space Z that contains no linear copy of c0, yet is bi-Lipschitz equivalent to `Z oplus_∞ c_0`. The same space contains a bi-Lipschitz image of every separable metric space.
View manuscript on GitHub → Open PDF (raw) →Family 325 The complete Crouzeix conjecture. 2 papers
Resolves the complete Crouzeix conjecture: for every bounded operator A on a complex Hilbert space and every finite matrix-valued polynomial P, one has ` P[A] ≤2sup(z W(A)) P(z) `, where `W(A)` is the numerical range. The constant 2 is sharp, independent of the matrix size, and valid in infinite dimensions. ([Lean](lean/docs/325.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M630
A direct proof of the complete Crouzeix inequality
We give a direct proof of the sharp constant-two numerical-range inequality for matrix-valued polynomials in all finite base and coefficient dimensions. This resolves the complete Crouzeix conjecture in its matrix formulation, including matrices whose numerical ranges are points or line segments.
View manuscript on GitHub → Open PDF (raw) →M631
The complete Crouzeix theorem: optimal similarity and a common positive boundary representation
We resolve the complete Crouzeix conjecture by proving the sharp constant-two numerical-range inequality for every bounded operator on a complex Hilbert space and every matrix-valued polynomial. No separability assumption is needed. The closure of the numerical range is a complete 2-spectral set, and the bound extends to finite matrix-valued functions holomorphic near that closure.
View manuscript on GitHub → Open PDF (raw) →Family 326 The cotype–cotype conjecture under the approximation property. 1 paper
Resolves the cotype–cotype conjecture for real Banach spaces with the approximation property. Such a nonzero space is K-convex if and only if both it and its dual have finite Rademacher cotype, with possibly different exponents. Equivalently, these cotype assumptions force nontrivial Rademacher type. ([Lean](lean/docs/326.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M632
The cotype–cotype conjecture under the approximation property
We prove the cotype–cotype conjecture under the ordinary approximation property. A nonzero real Banach space with this property is K-convex if and only if both the space and its dual have finite Rademacher cotype, possibly with different exponents.
View manuscript on GitHub → Open PDF (raw) →Family 327 Markov type characterizes superreflexivity. 1 paper
Proves that every real Banach space with Markov type p for some p > 1 admits an equivalent uniformly convex norm, answering Naor's renorming question. Together with the known converse, this characterizes superreflexivity by nontrivial Markov type. ([Lean](lean/docs/327.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M633
Nontrivial Markov Type Forces Superreflexivity
We prove that every real Banach space with Markov type p > 1 is superreflexive. Together with the known converse, this characterizes superreflexivity by nontrivial Markov type. This answers Naor's question: every real Banach space with nontrivial Markov type admits an equivalent uniformly smooth norm.
View manuscript on GitHub → Open PDF (raw) →Family 328 Nonexpansive fixed points in reflexive Banach spaces. 1 paper
Resolves Kirk's reflexive-space fixed-point problem: every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. The result uses the original norm, without assuming uniform convexity. ([Lean](lean/docs/328.md))
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Primary manuscript on GitHub → Family index on GitHub →M634
Fixed Points of Nonexpansive Maps in Reflexive Banach Spaces
Every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. This resolves the reflexive-space fixed point problem for the given norm.
View manuscript on GitHub → Open PDF (raw) →Family 329 A counterexample to metric-entropy duality. 1 paper
Disproves Pietsch's dimension-free duality conjecture for metric entropy. Origin-symmetric convex bodies violate every proposed choice of universal constants in the conjectured comparison between covering numbers and those of the polar bodies, even when the covering body is a cube. ([Lean](lean/docs/329.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M635
Counterexamples to the duality conjecture for metric entropy
We disprove Pietsch's dimension-free duality conjecture for metric entropy. For every proposed pair of universal constants, we construct origin-symmetric convex bodies that violate the corresponding covering-entropy inequality, already when the covering body is a cube.
View manuscript on GitHub → Open PDF (raw) →Family 330 A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces. 1 paper
Constructs a countable uniformly discrete metric space whose real Lipschitz-free Banach space has the approximation property but not the bounded approximation property, answering Kalton's question negatively. Finite-rank operators approximate the identity on every compact set, but their norms cannot share a finite bound. ([Lean](lean/docs/330.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M636
A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces
We construct a countable uniformly discrete metric space whose real Lipschitz-free space has the approximation property but fails the bounded approximation property, answering Kalton's question negatively. The identity can be approximated on every compact set by finite-rank operators, but no uniform bound on their norms is possible.
View manuscript on GitHub → Open PDF (raw) →Family 331 Reflexive midpoint convexity and diamond distortion. 7 papers
Constructs a real reflexive Banach space with an asymptotically midpoint uniformly convex norm but no equivalent asymptotically uniformly convex norm, extending Baudier's separation to reflexive spaces. In the same space, depth-k countably branching diamonds require distortion at least `√(1+k/12)`, so midpoint uniform convexity does not force uniformly bounded diamond distortion even under reflexivity. ([Lean](lean/docs/331.md))
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Primary manuscript on GitHub → Family index on GitHub →M637
Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space
We construct a separable reflexive real Banach space whose given norm is asymptotically midpoint uniformly convex but which admits no asymptotically uniformly convex equivalent norm. Its averaged midpoint modulus is at least `√(1+t²/12)-1`, and the countably branching diamond of depth k has distortion at least `√(1+k/12)` in this space. This gives a negative answer to the reflexive diamond converse for asymptotic uniform convexifiability.
View manuscript on GitHub → Open PDF (raw) →M638
Midpoint lenses in segment spaces
For a real segment-forest dual with unbounded finite component heights and the real infinite-height coordinate predual, we bound the tail of an arbitrary displacement in a symmetric lens by `2√(R²⁻ |x |²)`, where R is the lens radius and x is its finitely supported center. Both given norms are asymptotically midpoint uniformly convex, although neither space admits an equivalent asymptotically uniformly convex norm. The finite-height forest dual is reflexive.
View manuscript on GitHub → Open PDF (raw) →M639
Distortion of countably branching diamonds from midpoint and tree energies
We derive quantitative distortion bounds for countably branching diamond graphs from midpoint estimates and direct tree energies. In the dual of a finite-height segment forest, every distortion-D embedding of the depth-k diamond satisfies `D²≥1+k/4`. The same bound holds in the infinite-height coordinate predual. We also obtain power-type distortion bounds for path-cost and recursive tree norms.
View manuscript on GitHub → Open PDF (raw) →M640
Exact asymptotic moduli in a Daugavet subspace of L1
For an infinite-dimensional real subspace of L1 whose unit ball is totally bounded in measure and whose norm has the Daugavet property, we compute the averaged midpoint and one-sided asymptotic moduli at every unit center. They are `max t/2,t-1 ` and `max 0,t-2 `, respectively. The same weak-neighborhood geometry excludes every equivalent asymptotically uniformly convex norm. A quantitative realization of the Kadets–Werner construction supplies a space with both hypotheses.
View manuscript on GitHub → Open PDF (raw) →M641
Midpoint convexity from bounded tree potentials and path costs
Four real Banach spaces defined by bounded tree potentials satisfy the averaged asymptotic midpoint bound ` widehatδ(t)≥√(1+t²/4)-1` for `0 t 1`, while none admits an asymptotically uniformly convex (AUC) renorming. On finite-height trees, the globally constrained norm equals the least additive cost of a Hilbert vector and root paths. The quadratic path and segment-start outer Hilbert sums are reflexive and asymptotically midpoint uniformly convex, and admit no equivalent AUC norm.
View manuscript on GitHub → Open PDF (raw) →M642
Independent products in real L1: asymptotic midpoint convexity without AUC renormings
For a countably branching tree, the closed real L1 spans of products of independent exponential or Gaussian-square multipliers along its paths have positive averaged asymptotic midpoint moduli and admit no equivalent asymptotically uniformly convex norm. The renorming obstruction holds for every positive nonconstant mean-one multiplier with finite second moment.
View manuscript on GitHub → Open PDF (raw) →M643
Midpoint convexity from two recursive potentials
We study tree norms computed by two least nonnegative fields whose difference is the vector. For Euclidean child aggregation, two root-sum spaces have an averaged asymptotic midpoint modulus of at least `t³/128` for `0 t 1`, including a reflexive joining-root space. A reflexive construction with a fixed zero root also satisfies a homogeneous cubic estimate. These spaces admit no equivalent asymptotically uniformly convex norm. We also obtain sixth-power and cubic estimates when the aggregation exponent depends on the height of a finite component.
View manuscript on GitHub → Open PDF (raw) →Family 332 Metric Markov cotype of ℓ1 and Hilbert-space Lipschitz extension. 1 paper
Proves that real ℓ1 has metric Markov cotype two, answering Mendel and Naor's question. Consequently, every Lipschitz map from an arbitrary subset of a real Hilbert space into ℓ1 extends to the whole space with a universal multiplicative loss in its Lipschitz constant, resolving Ball's extension problem for this target.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M644
Metric Markov Cotype Two of ℓ1
We prove that the real Banach space ℓ1 has metric Markov cotype two, answering a question of Mendel and Naor. As a consequence, every Lipschitz map from an arbitrary subset of a real Hilbert space into ℓ1 extends to the whole Hilbert space with a universal multiplicative loss in its Lipschitz constant, resolving Ball's extension problem for this target.
View manuscript on GitHub → Open PDF (raw) →Family 333 Smooth isometric immersions of surfaces into ℝ4. 1 paper
Every closed smooth Riemannian surface admits a smooth isometric immersion into ℝ4, resolving the closed-surface form of the four-dimensional isometric-immersion problem. This includes nonorientable surfaces and metrics of arbitrary Gaussian curvature. ([Lean](lean/docs/333.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M645
Smooth isometric immersions of closed surfaces into Euclidean four-space
Every closed smooth Riemannian surface admits a smooth isometric immersion into Euclidean four-space, without an orientability assumption. This resolves the closed-surface form of the classical four-dimensional isometric-immersion problem.
View manuscript on GitHub → Open PDF (raw) →Family 334 A smooth surface metric with no local isometric immersion in ℝ3. 1 paper
Constructs a smooth positive-definite metric on `(-1,1)²` for which no neighborhood of the origin admits a smooth isometric immersion into…
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M646
A Smooth Metric with No Local Isometric Immersion into Three-Space
We construct a smooth positive-definite Riemannian metric on `(-1,1)²` that agrees with the Euclidean metric to every order at the origin, yet no neighborhood of the origin admits a smooth isometric immersion into Euclidean…
View manuscript on GitHub → Open PDF (raw) →Family 335 Gromov’s integral scalar-curvature bound for simplicial volume. 2 papers
Proves `∫_M(Scal_g⁻)n/² dV_g≥ a_n M ` for every closed connected oriented smooth n-manifold, n ≥ 3, and every smooth metric, with `a_n 0` depending only on dimension. Here `Scal_g⁻⁼max 0,-Scal_g ` and ` M ` is real simplicial volume. Also proves rational inessentiality under positive scalar curvature, resolving the Gromov–Lawson conjecture; every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M647
An integral scalar curvature bound for real simplicial volume
We prove the integral scalar-curvature inequality proposed by Gromov. For every dimension n ≥ 3, there is a constant `a_n 0`, depending only on n, such that ` ∫_M( Scal _g⁻)n/² dV_g≥ a_n |M |` for every closed connected oriented smooth n-manifold M and every smooth Riemannian metric g. Here ` Scal _g⁻:=max 0,- Scal _g `, and ` |M |` is real simplicial volume. The proof uses the nonnegative-scalar-curvature vanishing theorem of the companion paper on rational inessentiality.
View manuscript on GitHub → Open PDF (raw) →M648
Positive scalar curvature forces rational inessentiality
We prove that every closed connected oriented smooth manifold admitting strictly positive scalar curvature is rationally inessential: its rational fundamental class maps to zero under the classifying map. No spin or fundamental-group hypothesis is needed, and there is no upper dimension bound. This proves the Gromov–Lawson aspherical conjecture; moreover, every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat.
View manuscript on GitHub → Open PDF (raw) →Family 336 Spectral scalar curvature, Urysohn width, and macroscopic dimension. 3 papers
Every complete connected smooth boundaryless n-manifold, n ≥ 3, satisfying `-4 +Scal≥1` as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most `n-2` whose entire fibers have diameter bounded only by n in the original metric. This strengthens Gromov's width conclusion to spectral scalar curvature. Universal covers of closed positive-scalar-curvature manifolds also have continuous macroscopic dimension at most `n-2` for every n ≥ 2.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M649
Spectral scalar curvature and uniform Urysohn width
For every n ≥ 4, a complete connected smooth Riemannian n-manifold without boundary satisfying `-4 + Scal ≥1` as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most `n-2` whose entire fibers have diameter bounded only in terms of n. The bound is measured in the original metric. This extends the uniform Urysohn width theorem from a pointwise scalar-curvature lower bound to a spectral lower bound.
View manuscript on GitHub → Open PDF (raw) →M650
Spectral scalar curvature and Urysohn width in dimension three
Every connected complete smooth Riemannian three-manifold without boundary satisfying `-4 + Scal ≥λ 0` as a quadratic-form inequality admits a continuous map to a graph whose entire fibers have diameter at most `500/ sqrtλ` in the original metric. No orientability, spin, compactness, or bounded-geometry assumption is required.
View manuscript on GitHub → Open PDF (raw) →M651
Positive scalar curvature and uniform codimension-two width
We prove the quantitative continuous form of Gromov's scalar-curvature conjecture in every dimension n ≥ 4. Every complete connected smooth boundaryless n-manifold with scalar curvature at least one admits a continuous map to a simplicial complex of dimension at most `n-2` whose entire fibers have diameter bounded only in terms of n. We also obtain the continuous macroscopic-dimension conclusion for universal covers of closed manifolds with positive scalar curvature in every dimension n ≥ 2.
View manuscript on GitHub → Open PDF (raw) →Family 337 Sharp Cartan–Hadamard isoperimetry and rigidity. 1 paper
Proves generalized Cartan–Hadamard isoperimetry in every dimension: in a complete simply connected manifold with sectional curvature at most κ ≤ 0, every finite-volume finite-perimeter set satisfies the sharp comparison with the equal-volume model ball. Bounded positive-volume equality regions for κ = 0 are Euclidean balls. Also proves sharp Euclidean filling bounds for compactly supported integral n-cycles, n ≥ 2, in arbitrary proper CAT `(0)` spaces. ([Lean](lean/docs/337.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M652
Generalized Cartan–Hadamard isoperimetry and Euclidean equality rigidity
We resolve the generalized Cartan–Hadamard isoperimetric conjecture in every dimension. In a complete simply connected smooth manifold with sectional curvature at most κ ≤ 0, every finite-volume set of finite ambient perimeter has perimeter at least that of the equal-volume ball in curvature κ. For bounded positive-volume sets, equality in the Euclidean comparison holds precisely when the set agrees up to null sets with an open region isometric, with its induced metric, to a round Euclidean ball.
View manuscript on GitHub → Open PDF (raw) →Family 338 Yau's uniformization conjecture. 1 paper
Proves Yau's uniformization conjecture: every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to ℂn.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M653
Uniformization of complete Kähler manifolds with positive bisectional curvature
We prove that every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to complex Euclidean space. This resolves Yau's uniformization conjecture positively in every complex dimension.
View manuscript on GitHub → Open PDF (raw) →Family 339 Katok's entropy rigidity conjecture. 1 paper
Proves Katok's entropy rigidity conjecture for closed connected Riemannian manifolds of dimension at least three with strictly negative sectional curvature: normalized Liouville measure maximizes entropy for the geodesic flow if and only if the metric is locally symmetric.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M654
Entropy equality and local symmetry in negative curvature
For every closed connected smooth Riemannian manifold of dimension at least three with strictly negative sectional curvature, we prove that normalized Liouville measure has maximal entropy for the unit-speed geodesic flow if and only if the metric is locally symmetric. This resolves Katok's entropy rigidity conjecture positively in these dimensions, including all rank-one symmetric types at arbitrary scale.
View manuscript on GitHub → Open PDF (raw) →Family 340 A counterexample to the nearby Lagrangian conjecture. 1 paper
Disproves the unrestricted nearby Lagrangian conjecture. For some sufficiently large even N, constructs a closed exact embedded Lagrangian in `T*(S⁹× SN⁻¹)` that is diffeomorphic to the base but not Hamiltonian isotopic to its zero section.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M655
A counterexample to the nearby Lagrangian conjecture
We disprove the unrestricted nearby Lagrangian conjecture. For some sufficiently large even integer N, we construct a closed exact smoothly embedded Lagrangian in `T*(S⁹× SN⁻¹)` that is diffeomorphic to the base but is not Hamiltonian isotopic to the zero section.
View manuscript on GitHub → Open PDF (raw) →Family 341 Donaldson's hypersymplectic deformation conjecture. 1 paper
Proves Donaldson's hypersymplectic deformation conjecture in a cohomology-preserving form. Every positive triple of smooth closed two-forms on a closed connected oriented four-manifold, normalized by `∫ω_i wedgeω_j=δ(ij)`, deforms through positive closed triples to a hyperkähler triple while preserving all three cohomology classes. Any positive triple can first be normalized by a constant linear change.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M656
Deforming hypersymplectic four-manifolds to hyperkähler triples
Every smooth normalized positive triple of closed two-forms on a closed connected oriented four-manifold admits a smooth deformation, with each cohomology class fixed, to a hyperkähler triple. This resolves Donaldson's hypersymplectic deformation conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 342 Donaldson's tamed-to-compatible conjecture. 1 paper
Proves Donaldson's tamed-to-compatible conjecture: every smooth almost complex structure on a closed four-manifold that is tamed by a symplectic form admits a compatible symplectic form. The almost complex structure stays fixed; the form's cohomology class may change. ([Lean](lean/docs/342.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M657
Taming implies compatibility on four-manifolds
We prove that every smooth almost complex structure on a closed four-manifold which is tamed by a symplectic form is compatible with a symplectic form. This gives a positive solution to Donaldson's tamed-to-compatible conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 343 Symplectic ball packing in higher dimensions. 1 paper
Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension `2n≥6`. Finitely many closed symplectic balls of capacities `R_1,R_k` embed disjointly into an open ball of capacity R exactly when `∑_iR_in Rn` and `R_i+R_j R` for every distinct pair i, j. ([Lean](lean/docs/343.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M658
Symplectic Ball Packings in Higher Dimensions
We prove that, for all integers n ≥ 3 and k ≥ 1 and all positive real capacities `R_1,R_k`, the closed standard symplectic `2n` -balls of these capacities embed disjointly into the interior of a ball of capacity R > 0 if and only if ` ∑(i=1)k R_in Rn, R_i+R_j R (i≠ j).` Capacity is π times the squared Euclidean radius, and each embedding is defined on a neighborhood of its closed source ball. This proves Siegel and Yao's Conjecture A.
View manuscript on GitHub → Open PDF (raw) →Family 344 The metric Blaschke conjecture. 1 paper
Proves the metric Blaschke conjecture: every closed connected Riemannian manifold of positive dimension whose injectivity radius equals its diameter is, up to scaling, a standard compact rank-one symmetric space.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M659
The metric Blaschke theorem
We prove the metric Blaschke conjecture: every connected closed smooth Riemannian manifold of positive dimension whose global injectivity radius equals its diameter is, up to scale, a standard compact rank-one symmetric space.
View manuscript on GitHub → Open PDF (raw) →Family 345 Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds. 1 paper
Proves that every smooth Riemannian metric on Sn, n ≥ 2, has infinitely many prime closed geodesics with pairwise distinct images. The same conclusion holds on every closed manifold admitting a finite smooth spherical cover and on every closed three-manifold, without orientability or nondegeneracy restrictions.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M660
Infinitely many closed geodesic images on every Riemannian sphere
We resolve the sphere case of the closed-geodesic infinitude problem: every smooth Riemannian metric on the standard sphere Sn, n ≥ 2, has infinitely many prime closed geodesics with pairwise distinct images. The two-sphere case is classical; the result in higher dimensions includes degenerate metrics. Using finite covers and theorems of Perelman and Rademacher–Taimanov, we also obtain the same conclusion for every smooth Riemannian metric on a nonempty closed smooth three-manifold.
View manuscript on GitHub → Open PDF (raw) →Family 346 Sharp singular-set bounds for stationary integral varifolds. 2 papers
Proves that every stationary integral m-varifold in a Euclidean open set has singular set of Hausdorff dimension at most `m-1`, a sharp bound in every positive dimension and codimension. On round spheres, the family also proves almost-everywhere regularity: the singular set has zero m-dimensional Hausdorff measure.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M661
A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold
The singular set of every stationary integral m-varifold in an open Euclidean set has Hausdorff dimension at most `m-1`, in every positive dimension and codimension. This sharp bound proves the Euclidean singular-set conjecture recorded by Brena, Decio, and De Lellis.
View manuscript on GitHub → Open PDF (raw) →M662
Almost-everywhere regularity of stationary integral varifolds
We resolve the almost-everywhere regularity conjecture of Brena, Decio, and De Lellis for stationary integral varifolds of arbitrary positive dimension and codimension in Euclidean open sets. The singular set has zero measure in the dimension of the varifold: near almost every support point, the varifold is a constant positive integer multiple of a smooth embedded minimal submanifold. The corresponding statement also holds on round spheres.
View manuscript on GitHub → Open PDF (raw) →Family 347 Counterexamples to stable-Morse and strong Arnold fixed-point bounds. 5 papers
Disproves stable-Morse lower bounds for nondegenerate Hamiltonian fixed points: on simply connected closed Kähler manifolds of real dimension 22, the deficit below the stable Morse number is unbounded. A separate Hamiltonian diffeomorphism of the complex quadric threefold has exactly three fixed points, fewer than the four critical points required of every smooth function. ([Lean](lean/docs/347.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M663
Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two
We disprove the stable Morse lower bound for nondegenerate Hamiltonian fixed points with examples in fixed real dimension twenty-two. For every integer m ≥ 1, we construct a simply connected closed Kähler manifold with stable Morse number `80+1968m` and a Hamiltonian diffeomorphism with exactly `80+1952m` fixed points, all nondegenerate and with contractible orbit loops. The deficit is therefore unbounded, and the fixed-point count is at most 127/128 of the stable Morse number.
View manuscript on GitHub → Open PDF (raw) →M664
Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number
We construct a simply connected closed Kähler manifold of real dimension 3332 and minimal Chern number one whose Hamiltonian fixed-point count attains the cyclic integral Floer bound while falling below the stable Morse number. The Hamiltonian diffeomorphism has exactly `1 872 232` fixed points, all nondegenerate and with contractible orbit loops, whereas the stable Morse number is `1 872 264`. Thus the stronger stable Morse bound fails by exactly 32, even when the cyclic integral bound is sharp.
View manuscript on GitHub → Open PDF (raw) →M665
Hamiltonian Fixed Points Below the Stable Morse Number
We disprove the stable Morse lower bound for nondegenerate Hamiltonian fixed points by constructing a simply connected closed Kähler manifold with sixteen fewer fixed points than its stable Morse number. All the corresponding periodic orbits are contractible. The construction combines a Hamiltonian involution with integral homology torsion at two different primes; its proof uses finite-dimensional Morse theory and complex blowups.
View manuscript on GitHub → Open PDF (raw) →M666
A nondegenerate counterexample to the Morse-number Arnold bound
We construct a smooth one-periodic Hamiltonian on a closed symplectic twelve-manifold whose time-one map has fewer fixed points than the manifold's ordinary Morse number. Every fixed point is nondegenerate and has a contractible Hamiltonian trajectory. This disproves the Morse-number form of the Arnold conjecture. The deficit can be arbitrarily large among twelve-dimensional examples.
View manuscript on GitHub → Open PDF (raw) →M667
Three fixed points on the symplectic quadric threefold
We construct a smooth Hamiltonian diffeomorphism of the complex quadric threefold with exactly three fixed points, at least one of which is degenerate. The critical number and the unit-inclusive rational cup length of this manifold are both four. Thus the example disproves the unrestricted critical-number and rational cup-length forms of the Arnold conjecture.
View manuscript on GitHub → Open PDF (raw) →Family 348 Nonnegative-curvature Einstein classification and an L2 topological gap. 3 papers
Classifies closed connected Einstein four-manifolds with positive Einstein constant and nonnegative sectional curvature: up to scaling, their universal Riemannian covers are the round S4, Fubini–Study `CP²`, or a product of equal round two-spheres. A closed simply connected nonnegatively curved four-manifold is diffeomorphic to one of these whenever the scale-invariant L2 norm of its trace-free Ricci curvature lies below a universal positive constant. ([Lean](lean/docs/348.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M668
Zero-Plane Rigidity for Einstein Four-Manifolds
We prove that a closed Einstein four-manifold with positive Einstein constant, nonnegative sectional curvature, and a zero-curvature plane has universal Riemannian cover isometric to a product of two round two-spheres. Under the normalization ` Ric =3g`, both spheres have radius `1/ sqrt3`. The proof extends coupled estimates for the two Weyl curvature blocks to the boundary of the sectional-curvature cone and determines their equality case.
View manuscript on GitHub → Open PDF (raw) →M669
An L² Einstein Gap for Nonnegatively Curved Four-Manifolds
We prove a universal, scale-invariant L2 gap for the trace-free Ricci tensor on simply connected closed four-manifolds with nonnegative sectional curvature. If the trace-free Ricci energy is sufficiently small, the manifold is diffeomorphic to S4, `CP²`, or `S²× S²`. The proof uses the classification of positive-Einstein, nonnegatively curved four-manifolds supplied by the companion zero-plane rigidity theorem, stated explicitly as the classification premise of our result.
View manuscript on GitHub → Open PDF (raw) →M670
Positively curved Einstein four-manifolds
We prove the classification conjecture for connected smooth closed Einstein four-manifolds with strictly positive sectional curvature. Up to positive scaling and isometry, every such manifold is the round four-sphere, the complex projective plane with its Fubini–Study metric, or real projective four-space with its round metric. No orientability assumption is needed.
View manuscript on GitHub → Open PDF (raw) →Family 349 The Solomon–Yau least-volume conjecture. 1 paper
Proves the Solomon–Yau least-volume conjecture for minimal hypersurfaces of round spheres. For every m ≥ 2, a closed connected minimal immersion into the unit sphere `Sm⁺¹` with non-totally-geodesic image has volume at least that of the smallest minimal Clifford product, counting covering multiplicity.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M671
The Solomon–Yau least-volume theorem
We prove the Solomon–Yau least-volume conjecture for minimal hypersurfaces of unit round spheres: in every dimension m ≥ 2, a closed connected minimal immersion with non-totally-geodesic image has volume at least the smallest m-dimensional minimal Clifford product. Volume is measured on the domain, so covering multiplicities are included.
View manuscript on GitHub → Open PDF (raw) →Family 350 Yau’s nodal bounds: surfaces and higher dimensions. 3 papers
Proves the sharp `C sqrtλ` upper bound for nodal length on every fixed smooth closed surface, completing Yau's conjecture there. The upper bound fails for fixed smooth metrics in dimensions three and four, including metrics on S3 arbitrarily close to round. In dimension five, nodal measure can grow faster than `λ¹/²⁺ε_⁰` for some fixed `ε_0 0`, ruling out even arbitrarily small power losses. ([Lean](lean/docs/350.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M672
Sharp nodal length on smooth surfaces
We prove that a nonzero real Laplace eigenfunction with eigenvalue λ > 0 on a fixed smooth closed connected Riemannian surface has nodal length at most `C sqrtλ`. Together with the known lower bound, this proves Yau's conjecture in this setting.
View manuscript on GitHub → Open PDF (raw) →M673
Smooth counterexamples to Yau's nodal upper bound in dimensions three and four
We construct a smooth metric on the three-sphere, arbitrarily close to the round metric in the smooth topology, and a smooth metric on `S²× T²` for which sequences of exact real Laplace eigenfunctions have unbounded nodal measure divided by the square root of the eigenvalue. Each sequence belongs to one fixed metric. Thus the upper-bound part of Yau's nodal conjecture fails for smooth metrics in dimensions three and four.
View manuscript on GitHub → Open PDF (raw) →M674
Power-law violations of Yau's nodal upper bound
We construct a smooth Riemannian metric on `S⁴× S¹` and a sequence of real Laplace eigenfunctions whose nodal four-volume grows faster than `λ¹/²⁺ε_⁰` for one fixed `ε_0 0`. This disproves the smooth upper-bound assertion in Yau's nodal-set conjecture and the proposed bound with an arbitrarily small positive power loss.
View manuscript on GitHub → Open PDF (raw) →Family 351 Scalar curvature and finite-time Ricci-flow singularities. 3 papers
Proves that a smooth Ricci flow on a closed four-manifold extends past any finite time at which scalar curvature remains uniformly bounded. A higher-dimensional counterexample has bounded scalar curvature but unbounded full curvature at its finite maximal time, disproving the unrestricted scalar-curvature extension conjecture.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M675
A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup
We disprove the scalar-curvature extension conjecture in its unrestricted all-dimensions form. In sufficiently high dimension, we construct a Ricci flow on a closed manifold whose scalar curvature remains uniformly bounded while full curvature diverges at a finite maximal time. In one fixed sufficiently high dimension, the examples have two-sided power-law curvature blowup with arbitrarily large exponents.
View manuscript on GitHub → Open PDF (raw) →M676
Bounded scalar curvature and smooth extension of four-dimensional Ricci flow
We prove that a smooth Ricci flow on a closed real four-manifold extends on the same manifold through every finite time at which its scalar curvature remains uniformly bounded. This resolves the scalar-curvature extension problem in dimension four.
View manuscript on GitHub → Open PDF (raw) →M677
Path selection and an elliptic inequality on degenerating Ricci-flat trees
We prove a sequential elliptic inequality for the renormalized Einstein–Hilbert functional on a fixed finite tree of four-dimensional Ricci-flat spaces. As the joining lengths diverge and the scale-neutral weighted Ricci error M tends to zero, the functional satisfies `E=o(M)`. The root end has decay exponent greater than one, and every joined quotient group is nontrivial. We also prove the multivariable path-selection theorem for asymptotic expansions used to obtain this estimate.
View manuscript on GitHub → Open PDF (raw) →Family 352 A finite-time singularity of Calabi flow. 1 paper
Disproves Chen's smooth long-time existence conjecture for Calabi flow by constructing a smooth `U(10)` -invariant Kähler metric on ` CP ¹⁰` whose flow develops a finite-time singularity. The metric lies in the Fubini–Study class, so the failure occurs even in a class containing a constant-scalar-curvature metric.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M678
A finite-time singularity of Calabi flow on projective space
We construct a smooth Kähler metric in the Fubini–Study class of ` CP ¹⁰` whose Calabi flow develops unbounded scalar curvature in finite time. This disproves Chen's smooth long-time existence conjecture, even in a class containing a constant-scalar-curvature metric.
View manuscript on GitHub → Open PDF (raw) →Family 353 Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample. 2 papers
Proves that every smooth locally uniformly convex affine-maximal graph of dimension three through nine, complete for its induced Euclidean metric, is an elliptic paraboloid. A smooth entire nonquadratic example in dimension ten makes this range sharp. In dimensions three through nine, the paraboloid classification also holds for connected open locally uniformly convex affine-maximal hypersurfaces complete for the affine Berwald–Blaschke metric. ([Lean](lean/docs/353.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M679
A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten
We construct a smooth nonquadratic entire graph in dimension ten that solves the classical affine maximal equation and has positive-definite Hessian everywhere. This gives a smooth counterexample to the entire-graph affine Bernstein assertion in dimension ten. Hessian positivity is pointwise; no global uniform lower bound or completeness of the Berwald–Blaschke metric is asserted.
View manuscript on GitHub → Open PDF (raw) →M680
The affine Bernstein theorem in dimensions three through nine
We prove the Euclidean-complete affine Bernstein conjecture in dimensions three through nine: a smooth locally uniformly convex affine maximal graph is an elliptic paraboloid whenever its induced Euclidean metric is complete. The same conclusion holds, without an initial graph assumption, for connected smooth open (noncompact and without boundary) affine-complete locally uniformly convex immersed hypersurfaces that are classically affine maximal in these dimensions.
View manuscript on GitHub → Open PDF (raw) →Family 354 The isoperimetric profile of the cubic three-torus. 1 paper
Determines the isoperimetric profile of the unit cubic flat three-torus and classifies every finite-perimeter minimizer: balls, circular tubes around shortest closed geodesics, coordinate slabs, and their complements. The transition volumes are `4π/81` and `1/π`, with exactly the adjacent two types minimizing at each transition. ([Lean](lean/docs/354.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M681
The Isoperimetric Conjecture for the Cubic Flat Three-Torus
We prove the isoperimetric conjecture for the cubic flat three-torus and classify all minimizers, including the equality cases at the transition volumes `4π/81` and `1/π`. The minimizing regions are balls, circular tubes about shortest closed geodesics, coordinate slabs, and their complements.
View manuscript on GitHub → Open PDF (raw) →Family 355 Unique tangent flows at the first surface singularity. 1 paper
Proves the first-singular-time case of tangent-flow uniqueness for smooth compact connected embedded surfaces without boundary in ℝ3. At every singular point, all fixed-center backward tangent flows agree as area measures at every negative time in the original ambient coordinates, without mean-convexity or a prescribed tangent model.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M682
Uniqueness of tangent flows at the first singular time of embedded surface mean-curvature flow
We prove that, at each singular point of the first singular time of the mean-curvature flow of a smooth compact connected embedded surface without boundary in ℝ3, all fixed-center rescalings converge locally smoothly on compact negative-time intervals to one multiplicity-one homothetic self-shrinker flow. The limit is unique in the original ambient coordinates, including its position and axes. No mean-convexity assumption or prescribed tangent model is required.
View manuscript on GitHub → Open PDF (raw) →Family 356 Gigli’s characterization of Alexandrov curvature. 2 papers
Proves Gigli's conjecture: in every integer dimension n ≥ 2, Alexandrov curvature at least κ is characterized by the full-support ` RCD ((n-1),n)` condition with reference measure ` Hn` and distributional sectional curvature at least κ in the original global test classes. The RCD condition is unreduced. ([Lean](lean/docs/356.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M683
Gigli’s distributional curvature characterization of Alexandrov spaces
For every integer n ≥ 2 and κ ∈ ℝ, we prove that a complete separable metric space is an n-dimensional Alexandrov space of curvature at least κ if and only if, with reference measure ` Hn`, it is a full-support `RCD((n-1),n)` space whose distributional sectional curvature is at least κ in Gigli's original global test classes. This resolves Gigli's characterization conjecture in dimensions at least two.
View manuscript on GitHub → Open PDF (raw) →M684
Weak Hessian bounds along every geodesic in RCD spaces
On a full-support `RCD(K,N)` space with `1 N ∞`, we prove that a bounded globally Lipschitz function whose distributional Hessian is bounded above by a bounded continuous function satisfies the corresponding second-derivative inequality along every minimizing geodesic.
View manuscript on GitHub → Open PDF (raw) →Family 357 Bi-Lipschitz coordinates at every regular RCD point. 1 paper
Proves that every regular point of a noncollapsed ` RCD (K,n)` space, for K ∈ ℝ and integer n ≥ 2, has an open neighborhood bi-Lipschitz to an open subset of ℝn. Regularity requires all pointed tangents to be Euclidean, the reference measure is exactly ` Hn`, and the chart compares ambient distances with a point-dependent finite constant.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M685
Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces
We resolve the regular-point bi-Lipschitz conjecture in the noncollapsed setting. For every integer n ≥ 2 and every real K, every regular point of a noncollapsed `RCD(K,n)` space has an open neighborhood bi-Lipschitz homeomorphic to an open subset of ℝn. The bi-Lipschitz constant depends only on n, and the neighborhood uses the restricted ambient distance.
View manuscript on GitHub → Open PDF (raw) →Family 358 A three-manifold without conjugate points or nonpositive curvature. 1 paper
Constructs a closed connected orientable smooth three-manifold that admits a metric without conjugate points but no metric of nonpositive sectional curvature. This answers negatively, already in dimension three, whether the first metric-existence property implies the second. ([Lean](lean/docs/358.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M686
A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric
We construct a closed connected orientable smooth three-manifold that admits a smooth Riemannian metric without conjugate points but admits no smooth Riemannian metric of nonpositive sectional curvature.
View manuscript on GitHub → Open PDF (raw) →Family 359 Negative Kähler curvature without bounded holomorphic coordinates. 2 papers
Constructs a contractible domain in ℂ3 with a complete negatively pinched Kähler metric but no bounded holomorphic coordinates, disproving bounded-domain uniformization in this setting. A higher-dimensional example has sectional curvature at most −1 and only constant bounded holomorphic functions; its curvature is not bounded below. ([Lean](lean/docs/359.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M687
A negatively pinched Kähler threefold without bounded holomorphic coordinates
We construct a contractible domain in complex dimension three with a complete Kähler metric whose real sectional curvatures lie between two finite negative constants. It admits no bounded holomorphic map to ℂ3 with nowhere-vanishing Jacobian and is therefore not biholomorphic to a bounded domain. This gives a negative answer to the negatively pinched Kähler uniformization question.
View manuscript on GitHub → Open PDF (raw) →M688
One-sided negative sectional curvature and the holomorphic Liouville property
We construct, in some sufficiently large fixed finite complex dimension m, a domain in ℂm diffeomorphic to ` R²m` that admits a complete Kähler metric with real sectional curvature at most −1 and has only constant bounded holomorphic functions. Its sectional curvatures are unbounded below. The construction gives a negative answer to the one-sided bounded-holomorphic-function question.
View manuscript on GitHub → Open PDF (raw) →Family 360 Weak MTW curvature gives convexity and regular optimal transport. 2 papers
On every closed connected Riemannian manifold of dimension at least two satisfying weak Ma–Trudinger–Wang curvature, all tangent injectivity domains are convex, resolving Villani's conjecture in this setting. For squared-distance transport between measurable probability densities bounded above and away from zero, the optimal map and its inverse are Hölder continuous. ([Lean](lean/docs/360.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M689
Global Support and Convex Injectivity Domains under Weak MTW
We prove that weak Ma–Trudinger–Wang curvature on a smooth, connected, compact Riemannian manifold of dimension at least two without boundary implies convexity of every tangent injectivity domain, resolving Villani's conjecture in this setting. Conjugate cut points are allowed. More generally, every ordinary subgradient of a squared-distance cost potential is a minimizing velocity with a global supporting mountain. No density hypothesis is used.
View manuscript on GitHub → Open PDF (raw) →M690
Uniform Bi-Holder Transport from Weak MTW
We prove that the weak Ma–Trudinger–Wang condition on a fixed smooth connected compact boundaryless Riemannian manifold of dimension at least two implies a common Hölder estimate for optimal transport maps and their inverses over the entire class of probability densities with fixed positive upper and lower bounds. The maps have homeomorphic representatives, conjugate cut points are allowed, and no density regularity is assumed.
View manuscript on GitHub → Open PDF (raw) →Family 361 Failure of integer-degree harmonic dimension comparison. 2 papers
Disproves Yau's proposed Euclidean dimension bound for harmonic functions of integer growth on manifolds with nonnegative Ricci curvature. For every sufficiently large integer k, a complete smooth metric on ℝ3 has at least `(k+2)²` independent harmonic functions of growth at most k, exceeding the Euclidean count `(k+1)²`. The metric may depend on k. ([Lean](lean/docs/361.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M691
A counterexample to integer-degree harmonic dimension comparison
For some even n ≥ 8 and integer k ≥ 2, we construct a complete smooth metric on ℝn with nonnegative Ricci curvature whose space of real harmonic functions of pointwise polynomial growth at most k has dimension larger than the Euclidean harmonic-polynomial dimension. The metric is Euclidean near the origin, has asymptotic volume ratio strictly between zero and one, and has nonunique tangent cones at infinity. This answers the integer-degree form of Yau's dimension comparison question in the negative.
View manuscript on GitHub → Open PDF (raw) →M692
A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison
For every `4/9 v 1` and `1 c 9v/4`, all sufficiently large integers k admit a complete smooth metric on ℝ3 with nonnegative Ricci curvature, asymptotic volume ratio v, and at least `c(k+1)²` linearly independent real harmonic functions of pointwise growth at most k. This answers Yau's integer-degree dimension comparison question negatively in dimension three, with a fixed-factor excess over the Euclidean count.
View manuscript on GitHub → Open PDF (raw) →Family 363 Nonuniqueness with local conservation for the hard-sphere Boltzmann equation. 2 papers
Constructs two distinct global entropy solutions of the three-dimensional periodic hard-sphere Boltzmann equation from the same nonnegative initial density, with bounded velocity support and finite mass, energy and absolute entropy. Both are strongly continuous in L1 and satisfy exact local conservation of mass, momentum and kinetic energy. ([Lean](lean/docs/363.md))
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Primary manuscript on GitHub → Family index on GitHub →M694
Nonuniqueness with local conservation for the hard-sphere Boltzmann equation
We prove nonuniqueness for the three-dimensional periodic hard-sphere Boltzmann equation among global entropy solutions satisfying exact local conservation of mass, momentum, and kinetic energy. We construct one nonnegative initial density with bounded velocity support and finite mass, energy, and absolute entropy that gives rise to two distinct such solutions. Both are strongly continuous in L1, and their collision gain and loss terms are integrable with every polynomial velocity weight on every bounded time interval.
View manuscript on GitHub → Open PDF (raw) →M695
Nonuniqueness for the periodic hard-sphere Boltzmann equation
We prove nonuniqueness for the periodic hard-sphere Boltzmann equation by constructing two distinct global renormalized solutions with the same nonnegative initial density on ` T³× R³`. This density has bounded velocity support and finite mass, energy, and absolute entropy. Both solutions conserve local mass and total momentum and satisfy the global energy and entropy-dissipation inequalities. On a common initial interval, they are strongly continuous in L1, and their collision gains and losses are integrable.
View manuscript on GitHub → Open PDF (raw) →Family 364 Kinetic limits and fluctuations over the Boltzmann lifespan. 2 papers
Derives the nonlinear Boltzmann equation from three-dimensional grand-canonical Newtonian gases throughout every regular kinetic interval with uniform Gaussian decay. Stable finite-range radial potentials may have attractive wells and a singular repulsive core; initial pair exclusion and spatially summable Gaussian density and gradient bounds are assumed. A companion gives finite-dimensional hard-sphere Gaussian fluctuations, centered at the exact microscopic expectation and governed by the linear fluctuating Boltzmann equation.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M696
The Boltzmann–Grad limit for stable radial potentials on regular kinetic intervals
We derive the nonlinear Boltzmann equation from a grand-canonical Newtonian gas with initial pair exclusion. We treat stable, finite-range radial potentials that are C2 except for an allowed repulsive singularity at the origin, and C1 initial probability densities with spatially summable Gaussian bounds on the density and its spatial gradient.
View manuscript on GitHub → Open PDF (raw) →M697
Hard-sphere fluctuations on the regular Boltzmann lifespan
We prove a finite-dimensional central limit theorem away from equilibrium for a deterministic grand-canonical hard-sphere gas in three dimensions. The initial probability density is smooth, with spatially summable Gaussian velocity bounds on the density and its spatial gradient. The limit holds on every finite interval on which the classical Boltzmann solution has uniform Gaussian velocity decay. The empirical measure is centered by its exact microscopic expectation.
View manuscript on GitHub → Open PDF (raw) →Family 365 Joint metric and connection recovery from one boundary patch. 3 papers
Zero-frequency measurements on any nonempty open boundary patch determine a smooth metric and smooth unitary connection on a trivial Hermitian rank-two bundle over a compact connected manifold of dimension at least three, up to diffeomorphism and gauge fixed on that patch. Inputs and observations use the same patch. In contrast, distinct uniformly positive bounded measurable scalar conductivities on a three-dimensional ball can have identical full-boundary data. ([Lean](lean/docs/365.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M698
Determination of a metric and a unitary connection from one boundary patch
We prove that zero-frequency boundary measurements on any nonempty open boundary patch determine both a smooth Riemannian metric and a smooth unitary connection on the trivial Hermitian rank-two bundle over a compact connected smooth manifold of dimension at least three with smooth boundary. Both inputs and observations are restricted to the same patch. The metric and connection are determined up to a diffeomorphism and a unitary gauge that restrict to the identity on the measured patch.
View manuscript on GitHub → Open PDF (raw) →M699
Smooth anisotropic uniqueness in the Calderón problem from one boundary patch
We resolve the smooth anisotropic Calderón uniqueness problem with arbitrary same-patch measurements. A smooth Riemannian metric on a compact connected manifold of dimension at least three is determined, up to a diffeomorphism fixing the measured patch, by its zero-frequency Dirichlet-to-Neumann energy form with both input and observation on any nonempty open boundary patch.
View manuscript on GitHub → Open PDF (raw) →M700
Nonuniqueness for bounded measurable scalar conductivities in three dimensions
We construct two distinct uniformly positive bounded measurable scalar conductivities on a ball in ℝ3 with the same full Dirichlet-to-Neumann operator. Both conductivities equal one near the boundary. This gives nonuniqueness in the scalar Calderón problem at bounded measurable regularity.
View manuscript on GitHub → Open PDF (raw) →Family 366 The planar Mumford–Shah regularity conjecture and local weak-L4 gradient bounds. 1 paper
Resolves the interior regularity conjecture for reduced absolute planar Mumford–Shah minimizers with bounded fidelity data. Locally, the closed discontinuity set is a `C¹,α` arc, a regular crack tip, or three arcs meeting at `120° `; only finitely many global connected components meet any compact interior region.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M701
Interior regularity of planar Mumford–Shah minimizers
We prove the interior regularity assertion of the planar Mumford–Shah conjecture for reduced absolute minimizers with bounded fidelity data. Every interior point of the closed discontinuity set has a neighborhood consisting of a `C¹,α` arc, an arc ending at that point, or three such arcs meeting at 120 degrees. Only finitely many global connected components meet any relatively compact open set.
View manuscript on GitHub → Open PDF (raw) →Family 367 The critical dimension for the one-phase Bernoulli problem. 1 paper
Establishes seven as the first dimension admitting a nonflat, one-homogeneous global minimizer of the one-phase Bernoulli energy. Consequently, minimizing free boundaries are smooth through dimension six, and their singular sets have dimension at most `n-7` in higher dimensions. ([Lean](lean/docs/367.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M702
The critical dimension for one-phase Bernoulli minimizers
We prove that seven is the critical dimension for the one-phase Bernoulli problem: every nonzero one-homogeneous global minimizer in dimensions at most six is flat, while a nonflat one-homogeneous global minimizer exists in dimension seven. It follows that the interior free boundary of a local minimizer is smooth in dimensions at most six. In dimension n ≥ 7, its singular set has Hausdorff dimension at most `n-7`, and this bound is sharp. In dimension seven, the singular set is locally finite.
View manuscript on GitHub → Open PDF (raw) →Family 368 The three-dimensional Ball–Evans approximation problem. 2 papers
Resolves the three-dimensional Ball–Evans approximation problem: every `W¹,p` homeomorphism between arbitrary bounded domains in ℝ3, for `1≤ p ∞`, is a strong `W¹,p` limit of smooth diffeomorphisms onto the same target.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M703
Strong diffeomorphic approximation in three dimensions for 1≤p≤2
We resolve the three-dimensional Ball–Evans approximation problem for `1≤ p≤2`. Every `W¹,p` homeomorphism between arbitrary bounded domains in ℝ3 is a strong `W¹,p` limit of smooth diffeomorphisms onto the same target.
View manuscript on GitHub → Open PDF (raw) →M704
Strong diffeomorphic approximation in three dimensions for p>2
We resolve the three-dimensional Ball–Evans approximation problem for every finite p > 2. Every `W¹,p` homeomorphism between arbitrary bounded domains in ℝ3 can be approximated strongly in `W¹,p` by smooth diffeomorphisms onto the same target.
View manuscript on GitHub → Open PDF (raw) →Family 369 The hot spots conjecture for simply connected planar domains. 1 paper
Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed. ([Lean](lean/docs/369.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M705
Strict hot spots and absence of interior critical points on smooth simply connected planar domains
We prove the strict hot spots conjecture for smooth bounded simply connected planar domains. More precisely, every nonzero eigenfunction for the first positive Neumann eigenvalue has nonvanishing gradient in the interior, so all its global maxima and minima lie on the boundary. This holds even when the eigenvalue is multiple.
View manuscript on GitHub → Open PDF (raw) →Family 370 The Lane–Emden and Hénon–Lane–Emden conjectures. 1 paper
Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 2, `p,q 0` and real A, B, the system `- u=|x|A vp`, `- v=|x|B uq` has no positive entire solution when `(n+A)/(p+1)+(n+B)/(q+1) n-2`, with solutions continuous at the origin and classical elsewhere. No symmetry or growth assumption is needed. Known radial existence gives the exact existence criterion for n ≥ 3 and `A,B -2`. ([Lean](lean/docs/370.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M706
The Subcritical Hénon–Lane–Emden Conjecture
We prove the subcritical Hénon–Lane–Emden conjecture: for every dimension n ≥ 2, positive powers p, q, and real weights A, B, the system has no strictly positive entire solution when `(n+A)/(p+1)+(n+B)/(q+1) n-2`. Solutions need only be continuous at the origin and classical elsewhere, without a condition at infinity.
View manuscript on GitHub → Open PDF (raw) →Family 371 Stable blowup for the defocusing Schrödinger equation. 1 paper
For a sufficiently large odd nonlinearity power, constructs a nonempty open set of initial data in `Hk( T¹²)`, with k > 8, whose solutions of the scalar defocusing nonlinear Schrödinger equation blow up in finite time. Thus finite-time blowup is stable under Sobolev perturbations in this supercritical regime. ([Lean](lean/docs/371.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M707
Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus
We prove stable self-similar finite-time blowup for a supercritical defocusing nonlinear Schrödinger equation on the twelve-dimensional torus. For a sufficiently large odd power, the blowup initial data contain a nonempty open set in a high Sobolev space. As a consequence, Gaussian Fourier initial data of arbitrarily high Sobolev regularity can have positive probability of finite-time blowup.
View manuscript on GitHub → Open PDF (raw) →Family 372 Global uniqueness in smooth isotropic elasticity. 1 paper
Proves that full static boundary displacement-to-traction data determine both smooth real Lamé moduli on every bounded connected smooth domain in ℝ3, provided μ > 0 and `3λ+2μ 0` on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required. ([Lean](lean/docs/372.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M708
Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem
We prove that the full static displacement-to-traction map uniquely determines both real smooth Lamé moduli on every bounded connected smooth domain in ℝ3, provided μ > 0 and `3λ+2μ 0` on the closure. This resolves the smooth three-dimensional isotropic elastic Calderón uniqueness problem under these positivity assumptions.
View manuscript on GitHub → Open PDF (raw) →Family 373 Nonattainment of the three-marginal Coulomb Monge problem. 1 paper
An optimal three-particle Coulomb configuration need not be a deterministic function of the first particle, even for smooth identical spatial densities. The Monge and Kantorovich infima agree, but the Monge infimum is not attained. The same phenomenon occurs for every inverse-power Riesz exponent in each dimension at least two, with a suitable density in each case. ([Lean](lean/docs/373.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M709
A counterexample to the Monge ansatz for the three-marginal Coulomb cost
We construct a smooth compactly supported probability density on ℝ3, with a smooth compactly supported square root, for which the three-marginal Coulomb transport minimum is not attained by any pair of measure-preserving Borel maps. Nevertheless, the Monge and Kantorovich infima agree: we construct preserving maps whose costs approach the Kantorovich minimum. For every d ≥ 2 and s >
View manuscript on GitHub → Open PDF (raw) →Family 374 Sharp one-third stability of Brenier maps. 1 paper
For uniform source measure ρ on a compact convex body with interior in dimension at least two, quadratic optimal transport maps satisfy ` |T_μ-T_ν |(L²(ρ))≤ C W_2(μ,ν)¹/³` uniformly over targets in a fixed compact set. The exponent is sharp, even for three-atom targets, disproving Letrouit's conjectured square-root bound. ([Lean](lean/docs/374.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M710
Sharp One-Third Stability of Brenier Maps
For the uniform probability measure ρ on a compact convex body in ℝd, d ≥ 2, quadratic optimal transport maps are one-third Hölder continuous in `L²(ρ)` with respect to the target's 2-Wasserstein distance. The constant is uniform over all targets supported in a fixed compact set, and the exponent is optimal. A three-atom family on a fixed cube disproves Letrouit's conjectured uniform square-root estimate.
View manuscript on GitHub → Open PDF (raw) →Family 375 De Giorgi's conjecture in dimension eight. 1 paper
Proves De Giorgi's conjecture at its sharp dimension-eight endpoint.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M711
A positive resolution of De Giorgi's conjecture in dimension eight
We resolve De Giorgi's conjecture positively in dimension eight.
View manuscript on GitHub → Open PDF (raw) →Family 376 Universal computation in forced Navier–Stokes flows. 9 papers
Constructs viscous incompressible flows starting from rest on a fixed flat three-dimensional domain that perform universal computation under smooth external forcing. A terminating compiler turns a Turing machine and input into a finite program for the force, so a designated particle reaches a fixed region exactly when the machine halts. The viscosity is fixed, positive and computable. ([Lean](lean/docs/376.md))
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
Primary manuscript on GitHub → Family index on GitHub →M712
Finite Instructions and Solenoidal Shear Flows
We realize finite reciprocal affine instruction maps by smooth incompressible shear flows on a flat three-torus. Applied to a reversible one-head recorder with a finite transition table, this gives a complete machine-to-fluid construction: a fixed particle enters a fixed open strip exactly when a given machine halts, with zero initial velocity, zero pressure, and a solenoidal mean-zero force that is periodic after initialization. The construction acts on full closed rectangles and controls every intermediate trajectory.
View manuscript on GitHub → Open PDF (raw) →M713
A Fixed Particle Test for Computation in a Forced Viscous Flow
We construct smooth external forces for incompressible Navier–Stokes flow on a fixed flat three-torus at any fixed positive computable viscosity, from zero initial velocity, such that a fixed particle enters a fixed open set exactly when a prescribed Turing machine halts. Every mixed derivative of the force and velocity is bounded and square-integrable in time in spatial supremum norm. The construction uses the machine's ordinary instructions, records their history in a third coordinate, and compensates for finer spatial gates by longer time steps.
View manuscript on GitHub → Open PDF (raw) →M714
Universal Computation with Eventually Stationary Navier–Stokes Forcing
At every fixed positive computable viscosity, we construct smooth mean-zero forces on the flat three-torus that become stationary after time one and make a fixed particle, initially in a fluid at rest, enter a fixed open set exactly when a prescribed Turing machine halts. The force has bounded derivatives of every order and a finite effective description. A reversible recording table is realized on whole planar rectangles by Hamiltonian motions, then driven by a mean-zero spatial clock.
View manuscript on GitHub → Open PDF (raw) →M715
Geometric Programs for Solenoidal Forcing
We construct smooth solenoidal mean-zero forces, periodic from time zero, for which a fixed particle on a flat three-torus reaches a fixed open strip exactly when a given machine halts. The initial fluid velocity and the pressure are zero. We also realize positive diagonal maps on coding sheets by incompressible shears, with explicit normal compensation for changes of planar area, and reciprocal maps on whole boxes of positive thickness.
View manuscript on GitHub → Open PDF (raw) →M716
Prefix Instructions and Incompressible Flows
Finite prefix instructions may change area and erase information. We give explicit history processors and smooth incompressible motions that retain that information and realize every instruction on its full domain. A first application assigns each machine and finite input a smooth mean-zero force on the flat unit three-torus, at any fixed positive computable viscosity. The solution starts from rest; a fixed particle enters a fixed open strip exactly when the machine halts. The force repeats with period one after an initial loading interval.
View manuscript on GitHub → Open PDF (raw) →M717
Computation under Rapidly Vanishing Navier–Stokes Forcing
At any fixed positive computable viscosity, smooth forces can make a fixed fluid particle detect the halting of an arbitrary machine, starting from rest, while every mixed derivative of the force and velocity decreases faster than every inverse power of time. We give three complete memory constructions: compact moving curls, alternating fractional coordinates, and a periodic lattice on the flat three-torus. Each machine step takes one unit of physical time.
View manuscript on GitHub → Open PDF (raw) →M718
Velocity-Field Detection of Computation in Forced Navier–Stokes Flows
We construct smooth forces for three-dimensional incompressible Navier–Stokes flow, from rest at any fixed positive computable viscosity, whose velocity field detects whether a prescribed machine halts. On the unit flat torus, a pointwise test of the third velocity component uses successively shorter stirring intervals in a fixed spatial region. On ` R²× T`, an integral test uses an expanding array of translation regions and a force with globally bounded mixed derivatives.
View manuscript on GitHub → Open PDF (raw) →M719
Scalar Potentials and Slow Clocks for Forced Fluid Computation
We construct smooth forces of fixed compact spatial support for three-dimensional incompressible Navier–Stokes flow from rest whose particle at the origin detects halting by entering a fixed half-space. Every mixed derivative of the force and velocity decays at rate `O((1+t)⁻¹⁻j)` for time order j. Three scalar potentials lift a coded rectangle, perform its instruction, and lower its image; an unbounded logarithmic clock supplies the decay.
View manuscript on GitHub → Open PDF (raw) →M720
Incompressible Box Transport and Finite Computation
We realize finite positive diagonal affine maps of determinant one by effective smooth incompressible flows on neighborhoods of entire closed rational solid boxes. The source and target families are each disjoint, but may overlap each other. The construction uses localized curls, evacuation to storage and obstacle detours. A balanced three-stack recorder then assigns every machine and finite input a smooth Navier–Stokes force with one compact spatial support, periodic after a loading interval, at any fixed positive computable viscosity.
View manuscript on GitHub → Open PDF (raw) →Family 377 Interior `C¹,α` regularity for infinity-harmonic functions. 1 paper
Proves uniform interior `C¹,α_d` regularity for bounded infinity-harmonic functions in every dimension d ≥ 3, for a positive exponent depending only on dimension. The gradient's supremum norm and Hölder seminorm on the half unit ball are bounded by a dimension-dependent constant times the oscillation on the unit ball. The exponent is not explicit.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
Primary manuscript on GitHub → Family index on GitHub →M721
Uniform Interior `C¹,α` Estimates for Infinity-Harmonic Functions
We prove a uniform interior `C¹,α_d` estimate for bounded infinity-harmonic functions in every dimension d ≥ 3. For some `α_d (0,1/3]` depending only on the dimension, the gradient's supremum norm and αd-Hölder seminorm on B1/2 are bounded by a dimension-dependent constant times the oscillation on B1. The exponent is not explicit, and no endpoint or boundary regularity claim is made.
View manuscript on GitHub → Open PDF (raw) →All manuscripts (flat list)
M001 · Family 001 Milne's rationality conjecture for abelian varieties
We prove Milne's rationality conjecture for abelian varieties, including residue characteristic 2. After good reduction, the pairing of a rational Hodge class with any complementary product of divisor classes on the reduction is the same rational number in every prime-to-p realization and in crystalline cohomology. Using the Hodge theorem for CM abelian varieties, we also show that every such specialized Hodge class is represented by a single rational algebraic cycle in all these realizations.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M002 · Family 002 Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank
We prove the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one at some prime q. The analytic and Mordell–Weil ranks equal that corank, and the Tate–Shafarevich group is finite. The formula includes all prime factors and requires no additional hypotheses on reduction, rational torsion, isogenies, complex multiplication, or residual Galois representations.
PQC lens (via Family 002): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M003 · Family 002 The Selmer converse for elliptic curves at every prime
We prove the Selmer converse in coranks zero and one for every elliptic curve over ℚ and every prime p: if the full p-power Selmer group has ℤp-corank `r 0,1 `, then the analytic and Mordell–Weil ranks both equal r, and the entire Tate–Shafarevich group is finite. As an application at the additive prime 3, we prove that for every prime `ℓ≡4,7,8 pmod9`, the cubic `X³⁺Y³⁼ℓ Z³` has analytic and Mordell–Weil rank one and finite Tate–Shafarevich group. In particular, every such ℓ is a sum of two rational cubes.
PQC lens (via Family 002): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M004 · Family 002 The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one
We prove the two-primary Birch and Swinnerton-Dyer leading-term formula for every elliptic curve over the rationals whose two-power Selmer group has corank at most one. In this range, the algebraic rank, analytic rank, and Selmer corank are equal, and the Tate–Shafarevich group is finite. Combined with the quadratic-twist Selmer distribution, this gives the exact two-primary formula for a density-one set of signed squarefree twists of each fixed curve, ordered by absolute value; the common rank is zero or one, with each value having density one half.
PQC lens (via Family 002): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M005 · Family 003 The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane `Re s 7/8`
We prove that all finite-order Hecke L-functions over ` Q(√(-3))` and all Dirichlet L-functions are zero-free in the half-plane `Re s 7/8`, with the principal pole at s = 1…
PQC lens (via Family 003): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M006 · Family 003 The Quasi-Riemann Hypothesis (alternate 11/12 proof)
We establish the quasi-Riemann hypothesis by proving that every Dirichlet L-function, including Riemann's zeta function, has no zeros in the half-plane ` Re s 11/12`.
PQC lens (via Family 003): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M007 · Family 003 Uniform exclusion of Landau–Siegel zeros
We prove the uniform exclusion of Landau–Siegel zeros. There is an absolute constant c > 0 such that every real zero `β (0,1)` of every primitive nonprincipal real Dirichlet L-function of conductor q ≥ 3 satisfies `(1-β)log q≥ c`.
PQC lens (via Family 003): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M008 · Family 004 Hilbert’s tenth problem over the rational numbers
We give a negative answer to Hilbert's tenth problem over the rational numbers: no algorithm decides whether a polynomial with integer coefficients has a rational zero. The number of variables is part of the input.
PQC lens (via Family 004): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M009 · Family 004 A pointwise 2-converse for elliptic curves with rational two-torsion
We prove a pointwise 2-converse for elliptic curves over ` Q` with nonzero rational two-torsion: if the `2^∞` -Selmer corank is zero or one, then the analytic rank and Mordell–Weil rank equal that corank, and the Shafarevich–Tate group is finite. The result allows arbitrary reduction at 2.
PQC lens (via Family 004): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M010 · Family 005 Catalan's constant is irrational
We prove that Catalan's constant `G=∑(j≥0)(-1)j/(2j+1)²`…
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M011 · Family 006 Goldfeld's analytic density conjecture and the 2-converse for elliptic curves
We prove Goldfeld's analytic density conjecture: for every elliptic curve E over ℚ, the quadratic twists of E with analytic rank zero and one each have density 1/2 among signed squarefree twist parameters ordered by absolute value. We also prove the low-corank 2-converse: if the `2^∞` -Selmer corank of E is zero or one, then it equals the analytic and Mordell–Weil ranks, and the Tate–Shafarevich group is finite.
PQC lens (via Family 006): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M012 · Family 006 The mean analytic rank of quadratic twists of elliptic curves
For every elliptic curve over ℚ, we prove that the average analytic rank of its quadratic twists tends to 1/2 when signed squarefree twist parameters are ordered by absolute value. This resolves Goldfeld's mean analytic-rank conjecture in this counting convention.
PQC lens (via Family 006): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M013 · Family 007 Ordinary two-point correlations of multiplicative functions
We prove the ordinary two-point Chowla conjecture. For every fixed pair of nonproportional affine forms, the Liouville correlation has a power-of-logarithm saving at every cutoff, with an absolute exponent. We also prove the binary corrected Elliott conjecture for ordinary averages of complex multiplicative functions of modulus at most one, under uniform nonpretentiousness of at least one original factor. This qualitative conclusion holds in fixed residue classes and for fixed nonproportional affine forms.
PQC lens (via Family 007): Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
View manuscript on GitHub → Open PDF (raw) →M014 · Family 008 The Deligne-Drinfeld conjecture
We prove the Deligne–Drinfeld conjecture: the rational Grothendieck–Teichmüller Lie algebra is freely generated by one element in every odd weight at least three, with the corresponding isomorphism after weight completion.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M015 · Family 009 Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory
We prove a mod-ℓ Bogomolov–Pop reconstruction theorem for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from ℓ. The groups `K M_1/ℓ` and `K M_2/ℓ`, together with their full bilinear product, determine the perfect closure and its constant field. Every compatible isomorphism of these data is induced by a field isomorphism up to a single scalar in ` F_ℓ^×`, with only Frobenius ambiguity in the field isomorphism in positive characteristic.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M016 · Family 009 Reconstruction from Milnor K-theory modulo the characteristic
Let K and L be finitely generated extensions of transcendence degree at least two over algebraically closed fields of characteristic p. We prove that every isomorphism of their first Milnor K-groups modulo p preserving the degree-two Steinberg relations is a nonzero scalar multiple of the map induced by a unique field isomorphism. The proof recovers projective lines over the subfields of pth powers by an elementary calculation with derivations.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M017 · Family 009 The Bogomolov-Pop reconstruction theorem
We prove the Bogomolov–Pop reconstruction conjecture for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from ℓ. The pro-ℓ abelian-by-central datum determines the perfect closure and its constant field, with precisely the Frobenius and ℓ-adic unit ambiguities.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M018 · Family 010 Unrestricted pro-modularity at the prime two
Every continuous, odd, absolutely irreducible two-dimensional 2-adic representation of `G( Q)` that is unramified outside finitely many finite primes occurs in the full completed Hecke algebra at some odd tame level. The level may contain auxiliary tame primes, and scalar and reducible residual representations are included. This is a completed-Hecke occurrence result, with no de Rham hypothesis; it does not assert classical modularity.
PQC lens (via Family 010): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M019 · Family 010 The Dimension of the Two-Adic Hecke Algebra at Odd Level
For every odd positive integer N, every irreducible component of the full two-adic Hecke algebra of level ` Gamma_1(N)` has Krull dimension four. This proves the p = 2 case of Emerton's dimension conjecture, including all residual components.
PQC lens (via Family 010): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M020 · Family 010 Fontaine–Mazur modularity at the prime 2
We prove that every continuous, irreducible, odd two-dimensional 2-adic representation of `G( Q)`, unramified outside finitely many primes and de Rham at 2 with distinct Hodge–Tate weights, is modular up to Tate twist. This resolves the odd, regular two-dimensional Fontaine–Mazur conjecture over ℚ at 2, including all residual representations.
PQC lens (via Family 010): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M021 · Family 011 Weighted dilation graphs, smooth shifted primes and totient fibers
We prove Erdős's conjecture on the largest fibers of Euler's totient function: for every ε > 0, infinitely many positive integers n have more than `n¹⁻ε ` preimages. We also show that, for every fixed δ > 0, there are at least `x¹⁻o(¹)` primes p in `2x p≤5x` whose predecessors have no prime factor exceeding xδ.
PQC lens (via Family 011): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M022 · Family 011 The Poisson-Dirichlet law for prime predecessors
For a prime p chosen uniformly from `3≤ p≤ x`, list the prime factors of `p-1` in decreasing order, with multiplicity. As `x→∞`, their logarithms, divided by `log(p-1)`, converge in every finite joint distribution to the Poisson–Dirichlet distribution with parameter one. This proves the conjecture of Ford, Konyagin and Luca.
PQC lens (via Family 011): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M023 · Family 011 Prime Predecessors with an Even Number of Prime Factors
We prove that there are infinitely many primes p for which `p-1` is squarefree and has an even number of prime factors. Equivalently, there are infinitely many primes p with `μ(p-1)=1`.
PQC lens (via Family 011): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M024 · Family 012 The joint Dickman law for consecutive integers
Let `P⁺(n)` denote the largest prime factor of n. We prove that `log P⁺(n)/log n` and `log P⁺(n⁺¹)/log n` are asymptotically independent in ordinary natural density, with Dickman marginals. This resolves the Erdős–Pomerance joint Dickman conjecture positively and implies that the ordering `P⁺(n) P⁺(n⁺¹)` has natural density 1/2.
PQC lens (via Family 012): Affects polynomial or integer factorization—core to RSA/DSA analysis and to comparing classical vs quantum factoring.
View manuscript on GitHub → Open PDF (raw) →M025 · Family 013 The additive indecomposability of the primes
We prove Ostmann's inverse Goldbach conjecture: no set differing from the primes by finitely many elements can be written as `A+B`, where A and B are sets of nonnegative integers with at least two elements each.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M026 · Family 014 Global Arthur Enhancements of Cuspidal Excursion Parameters
Under the finite-level Ramanujan–Arthur decomposition stated in Theorem 1.1, we construct a global Arthur enhancement for every occurring cuspidal excursion parameter at a specified full finite level for a split connected semisimple group over a global function field. A single algebraic SL2 and a commuting Weil centralizer map recover the given parameter by diagonal specialization on the entire Weil group, including inertia. The result does not assert ellipticity, Arthur-packet classification, or a multiplicity formula.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M027 · Family 014 Rationality of the Canonical Unramified Arthur Filtration
Under the characteristic hypotheses of restricted geometric Langlands theory, we prove that the canonical Arthur filtration on finitely supported unramified automorphic functions is defined over ℚ for split connected semisimple groups. The result includes the noncuspidal part and every closed invariant nilpotent support, establishing the rationality conjecture of Gaitsgory–Lafforgue–Raskin in this setting.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M028 · Family 014 Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level
We prove rational and `Q̄_ℓ` decompositions of cuspidal automorphic functions for split semisimple groups over global function fields into subspaces indexed by nilpotent orbits of the dual group. The decompositions hold at every full finite level, with arbitrary divisor multiplicities. Each summand is governed by the same orbit at every unramified place and every complex embedding. At trivial level this proves Conjectures 3.4.5 and 3.4.6 of Gaitsgory–Lafforgue–Raskin.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M029 · Family 014 Temperedness at ramified places for globally generic exceptional groups
We prove the generalized Ramanujan conjecture for globally generic cuspidal automorphic representations of split connected adjoint exceptional groups over global function fields. Every local component is tempered, without restrictions on characteristic or ramification depth. The proof extends the unramified Ramanujan theorem of a companion paper to all ramified places.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M030 · Family 014 The Restricted Geometric Langlands Equivalence in Positive Characteristic
We prove the `Q̄_ℓ` -linear restricted geometric Langlands equivalence for smooth projective connected curves and connected reductive groups in characteristic p > 0, with ℓ ≠ p, in two regimes. Over ` F _q`, we assume the four characteristic conditions of the restricted theory stated below. Over arbitrary algebraically closed fields, we assume these conditions and additionally that p is very good and does not divide the Weyl-group order.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M031 · Family 014 Constructible tame Hecke eigensheaves in positive characteristic
We construct nonzero locally constructible perverse Hecke eigensheaves with Borel level at one marked point on a smooth projective curve of genus at least two over an algebraic closure of a finite field. The parameter is a Zariski-dense geometric ℓ-adic local system with tame regular-unipotent monodromy. The group is simple and simply connected and satisfies four explicit Lie-theoretic characteristic hypotheses.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M032 · Family 014 Tame Hecke Eigensheaves with Several Marked Points
For SLn in characteristic p > n, we construct nonzero locally constructible perverse Hecke eigensheaves with Borel level at two or more marked points on a smooth projective curve of genus at least two over an algebraic closure of a finite field. The parameter is a Zariski-dense geometric ℓ-adic PGLn-local system with unipotent tame monodromy; its nilpotent logarithms may have any Jordan type, including zero.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M033 · Family 014 Frobenius Structures on Tame Hecke Eigensheaves
We construct nonzero perverse Weil Hecke eigensheaves for SLn with Borel level at one marked point, after a finite extension of the field of constants. The parameter is a geometrically dense arithmetic PGLn-local system with tame regular-unipotent monodromy on a once-punctured curve of genus at least two over a finite field of characteristic p > n.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M034 · Family 015 Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type
We prove the packet form of the higher-dimensional Duke equidistribution problem for totally real fields of any fixed prime degree at least five, allowing arbitrary local homothety types of full lattices. As the multiplier-order discriminant tends to infinity, the volume-weighted packet measures converge to Haar probability measure, with no escape of mass.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M035 · Family 015 Equidistribution of primitive quartic torus packets for arbitrary orders
Let K range over totally real quartic fields with no proper intermediate field, and let ` O` be any order in K. We prove that the packets of periodic diagonal orbits attached to invertible ` O` -ideal classes, weighted by orbit volume, equidistribute with no escape of mass as `| Disc ( O)|` tends to infinity. The proof combines measure rigidity with a cubic-resolvent estimate that remains uniform at primes dividing the order index.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M036 · Family 015 Equidistribution of Primitive Sextic Torus Packets
We prove that the complete ideal-class packets of maximal orders in totally real sextic fields with no proper intermediate fields become equidistributed, with no escape of mass, in the space of unimodular lattices as their field discriminants tend to infinity. The limit is Haar probability measure. Each packet includes all coordinate-sign translates and is weighted by diagonal-orbit volume.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M037 · Family 016 The abelian Zilber–Pink conjecture
We prove the abelian Zilber–Pink conjecture over `Q̄`. Every irreducible subvariety of an abelian variety has only finitely many maximal atypical subvarieties, where atypicality is measured inside its smallest containing torsion coset.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M038 · Family 016 The E×CM component of Zilber–Pink for curves in A2
We prove the `E×CM` component of Zilber–Pink for Hodge-generic algebraic curves in ` A_2` over `Q̄`. Each such curve contains only finitely many points whose abelian surface is isogenous to a product of elliptic curves with at least one factor having complex multiplication.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M039 · Family 016 Quaternionic division points on curves in the Siegel threefold
We prove the quaternionic-division component of Zilber–Pink for curves in ` A_2` over `Q̄`. A Hodge-generic algebraic curve contains only finitely many points whose full geometric rational endomorphism algebra is an indefinite quaternion division algebra over ℚ. No boundary or reduction hypothesis is required.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M040 · Family 016 Elliptic squares and Zilber–Pink for curves in A2
We prove that every Hodge-generic algebraic curve in ` A_2` over `Q̄` contains only finitely many points whose abelian surface is isogenous to the square of an elliptic curve without complex multiplication (CM). Combining this result with the companion `E×CM` and quaternionic-division finiteness theorems, we prove the curve case of Zilber–Pink in ` A_2` over `Q̄`, without a boundary hypothesis.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M041 · Family 017 The irrationality exponent of pi is 2
We prove the conjecture that the irrationality exponent of π is 2. As a consequence, the classical Flint–Hills series `∑(n≥1)1/(n³sin² n)` converges, with angles in radians.
PQC lens (via Family 017): Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
View manuscript on GitHub → Open PDF (raw) →M042 · Family 018 The Margulis–Platonov conjecture over global function fields
We prove the Margulis–Platonov conjecture over every global function field, including characteristic two. For an absolutely almost simple simply connected algebraic group, every noncentral abstract normal subgroup of its rational points is the inverse image of an open normal subgroup of the finite product of its anisotropic local groups.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M043 · Family 018 The Margulis–Platonov conjecture over number fields
We prove the Margulis–Platonov conjecture over number fields. For an absolutely almost simple simply connected group, every noncentral abstract normal subgroup of its rational points is the inverse image of an open normal subgroup of the product of its anisotropic nonarchimedean local groups.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M044 · Family 019 Étale covers with a prescribed exterior sheet
Let X be a smooth proper hyperbolic curve over an algebraic closure of a p-adic local field. Given finitely many disjoint small open disks, we construct one connected finite étale cover with a sheet isomorphic to their entire exterior and with degree divisible by p on every connected component above each disk.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M045 · Family 019 The p-adic section conjecture
We prove the local p-adic section conjecture for smooth proper geometrically connected curves of genus at least two, over every finite extension of ℚp. Combined with established finite-descent theorems, this also yields the global section conjecture for smooth proper geometrically connected curves of genus at least two over number fields when their finite-cover descent locus equals their rational points; this includes `X_0(N)` and `X_1(N)` of genus at least two over ℚ.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M046 · Family 020 Squarefree values of quartics and power-free values of polynomials
We prove that every irreducible integer quartic with no fixed prime-square divisor takes squarefree values with the predicted positive Euler-product density. More generally, we obtain the corresponding `(d-2)` -power-free density in degrees `4≤ d≤8`. The proof combines number-field factorization, determinant estimates with adaptive auxiliary primes, and explicit low-degree geometry. Together with Browning's theorem for higher degrees, this gives the `(d-2)` -power-free density for every d ≥ 4.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M047 · Family 021 A quadratic bound for Jacobsthal's function
Let `h(k)` be the least integer such that every interval of `h(k)` consecutive integers contains an integer coprime to any prescribed positive integer having at most k distinct prime divisors. We prove `h(k) k²/(loglog(3k))²`, giving an affirmative answer to Jacobsthal's quadratic-bound question.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M048 · Family 022 The Weak Inhomogeneous Duffin–Schaeffer Conjecture
We prove the weak inhomogeneous Duffin–Schaeffer conjecture. For every fixed γ ∈ ℝ and finite-valued `: N→[0,∞)`, divergence of `∑(q≥1)φ(q) (q)/q` implies ` |qx-γ | (q)` for infinitely many q, for almost every x. Here ϕ is Euler's totient and ` |· |` is distance to the nearest integer. Numerators need not be reduced, and the shift satisfies no Diophantine restriction.
PQC lens (via Family 022): Affects polynomial or integer factorization—core to RSA/DSA analysis and to comparing classical vs quantum factoring.
View manuscript on GitHub → Open PDF (raw) →M049 · Family 023 An unconditional first moment for cubic Gauss sums
We prove Patterson's first-moment conjecture for normalized cubic Gauss sums over all primary Eisenstein primes, unconditionally. The sharp-cutoff main term is `(6/5)c_*X⁵/⁶/log X`, where `c_*=(2π)²/³/(3 (2/3))`. For every fixed nonzero angular Fourier mode of the prime argument, we also prove cancellation at the first-moment scale.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M050 · Family 024 An asymptotic formula for the number of totients
Let `V(x)` count the distinct values of Euler's totient function up to x. We give an explicit asymptotic equivalent for `V(x)`. Its coefficient is a uniform limit of functions defined from finite arithmetic data. We also prove that `V(cx)/V(x)→ c` as `x→∞` for every fixed c > 0, answering a question of Erdős and Hall.
PQC lens (via Family 024): Affects polynomial or integer factorization—core to RSA/DSA analysis and to comparing classical vs quantum factoring.
View manuscript on GitHub → Open PDF (raw) →M051 · Family 025 Short Egyptian fractions
We prove a conjecture of Erdős: for every sufficiently large integer b, every rational number `a/b` with `1≤ a b` is a sum of `O(loglog b)` distinct positive unit fractions, with an absolute implied constant. This order is best possible when the numerator varies. We also show that both the number of expansions of 1 with exactly k distinct terms and the least integer at least 2 that never occurs as a denominator in such an expansion grow doubly exponentially in k: their double logarithms have order k.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M052 · Family 026 Positive lower density of large prime gaps
For every fixed C > 0, we prove that a positive proportion of consecutive prime gaps exceed `Clog p`, where p is the smaller prime. The proportion is bounded below for every sufficiently large initial segment of the prime sequence, with a constant depending on C. It follows that the indices at which `p_n/n` increases have positive lower asymptotic density, answering a question of Erdős and Prachar.
PQC lens (via Family 026): Affects polynomial or integer factorization—core to RSA/DSA analysis and to comparing classical vs quantum factoring.
View manuscript on GitHub → Open PDF (raw) →M053 · Family 027 Integral points on character varieties of curves
We prove potential Zariski density of integral points on SLr-character varieties of smooth complex curves in every rank. The result allows prescribed quasi-unipotent boundary monodromy, including exact nonsemisimple conjugacy classes, and holds on every component over the full ring of integers of one finite extension. Here integrality is measured in the ambient character variety, while the exact boundary conditions are imposed on the complex representation.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M054 · Family 028 Bounded-Step Walks on Gaussian Primes
We prove the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have bounded steps. More strongly, for each fixed finite step bound, the connected components of the Gaussian-prime graph have uniformly bounded size. This bound applies to every starting prime, including primes on the coordinate axes, and is nonexplicit. The proof constructs a finite periodic sieve obstruction using geometric sampling and information-theoretic estimates.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M055 · Family 029 Primitive roots for every admissible integer base
We prove the infinitude assertion in Artin's primitive root conjecture: for every integer a that is neither −1 nor a square, there are at least `c_a x/(log x)²` primes in `(x,2x)` with primitive root a, for some `c_a 0` and every sufficiently large x.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M056 · Family 029 Simultaneous primitive roots: a conditional lower bound for prime bases
For every fixed finite set of distinct positive primes, we prove that at least `cx/(log x)²` primes in `(x,2x)` have every member of the set as a primitive root, for some c > 0 and all sufficiently large x. The result assumes four explicitly stated analytic and sieve inputs from the companion paper on primitive roots for admissible integer bases.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M057 · Family 030 Modularity of elliptic curves over imaginary quadratic fields
We prove the modularity conjecture for elliptic curves over imaginary quadratic fields: every such curve is modular, with matching local parameters at every place.
PQC lens (via Family 030): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M058 · Family 031 Open Homomorphisms of Global Solvably Closed Galois Groups
We prove Uchida's conjecture on open homomorphisms of Galois groups. Every continuous open homomorphism between Galois groups of solvably closed Galois extensions of number fields is induced by a unique equivariant field embedding in the opposite direction. No restriction on the kernel or additional compatibility hypothesis is required.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M059 · Family 032 The rational Hodge conjecture for products of K3 surfaces
We prove the rational Hodge conjecture for every finite product of projective complex K3 surfaces: every rational Hodge class is algebraic. The factors may be distinct or repeated, with no restrictions on their Picard numbers, periods, or endomorphism fields.
PQC lens (via Family 032): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M060 · Family 032 Algebraicity of Kuga–Satake Correspondences for K3 Surfaces
We prove that the Kuga–Satake correspondence is algebraic for every smooth projective complex K3 surface. More precisely, the prescribed embedding of its transcendental cohomology into the cohomology of its Kuga–Satake abelian variety is induced by a rational algebraic cycle, with the fixed normalization and full even-Clifford target. The result also holds for isogenous Kuga–Satake models with the transported embedding.
PQC lens (via Family 032): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M061 · Family 032 The rational Hodge conjecture for CM abelian varieties
We prove the rational Hodge conjecture for complex abelian varieties with complex multiplication: every rational Hodge class on such a variety is a rational linear combination of algebraic cycle classes. As consequences, we obtain the generalized Hodge conjecture for CM abelian varieties, the Tate conjecture for abelian varieties over finite fields, and the Hodge standard conjecture for abelian varieties in arbitrary characteristic.
PQC lens (via Family 032): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M062 · Family 032 Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus
We prove the rational Hodge and generalized Hodge conjectures in every cohomological degree of every self-power of a projective complex K3 surface whose transcendental quadratic space, with its cup-product form, admits a rational isometric embedding in ` U( Q) oplus² -1 rangle⁴`.
PQC lens (via Family 032): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M063 · Family 032 Weil classes and Hodge classes on abelian powers
We prove the rational Hodge conjecture in every codimension on every self-power of a complex abelian sixfold with an imaginary-quadratic action and a compatible polarization whose rational homological Hermitian form is hyperbolic of signature `(3,3)`. We also prove it in every codimension on every self-power of a complex abelian variety of dimension at most five admitting an imaginary-quadratic action. Both results include nonsimple varieties and special periods with additional endomorphisms.
PQC lens (via Family 032): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M064 · Family 032 Abelian covers, Gale correspondences, and the Hodge conjecture for powers
We prove the rational Hodge conjecture on every self-power of the Jacobian at each tensor Hodge-generic point of the full marked variation of a connected abelian cover of curves. This holds in every base genus and for every compatible branching pattern. For any CM abelian variety, the conclusion also holds for every self-power of its product with the Jacobian, on the same Hodge-generic locus.
PQC lens (via Family 032): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M065 · Family 032 Algebraicity of Weil classes on split abelian eightfolds
We prove that every rational Weil class on a split abelian eightfold of Weil type is algebraic. The result holds for every imaginary quadratic field, every compatible polarization type, and every member of the split family, including those with additional endomorphisms. Thus the full two-dimensional rational Weil space in codimension four is generated by algebraic cycle classes.
PQC lens (via Family 032): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M066 · Family 032 A Conditional Reduction for Algebraic Kuga–Satake Correspondences
For a polarized K3 surface whose primitive cohomology has full orthogonal Hodge group, one algebraic correspondence inducing a nonzero map from that cohomology to the second cohomology of an abelian variety suffices to recover the prescribed full Kuga–Satake correspondence. We give an equivalent condition using holomorphic one-forms on a generically finite surface cover.
PQC lens (via Family 032): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M067 · Family 033 Orbifold and logarithmic Iitaka subadditivity
We prove Campana's orbifold Iitaka subadditivity conjecture for rational simple normal crossing boundaries on compact manifolds in Fujiki class ` C`, including coefficient one. Ordinary and logarithmic subadditivity follow.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M068 · Family 033 Logarithmic Kodaira dimension and whole-fiber variation
We prove the logarithmic Iitaka–Viehweg inequality for projective surjective morphisms with connected fibers between smooth complex quasi-projective varieties whose base has nonnegative logarithmic Kodaira dimension. The variation measures the birational field of definition of the whole geometric generic fiber. This resolves Popa's logarithmic variation conjecture positively.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M069 · Family 033 The reverse logarithmic Kodaira inequality and additivity
We prove the reverse logarithmic Kodaira inequality for a surjective connected-fiber morphism `f:(X,E)→(Y,D)` of smooth projective reduced simple-normal-crossing pairs, with ` Supp (f*D) Supp E`, such that X and every boundary stratum are smooth over `Y Supp D`. Together with logarithmic subadditivity, the inequality gives additivity, including both negative-infinity cases. This resolves Popa's logarithmic additivity conjecture positively in the projective reduced-SNC, stratum-smooth setting.
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View manuscript on GitHub → Open PDF (raw) →M070 · Family 033 Projective Hodge lines and ordinary Iitaka subadditivity
We prove the ordinary Iitaka subadditivity conjecture for surjective projective morphisms with connected fibers between smooth connected projective varieties over algebraically closed fields of characteristic zero. If F is the geometric generic fiber of `f:X→ Z`, then ` (X)≥ (F)+ (Z)`.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M071 · Family 033 B-semiampleness for compact log-smooth Kähler fibrations
We prove the compact log-smooth Kähler case of b-semiampleness. Let `f:Y→ X` be a surjective holomorphic map with connected fibers between smooth compact connected Kähler manifolds, and let Δ be an effective rational divisor with simple normal crossing support and coefficients in `[0,1]`, with `K_Y+ ( Q)f*L` for `L Pic (X)( Q)`.
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View manuscript on GitHub → Open PDF (raw) →M072 · Family 034 Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity
Assume logarithmic Iitaka subadditivity for surjective morphisms with connected fibers between smooth projective complex varieties with compatible reduced simple normal crossing boundaries. We prove log abundance for normal irreducible compact Kähler spaces in every dimension: for a log canonical pair `(X, )` with effective rational boundary and `K_X+ ` ℚ-Cartier, analytic nefness of `K_X+ ` implies semiampleness.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M073 · Family 034 Uniform indices for semi-log-canonical log Calabi–Yau pairs
We prove a uniform index theorem for connected projective semi-log-canonical log Calabi–Yau pairs in every fixed dimension at least four over an algebraically closed field of characteristic zero. For boundary coefficients in a fixed finite rational set, a single multiple of the log canonical divisor is Cartier and linearly trivial. The multiple depends only on the dimension and coefficient set, not on the number of irreducible components.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M074 · Family 034 Conditional good minimal models for compact Kähler fourfolds
Assuming orbifold Iitaka subadditivity, the specified pseudo-effective fourfold minimal model program, and abundance for nef fourfold adjoints of nonnegative Kodaira dimension, we prove the existence of good minimal models for globally strongly ℚ-factorial compact Kähler klt fourfold pairs with effective rational boundary and analytically pseudo-effective actual ℚ-Cartier adjoint. The additional step is nonvanishing.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M075 · Family 034 Log abundance in characteristic zero
We prove the rational-boundary log abundance conjecture in every dimension over algebraically closed fields of characteristic zero: every nef ℚ-Cartier log canonical divisor on a projective log canonical pair with effective rational boundary is semiample. Over ℂ, the proof also establishes canonical nonvanishing for smooth projective varieties in every dimension.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M076 · Family 034 Minimal metrics and interior injectivity for nef adjoints
Let `(H, )` be a projective complex klt pair with effective rational boundary and nef ℚ-Cartier adjoint. On any projective log resolution, minimal semipositive metrics on the pulled-back adjoint exist and have zero Lelong numbers everywhere. On smooth projective complex varieties, we also prove an H1-injectivity theorem for rational interior boundaries with simple-normal-crossing support when both endpoint bundles carry zero-Lelong semipositive metrics.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M077 · Family 034 Fourfold nonvanishing by minimal metrics and moving jets
We prove canonical nonvanishing for smooth connected projective complex fourfolds: if KX is pseudo-effective, then `H⁰(X,mK_X)≠0` for some positive integer m.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M078 · Family 034 Lifting sections from the reduced support of an adjoint
For a projective ℚ-factorial dlt pair with effective rational boundary over an algebraically closed field of characteristic zero, we prove that the adjoint has positive Iitaka dimension whenever a nonzero effective Cartier multiple is supported on the coefficient-one boundary and restricts to a semiample line bundle on its whole reduced support. This gives log abundance after nonvanishing in dimension at most four over ℂ.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M079 · Family 034 Schnell fiber spaces and good canonical models
We prove the Campana–Peternell inequality ` (X)≥ (D)` for a smooth connected projective complex variety X and an effective Cartier divisor D whenever `m_0K_X-D` is pseudo-effective for some positive integer m0. For an algebraic fiber space `f X→ Y` between smooth connected projective complex varieties, the hypothesis that `m_0K_X-f*H` is pseudo-effective with H ample Cartier gives ` (X)= (F)+dim Y` for a very general smooth fiber F, as well as nonzero sections of `mK_X-f*H` for all sufficiently large divisibl…
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M080 · Family 034 Uniform log Iitaka fibrations and bounded moduli denominators
For normal projective log canonical pairs over algebraically closed fields of characteristic zero, of fixed dimension d ≥ 5 and with effective boundary coefficients in a fixed finite rational set, we prove that one complete rounded pluricanonical system generates the full Iitaka field whenever the ℚ-Cartier log canonical divisor has nonnegative Kodaira dimension. Its degree depends only on the dimension and coefficient set.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M081 · Family 034 Uniform Pluricanonical Iitaka Fibrations
We prove the effective Iitaka fibration conjecture in characteristic zero. For each dimension, one pluricanonical degree defines the Iitaka fibration of every smooth integral projective variety of that dimension and nonnegative Kodaira dimension over an algebraically closed field. The associated sections generate the full Iitaka function field.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M082 · Family 034 Relative denominators and effective systems for log Calabi-Yau fibrations
We prove uniform denominator and effective-system bounds for log Calabi-Yau fibrations from projective log canonical complex pairs of dimension at most four onto positive-dimensional bases, with boundary coefficients in a fixed finite rational set. The bounds give a uniform trivializing degree and a uniform b-Cartier multiple of the moduli b-divisor. When the base divisor is big, a uniform complete rounded adjoint system has section ratios generating the full base function field.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M083 · Family 034 Arithmetic Stein-degree bounds for log Calabi–Yau pairs
Fix d ≥ 1 and t > 0. Let `(X,B)` be an ordinary projective log canonical ℚ-pair of dimension d over a characteristic-zero field k, with X normal and integral, `H⁰(X, O_X)=k`, B effective, and `K_X+B ( Q)0`. We prove that every prime component S of B with coefficient at least t satisfies `[k_S:k]≤ N(d,t)`, where kS is the relative algebraic closure of k in `k(S)`.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M084 · Family 034 Uniform effective log Iitaka fibrations for fourfolds
We prove the finite-rational-coefficient case of the effective log Iitaka conjecture in dimension four. For normal projective log canonical complex fourfolds with boundary coefficients in a fixed finite rational set and pseudo-effective rational Cartier adjoint, one uniform degree makes the complete rounded reflexive system nonempty and its section ratios generate the full Iitaka field. We also prove a uniform canonical index bound for projective klt complex fourfolds with rationally trivial canonical divisor.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M085 · Family 034 Abundance after nonvanishing for compact Kähler fourfolds
We prove semiampleness of the actual ℚ-Cartier adjoint `K_X+ ` of a normal connected compact Kähler klt fourfold whenever it is analytically nef and some positive Cartier multiple has a nonzero section. The boundary is effective and rational; the fourfold need not be projective or ℚ-factorial. In Iitaka dimension zero, a positive Cartier multiple is the trivial holomorphic line bundle.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M086 · Family 035 Log abundance in numerical dimension one for threefolds in positive characteristic
We prove the numerical-dimension-one case of log abundance for threefolds over algebraically closed fields of characteristic p > 3. If `(X,B)` is a projective log canonical threefold pair with effective rational boundary, and `K_X+B` is ℚ-Cartier, nef, and of numerical dimension one, then `K_X+B` is semiample. Neither terminality nor ℚ-factoriality of X is required.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M087 · Family 035 Abundance in numerical dimension one for terminal threefolds in positive characteristic
Let X be a projective ℚ-factorial terminal threefold over an algebraically closed field of characteristic p > 3. We prove the numerical-dimension-one case of abundance: if KX is nef with `ν(K_X)=1`, then KX is semiample and ` (X,K_X)=1`.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M088 · Family 036 Numerical semiampleness of nef adjoint classes on compact Kähler manifolds
Let X be a smooth connected compact Kähler manifold, let B be an effective rational simple normal crossing divisor with coefficients less than one, and let M be a nef rational holomorphic line bundle on X. If `K_X+B` is pseudo-effective and `K_X+B+M` is nef, we prove that its first Chern class in real Bott–Chern cohomology is represented by a semiample rational line bundle. This numerical statement allows a flat change of line bundle; it does not assert semiampleness of the original adjoint.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M089 · Family 036 Numerical Semiampleness of Nef Adjoint Divisors
We prove the Generalised Abundance Conjecture: if `(X,B)` is a projective klt ℚ-pair over an algebraically closed field of characteristic zero, `K_X+B` is pseudo-effective, M is a nef ℚ-Cartier divisor on X, and `K_X+B+M` is nef, then `K_X+B+M` is numerically equivalent to a semiample ℚ-Cartier divisor on X.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M090 · Family 036 Minimal models and Mori fibre spaces for generalized log canonical Q-pairs
We resolve the existence form of the minimal-model conjecture for projective generalized log canonical ℚ-pairs over algebraically closed fields of characteristic zero. Such a pair admits a minimal model when its adjoint divisor is pseudo-effective, and a Mori fibre space otherwise, while keeping the nef ℚ-Cartier b-divisor fixed. Taking the nef part to be zero gives the corresponding result for ordinary log canonical ℚ-pairs.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M091 · Family 036 Minimal models in numerical dimension one
We resolve the numerical-dimension-one case of the minimal-model conjecture for smooth connected complex projective varieties of dimension at least three. If KX is pseudo-effective and ` kappa_σ(X,K_X)=1`, with κσ defined by section growth with a fixed ample twist, then X admits a projective ℚ-factorial terminal minimal model.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M092 · Family 037 The ordinary-double-point gap in every dimension
We prove the ordinary-double-point gap conjecture for boundary-zero complex algebraic klt germs in every dimension: a singular n-dimensional germ has normalized volume at most `2(n-1)n`, with equality precisely at an analytic ordinary double point.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M093 · Family 037 The normalized-volume gap in dimension four
We prove that every singular complex algebraic klt fourfold germ with zero boundary has normalized volume at most 162, with equality precisely for an analytic ordinary double point. This resolves the ordinary-double-point volume-gap conjecture in dimension four.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M094 · Family 038 Fujita's freeness conjecture
We prove Fujita's freeness conjecture. If X is a smooth complex projective variety of dimension n and L is an ample line bundle, then `K_X+mL` is globally generated for every integer `m≥ n+1`.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M095 · Family 039 Nagata's conjecture for plane curves
We prove that a nonzero effective plane curve of degree d at r ≥ 10 very general complex points has total multiplicity strictly less than `d r`. The inequality holds simultaneously for all curves, including reducible and nonreduced curves, and establishes Nagata's conjecture in its strict, nonhomogeneous form.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M096 · Family 039 Maximal Seshadri constants on arbitrary polarized surfaces
We prove that, for every smooth integral complex projective surface S and every ample line bundle L, the multipoint Seshadri constant at r very general points equals `√(L²/r)` for every sufficiently large integer r. This resolves positively the qualitative Nagata–Biran conjecture for surfaces.
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View manuscript on GitHub → Open PDF (raw) →M097 · Family 039 Maximal Multipoint Seshadri Constants in Higher Dimensions
Let L be an ample line bundle on a smooth integral complex projective variety X of dimension n ≥ 3. We prove that there is a threshold `r_0=r_0(X,L)` such that for every integer `r≥ r_0`, the ordinary multipoint Seshadri constant at r very general points equals the volume bound `(Ln/r)¹/n`. This establishes the qualitative Nagata–Biran–Szemberg assertion in these dimensions.
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View manuscript on GitHub → Open PDF (raw) →M098 · Family 039 Maximal multipoint Seshadri constants in positive characteristic
Let L be an ample line bundle on a smooth integral projective variety X over an algebraically closed field of positive characteristic, with `dim X=n≥3`. We prove that there is a threshold `r_0=r_0(X,L)` such that for every integer `r≥ r_0`, the ordinary multipoint Seshadri constant at the geometric generic tuple of r points equals the volume bound `(Ln/r)¹/n`. The same conclusion holds in dimension two. This establishes the positive-characteristic form of the qualitative Nagata–Biran–Szemberg assertion at geometric generic tuples.
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View manuscript on GitHub → Open PDF (raw) →M099 · Family 040 Bloch’s conjecture for surfaces with p_g=0
We prove Bloch's conjecture for smooth connected projective complex surfaces with `p_g=0`: the Albanese homomorphism on integral degree-zero zero-cycles is an isomorphism.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M100 · Family 041 Projective-space bases of Lagrangian fibrations
We prove that the normal projective base of a projective Lagrangian fibration from a compact irreducible holomorphic symplectic Kähler manifold is projective space. This resolves the projective-space base conjecture for such fibrations in every dimension and deformation type.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M101 · Family 041 The strong hyperkähler SYZ conjecture
We prove the strong hyperkähler SYZ conjecture: every holomorphic line bundle with nonzero nef isotropic first Chern class on a compact irreducible holomorphic symplectic Kähler manifold is semiample.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M102 · Family 042 Every complex K3 surface is Oka
We prove that every complex K3 surface X is Oka, resolving the K3 Oka conjecture. Equivalently, for every m ≥ 1, every holomorphic map to X from a neighborhood of a compact convex set in ℂm can be approximated uniformly on that set by entire maps ` Cm→ X`.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M103 · Family 043 P=W in composite rank for fixed determinant
We prove the P = W conjecture on the full rational cohomology of fixed-determinant, trace-free Higgs moduli spaces in composite rank and coprime degree, for smooth projective complex curves of genus at least two. Together with the established prime-rank cases, this gives the equality in every coprime rank.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M104 · Family 044 The equivariant cohomological Hikita conjecture for arbitrary quivers
We prove the equivariant cohomological Hikita conjecture for arbitrary finite quivers, including loops and multiple arrows. For any dimension and framing vectors, any commuting flavor torus, and a stability character whose semistable locus is stable and has free gauge action, the equivariant cohomology of the Nakajima variety is canonically isomorphic to the coordinate ring of the scheme-theoretic fixed locus of the stability cocharacter on the flavor-deformed Coulomb branch.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M105 · Family 046 A projective fourfold with large fundamental group and non-Stein universal cover
We construct a smooth projective complex fourfold with large fundamental group whose universal cover is not Stein. The universal cover contains no positive-dimensional compact complex-analytic subvariety, so this disproves Shafarevich's holomorphic-convexity conjecture even under the large-fundamental-group hypothesis.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M106 · Family 046 A surface counterexample to Shafarevich holomorphic convexity
We construct a smooth connected projective complex surface whose universal cover is not holomorphically convex. This disproves the Shafarevich conjecture on holomorphic convexity in complex dimension two.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M107 · Family 047 An explicit failure of complex affine-space cancellation
We construct an explicit integral complex affine fourfold X with `X× A¹ A⁵` but `X A⁴`. This gives a negative answer to Zariski's affine-space cancellation problem over ℂ in dimension four. The same construction disproves the Stable Coordinate Conjecture in ambient dimension five and yields smooth 𝔸3-fibrations over 𝔸1 and 𝔸2 that are not Zariski-locally trivial. These fibrations disprove the Dolgachev–Weisfeiler affine-fibration conjecture over these bases.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M108 · Family 048 A singular normal affine surface with free tangent sheaf
We construct a singular normal affine complex surface whose tangent sheaf is free of rank two. This disproves the Lipman–Zariski conjecture in characteristic zero.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M109 · Family 049 A stable coordinate that is not a coordinate in four variables
We construct an explicit degree-five polynomial over ` C` that is not a coordinate in four variables but becomes one after adjoining a single variable. This gives a counterexample to the stable coordinate conjecture in four variables. Every fiber is isomorphic to affine three-space, yet its embedding in affine four-space is not rectifiable. Thus the example also disproves the Abhyankar–Sathaye embedding conjecture in ambient dimension four.
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View manuscript on GitHub → Open PDF (raw) →M110 · Family 049 An explicit noncoordinate polynomial with affine three-space zero fibre
We construct an explicit counterexample to the Abhyankar–Sathaye conjecture: a noncoordinate polynomial in four complex variables whose zero fibre is affine three-space. Adjoining variables gives counterexamples in every ambient dimension at least four.
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View manuscript on GitHub → Open PDF (raw) →M111 · Family 050 Ample rank-two bundles on the quadric surface without Griffiths-positive metrics
We give counterexamples to Griffiths' conjecture in rank two on the quadric surface ` P¹× P¹`. We construct an explicit bundle G whose coordinatewise power pullbacks, tensored with ` O(1,1)`, are ample for every positive power, but admit no smooth strictly Griffiths-positive Hermitian metric for all sufficiently large powers.
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View manuscript on GitHub → Open PDF (raw) →M112 · Family 051 Canonical ampleness of compact hyperbolic Kähler manifolds
We prove that every compact connected Kähler manifold of positive complex dimension containing no nonconstant entire curve has ample canonical bundle and is projective. This resolves positively Kobayashi's canonical-ampleness conjecture in the smooth compact Kähler category.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M113 · Family 052 Universal-cover splitting for compact Kähler manifolds
We prove the two-summand form of Beauville's compatible splitting conjecture. If the tangent bundle of a compact connected Kähler manifold decomposes into two integrable holomorphic subbundles of positive rank, then its ordinary universal cover admits a product decomposition whose factor tangent bundles are the lifted specified summands.
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View manuscript on GitHub → Open PDF (raw) →M114 · Family 052 Integrability of split tangent bundles on rationally connected manifolds
We prove that both summands of every specified holomorphic splitting of the tangent bundle of a smooth rationally connected projective complex manifold into two positive-rank subbundles are integrable. This proves Höring's conjecture on rationally connected projective manifolds. Höring's product theorem then gives a product decomposition compatible with the specified splitting.
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View manuscript on GitHub → Open PDF (raw) →M115 · Family 053 A high-arity counterexample to Pixton completeness in Chow
We disprove the Chow and rational-cohomological forms of Pixton's completeness conjecture for his original relation system. We construct a formal tautological class outside the original Pixton relation span whose image is zero in the Chow ring and in rational cohomology. The example has genus `10⁶⁰` and `3 10⁶⁰ 3` markings.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M116 · Family 054 Irrational cubic fourfolds with geometric K3 categories
For every sufficiently large admissible Hassett discriminant, we prove that a very general cubic fourfold of that discriminant is irrational, although its Kuznetsov component is equivalent to the ordinary derived category of a projective K3 surface. The discriminant threshold is ineffective. This disproves Kuznetsov's rationality conjecture. The same cubics have associated untwisted polarized K3 surfaces in the Hodge-theoretic sense, so they also disprove the sufficiency direction of the associated-K3 rationality prediction.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M117 · Family 055 A Gepner stability condition on every smooth quintic threefold
We prove Toda's normalized quintic Gepner conjecture. On every smooth complex quintic threefold, we construct a numerical Bridgeland stability condition for which tensoring by the hyperplane bundle, followed by the spherical twist at the structure sheaf, increases phase by 2/5.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M118 · Family 055 Prescribed large-volume charges on threefolds with trivial canonical bundle
Let X be a smooth projective complex threefold with trivial canonical bundle. We construct numerical Bridgeland stability conditions with the exact ordinary and square-root-Todd central charges at every sufficiently large volume. One volume threshold works on an open set of real twists and ample directions. The resulting stability conditions have the support property on the full numerical Grothendieck group and stable point sheaves.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M119 · Family 056 Termination of generalized log canonical flips on compact Kähler fourfolds
Every sequence of generalized log canonical flips on a normal irreducible globally Weil ℚ-factorial compact Kähler fourfold terminates, provided the flips are projective small diagrams with the stated opposite ample signs. The boundary is rational, and the nef b-divisor is fixed and represented by an analytically nef ℚ-Cartier divisor on a projective modification. No scaling rule or pseudo-effectivity assumption is required. The theorem concerns existing flip sequences; it does not assert the existence of all contractions or flips.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M120 · Family 056 Termination of generalized-canonical flips on compact Kähler fourfolds
Every sequence of generalized-canonical flips on compact Kähler fourfolds with rational boundary and fixed rational analytically nef b-divisor terminates when the small morphisms are projective. We prove this for normal globally Weil ℚ-factorial models with boundary coefficients less than one, allowing exceptional log discrepancies equal to one. No scaling rule or pseudo-effectivity assumption is required. The theorem applies to existing projective small diagrams with the specified opposite ample signs.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M121 · Family 056 Termination for projective log canonical fourfolds with rational boundary
We prove that every permitted minimal model program for a projective log canonical fourfold with rational boundary over an algebraically closed field of characteristic zero terminates. The result allows arbitrary negative extremal rays and mixed birational steps on the given models, without pseudo-effectivity or initial ℚ-factoriality.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M122 · Family 056 Finite ordinary minimal model programs on compact Kähler fourfolds
We prove that every maximal ordinary negative-ray program starting from a compact Kähler klt fourfold pair with effective rational boundary in the global Weil-divisor ℚ-factorial category terminates. It ends at a nef model when the adjoint is pseudo-effective and at a projective Mori fibre space otherwise.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M123 · Family 056 Finite ordinary minimal model programs on compact Kähler fourfolds
We prove that a globally Weil-ℚ-factorial compact Kähler klt fourfold pair with effective rational boundary and canonical rational line bundle admits a finite ordinary minimal model program starting on the given pair. It ends at a nef model when the adjoint class is pseudo-effective and at a Mori fibre space otherwise.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M124 · Family 057 The abelianity conjecture for special compact Kähler manifolds
We prove that the fundamental group of every special compact Kähler manifold is virtually abelian, resolving Campana's abelianity conjecture in all dimensions. In particular, every smooth compact connected Kähler manifold of Kodaira dimension zero has virtually abelian ordinary fundamental group.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M125 · Family 057 Two-step monodromy of special quasi-projective varieties
We give an independent proof that every complex linear representation of the ordinary fundamental group of a connected smooth special complex quasi-projective variety has virtually nilpotent image of class at most two. This conclusion was previously announced by Cao–Deng–Hacon–Păun. We also construct special open surfaces whose general quasi-Albanese fibres are not special.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M126 · Family 057 A conditional abelianity theorem for special fourfold pairs with a half-weight divisor
Using the abelianity theorem for special compact Kähler manifolds in every dimension, we prove virtual abelianity of the entire orbifold fundamental group of the special order-two root orbifold associated with a pair `(X, tfrac12D)`, where X is a smooth projective fourfold and D is a nonempty smooth connected divisor. We construct a special smooth projective eightfold whose fundamental group surjects onto the required group.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M127 · Family 058 Semialgebraic universal covers of normal projective varieties
We prove the Kollár–Pardon conjecture: the universal cover of a connected normal projective complex variety is biholomorphic to a semialgebraic open subset of a projective variety if and only if it is a product of a bounded symmetric domain, a complex affine space, and a simply connected normal projective variety.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M128 · Family 058 Symmetry of semialgebraic bounded domains with compact quotient
Every nonempty connected semialgebraic bounded open subset of a complex affine variety admitting a properly discontinuous cocompact group of biholomorphisms is smooth and biholomorphic to a bounded symmetric domain. This answers the bounded-domain question of Kollár and Pardon affirmatively.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M129 · Family 059 Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three
We give a negative answer to the embedded Zariski multiplicity conjecture. We construct two reduced hypersurface germs that are ambiently homeomorphic but have multiplicities two and three. Both have isolated singularities and lie in a common complex affine space of dimension divisible by eight. Their defining function germs are also topologically right equivalent, and the same examples answer Arnold's corank problem negatively for ambient topological equivalence.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M130 · Family 059 Ambiently homeomorphic isolated hypersurface germs in ℂ⁴ with multiplicities four and five
We give a negative answer to Zariski's multiplicity question in four complex variables. We construct two reduced convergent holomorphic function germs with isolated critical points and multiplicities four and five whose zero-set germs are ambiently homeomorphic.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M131 · Family 060 Global Spherical Shells on Minimal Surfaces of Class VII
We prove the Global Spherical Shell conjecture: every connected minimal compact complex surface of class VII with positive second Betti number contains a global spherical shell. This is a holomorphically embedded neighborhood of the standard three-sphere in ` C² 0 ` whose complement is connected.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M132 · Family 062 Contact Fano manifolds and the LeBrun–Salamon conjecture
We resolve the contact-Fano homogeneity conjecture and the Riemannian LeBrun–Salamon conjecture positively. Every smooth connected complex projective contact Fano manifold of complex dimension at least three, with its given contact distribution, is contact-isomorphic to the adjoint variety of a simple complex Lie algebra. Consequently, every closed connected smooth positive quaternionic-Kähler manifold of real dimension `4m≥8` is homothetic to a compact symmetric Wolf space.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M133 · Family 063 The generalized Mukai conjecture
We prove the generalized Mukai conjecture.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M134 · Family 064 Topological triviality of mu-constant families of surface singularities
We prove that every holomorphic one-parameter family of isolated hypersurface singularities in ℂ3 with constant Milnor number is topologically right-trivial. This gives a positive answer to the surface case of the μ-constant problem. The trivialization fixes the parameter and the origin section, and preserves the defining functions.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M135 · Family 065 Virasoro Constraints under Projectivization
We prove that full ordinary descendant Virasoro constraints pass from a smooth projective complex base to the projectivization of any algebraic vector bundle of rank at least two. The bundle need not split and satisfies no positivity requirement. Assuming the full constraints on the base, the conclusion includes every genus, each individual integral curve class, and all cohomology insertions, including primitive and odd classes. The result also applies successively to towers of projective bundles.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M136 · Family 065 Virasoro Constraints for Projective Complete Intersections
We prove the Virasoro conjecture for the ordinary descendant Gromov–Witten theory of smooth complete intersections in projective space, in every genus and curve class and with arbitrary cohomology insertions. This includes primitive and odd cohomology classes, with no semisimplicity assumption.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M137 · Family 066 Bounded klt complements for Fano contractions
For fixed dimension d and positive rational ϵ, we prove that every ϵ-log-canonical Fano contraction over an algebraically closed field of characteristic zero admits, near each closed base point, a klt complement of index bounded only by d and ϵ. Over ℂ, we also obtain monotone klt complements for Fano type pairs with nef anti-log-canonical divisor and coefficients in a fixed finite rational set. This proves the finite-rational-coefficient form of Shokurov's bounded-klt-complement conjecture.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M138 · Family 066 Uniform Cartier sections for Fano type contractions
We prove the Cartier-divisor conjecture of Birkar and Shokurov for rational boundaries in characteristic zero and for real boundaries over ℂ. For an ϵ-lc Fano type contraction with positive-dimensional base and nef negative log canonical class, a Cartier divisor through any prescribed base point can be chosen with pullback log canonical threshold bounded below in terms of the dimension and ϵ alone.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M139 · Family 067 The Campana–Peternell conjecture in dimension six
We prove that every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous, resolving the Campana–Peternell conjecture in complex dimension six. As a consequence, a connected compact Kähler manifold X with nef holomorphic tangent bundle and `dim( C)X- q(X)≤6` has ordinary universal cover `F× C q(X)`, where F is a rational homogeneous manifold and ` q(X)` is the maximal irregularity of a connected finite étale cover.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M140 · Family 068 Anticanonical nonvanishing from smooth semipositivity
Every smooth connected projective complex variety with smoothly semipositive anticanonical bundle has a nonzero section of some positive anticanonical power.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M141 · Family 068 Invariant anticanonical indices and conversion of twisted differentials
For a holomorphic action of a compact torus on a compact complex manifold, we prove that the invariant Euler characteristic of naturally linearized anticanonical powers is polynomial on a divisible progression, with its actual value at exponent zero. Independently, on a smooth projective variety with smoothly semipositive anticanonical bundle, we remove a fixed pseudoeffective error from an unbounded sequence of effective twists.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M142 · Family 068 Bounded anticanonical metrics on klt pairs and torus quotients
Let `(W,D)` be a projective ℚ-factorial klt pair with effective rational boundary. We prove that a positive Cartier multiple of `-(K_W+D)` has a nonzero section if this divisor is nef, its pullback to a resolution admits a semipositive metric with locally bounded weights, and the resolution has nonzero structure-sheaf Euler characteristic. Consequently a smooth rationally connected projective variety with smoothly semipositive anticanonical bundle has a nonzero naturally invariant anticanonical plurisection for every algebraic torus action.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M143 · Family 068 Cohomological transfer and equivariant anticanonical sections
Let X be a smooth projective complex variety with smoothly semipositive anticanonical bundle. For any torus linearization of that bundle, we show that invariant sections in unbounded degrees with one fixed negative pseudoeffective error yield an invariant section in a positive untwisted degree. Combining the natural invariant index with compact-monodromy structure, we deduce that every such X has a nonzero section of some positive anticanonical power.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M144 · Family 068 Metric descent and rank-preserving contractions
From a smooth semipositive anticanonical metric on a smooth projective complex variety, we construct an unweighted integrable semipositive metric on a corrected anticanonical line of a smooth base, using a normal equidimensional toroidal model and full generic adjoint rank one at exponent zero. The explicit rational boundary correction pulls back to an exceptional divisor.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M145 · Family 068 Integrable metrics and effectivity with controlled boundary
Let Y be smooth projective and let C be an effective integral divisor. If `-K_Y+C` admits a metric with a global strict curvature lower bound for which the canonical section of C is locally square integrable, every pseudoeffective rational divisor becomes rationally effective after an effective correction supported on C.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M146 · Family 068 Exact orders and invariant anticanonical linear systems
On a smooth connected projective complex variety with smoothly semipositive anticanonical bundle, we prove that asymptotically zero valuation on the curvature-null face is attained by an effective rational anticanonical divisor. The fixed auxiliary line bundle is arbitrary.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M147 · Family 069 Global quantum geometric Langlands at irrational level
We prove the unramified quantum geometric Langlands equivalence for every connected simple complex algebraic group G and every level `c C Q`, including non-real levels. For every smooth projective connected complex curve X, it identifies the full twisted D-module categories on ` Bun _G(X)` and ` Bun (G^ )(X)` at the dual shifted levels c and `-1/(rc)`, where r is the lacing number. The equivalence uses the given global forms and includes all connected components.
PQC lens (via Family 069): Directly shapes quantum threat models, algorithm timelines, or query-complexity assumptions used in security proofs.
View manuscript on GitHub → Open PDF (raw) →M148 · Family 071 Removable Boundaries and Rigidity of Circle Domains
We prove that a circle domain with conformally removable boundary is conformally rigid, with no restriction on the number of complementary components. This establishes the removability-to-rigidity direction of the He–Schramm Conjecture.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M149 · Family 071 Koebe's Circle-Domain Conjecture
We prove that every domain in the Riemann sphere is conformally equivalent to a circle domain, resolving Koebe's circle-domain conjecture positively.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M150 · Family 072 Brennan's conjecture and sharp inverse-square integral means
We prove Brennan's conjecture: if a simply connected plane domain admits a conformal bijection φ onto the unit disk, then `| '|s` is area-integrable for every `4/3 s 4`.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M151 · Family 072 A strict inverse-first-power bound for univalent functions
We prove a uniform upper bound for inverse-first-power integral means of normalized univalent disk maps with exponent strictly below 1/4.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M152 · Family 073 The Falconer distance conjecture in all dimensions
We resolve the Falconer distance conjecture in every dimension. For every integer d ≥ 2, a compact subset of ℝd with Hausdorff dimension greater than `d/2` determines a set of Euclidean distances of positive Lebesgue measure.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M153 · Family 074 The Kakeya maximal conjecture in three dimensions
We prove the Kakeya maximal conjecture in three dimensions. For every ε > 0, the maximal average over unit tubes of radius δ maps `L³( R³)` to `L³(S²)` with norm at most `C_εδ⁻ε`.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M154 · Family 074 Every four-dimensional Kakeya set has full Hausdorff dimension
We prove the four-dimensional Hausdorff-dimension Kakeya conjecture: every subset of ℝ4 containing a unit line segment in every direction has Hausdorff dimension four. No compactness or regularity assumption is imposed on the set or its witnessing line family.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M155 · Family 075 Almost-everywhere Fourier convergence in L log L
We prove that the ordinary symmetric Fourier partial sums of every complex-valued function in `Llog L` on the circle converge almost everywhere to the function along the full sequence. This establishes the classical `Llog L` sufficiency conjecture.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M156 · Family 076 Ultraflat real Littlewood polynomials
For every `ε (0,1)` and every sufficiently large integer N, there is a polynomial of length N with coefficients in ` -1,1 ` whose modulus lies between `(1-ε) N` and `(1+ε) N` everywhere on the unit circle. Thus real Littlewood polynomials can be ultraflat through every sufficiently large integer length. The signs may be chosen separately at each length.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M157 · Family 076 Nearly minimal maxima and positive minima of Littlewood polynomials
For every η > 0 and every sufficiently large integer N, there is a polynomial with N consecutive coefficients in ` -1,1 ` whose modulus lies between ` N/16` and `(1+ ) N` everywhere on the unit circle.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M158 · Family 076 Asymptotically minimal maxima of real Littlewood polynomials
We prove that the minimum possible maximum modulus on the unit circle of a polynomial with N consecutive real coefficients in ` -1,1 ` is `(1+o(1)) N`, as N tends to infinity through all integers. This disproves the real-sign analogue of Erdős's fixed relative-gap conjecture. As a consequence, the largest binary merit factor at length N tends to infinity through all integer lengths, disproving Turyn's conjecture.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M159 · Family 077 Elliptic capacity propagation and Fourier restriction to the sphere
We prove the bounded-data Fourier restriction conjecture for the sphere in three dimensions: the Fourier extension operator maps `L^∞(S²)` boundedly into `Lp( R³)` for every p > 3. This is the full conjectured open range, and the threshold p = 3 is sharp.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M160 · Family 077 Diagonal Fourier extension for positively curved surfaces in three dimensions
We prove the diagonal Fourier extension conjecture for compact smooth positively curved surfaces ` R³`, including surfaces with smooth boundary. For every `3 p ∞`, the extension operator maps `Lp( )` boundedly into `Lp( R³)`. The compact paraboloid case also yields free Schrödinger local smoothing in two spatial dimensions for every `3 p ∞` and Sobolev order `s 2-6/p`.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M161 · Family 078 Bochner–Riesz multipliers in three dimensions
We prove the three-dimensional Bochner–Riesz conjecture in its strict-order formulation. The main result is boundedness of the Bochner–Riesz multipliers on `L³( R³)` for every positive order. Interpolation and duality then give the full conjectured strict range.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M162 · Family 079 Critical local smoothing for the three-dimensional wave equation
We prove the critical L3 local smoothing estimate for the wave equation in three spatial dimensions, with every positive Sobolev loss. This resolves Sogge's Euclidean local smoothing conjecture in dimension three.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M163 · Family 080 Endpoint convergence for the planar Schrodinger equation
We resolve the planar Sobolev endpoint of Carleson's convergence problem: for initial data in `H¹/³( R²)`, the Schrödinger evolution converges almost everywhere to the initial data as `t downarrow0`. The evolution is defined by taking the Gaussian regularization limit first; on one full-measure spatial set, this limit exists for every `0 t 1`, and the resulting evolution is continuous at t = 0.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M164 · Family 080 Endpoint pointwise convergence for the Schrodinger equation in higher dimensions
We resolve the Sobolev endpoint of Carleson's pointwise convergence problem in every dimension n ≥ 3. For initial data in `Hn/(²(n⁺¹)( Rn)`, the free Schrödinger evolution converges almost everywhere to the initial data as `t downarrow0`. The evolution is defined by removing Gaussian regularization on one full-measure spatial set, uniformly over the interval `0 t 1`.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M165 · Family 081 Riesz transforms and uniform rectifiability in higher codimension
We prove that an n-Ahlfors–David regular Radon measure on ℝd is uniformly n-rectifiable if its n-dimensional Riesz transform is uniformly bounded from scalar `L²(μ)` to vector-valued `L²(μ)` over all positive hard truncations, for integers d ≥ 4 and `2≤ n≤ d-2`. This gives a positive answer to the David–Semmes Riesz-transform question in its remaining higher-codimension range. The conclusion gives uniform big pieces of Lipschitz images of Euclidean balls.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M166 · Family 082 Annular variation of the triangular Hilbert transform at the symmetric point
We prove the annular r-variation estimate for the triangular Hilbert transform from complex `L³× L³` to L3/2 for every r > 2. The partitions may depend on the output point and range over all positive scales. The estimate yields the two-endpoint maximal bound and joint almost-everywhere and norm principal values, and resolves the symmetric scalar triangular Hilbert transform problem.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M167 · Family 082 An L³ bound for the dyadic triangular Hilbert form
We prove a uniform `L³× L³× L³` estimate for the dyadic triangular Hilbert form with unrestricted real inputs. The sum of the absolute local contributions over any finite set of scales is bounded by forty times the product of the input norms. In particular, the bound allows coefficients of modulus at most one to vary independently among admissible interval triples.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M168 · Family 082 The maximal triangular Hilbert transform at the symmetric point
For arbitrary complex inputs in `L³( R²)`, we prove the pointwise maximal `L³× L³→ L³/²` estimate for the triangular Hilbert transform, with the supremum over both hard truncation endpoints. The estimate yields joint almost-everywhere and L3/2 convergence as the lower endpoint tends to zero and the upper endpoint tends to infinity. This also proves the conjectured scalar estimate at the symmetric point.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M169 · Family 083 A uniform Hilbert transform estimate for Lipschitz directions
We prove a uniform L2 bound for the Hilbert transform along a Lipschitz unit vector field in the plane, with integration restricted to a fixed absolute multiple of the reciprocal Lipschitz seminorm. The bound is uniform in the inner truncation and holds for fields depending on both coordinates. This gives an affirmative answer to Stein's weak- `(2,2)` conjecture.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M170 · Family 084 The geometric case of the Erdős similarity conjecture
We prove the geometric-progression case of the Erdős similarity conjecture. For every fixed `q (0,1)` and every ` (0,1)`, we construct a compact set `E(q, ) [0,1]` of measure greater than `1- ` containing no nontrivial affine copy of ` qn:n≥1 `, for any translation and either sign of nonzero dilation. The set may depend on q; the result makes no simultaneous assertion for different ratios.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M171 · Family 084 The dyadic case of the Erdős similarity conjecture
We construct a compact subset of the unit interval, of measure arbitrarily close to one, that contains no affine copy of the dyadic sequence ` 2⁻n:n≥1 `. The conclusion holds for every translation and every nonzero real dilation, of either sign, proving the dyadic case of the Erdős similarity conjecture.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M172 · Family 085 An Endpoint Gradient Bound for the Centered Disk Maximal Operator
We prove the endpoint gradient bound ` | Mf |_1≤ C | f |_1` for every real-valued `f W¹,¹( R²)`, where `Mf(x)` is the supremum of the averages of `|f|` over disks centered at x, and C is an absolute constant. The maximal function belongs to `W¹,¹(loc)( R²)` and has a globally integrable weak gradient. This gives a positive resolution of the planar centered-disk case of the endpoint question of Hajłasz and Onninen.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M173 · Family 086 An L3 bound for the trilinear Hilbert transform
We prove that the trilinear Hilbert transform with fixed slopes 1, 2, 3 maps `L³( R)× L³( R)× L³( R)` to `L¹( R)`, resolving this case of the trilinear Hilbert transform conjecture.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M174 · Family 087 The symmetric Mahler conjecture and its equality cases
We resolve the symmetric Mahler conjecture positively, including its equality classification. Every origin-symmetric convex body in ℝn has volume product at least `4n/n!`, with equality exactly for invertible linear images of Hanner polytopes.
PQC lens (via Family 087): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M175 · Family 087 The Mahler Conjecture for General Convex Bodies
We resolve the Mahler conjecture for general convex bodies positively. For every convex body `K Rn`, n ≥ 1, with Santaló point `s(K)`, `|K| |(K-s(K))^ |≥ (n+1)n⁺¹/(n!)²`, with equality exactly for simplices.
PQC lens (via Family 087): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M176 · Family 087 Symplectic Balls in Symmetric Polar Products
For every integer n ≥ 2 and every origin-symmetric convex body `K Rn`, we prove that the Gromov width of ` int K× int K^ ` is…
PQC lens (via Family 087): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M177 · Family 088 Petty’s projection-volume conjecture in dimensions at least four
We prove that ellipsoids uniquely minimize the volume of the projection body among convex bodies of fixed volume in every dimension at least four. This proves Petty's projection-volume conjecture in these dimensions.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M178 · Family 088 A product counterexample to the simplex maximum for projection-body volume
The product of two ten-dimensional simplices has larger normalized projection-body volume than a twenty-dimensional simplex. This gives a counterexample to Brannen's proposed simplex maximum.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M179 · Family 089 Planar Graph Metrics Embed into L1 with Constant Distortion
We prove that every finite connected planar graph with arbitrary positive real edge lengths embeds into real L1 with a universal distortion bound. This resolves the planar embedding conjecture positively.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M180 · Family 089 L1 Embeddings of Graphs of Bounded Treewidth
For every fixed treewidth bound, the shortest-path metrics of finite connected graphs with arbitrary positive real edge lengths embed into real L1 with uniformly bounded distortion. This resolves the bounded-treewidth case of the Gupta–Newman–Rabinovich–Sinclair conjecture positively.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M181 · Family 090 An atomic certificate for triangular-lattice universal optimality
We prove that the density-one triangular lattice minimizes the lower energy per particle for every nonnegative completely monotone function of squared distance, among all locally finite planar configurations of centered disk density one. The comparison includes infinite energies. The proof constructs sharp Gaussian Fourier minorants using an atomic interpolation certificate.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M182 · Family 090 Universal optimality of the triangular lattice
We prove universal energy minimality for the triangular lattice in the plane. Among locally finite configurations of centered density one, it minimizes the lower limit of centered-ball energy averages for every nonnegative completely monotone function of squared distance, including when the energy is infinite. We also prove triangular minimality for planar logarithmic and Riesz renormalized energies, with `0 s 2` in the Riesz case, and the corresponding jellium minima.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M183 · Family 090 A sharp Fourier certificate for planar circle packing
We resolve the planar Cohn–Elkies sharpness conjecture: the two-point Fourier bound attains the optimal circle-packing density `π/(2 sqrt3)`. We construct a radial Schwartz certificate and prove its global sign conditions using rigorous interval arithmetic and analytic estimates. The same certificate recovers the classical uniqueness of the triangular packing among periodic equality cases.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M184 · Family 090 Triangular minimality for planar Coulomb renormalized energy
We prove the Sandier–Serfaty conjecture: the triangular lattice of covolume one minimizes planar Coulomb renormalized energy over all admissible curl-free fields with a unit uniform background. Combined with Bétermin and Sandier's asymptotic formula, this also proves the Brauchart–Hardin–Saff conjecture for the linear term of optimal ordered-pair logarithmic energy on the unit two-sphere. The proof uses a direct Voronoi-cell comparison with rigorous interval arithmetic.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M185 · Family 091 The logarithmic Brunn–Minkowski conjecture
We prove the logarithmic Brunn–Minkowski conjecture for arbitrary origin-symmetric convex bodies in every dimension. The theorem also gives the symmetric Lp Brunn–Minkowski inequality for every `0 p 1`. Combined with Saroglou's transfer theorem and a support-subspace reduction, it yields the logarithmic inequality for every even log-concave Radon measure and the scalar-dilation `(B)` -conjecture.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M186 · Family 092 A single-lattice covering bound of order n log n
Every convex body in ℝn, n ≥ 2, admits a covering by translates along one full-rank lattice with density at most `Cnlog n`, for an absolute constant C. No symmetry or boundary regularity is assumed.
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View manuscript on GitHub → Open PDF (raw) →M187 · Family 092 Translative covering densities of order n log n
For every sufficiently large dimension n, we construct a centrally symmetric convex body whose translative covering density exceeds `c nlog n`, where c > 0 is absolute. This disproves the existence of a universal linear upper bound and matches the order of Rogers' upper bound.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M188 · Family 093 A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures
We prove that every centered log-concave probability measure with a Lebesgue density on ℝn and linear subgaussian parameter a satisfies ` Ent _μ(f²)≤ Ca²∫|Df|² dμ` for compactly supported smooth f, with one universal constant C. This resolves positively the dimension-free logarithmic Sobolev conjecture for subgaussian log-concave measures.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M189 · Family 094 Subpolynomial dimension reduction in Lp
For every fixed `1 p ∞` and D > 1, every n-point subset of a real Lp space embeds into `ℓ_pd` with distortion at most D and subpolynomial dimension `d=no(¹)`. The target has the same exponent p, and the embedding need not be linear.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M190 · Family 095 Hyperbolicity Cones Without Semidefinite Lifts
We prove that not every hyperbolicity cone is a spectrahedral shadow: some closed hyperbolicity cones admit no finite affine semidefinite lift, even with arbitrary real coefficients and any finite number of auxiliary variables. This disproves the Projected Lax Conjecture and hence the generalized Lax conjecture.
PQC lens (via Family 095): Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
View manuscript on GitHub → Open PDF (raw) →M191 · Family 095 A nonspectrahedral hyperbolicity cone
We construct a homogeneous polynomial of degree 16 in 23 real variables whose hyperbolicity cone has no representation by a finite homogeneous real symmetric linear matrix inequality. This disproves the geometric Generalized Lax conjecture.
PQC lens (via Family 095): Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
View manuscript on GitHub → Open PDF (raw) →M192 · Family 095 An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone
We construct an exact semidefinite lift of the explicit nonspectrahedral hyperbolicity cone in twenty-three variables defined in the companion paper. The lift is a homogeneous real symmetric pencil of size 100 with 307 auxiliary variables and represents the entire closed cone, including every point with singular X. Thus, although this cone has no semidefinite representation in its original coordinates, it admits one when auxiliary variables are allowed. The stated sizes are not claimed to be minimal.
PQC lens (via Family 095): Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
View manuscript on GitHub → Open PDF (raw) →M193 · Family 096 The Gaussian propeller bound in every dimension
We prove the Gaussian propeller conjecture: for every finite measurable partition of a Euclidean space, the sum of the squared lengths of its Gaussian first moments is at most `9/(8π)`. For dimension at least two and at least three cells, three planar sectors of angle `2π/3`, extended by an orthogonal Euclidean factor, attain the bound.
PQC lens (via Family 096): Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
View manuscript on GitHub → Open PDF (raw) →M194 · Family 097 The Euclidean Steinitz–Bergström theorem
Every prescribed-order finite sequence of vectors in the Euclidean unit ball of ℝd has one signing for which every signed prefix has norm at most `C d`, with C absolute and independent of the sequence length. Consequently, every indexed zero-sum family of unit-ball vectors admits an ordering with the same bound for its unsigned partial sums. This determines the Euclidean Steinitz constant up to absolute factors, `S_2(d)= ( d)`, and resolves the Euclidean Steinitz–Bergström conjecture.
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View manuscript on GitHub → Open PDF (raw) →M195 · Family 098 A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding
Every infinite-dimensional real Banach space contains a compact doubling subset that admits no bi-Lipschitz embedding into any finite-dimensional real normed space. The doubling constant has a universal bound, independent of the ambient Banach space. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space.
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View manuscript on GitHub → Open PDF (raw) →M196 · Family 099 Edit Distance in l1: Matching Bounds up to Constants in the Exponent
We determine the exponential scale of the least ℓ1 distortion of unit-cost edit distance on all strings of length at most d. For every sufficiently large d, uniformly over finite alphabets of size at least two, the distortion lies between `exp(c√(log d loglog d))` and `exp(C√(log d loglog d))` for absolute constants `c,C 0`. The lower bound already holds on binary strings of one common length. Thus the order of logarithmic distortion is sharp up to absolute constants.
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View manuscript on GitHub → Open PDF (raw) →M197 · Family 099 Finite-Circle Obstructions, Binary Codes, and Histogram Embeddings for Edit Distance
We give two finite-circle constructions of binary strings whose least ℓ1 distortion is `exp( (√(log d loglog d)))`, where d bounds their length. Both constructions supply words of one common length for every sufficiently large cap. Two direct binary coding arguments transfer the constructions with absolute distortion and logarithmic block width.
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View manuscript on GitHub → Open PDF (raw) →M198 · Family 099 Tree Constructions for the l1 Distortion of Binary Edit Distance
We give two independent constructions of binary words of one length at most d whose ordinary edit-distance metrics require ℓ1 distortion `exp( (√(log d loglog d)))` for every sufficiently large d. We also prove a constant-distortion binary conversion for one prescribed input length. Together with the companion upper embedding theorem, these lower bounds determine the order of logarithmic distortion uniformly over finite alphabets with at least two symbols.
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View manuscript on GitHub → Open PDF (raw) →M199 · Family 100 Finite angular cylinder covers below the half-area bound
A regular tetrahedron admits a finite cylinder covering with compact triangular perpendicular bases whose total area is less than half its minimum orthogonal projection area. This disproves the half-area cylinder-covering conjecture. By affine invariance, the same construction gives a counterexample to the directionwise normalized half-bound for every nondegenerate tetrahedron.
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View manuscript on GitHub → Open PDF (raw) →M200 · Family 100 Finite cylinder approximation of ruled sets
We approximate compact ruled families of segments by finitely many cylinders with square intercept tiles and perpendicular-base area at most their integral projection cost plus any positive error. The velocity field is C1, and its differential has opposite real eigenvalues whose magnitudes are strictly below the inverse segment half-length; both eigenvalues may vanish. The result includes square-zero differentials with unrestricted shear and fields that pass between the two regimes. It holds for arbitrary compact label sets.
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View manuscript on GitHub → Open PDF (raw) →M201 · Family 100 Slope-field perturbations of the two-cylinder covering
The two-cylinder covering of a regular tetrahedron can be perturbed to give finite covers with total perpendicular base area strictly below half its minimum projection area. These covers give negative answers to both the half-area question and the directionwise normalized half-bound conjecture. By affine invariance, the directionwise conclusion holds for every nondegenerate tetrahedron.
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View manuscript on GitHub → Open PDF (raw) →M202 · Family 100 Finite triangular approximation of radial sweeps
We prove that radially aligned segment sweeps admit finite cylinder covers with one triangular base for each interval of any tagged partition. As the mesh tends to zero, the total perpendicular base area converges to a weighted parameter area. An explicit application covers every regular tetrahedron with total base area below half its minimum projection area, giving negative answers to the half-area question and the directionwise normalized 1-Codimensional Cylinder Covering Conjecture.
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View manuscript on GitHub → Open PDF (raw) →M203 · Family 101 A sharp entropy bound and the simplex inequality for isotropic constants
We prove the strong isotropic constant conjecture: in each dimension, simplices are the unique maximizers of the isotropic constant among convex bodies. We also prove the sharp entropy bound `h(f)≥ m+ tfrac12logdet Cov (f)` for every log-concave probability density on ℝm, with equality precisely for invertible affine images of products of one-sided exponential laws.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M204 · Family 102 The Unique Games Theorem
We prove the Unique Games Conjecture. For every fixed `ε,δ (0,1/2)`, we give a deterministic polynomial-time reduction from 3SAT to Unique Games over a fixed finite alphabet, with completeness at least `1-ε` and soundness at most δ.
PQC lens (via Family 102): Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
View manuscript on GitHub → Open PDF (raw) →M205 · Family 102 A Direct Proof of Optimal Max-Cut Hardness
We prove that approximating Max-Cut on simple unweighted graphs within any fixed factor greater than the Goemans–Williamson constant is NP-hard.
PQC lens (via Family 102): Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
View manuscript on GitHub → Open PDF (raw) →M206 · Family 102 The Factor-Two Hardness Threshold for Vertex Cover
We prove that minimum Vertex Cover is NP-hard to approximate within every fixed factor below two, even on simple unweighted graphs.
PQC lens (via Family 102): Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
View manuscript on GitHub → Open PDF (raw) →M207 · Family 102 Constant-factor hardness of Min-UnCut
For every fixed C > 1, approximating Min-UnCut within factor C is NP-hard, even on simple undirected unweighted graphs.
PQC lens (via Family 102): Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
View manuscript on GitHub → Open PDF (raw) →M208 · Family 102 Constant-factor hardness of directed feedback vertex set
Approximating minimum directed feedback vertex set within any fixed constant factor is NP-hard, even on unweighted digraphs.
PQC lens (via Family 102): Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
View manuscript on GitHub → Open PDF (raw) →M209 · Family 103 Exact derandomization of logarithmic space: L = RL = BPL
We prove ` L= RL = BPL `, resolving the derandomization problem for polynomial-time randomized logarithmic space.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M210 · Family 104 Turn-Based Stochastic Mean-Payoff Games in Deterministic Quasipolynomial Time
We give a uniform deterministic quasipolynomial-time algorithm for finite turn-based stochastic mean-payoff games with signed integer rewards and rational chance-transition probabilities encoded in binary.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M211 · Family 104 Mean-payoff parity games in quasipolynomial time
We give a uniform deterministic quasipolynomial-time algorithm for mean-payoff parity games.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M212 · Family 104 Deterministic quasipolynomial-time mean-payoff games
We give a deterministic algorithm that computes the complete zero-threshold winning set of a finite mean-payoff game with arbitrary signed integer edge weights encoded in binary.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M213 · Family 104 Randomized quasipolynomial-time mean-payoff games
We give a randomized algorithm that computes the complete zero-threshold winning set of a finite mean-payoff game with arbitrary signed integer edge weights encoded in binary.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M214 · Family 105 Perfect completeness for 2-to-1 games
We prove the 2-to-1 Games Conjecture with perfect completeness. For every fixed rational `δ (0,1)`, it is NP-hard to distinguish satisfiable 2-to-1 games from games of value at most δ, with a fixed alphabet and an explicitly listed unweighted multiset of constraints.
PQC lens (via Family 105): Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
View manuscript on GitHub → Open PDF (raw) →M215 · Family 106 Hardness of finding large independent sets in three-colorable graphs
We prove that, for every fixed `0 δ 1/3`, it is NP-hard to distinguish three-colorable graphs from graphs in which every independent set has fewer than δ times the number of vertices. Consequently, for every fixed integer c ≥ 3, finding a proper c-coloring of a three-colorable graph is NP-hard.
PQC lens (via Family 106): Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
View manuscript on GitHub → Open PDF (raw) →M216 · Family 107 An Upper Bound of 9/4 for the Matrix Multiplication Exponent
We prove that the exponent of matrix multiplication over the complex numbers is at most 9/4.
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View manuscript on GitHub → Open PDF (raw) →M217 · Family 107 Complex Matrix Multiplication Below 2.258 and Rectangular Bounds
— secondary writeup Over every field of characteristic zero, we prove that the square matrix-multiplication exponent satisfies ω < 2.258, the dual exponent satisfies α > 0.465, and `ω(1,0.709,1) 2.092`. The strict square and k = 0.709 rectangular bounds also hold over every field except possibly in one finite set of positive characteristics, in the arithmetic-operation model.
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View manuscript on GitHub → Open PDF (raw) →M218 · Family 107 Staggered extraction for exact matrix multiplication over every field
We prove that the arithmetic exponent of square matrix multiplication over every fixed field satisfies ω < 2.371054886006746. This includes every positive characteristic.
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View manuscript on GitHub → Open PDF (raw) →M219 · Family 108 A cubic lower bound for border determinantal complexity of the permanent
We prove that the complex border determinantal complexity of the `m× m` permanent is ` (m³)`, allowing arbitrary affine-linear determinant representations and coefficientwise limits. It also gives cubic lower bounds for exact determinantal complexity and for the numbers of vertices and edges in affine-linear algebraic branching programs, including coefficientwise limits with a fixed vertex or edge budget.
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View manuscript on GitHub → Open PDF (raw) →M220 · Family 109 Integer multiplication below n log n
We give a deterministic algorithm that multiplies two n-bit integers in `O(n( n)¹⁻ )` worst-case time, with ` =2⁻¹⁸²`, on one fixed finite-alphabet Turing machine with a fixed finite number of one-dimensional tapes. The algorithm is exact for every input length and disproves the `nlog n` optimality conjecture of Schönhage and Strassen in this model.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M221 · Family 110 Squared-logarithmic randomized k-server on arbitrary metrics
We prove that randomized k-server has competitive ratio `O((log(k+1))²)` on every metric space against oblivious request sequences, matching the known worst-case lower bound. For each metric and initial configuration, one policy works for all finite request sequences, including on infinite and unbounded spaces. When the initial server positions are distinct, no additive term is needed.
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View manuscript on GitHub → Open PDF (raw) →M222 · Family 110 Uniform computation of the squared-logarithmic k-server bound
We construct a uniform randomized k-server algorithm on finite rational metrics with competitive ratio `O(log²(k⁺¹)` against oblivious request sequences. Preprocessing is polynomial in the input length, and per-request bit complexity is polynomial in that length and the binary request-counter length. The additive movement constant is finite and instance-dependent, but may be enormous. The construction uses the companion squared-logarithmic existence theorem.
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View manuscript on GitHub → Open PDF (raw) →M223 · Family 111 One Sample Suffices for Matroid Prophet Inequalities against an Almighty Adversary
We prove that one independent sample per element suffices for a constant-competitive prophet inequality on every finite matroid. The guarantee holds even when the arrival-order adversary observes all samples, all online values, and the algorithm's entire random seed. The rule needs no description of the value distributions and achieves the absolute competitive ratio `2⁻³¹⁰`.
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View manuscript on GitHub → Open PDF (raw) →M224 · Family 112 Beyond the Square-Root Exponent for Depth-Three Boolean Circuits
We construct a language in deterministic polynomial time whose n-bit membership function requires `2ω( n)` gates in an unbounded-fan-in OR–AND–OR circuit. The bound holds at every sufficiently large input length and counts all gates, including the bottom layer.
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View manuscript on GitHub → Open PDF (raw) →M225 · Family 113 A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs
We give a fully polynomial randomized approximation scheme (FPRAS) for counting perfect matchings in arbitrary finite simple undirected graphs, resolving the general-graph perfect-matching approximation problem. The algorithm returns zero with certainty when no perfect matching exists. Otherwise, it achieves relative error ε with failure probability at most δ in worst-case bit time polynomial in the input length, `ε ⁻¹`, and `logδ⁻¹`.
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View manuscript on GitHub → Open PDF (raw) →M226 · Family 113 Entropy and Face Dimension of the Perfect-Matching Polytope
We prove the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant. For every feasible vector x of perfect-matching edge marginals in a loopless labelled multigraph on `2m≥2` vertices, the maximum entropy `H(x)` of a matching law with marginals x satisfies ` F(x)-(2-2/m)B(x)≤ H(x)≤ F(x),` where `F(x)=-∑_e x_elog x_e` and `B(x)=-∑_e(1-x_e)log(1-x_e)`. This bound holds throughout the polytope, including its boundary.
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View manuscript on GitHub → Open PDF (raw) →M227 · Family 114 An FPRAS for Common Integer Polymatroid Bases with Binary Capacities
We give a fully polynomial randomized approximation scheme for counting common integer bases of two polymatroids with the same total rank, supplied by exact rank-value oracles. The total rank and capacities are encoded in binary, and each integer vector is counted once. On every execution, the number of oracle calls and the bit work outside the oracles are bounded by a fixed polynomial in the ground-set size, the binary input length, the inverse relative-error tolerance, and the logarithm of the inverse failure probability.
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View manuscript on GitHub → Open PDF (raw) →M228 · Family 114 Approximate counting of common bases of two matroids
We give a fully polynomial randomized approximation scheme for counting the common bases of two arbitrary matroids of the same rank, supplied by independence oracles. The algorithm requires no explicit representation of either matroid and has polynomial bounds on both oracle calls and bit operations on every execution.
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View manuscript on GitHub → Open PDF (raw) →M229 · Family 115 Exact Uniform Sampling of Contingency Tables with Arbitrary Margins
We give an exact uniform sampler for nonnegative integer contingency tables with arbitrary prescribed margins. It terminates almost surely and has expected bit complexity polynomial in both dimensions and the binary length of the margins. No positivity, balance, sparsity, or fixed-dimension assumption is required.
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View manuscript on GitHub → Open PDF (raw) →M230 · Family 115 An FPRAS for Cell-Bounded Contingency Tables
We give a fully polynomial randomized approximation scheme for counting nonnegative integer matrices with prescribed row sums, column sums, and individual entry bounds. Both dimensions vary, all numerical data are encoded in binary, and zero bounds are allowed. The algorithm uses only unbiased random bits and has a polynomial bound on its bit operations on every execution.
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View manuscript on GitHub → Open PDF (raw) →M231 · Family 116 Uniform Matrix Hitting Points in Every Positive Characteristic
We construct a single matrix substitution that detects every nonzero size-s division-free noncommutative formula in n variables over every field of a given positive characteristic. One deterministic machine, given a promised prime p in binary and n, s in unary, outputs matrices over 𝔽p of dimension `O(n³s⁶)` in polynomial bit time. The same tuple works with arbitrary extension-field coefficients, including in characteristic two.
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View manuscript on GitHub → Open PDF (raw) →M232 · Family 116 One Rational Matrix Hitting Point for Noncommutative Formulas
We construct, in deterministic polynomial bit time, one tuple of rational matrices that detects every nonzero polynomial computed by a noncommutative division-free formula of a prescribed size. The matrices have dimension `O(ns²)` for n variables and formula size s, and the same tuple works over every field of characteristic zero.
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View manuscript on GitHub → Open PDF (raw) →M233 · Family 116 Polynomial Hitting Lists for Noncommutative Rational Formulas
We construct, in deterministic polynomial bit time, a polynomial-size list of rational matrix tuples for noncommutative rational formulas over ℚ of bounded tree size. Every nonzero admissible formula has a defined, invertible value at one tuple, with no separate bounds on inverse nesting or rational constant heights. Matrix dimensions, entry bit lengths, and total output length are polynomially bounded.
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View manuscript on GitHub → Open PDF (raw) →M234 · Family 117 Constant-factor hardness of uniform sparsest cut
We prove that, for every fixed C > 1, approximating Uniform Sparsest Cut within factor C is NP-hard. The output graphs have nonnegative rational capacities and unit demand between every pair of distinct vertices.
PQC lens (via Family 117): Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
View manuscript on GitHub → Open PDF (raw) →M235 · Family 117 Near-square-root logarithmic integrality gaps for uniform sparsest cut
We construct uniform sparsest-cut instances whose Goemans–Linial semidefinite integrality gap is at least `c√(log n)/(loglog n)³` along a sequence `n→∞`. The demand is one between every pair of distinct vertices, and the capacities are nonnegative real numbers. This matches the Arora–Rao–Vazirani upper bound up to a power of `loglog n`.
PQC lens (via Family 117): Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
View manuscript on GitHub → Open PDF (raw) →M236 · Family 118 Additive hardness and unbounded configuration gaps in bin packing
We disprove the Modified Integer Round-Up Conjecture for bin packing by constructing instances with arbitrarily large additive gaps between the configuration-LP value and the integral optimum. We also prove that, for every fixed nonnegative integer c, distinguishing instances that fit in B bins from those requiring more than `B+c` bins is NP-hard. Both results hold with rational item sizes greater than 1/6, so each bin contains at most five items.
PQC lens (via Family 118): Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
View manuscript on GitHub → Open PDF (raw) →M237 · Family 119 Sharp binary-information contraction on the discrete cube
We prove sharp contraction of the information carried by a binary channel under independent symmetric noise on a uniform discrete cube. At fixed initial information, a noisy coordinate retains the most information. The Boolean specialization resolves the Courtade–Kumar conjecture and gives an output-entropy refinement. We also establish a stronger mean-dependent entropy-production bound.
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View manuscript on GitHub → Open PDF (raw) →M238 · Family 119 Hellinger contraction with arbitrary Boolean output bias
We prove the Hellinger conjecture for Boolean functions on the uniform discrete cube, with arbitrary output bias. For a Boolean function of mean m and every `ρ [-1,1]`, the loss `√(1-m²)⁻ E√(1-(T_ρ f)²)` is at most `1-√(1-ρ²)`, with equality for signed coordinates. The proof combines asymmetric dimension induction, a calibrated noise-semigroup energy estimate, and finite exact arithmetic certificates. The Hellinger inequality also yields the Courtade–Kumar information bound.
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View manuscript on GitHub → Open PDF (raw) →M239 · Family 120 Almost-Linear-Time Maximum-Cardinality Matching in General Graphs
We prove that maximum-cardinality matching in a simple undirected graph with n vertices and m edges can be found by one uniform randomized algorithm in `(n+m)¹⁺o(¹)` time. The bound holds on every computation path in a logarithmic-word model, and the algorithm returns an explicit maximum matching with probability at least 2/3. An explicit reduction gives the same time and probability guarantees for deciding whether a simple host graph has a spanning subgraph with prescribed valid vertex degrees, and for finding one when it exists.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M240 · Family 121 An Almost-Linear Approximation Scheme for Edit Distance
We give a uniform randomized approximation scheme for unit-cost edit distance. For every fixed rational `ε (0,1)`, it estimates the distance between arbitrary explicitly stored strings of total length N within a factor `1+ε` with probability at least 2/3, in worst-case expected time `N¹⁺o(¹)` on a logarithmic-word RAM. The algorithm supports polynomially bounded integer alphabets and returns zero deterministically on equal strings.
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View manuscript on GitHub → Open PDF (raw) →M241 · Family 122 Uniform quasipolynomial-time trace reconstruction
We give a uniform algorithm that reconstructs every binary string from independent deletion traces when its length and rational retention probability are known. For each fixed retention probability, both the number of traces and the bit complexity are quasipolynomial in the string length. More generally, we give an explicit sample bound uniform over all rational retention probabilities, with running time polynomial in the sample budget and the binary input length. If the deletion probability is at most `n⁻ε` for fixed ε >
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View manuscript on GitHub → Open PDF (raw) →M242 · Family 122 A latest-anchor induction with spectrally compact masks for worst-case trace reconstruction
We give an improved worst-case sample bound for reconstructing a string from independent deletion traces, with its length and retention probability known. For each fixed retention probability, the number of traces is quasipolynomial: the logarithm of the sample budget is `O((log n)³(¹⁺loglog(²n)⁶)`. If the deletion probability is at most `n⁻ε` for fixed ε > 0, polynomially many traces suffice. These bounds apply to binary strings and to strings of general symbols observed exactly.
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View manuscript on GitHub → Open PDF (raw) →M243 · Family 122 Quantitative lower bounds for trace reconstruction
Exact worst-case reconstruction of a binary word from independent deletion traces requires `n (loglog n)` samples for every fixed deletion probability `q (0,1)`, even with unrestricted computation and any fixed positive success probability. This gives a negative answer to the polynomial-sample question for binary trace reconstruction. More generally, when `q³log n→∞`, we prove a lower bound of `nclog(q^³log n)` samples for every fixed `0 c 1/(4log2)`. Here q is the known deletion probability, and all logarithms are natural.
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View manuscript on GitHub → Open PDF (raw) →M244 · Family 124 A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling
We give a uniform deterministic polynomial-time algorithm for scheduling unit-length jobs with arbitrary precedence constraints on three identical parallel machines. The algorithm constructs a schedule of minimum makespan and decides exactly whether all jobs can finish by a specified deadline. The proof reorganizes feasible schedules into intervals whose job sets have descriptions of bounded size. A dynamic program searches a family containing polynomially many such descriptions.
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View manuscript on GitHub → Open PDF (raw) →M245 · Family 125 Single-exponential recovery and bounded-price strictness for metric k-median
We give an exact-budget recovery algorithm for metric k-median with single-exponential dependence on the number of comparison clusters without accurate, distinct proxies in a supplied anchor solution. On positive integral metrics of polynomially bounded diameter, a sufficiently small total proxy error and logarithmically many such clusters yield a `(1+2/e+ε)` approximation in polynomial time with arbitrarily high success probability. We also prove bounded-price strictness for one compatible execution of the logarithmic-surplus construction.
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View manuscript on GitHub → Open PDF (raw) →M246 · Family 125 The approximation threshold for metric k-median
For every fixed ε > 0, we give a deterministic polynomial-time `(1+2/e+ε)` -approximation for finite rational metric k-median with specified candidate facilities, opening at most k facilities. Under `P≠ NP`, the infimum approximation factor in this model is therefore `1+2/e`.
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View manuscript on GitHub → Open PDF (raw) →M247 · Family 126 Exponential PSD rank of positively shifted matching matrices
For every fixed `0 ρ 1`, the matrix indexed by odd vertex sets U and perfect matchings M of Kn, with entries `|M capδ(U)|-1+ρ`, has real positive semidefinite rank `2 (n)` as even n tends to infinity. Here `δ(U)` is the edge cut of U. Consequently, every exact semidefinite lift of the perfect matching polytope has exponential size.
PQC lens (via Family 126): Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
View manuscript on GitHub → Open PDF (raw) →M248 · Family 127 Average sensitivity of polynomial threshold functions
For every n ≥ 1 and `1≤ d≤ n`, we prove that a polynomial threshold function of degree at most d on the uniform Boolean cube has average sensitivity at most `8d n`. This proves the asymptotic form of the Gotsman–Linial conjecture. The bound is uniform in both parameters and uses the convention ` sgn (0)=1`.
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View manuscript on GitHub → Open PDF (raw) →M249 · Family 128 A Polynomial-Time 2-Approximation for Shortest Common Superstring
We give a deterministic algorithm that, for every finite family of explicitly represented ordinary strings, outputs a common superstring of length at most twice the optimum in time polynomial in the total encoded input size, including symbol labels. The guarantee applies to the algorithm constructed here, not the classical maximum-overlap Greedy procedure.
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View manuscript on GitHub → Open PDF (raw) →M250 · Family 129 An exponential state lower bound for two-way nondeterministic complementation
We prove that two-way nondeterministic finite automata cannot be complemented with a polynomial number of states independent of the alphabet. For each n ≥ 4 we construct an n-state automaton over a finite alphabet whose complement requires at least ` tfrac12 2 (n⁻⁴)/¹²⁷ -1` states.
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View manuscript on GitHub → Open PDF (raw) →M251 · Family 129 An exponential two-way deterministic state lower bound for one-way liveness
One-way liveness on h points accepts a word of binary relations when their ordered product is nonempty. For every h ≥ 2, it has a nondeterministic automaton with `h+3` states and no left moves, whereas every equivalent s-state two-way deterministic automaton satisfies `4(s+2)²≥2 (h⁻²)/³¹ `. Partial transition rules, stay moves, and nonaccepting infinite computations are allowed.
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View manuscript on GitHub → Open PDF (raw) →M252 · Family 130 An explicit power saving for the exact discrete Fourier transform
We give a deterministic algorithm that computes the discrete Fourier transform at every length n in `O(n(log n)¹⁻¹⁰^ ⁻¹³ )` operations. The model uses exact complex arithmetic, unrestricted coefficients, specified Fourier roots, and unit-cost logarithmic-size indexing; scalar preparation and array organization are included.
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View manuscript on GitHub → Open PDF (raw) →M253 · Family 130 Finite tensor savings and exact Fourier circuits
We construct exact nonuniform Fourier circuits of size `o(nlog n)` along an unbounded sequence of lengths, counting every addition, subtraction, and scalar multiplication. This refutes the ` (nlog n)` lower bound in the unrestricted complex linear-circuit model. The construction uses a finite tensor saving: a tensor power of some invertible nonmonomial complex matrix can be computed with fewer matrix calls than the standard tensor-axis algorithm on the same coordinates, when invertible monomial maps are allowed freely between calls.
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View manuscript on GitHub → Open PDF (raw) →M254 · Family 131 Polynomial mixing of the switch chain for every graphical degree sequence
We prove the simple-undirected form of the Kannan–Tetali–Vempala conjecture: the switch chain on simple undirected graphs mixes in polynomial time for every graphical degree sequence. For a lazy chain that proposes switches uniformly on four vertices, the total-variation mixing time at distance 1/4 is at most `2n⁸`. We also give an exactly uniform sampler for every graphical labeled degree vector. It uses unbiased random bits, terminates almost surely, and has expected polynomial bit running time.
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View manuscript on GitHub → Open PDF (raw) →M255 · Family 132 A superquadratic separation between sensitivity and block sensitivity
We disprove the quadratic strengthening of the Sensitivity Conjecture by constructing nonconstant total Boolean functions whose block sensitivity grows faster than any constant multiple of sensitivity squared. In fact, for some fixed α > 2, our examples have unbounded block sensitivity and satisfy ` bs (f)≥ s(f)^α`.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M256 · Family 133 Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence
For k ≥ 4, we construct two uncolored graphs that are k-dimensional Weisfeiler–Leman equivalent exactly when a prescribed finite-domain choice system has no compatible choice. The system has `k+1` domains for joint refinement and k for separate-coordinate refinement. A successful choice is detected after two joint rounds or one separate round. Applied to sparse satisfiability, the reduction gives fixed-dimension `n (k)` time exclusions under positive-rate ETH, even for deciding equality of these early histograms.
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View manuscript on GitHub → Open PDF (raw) →M257 · Family 133 The complexity of identifying a graph by Weisfeiler–Leman refinement
We prove that deciding whether Weisfeiler–Leman refinement of an input dimension identifies a given graph is EXPTIME-complete. The input is a nonempty finite simple uncolored graph in adjacency-matrix form and a positive binary-encoded dimension. Identification quantifies over every comparison graph.
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View manuscript on GitHub → Open PDF (raw) →M258 · Family 133 Unconditional time lower bounds for Weisfeiler–Leman equivalence
For every sufficiently large fixed k, deciding whether two n-vertex graphs are k-Weisfeiler–Leman equivalent requires `n (k)` deterministic sequential time in the worst case. The bound holds at every sufficiently large graph order, even for simple connected uncolored graphs of diameter at most two, without a complexity assumption. Inputs are explicit adjacency matrices; the models are multitape Turing machines and sequential logarithmic-word RAMs with fixed polynomial-bit-time instructions.
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View manuscript on GitHub → Open PDF (raw) →M259 · Family 133 Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs
Deciding joint-update k-dimensional Weisfeiler–Leman equivalence is ` EXPTIME ` -complete when the two graphs are given by explicit adjacency matrices and k ≥ 2 is encoded in binary. The result holds even for connected simple uncolored graphs of equal positive order and maximum degree at most three.
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View manuscript on GitHub → Open PDF (raw) →M260 · Family 134 Finite Monoid Computations and a Uniform Generalized Star-Height Bound
Every regular language over a finite alphabet has a generalized regular expression of star height at most thirteen over that same alphabet. We prove this uniform bound by representing finite monoid computations as affine updates and recovering them through twelve successive split constructions.
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View manuscript on GitHub → Open PDF (raw) →M261 · Family 134 Generalized Star Height at Most Four
Every regular language over a finite alphabet has generalized star height at most four over that same alphabet. We give a complete construction using an affine correction that hides one interval product, a finite clock, and several scales for moving boundaries through periodic words.
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View manuscript on GitHub → Open PDF (raw) →M262 · Family 134 Generalized Star Height at Most Three
Every regular language over a finite alphabet has generalized star height at most three, with complement taken in the same free monoid. We express finite-monoid computations using a prefix code of word pieces.
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View manuscript on GitHub → Open PDF (raw) →M263 · Family 135 Homogeneous depth-five lower bounds for iterated matrix multiplication
Let ` IMM (n,n)` be the `(1,1)` entry of a product of n independent `n× n` matrices of variables. Over every field of characteristic zero, every syntactically homogeneous ` ` circuit computing ` IMM (n,n)` has at least `n n/⁴⁰⁰` gates for all sufficiently large n, with an absolute threshold independent of the field. Bottom linear forms may have arbitrary support, and arbitrary finite fan-in, fan-out, and gate sharing are allowed. Over every field, a block expansion gives such circuits with at most `n n⁺⁴` gates for n ≥ 2.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M264 · Family 136 The PCP-for-PPAD conjecture: a quasilinear reduction
We prove the quasilinear-size PCP-for-PPAD conjecture of Babichenko, Papadimitriou, and Rubinstein. There are fixed positive rational constants ε and δ and a deterministic polynomial-time reduction that transforms an End-of-Line instance of binary length N into a generalized circuit of total binary length `N(log N)O(¹)`.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M265 · Family 137 Simulating One-Tape Time in Two-Fifths-Power Space
We show that a fixed deterministic Turing machine with one writable tape and head can be simulated in `O(T²/⁵ polylog (T+2))` work-space bits when a binary time cap T ≥ 2 is supplied. The simulator computes the finite-control and halting outcome by time T; its running time is unrestricted. The result allows a fixed number of read-only input heads and requires a fixed accessor that supplies every initial writable and read-only symbol within distance T of the relevant head origin in polylogarithmic space.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M266 · Family 138 Subset Sum in Time `O(2⁰.⁴⁹n)`
We give a uniform randomized classical algorithm for Subset Sum with bounded error and worst-case running time `O(2⁰.⁴⁹n)` on polynomial-bit inputs in a word-RAM model, where n is the number of input integers. The time bound holds on every random execution.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M267 · Family 138 A Low-Space Algorithm for Worst-Case Subset Sum
We give a uniform classical randomized decision algorithm for worst-case Subset Sum. Under every fixed polynomial bound on input-integer bit length, it uses ` poly (n)2n/²` time and ordinary `O(2n/⁵)` writable words of `O(n+b)` bits, where b is the largest input bit length. Both resource bounds hold on every execution. The error is one-sided: the algorithm always rejects unsolvable instances and accepts each solvable instance with probability at least 2/3.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M268 · Family 139 Subpolynomial query complexity for well-conditioned log-concave sampling
For every fixed ε > 0, we give a sampling algorithm using at most `C_ε d^ε` exact first-order queries on every execution for C2 potentials on ℝd with a known minimizer and Hessian between Id and `2I_d`. The output has total-variation distance at most 1/10 from the target Gibbs law. Computation between queries is unrestricted. We also prove an ` (log d)` query lower bound for arbitrary randomized adaptive algorithms, determining the optimal dimension exponent to be zero.
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View manuscript on GitHub → Open PDF (raw) →M269 · Family 140 Memory and precision in noiseless Gaussian regression
For every fixed A > 0, a learner that retains at most `Ad²` bits between fresh exact Gaussian linear measurements needs ` Omega_A(dlog(1/ε))` measurements to estimate a uniformly random unit vector to angular error at most ϵ, for any `0 ε≤ 1/10`, with probability at least 2/3. The constant is absolute for `o(d²)` memory.
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View manuscript on GitHub → Open PDF (raw) →M270 · Family 140 Posterior replicas and conditional information in Gaussian regression
For a signal with density bounded by L relative to uniform probability on `Sd⁻¹`, we bound the information in a finite message W formed from exact Gaussian measurements, conditional on an independent projection revealed only to the analyst. For explicit row counts proportional to d, the bound is `O(H(W)/d+d+log(2+log L))`.
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View manuscript on GitHub → Open PDF (raw) →M271 · Family 140 Localization costs and information growth for exact Gaussian observations
For the image of a uniform cube under a spherical coordinate map, we prove that finite messages from t blocks of ` (d)` exact Gaussian measurements reveal only `O_A(dt)` information when each message has at most `exp(Ad²)` values, for fixed A. The same bound holds when each message is supplemented with a nested cell that restores the required geometric spread.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M272 · Family 140 Projection moments, positive cap domination, and Riesz estimates on the sphere
We prove moment estimates for exact random projections of finite measures whose mass is controlled on Euclidean balls, and derive positive domination by countable sums of spherical cap measures. For learners with `M=o(d²)` bits of memory, these estimates give three proofs that uniform-sphere average success at least 2/3 at angular accuracy `0 ε≤1/10` requires ` (dlog(1/ε))` noiseless Gaussian observations. The three proofs keep their different stopping and accuracy costs explicit.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M273 · Family 140 Replacing Gaussian observations in memory-constrained inference
Replacing the Gaussian rows used to select a finite message by independent rows increases the remaining conditional information by at most `Cd`, for a uniform spherical signal, message entropy at most d2, and the specified row dimensions proportional to d. As an application, we prove that learners with `M=o(d²)` persistent bits need `T= (dlog(1/ε))` exact observations to attain uniform-sphere angular success at least 3/5, for `0 ε≤1/10` and a deterministic finite horizon.
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View manuscript on GitHub → Open PDF (raw) →M274 · Family 140 Subsphere methods for memory-sample lower bounds in noiseless Gaussian regression
Let a finite-state streaming learner estimate a uniformly random unit vector from independent exact Gaussian linear measurements. We prove that `o(d²)` bits of persistent memory and angular success probability at least 2/3 require at least `2⁻¹⁶dlog_2(1/ε)` samples for all sufficiently large d, uniformly for `0 ε≤1/10`. The proof conditions each batch on its observed projection and controls the resulting random residual subsphere.
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View manuscript on GitHub → Open PDF (raw) →M275 · Family 141 Existential–universal real sentences in the counting hierarchy
We prove that the existential theory of the reals lies in the counting hierarchy. More generally, we show that the truth of existential–universal sentences over the reals can be decided in a fixed level of the counting hierarchy, even when the integer polynomials are given by arithmetic circuits.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M276 · Family 142 Deterministic Polynomial Factorization over Prime Fields
We give a uniform deterministic polynomial-time algorithm for complete factorization over prime fields. For a prime p in binary and a nonzero polynomial `f F_p[x]` given by its dense coefficient list, the algorithm computes the irreducible factors and their multiplicities using a number of bit operations polynomial in `(deg f+1)log p`. The proof uses the uniform Hecke zero-free theorem from the companion paper *Primitive roots for every admissible integer base*.
PQC lens (via Family 142): Affects polynomial or integer factorization—core to RSA/DSA analysis and to comparing classical vs quantum factoring.
View manuscript on GitHub → Open PDF (raw) →M277 · Family 143 Uniform bounds for planar polynomial limit cycles
For every degree, we prove that the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on that degree. This establishes the uniform boundedness assertion in the second part of Hilbert's sixteenth problem. The proof uses separation of asymptotic expansions on nested complex domains and a finite-dimensional counting argument.
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View manuscript on GitHub → Open PDF (raw) →M278 · Family 143 Two limit cycles for quintic Liénard systems
Every classical Liénard system ` x=y-F(x)`, ` y=-x`, with F an arbitrary real polynomial of degree at most five, has at most two geometrically distinct isolated periodic orbits, and the bound is attained. This proves the degree-five case of the Lins Neto–de Melo–Pugh conjecture.
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View manuscript on GitHub → Open PDF (raw) →M279 · Family 144 A smooth three-torus diffeomorphism with simple Lebesgue spectrum
We solve the probability-preserving form of Banach's simple Lebesgue-spectrum problem in smooth dynamics. We construct a C∞ diffeomorphism of the three-torus that preserves standard volume and whose Koopman operator has simple Lebesgue spectrum on the entire mean-zero L2 space.
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View manuscript on GitHub → Open PDF (raw) →M280 · Family 145 Rokhlin's multiple-mixing problem for one transformation
Every invertible mixing probability-preserving transformation is mixing of every finite order. Thus Rokhlin's multiple-mixing problem for a single transformation has an affirmative answer.
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View manuscript on GitHub → Open PDF (raw) →M281 · Family 146 Positive Metric Entropy for the Standard Map at Large Parameters
We prove that the standard sine map of the two-dimensional torus has positive metric entropy with respect to normalized area for every sufficiently large positive parameter. This gives a full parameter tail, and hence answers Sinai's positive-parameter-measure conjecture affirmatively.
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View manuscript on GitHub → Open PDF (raw) →M282 · Family 147 Continuous Phase Foliations Create Analytic Caustic Collars
For a smooth strictly convex planar billiard of positive curvature, a continuous foliation of a full grazing annulus by individually invariant essential curves forces an analytic boundary and a jointly analytic collar of smooth strictly convex caustics. The physical-collar rigidity theorem proved in the companion article then implies that the table is an ellipse.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M283 · Family 147 Rigidity of Smooth Billiards with a Continuous Caustic Collar
We prove that a smooth strictly convex planar billiard with positive curvature is an ellipse whenever a full neighborhood of its boundary is continuously foliated by smooth closed convex caustics. No differentiability across the leaves is assumed. This resolves the near-boundary Birkhoff conjecture in the stated smooth class.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M284 · Family 148 The entropy-rate dimension formula for self-similar measures on the line
We prove that the Hausdorff dimension of every finite real self-similar measure equals the minimum of one and its random-walk entropy rate divided by its Lyapunov exponent. Exact overlaps are allowed, and the contraction ratios may be unequal and negative. This resolves the entropy-rate dimension conjecture.
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View manuscript on GitHub → Open PDF (raw) →M285 · Family 149 Uniform Permanence in Weakly Reversible Mass-Action Systems
We prove the permanence conjecture for finite weakly reversible mass-action systems with fixed positive reaction rates. Every positive stoichiometric compatibility class admits one compact convex forward-invariant set that every positive trajectory in that class enters in finite time, even when the class is unbounded. The set depends only on the network, the rates, and the class; the entry time may depend on the initial state. Thus, after its entry time, every concentration satisfies the same positive lower and finite upper bounds throughout that class.
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View manuscript on GitHub → Open PDF (raw) →M286 · Family 149 Boundedness and persistence of weakly reversible mass-action systems
We prove the boundedness and persistence conjectures for finite weakly reversible mass-action systems with positive constant reaction rates. For every positive initial condition, the solution exists for all forward time, and every concentration remains bounded above and bounded away from zero. The bounds may depend on the initial condition, and no boundedness assumption is imposed on its stoichiometric compatibility class.
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View manuscript on GitHub → Open PDF (raw) →M287 · Family 150 Weak mixing of triangular billiards with an irrational angle
We prove that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized Liouville measure. Equivalently, the product of the flow with itself is ergodic.
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View manuscript on GitHub → Open PDF (raw) →M288 · Family 150 Ergodicity of triangular billiards with an irrational angle
We prove that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is ergodic for normalized area times uniform angular measure. No genericity or Diophantine condition is required. The flow is considered outside the null set of trajectories that hit a vertex.
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View manuscript on GitHub → Open PDF (raw) →M289 · Family 151 A C^1 Counterexample to the Entropy Conjecture
We disprove the general C1 self-map formulation of Shub's entropy conjecture. We construct a noninvertible C1 self-map of a compact smooth manifold without boundary whose topological entropy is zero, while its action on second real homology has eigenvalue 2. Thus the topological entropy is strictly smaller than the logarithm of the homological spectral radius.
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View manuscript on GitHub → Open PDF (raw) →M290 · Family 152 A zero-entropy system without a smooth positive-volume model
We construct an ergodic invertible transformation of a standard nonatomic probability space with zero Kolmogorov–Sinai entropy that has no smooth positive-volume model. More precisely, it is not measurably conjugate to any C∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. A single example excludes every finite dimension, including models on nonorientable manifolds and manifolds with smooth boundary.
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View manuscript on GitHub → Open PDF (raw) →M291 · Family 153 Arithmetic classification and non-Pisot singularity for Bernoulli convolutions
We give an arithmetic classification of singular and absolutely continuous unbiased Bernoulli convolutions for every parameter `λ (0,1)`. The criterion is expressed through one-sided approximation by explicitly defined finite sets of algebraic units; it is an infinite approximation condition, not a finite membership algorithm.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M292 · Family 154 Pointwise Multiple Ergodic Averages for Mixing Transformations
Let T be an invertible mixing probability-preserving transformation. For every integer n ≥ 2 and every fixed tuple of bounded measurable functions, we prove that the consecutive multiple ergodic averages of length n converge almost everywhere to the product of the integrals, as the averaging length tends to infinity through all positive integers. The probability space need not be standard, and no rate of mixing is required.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M293 · Family 154 Pointwise convergence of fourfold ergodic averages for mixing transformations
We prove that fourfold ergodic averages along the times `n,2n,3n,4n` converge almost everywhere to the product of the integrals for every invertible, bimeasurable, mixing probability-preserving transformation and every fixed choice of bounded measurable inputs. Convergence holds along all positive integer averaging lengths. No quantitative mixing rate is assumed, and the probability space need not be standard.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M294 · Family 154 Triple ergodic averages with distinct integer slopes
For every invertible mixing probability-preserving transformation, triple ergodic averages of bounded measurable functions along any three pairwise distinct nonzero integer slopes converge almost everywhere to the product of their integrals. The slopes may be positive or negative.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M295 · Family 154 Pointwise convergence of triple ergodic averages for mixing transformations
We prove pointwise convergence of triple ergodic averages for every invertible mixing probability preserving transformation T of an arbitrary probability space `(X, F,μ)`, with measurable inverse. For every triple of bounded measurable functions `f_1,f_2,f_3:X→ C`, ` frac1N∑(n=1)N f_1(Tnx)f_2(T²nx)f_3(T³nx) ∏(j=1)³∫_X f_j dμ` for μ-almost every x as `N→∞` through all positive integers.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M296 · Family 155 A translational tile with no fully periodic tiling in dimension three
We construct a finite translational tile in ℤ3 that admits tilings but no fully periodic tiling. Its unit-cube thickening has the same property in ℝ3, even when arbitrary real translations are allowed. This gives a negative resolution of the periodic tiling conjecture in dimension three.
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View manuscript on GitHub → Open PDF (raw) →M297 · Family 156 A nine-dimensional counterexample to Borsuk's covering assertion
The compact set of rank-one orthogonal projectors on ℝ4, with the Frobenius metric, cannot be covered by ten sets of strictly smaller diameter. It therefore gives a counterexample to Borsuk's conjecture in dimension nine.
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View manuscript on GitHub → Open PDF (raw) →M298 · Family 157 A counterexample to Hadwiger's conjecture
We disprove Hadwiger's conjecture by constructing arbitrarily large graphs whose chromatic number exceeds their Hadwiger number. The examples have independence number at most two, and even their ordinary fractional chromatic number exceeds their Hadwiger number. Thus they also disprove the fractional-coloring weakening discussed by Reed and Seymour.
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View manuscript on GitHub → Open PDF (raw) →M299 · Family 157 A counterexample to the Colin de Verdière chromatic conjecture
We disprove the Colin de Verdière chromatic conjecture by constructing graphs whose chromatic number exceeds their Colin de Verdière invariant by more than one. The examples have independence number at most two. In fact, their ordinary fractional chromatic number also exceeds their Colin de Verdière invariant by more than one.
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View manuscript on GitHub → Open PDF (raw) →M300 · Family 157 A linear list-coloring bound in terms of the Hadwiger number
We prove that every finite nonempty graph G satisfies ` (list)(G)≤ C h(G)` for an absolute integer C, where `h(G)` is the largest order of a clique minor. This resolves the Linear List Hadwiger conjecture affirmatively.
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View manuscript on GitHub → Open PDF (raw) →M301 · Family 158 The Euclidean plane is not five-colorable
We prove that every coloring of the Euclidean plane with five colors has a monochromatic unit-distance pair, with no regularity assumption on the color classes. Consequently, the chromatic number of the plane is either six or seven.
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View manuscript on GitHub → Open PDF (raw) →M302 · Family 159 Quasipolynomial Bounds for Arithmetic Progressions
We prove Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length.
PQC lens (via Family 159): Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
View manuscript on GitHub → Open PDF (raw) →M303 · Family 160 Quantitative Superexponential Bounds for van der Waerden Numbers
We prove that there are absolute constants c > 0 and K0 such that `W_r(k) kck lfloorlog_² r ` for every `k≥ K_0` and r ≥ 2. Consequently `W_r(k)¹/k→∞` for each fixed r ≥ 2, giving a quantitative positive resolution of Erdős's superexponential-growth question, including the two-color case.
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View manuscript on GitHub → Open PDF (raw) →M304 · Family 161 A counterexample to Sidorenko's conjecture
We disprove Sidorenko's conjecture with a bipartite graph on 35 vertices and 66 edges: its homomorphism density in some finite simple graph is smaller than the conjectured lower bound. The same connected graph also disproves the forcing conjecture: at one fixed density, asymptotically matching the edge and pattern densities does not imply quasirandomness.
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View manuscript on GitHub → Open PDF (raw) →M305 · Family 162 Balanced counterexamples to Ryser's conjecture at prime orders
For every sufficiently large prime q, we construct a finite intersecting `(q+1)` -partite `(q+1)` -uniform hypergraph with covering number `q+1` and exactly `q+1` nonisolated vertices in each part. This disproves Ryser's covering conjecture, even for intersecting hypergraphs with equally sized parts.
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View manuscript on GitHub → Open PDF (raw) →M306 · Family 162 A counterexample to Ryser's covering conjecture
For every sufficiently large prime `s≡2 pmod3` and every sufficiently large odd integer n, with the threshold depending on s, we construct an intersecting `(sn⁺¹)` -partite `(sn⁺¹)` -uniform hypergraph with covering number `sn⁺¹`. This disproves Ryser's covering conjecture in its intersecting case.
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View manuscript on GitHub → Open PDF (raw) →M307 · Family 164 Monochromatic finite sums and products in the positive integers
We prove Hindman's finite sums and products conjecture: for every finite coloring of the positive integers and every positive integer k, there is a k-element set whose nonempty subset sums and nonempty subset products all have the same color.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M308 · Family 165 The crossing number of complete graphs
We prove the Harary–Hill conjecture: for every positive integer n, the ordinary crossing number of the complete graph Kn is ` frac14≤ft n2 ≤ft n-1 2 ≤ft n-2 2 ≤ft n-3 2.`
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View manuscript on GitHub → Open PDF (raw) →M309 · Family 165 The crossing number of complete bipartite graphs
We prove the Zarankiewicz crossing-number conjecture, resolving Turán's brickyard problem. For all positive integers m, n, the ordinary crossing number of the complete bipartite graph `K(m,n)` is ` ≤ft m2 ≤ft m-1 2 ≤ft n2 ≤ft n-1 2.`
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View manuscript on GitHub → Open PDF (raw) →M310 · Family 166 The higher-dimensional Erdős distinct-distances conjecture
For every fixed integer d ≥ 3, we prove that every set of n ≥ 2 distinct points in ℝd determines at least `c_d n²/d` distinct distances, where `c_d 0` depends only on d. This resolves the higher-dimensional Erdős distinct-distances conjecture positively.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M311 · Family 167 The weak pinned planar distance theorem
We prove the weak pinned Erdős distinct-distance conjecture. For every fixed ε > 0, all but `o(n)` points of any n-point planar set determine at least `n¹⁻ε` distinct nonzero distances.
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View manuscript on GitHub → Open PDF (raw) →M312 · Family 167 A power saving for planar unit distances
We prove a power saving for the planar unit-distance problem: for some absolute `β 4/3`, every set of n points in the Euclidean plane determines `O(n^β)` unordered pairs at unit distance.
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View manuscript on GitHub → Open PDF (raw) →M313 · Family 168 Combinatorial invariance of Kazhdan–Lusztig polynomials
We prove that an isomorphism of Bruhat intervals in arbitrary Coxeter systems preserves their equal-parameter Kazhdan–Lusztig polynomials. This resolves the full combinatorial invariance conjecture positively.
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View manuscript on GitHub → Open PDF (raw) →M314 · Family 169 Elementary positivity of chromatic quasisymmetric functions
We prove that the chromatic quasisymmetric function of every natural unit interval graph is elementary-positive over ` N[q]`. This resolves the elementary-positivity part of the Shareshian–Wachs conjecture.
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View manuscript on GitHub → Open PDF (raw) →M315 · Family 170 The sharp logarithmic exponent of r(5,t)
We determine the sharp logarithmic exponent of the off-diagonal Ramsey number `r(5,t)`: ` r(5,t)= t⁴ (log t)³⁺o(¹) (t ∞).`
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View manuscript on GitHub → Open PDF (raw) →M316 · Family 170 Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers
For every fixed integer s ≥ 6, we determine the sharp logarithmic exponent of the off-diagonal Ramsey number: ` r(s,t)= ts⁻¹ (log t)s⁻²⁺o(¹) (t ∞).`
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View manuscript on GitHub → Open PDF (raw) →M317 · Family 171 The hypercube Ramsey number has linear order
We prove that the two-color Ramsey number of the n-dimensional binary cube is at most `C2n`, where C is an absolute constant. This resolves positively the hypercube Ramsey conjecture of Burr and Erdős.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M318 · Family 172 A classification of finite Euclidean Ramsey configurations
We classify finite Euclidean Ramsey configurations by a necessary and sufficient tensor condition over their coordinate fields. The Ramsey property here concerns monochromatic congruent copies at the original scale under arbitrary finite colorings. The criterion shows that every nonempty subtransitive set and every nonempty set of at most five points on a circle is Ramsey. In particular, some Ramsey cyclic quadrilaterals are not subtransitive, disproving the necessity direction of the Leader–Russell–Walters conjectured characterization.
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View manuscript on GitHub → Open PDF (raw) →M319 · Family 173 A proof of Seymour’s second-neighborhood conjecture
We prove that every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance two as at directed distance one. This resolves Seymour's second neighborhood conjecture positively.
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View manuscript on GitHub → Open PDF (raw) →M320 · Family 174 The strong thin tree conjecture
We prove that every finite loopless k-edge-connected multigraph on at least two vertices, with k ≥ 1, has a spanning tree meeting each cut in at most `C/k` times the size of the cut, where C is a universal constant. This resolves the strong thin tree conjecture.
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View manuscript on GitHub → Open PDF (raw) →M321 · Family 174 A polynomial-time construction of strong thin trees
We give a deterministic polynomial-time construction of strong thin trees. Given a finite k-edge-connected loopless multigraph on at least one vertex, the algorithm constructs a spanning tree meeting every cut in at most a `C/k` fraction of its edges, for a universal constant C. The running time is polynomial in the binary input length, including when parallel-edge multiplicities are encoded in binary.
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View manuscript on GitHub → Open PDF (raw) →M322 · Family 175 Graph Decompositions at the Integral Expectation Threshold
We prove the graph-decomposition conjecture of Ascoli, He, Park, and Talagrand. Every graph admits a partition into a universally bounded number of fixed edge pieces, each having ordinary containment threshold at most a universal constant times the original graph's integral expectation threshold. The partition is chosen before sampling the random host, and the separate embeddings of the pieces need not agree on shared vertices.
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View manuscript on GitHub → Open PDF (raw) →M323 · Family 175 Integral and fractional expectation thresholds are equivalent
We prove Talagrand's conjecture that integral and fractional expectation thresholds are within a universal constant factor, with the same covering budget.
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View manuscript on GitHub → Open PDF (raw) →M324 · Family 175 Talagrand’s discrete-convexity conjecture
We prove Talagrand's discrete-convexity conjecture. There is a universal integer k such that, whenever an arbitrary family has Bernoulli product measure at least `1-1/k`, the sets not contained in a union of k members admit a cover of total cost at most 1/2 at the same density.
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View manuscript on GitHub → Open PDF (raw) →M325 · Family 176 The second Kahn–Kalai conjecture
We prove the second Kahn–Kalai conjecture. For every finite simple graph H with h ≥ 1 edges and at most n vertices, the threshold for `G(n,p)` to contain an ordinary copy of H is at most `C p( E)(n,H)(1+log_2 h)`, where C is universal. Here `p( E)(n,H)` is the least density at which every subgraph of H has expected copy count at least one half.
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View manuscript on GitHub → Open PDF (raw) →M326 · Family 177 Bounded-degree coboundary expanders in every dimension
For every integer d ≥ 3, we construct arbitrarily large finite d-dimensional simplicial complexes with uniformly bounded vertex degrees and uniform 𝔽2 coboundary expansion in every degree below d. Together with the known graph and two-dimensional cases, this establishes the existence of bounded-degree 𝔽2 coboundary expanders in every positive dimension.
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View manuscript on GitHub → Open PDF (raw) →M327 · Family 178 Deterministic nonbipartite Ramanujan graphs in every fixed degree
For every fixed integer d ≥ 3, we give a deterministic algorithm that constructs a simple nonbipartite d-regular Ramanujan graph on every sufficiently large even number n of vertices. It outputs the full adjacency list in polynomially many bit operations, with an exponent that may depend on d. Every nonconstant adjacency eigenvalue lies strictly between `-2√(d-1)` and `2√(d-1)`.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M328 · Family 179 The circulant Hadamard conjecture
We prove the circulant Hadamard conjecture: a real circulant Hadamard matrix has order 1 or 4. As a consequence, Barker sequences of length greater than one exist exactly at lengths 2, 3, 4, 5, 7, 11, 13, proving the Barker-sequence conjecture.
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View manuscript on GitHub → Open PDF (raw) →M329 · Family 180 Paired states and Hamiltonian cycles in cubic bipartite planar graphs
We prove Barnette's conjecture: every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle.
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View manuscript on GitHub → Open PDF (raw) →M330 · Family 181 A linear cycle-and-edge decomposition of every graph
We prove that every finite simple undirected graph on n vertices has an edge partition into at most `Cn` simple cycles and single edges, for an absolute constant C. This resolves the Erdős–Gallai cycle decomposition conjecture positively.
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View manuscript on GitHub → Open PDF (raw) →M331 · Family 182 A power saving for intersective polynomial differences with an exponent depending only on the degree
An integer polynomial is intersective if it has a root modulo every positive integer. For each degree k ≥ 2, we prove that there is an exponent `c_k 0` such that every set `A 1,N ` whose differences avoid all nonzero values `h(1),h(2), `, where h is an intersective polynomial of degree k with positive leading coefficient, satisfies `|A|=O_h(N¹⁻c_k)`. The implied constant may depend on h, but the power-saving exponent depends only on its degree.
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View manuscript on GitHub → Open PDF (raw) →M332 · Family 182 A Power Saving for Polynomial Differences at Prime Arguments
Let h be a fixed integer polynomial of degree at least two with positive leading coefficient, having a unit root modulo every positive integer. We prove that any set `A 1,N ` whose differences avoid all nonzero values `h(p)` at primes satisfies `|A|≤ C_hN¹⁻c_h`, where `c_h 0` and `C_h≥1` depend only on h. Thus the local unit-root condition gives a fixed power saving even when polynomial arguments are restricted to primes.
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View manuscript on GitHub → Open PDF (raw) →M333 · Family 182 A power saving for square-difference-free sets
We prove that there are absolute constants c > 0 and C < ∞ such that every set `A 1,N ` with no nonzero square difference satisfies `|A|≤ C N¹⁻c`. This answers the fixed-power question posed by Green and Sawhney.
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View manuscript on GitHub → Open PDF (raw) →M334 · Family 183 A power saving for planar halving lines
There are absolute constants ε > 0 and C such that every sufficiently large even n-point set in the plane with no three collinear has at most `Cn⁴/³⁻ε` unordered halving pairs. This gives a power saving over the classical `O(n⁴/³)` bound for planar halving lines. The proof is nonquantitative and does not supply explicit constants.
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View manuscript on GitHub → Open PDF (raw) →M335 · Family 184 Correspondence coloring graphs with a forbidden clique
For every fixed integer r ≥ 4, we prove that every Kr-free graph of sufficiently large maximum degree Δ has correspondence chromatic number `O_r( /log )`. This resolves the Alon–Krivelevich–Sudakov coloring conjecture in the stronger correspondence-coloring form. The same bound, with a constant depending on F, holds when any fixed graph F is excluded as an ordinary subgraph. Ordinary and list coloring satisfy the same bounds.
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View manuscript on GitHub → Open PDF (raw) →M336 · Family 184 A logarithmic independence bound for clique-free graphs
For every fixed integer r ≥ 4, every Kr-free graph on n vertices with average degree d ≥ 2 has an independent set of size at least `c_r nlog d/d`, where `c_r 0` depends only on r. This proves the fixed-clique-size independence conjecture of Ajtai, Erdős, Komlós and Szemerédi.
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View manuscript on GitHub → Open PDF (raw) →M337 · Family 185 A Counterexample to the Infinite Matroid Packing/Covering Conjecture
We construct in ZFC two self-dual partitional matroids on a countably infinite common ground set that admit neither a packing/covering partition nor an intersection witness. This disproves the unrestricted infinite matroid packing/covering and intersection conjectures and answers Joó's question for two partitional matroids negatively. The examples are neither finitary nor cofinitary.
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View manuscript on GitHub → Open PDF (raw) →M338 · Family 186 A uniform influence bound for hypergraph properties
For every fixed integer r ≥ 3, we prove that every relabeling-invariant Boolean property of simple r-uniform hypergraphs on n vertices satisfies ` Var _p(f)≤ C_r I_p(f)/(log n)r/(r⁻¹)`. The constant depends only on r, and the bound holds uniformly for all `0 p 1` without a monotonicity assumption. For increasing properties, it gives the corresponding threshold-width bound with exponent `r/(r-1)`, proving the hypergraph threshold-width conjecture of Friedgut and Kalai.
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View manuscript on GitHub → Open PDF (raw) →M339 · Family 186 A Sharp Threshold Bound for Monotone Graph Properties
We prove the Friedgut–Kalai sharp-threshold conjecture. For every integer n ≥ 2, every nontrivial increasing family of graphs on n vertices invariant under all vertex permutations, and every `0 ε 1/2`, the edge probabilities at which its probability equals ε and `1-ε` differ by at most `Clog(1/(2ε))/(log n)²`, for a universal constant C.
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View manuscript on GitHub → Open PDF (raw) →M340 · Family 187 Snaky in 21 Maker moves
We prove that Maker can achieve the Snaky hexomino within 21 actual Maker moves against arbitrary legal Breaker play on the initially empty infinite square board. The same bound holds on a `17×17` square; in fact, Maker can confine its claims to a fixed 251-cell board.
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View manuscript on GitHub → Open PDF (raw) →M341 · Family 188 The sharp terminal leave in random triangle removal
Starting from the complete graph on n vertices, repeatedly remove the three edges of a uniformly chosen remaining triangle. We prove that the number of edges left at termination, divided by n3/2, converges in L2 to `1/(2 sqrt2)`. This proves the triangle case of the sharp-constant conjecture of Joos and Kühn. In particular, the same limit holds in probability and for the normalized expectation.
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View manuscript on GitHub → Open PDF (raw) →M342 · Family 189 Cycle–clique Ramsey numbers
We prove that `R(C_m,K_n)=(m-1)(n-1)+1` for every pair of integers `m≥ n≥3` other than `(m,n)=(3,3)`, for which `R(C_3,K_3)=6`. This establishes the cycle–clique conjecture of Erdős, Faudree, Rousseau and Schelp. The proof combines expansion in a minimal counterexample with a large-clique lemma and an optimization of paths joining clique vertices. These arguments reduce the remaining cases to `3, 099` finite parameter-pattern instances, which are excluded by two exact implementations of proved inference rules.
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View manuscript on GitHub → Open PDF (raw) →M343 · Family 190 Polynomial removal fails for ordered binary matrices
We construct a fixed `66×66` binary matrix for which ordered matrix removal has no polynomial bound. This disproves the polynomial ordered binary matrix-removal conjecture. Ordered copies preserve the separate row and column orders and match both zeros and ones; removal permits changing entries in either direction.
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View manuscript on GitHub → Open PDF (raw) →M344 · Family 191 A power improvement in the Heilbronn triangle lower bound
There are absolute constants `,c_1 0` such that, for every sufficiently large integer n, one can choose n points in the unit square so that every triangle they determine has area at least `c_1n⁻²⁺ `. Thus the almost n−2 upper-bound formulation of Heilbronn's triangle problem is false. The exponent η is fixed but extremely small.
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View manuscript on GitHub → Open PDF (raw) →M345 · Family 192 Unbounded Violations of the Square-Root Degree Bound
We disprove the Gopalan–Servedio square-root conjecture, even up to an arbitrary constant factor. For every real C > 0, there is a nonconstant Boolean function `f: -1,1 n→ -1,1 ` on a finite sign cube such that ` ∑(i=1)n f( i ) C√(deg(f)).` Here ` f( i )` is the linear Fourier coefficient associated with the ith input, and `deg(f)` is the degree of the real multilinear polynomial representing f.
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View manuscript on GitHub → Open PDF (raw) →M346 · Family 193 Positivity of Serre's Intersection Multiplicity
We prove Serre's positivity conjecture, including ramified mixed characteristic. If two nonzero finitely generated modules over a regular local ring have tensor product of finite length and complementary dimensions, then their intersection multiplicity is strictly positive.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M347 · Family 194 Lech's multiplicity conjecture
We prove Lech's multiplicity conjecture: Hilbert–Samuel multiplicity cannot decrease under a flat local homomorphism of nonzero Noetherian local rings. The result holds in arbitrary dimension, with no restrictions on the residue fields or characteristics.
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View manuscript on GitHub → Open PDF (raw) →M348 · Family 195 A Complete Local Domain without a Small Cohen–Macaulay Module
We construct a three-dimensional complete Noetherian normal local domain containing ℂ, with residue field ℂ, that has no nonzero finitely generated maximal Cohen–Macaulay module. This disproves the domain form of the small Cohen–Macaulay module conjecture.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M349 · Family 196 A Torsion-Free Group Algebra with Zero Divisors
We disprove Kaplansky's zero-divisor conjecture by constructing a finitely presented torsion-free group G for which ` F_2[G]` has nonzero zero divisors. The group admits a finite two-dimensional classifying space.
PQC lens (via Family 196): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M350 · Family 197 A Torsion-Free Group Algebra That Is Not Directly Finite
We construct a finitely presented torsion-free counterexample to Kaplansky's direct finiteness conjecture over the field of two elements. The group admits a finite two-dimensional classifying complex.
PQC lens (via Family 197): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M351 · Family 197 A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two
We disprove Kaplansky's direct-finiteness conjecture by constructing a finite field K of characteristic two, a finitely presented group G, and finite sums `a,b K[G]` with `ab=1` but `ba≠1`. The group G is nonsofic. The same elements define a cellular automaton on KG that is injective but not surjective, disproving Gottschalk's surjunctivity conjecture.
PQC lens (via Family 197): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M352 · Family 197 A Counterexample to the Group-Ring Determinant Conjecture
We disprove the unrestricted group-ring Determinant Conjecture. We construct a finitely generated group G and a square matrix over ` Z[G]` that is invertible over ` Q[G]` and has Fuglede–Kadison determinant strictly between zero and one. The logarithmic integral defining the determinant is finite.
PQC lens (via Family 197): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M353 · Family 197 A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic
We construct a counterexample to Kaplansky's direct-finiteness conjecture in odd characteristic. For one specified odd prime p, we obtain a field K of order p4, a finitely generated group G containing torsion, and finite sums `a,b K[G]` with `ab=1` but `ba≠1`. The same elements define a cellular automaton on KG that is injective but not surjective.
PQC lens (via Family 197): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M354 · Family 198 An algebra of infinite little finitistic dimension
We construct a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture.
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View manuscript on GitHub → Open PDF (raw) →M355 · Family 199 An explicit counterexample to the Auslander-Reiten conjecture
We disprove the Auslander–Reiten conjecture for Artin algebras. We construct a finite-dimensional algebra Λ over `k= F_2(q,H_1,H_2)` and a finite-dimensional nonprojective left module Z such that ` Ext i_ (Z,Z)= Ext i_ (Z, )=0` for every i > 0. The module is Gorenstein-projective, so the same example also disproves the Gorenstein-projective conjecture. Both counterexamples persist after every extension of k.
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View manuscript on GitHub → Open PDF (raw) →M356 · Family 199 A counterexample to Tachikawa's second conjecture
We disprove Tachikawa's second conjecture by constructing a finite-dimensional symmetric algebra over `k= F_2(q,H_1,H_2)` with a finite-dimensional nonprojective module whose self-extension groups vanish in every positive degree. The associated endomorphism algebra also gives counterexamples to the classical, generalized, and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. These conclusions persist after every extension of k.
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View manuscript on GitHub → Open PDF (raw) →M357 · Family 200 The Artinian Lex-Plus-Powers Betti Theorem
We prove the Eisenbud–Green–Harris and lex-plus-powers conjectures over every characteristic-zero field for homogeneous regular sequences of any positive length, with degrees at least two. For every homogeneous ideal containing such a sequence, the corresponding lex-plus-powers ideal has the same Hilbert function and at least as large a graded Betti number in every homological and internal degree.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M358 · Family 200 Commuting Division-Coefficient Forms and the Artinian Eisenbud–Green–Harris Conjecture
We prove the Eisenbud–Green–Harris conjecture over every characteristic-zero field for homogeneous regular sequences of any positive length, with degrees at least two. Thus every homogeneous ideal containing such a sequence has the Hilbert function of a monomial ideal containing the corresponding pure powers.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M359 · Family 201 A Counterexample to Kurosh’s Division-Ring Problem
We construct a countable division ring of characteristic zero that is algebraic over its center, generated by two elements as an algebra over that center, and infinite-dimensional over it. This gives a negative answer to the Kurosh problem for division rings.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M360 · Family 202 The Blockwise Alperin Weight Conjecture
We prove the numerical blockwise Alperin weight conjecture for every finite group and every prime p. For each p-block B, the number of irreducible Brauer characters in B equals the number of conjugacy classes of B-weights.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M361 · Family 203 Donovan's Conjecture over Algebraically Closed Fields
We prove Donovan's conjecture over every algebraically closed field K of characteristic p > 0. For each fixed K and bound on defect-group order, blocks of finite groups represent only finitely many K-linear Morita equivalence classes. The defect groups need not be abelian, and the result includes p = 2.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M362 · Family 203 Integral Donovan Finiteness over Witt Vectors
We prove integral Donovan finiteness: for every prime p and positive integer M, the blocks of all finite groups with defect groups of order at most M have only finitely many Morita equivalence classes over `W( F _p)`. The defect groups need not be abelian, and the result includes p = 2. The same bounded-defect finiteness holds over each fixed complete discrete valuation ring of characteristic zero with algebraically closed residue field of characteristic p, including ramified rings.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M363 · Family 204 Tensor saturation for even spin groups
We prove the saturation conjecture for ` Spin (2n)`, n ≥ 2. If three dominant integral weights sum to an element of the root lattice, then the existence of a nonzero tensor invariant after a positive integral dilation implies the existence of one at the original weights.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M364 · Family 205 Universal Tensor Squares for Symmetric Groups
For every positive integer n other than 2, 4, and 9, we prove that some irreducible complex representation of Sn has a tensor square containing every irreducible representation. This resolves the tensor square conjecture for symmetric groups affirmatively.
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View manuscript on GitHub → Open PDF (raw) →M365 · Family 205 A Cyclic Polytabloid Proof of Saxl's Conjecture
For every staircase partition, we prove that the tensor square of the corresponding irreducible complex representation of the symmetric group contains every irreducible representation of that group. This proves Saxl's conjecture.
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View manuscript on GitHub → Open PDF (raw) →M366 · Family 206 Finite congruence lattices: characterization and undecidability
We give an explicit colored-graph characterization of the finite nonempty lattices that occur as full congruence lattices of finite algebras, and prove that deciding this representation property is undecidable. In particular, the finite lattice representation problem has a negative answer. We also prove that recognition of full subgroup intervals in finite groups is undecidable.
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View manuscript on GitHub → Open PDF (raw) →M367 · Family 206 A negative solution to the finite lattice representation problem
We give a negative solution to the finite lattice representation problem. We prove that there is a finite nonempty lattice that is not the full congruence lattice of any finite nonempty algebra of any finite signature.
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View manuscript on GitHub → Open PDF (raw) →M368 · Family 207 The ℓ¹-Bass Conjecture for Discrete Groups
We prove the ℓ1-Bass conjecture for every discrete group. The Hattori–Stallings trace of every idempotent matrix over the complex ℓ1 group algebra is supported on finitely many conjugacy classes of finite-order elements.
PQC lens (via Family 207): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M369 · Family 207 The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture
We prove the complex group-ring Bass trace conjecture for every discrete group: the Hattori–Stallings trace of a finitely generated projective module over its complex group ring is supported on conjugacy classes of finite-order elements. As a consequence, for every torsion-free group G and every commutative unital domain R of characteristic zero, the only idempotents in `RG` are 0 and 1. This proves Kaplansky's idempotent conjecture in characteristic zero.
PQC lens (via Family 207): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M370 · Family 208 Fiber functors for finite symmetric tensor categories in positive characteristic
We prove the finite case of the Benson–Etingof–Ostrik conjecture: every finite symmetric tensor category over an algebraically closed field k of characteristic p > 0 admits a k-linear exact faithful strong symmetric monoidal functor to a finite higher Verlinde category ` Ver (pn)(k)`. The result includes characteristic two, and the level n may depend on the category.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M371 · Family 209 An integral counterexample to Gersten's conjecture
We construct a two-dimensional ramified regular local ring A of mixed characteristic `(0,5)` for which `K_5(A)→ K_5( Frac A)` has a nonzero kernel. This disproves Gersten's conjecture in its unrestricted integral form.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M372 · Family 209 An integral degree-three Gersten counterexample
We construct a two-dimensional ramified regular local ring A in mixed characteristic `(0,5)` for which the integral map `K_3(A)→ K_3( Frac A)` has nonzero kernel. This gives a negative answer to the unrestricted integral Gersten conjecture.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M373 · Family 210 Foulkes' conjecture for the sixth symmetric power
We prove the sixth-symmetric-power case of Foulkes' conjecture. For every integer b ≥ 6 and every finite-dimensional complex vector space V, there is a ` GL (V)` -equivariant injection ` Sym ⁶( Sym b V) Sym b( Sym ⁶ V)`.
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View manuscript on GitHub → Open PDF (raw) →M374 · Family 210 Quadratic stabilization of the canonical Foulkes–Howe map
For every finite-dimensional complex vector space V, we prove that the canonical Foulkes–Howe map ` Sym b( Sym a V) Sym a( Sym b V)` is surjective whenever a ≥ 2 and `b≥ a(a-1)`. This gives a quadratic stabilization bound independent of `dim V`.
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View manuscript on GitHub → Open PDF (raw) →M375 · Family 211 Random Walks on Critical FK–Ising Maps and Liouville Brownian Motion
We prove that stationary random walks on critical spherical FK–Ising planar maps converge to Liouville Brownian motion on the ordinary unit-area ` sqrt3` -quantum sphere, using the geometric and electrical results of the spectral companion. The walk chooses uniformly among all incident half-edges, retaining loops and multiple edges. With the corner measure as the stationary law, the deterministic time acceleration is exactly the number of map edges.
PQC lens (via Family 211): Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
View manuscript on GitHub → Open PDF (raw) →M376 · Family 211 Spectral convergence for critical FK–Ising planar maps
Using the conformal and metric-measure companion results and the stated Brownian/Liouville quantum gravity inputs, we prove spectral convergence for critical spherical FK–Ising maps to Liouville Brownian motion on the ordinary unit-area ` sqrt3` -quantum sphere. The discrete walk has total attempt rate one, uses every map edge including loops and multiplicities, and has the corner measure as its stationary law.
PQC lens (via Family 211): Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
View manuscript on GitHub → Open PDF (raw) →M377 · Family 211 A Linear Clock for Random Walk on Tree-Weighted Planar Maps
We prove that stationary random walk on a planar map sampled with weight equal to its number of spanning trees converges to Liouville Brownian motion on the unit-area ` sqrt2` -Liouville quantum sphere. The convergence retains the conditional path law jointly with the measured metric space. The walk chooses uniformly among all incident half-edges and starts from the stationary degree measure. For total attempt rate one and the continuum Dirichlet form with factor 1/2, the time acceleration is exactly the number of map edges.
PQC lens (via Family 211): Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
View manuscript on GitHub → Open PDF (raw) →M378 · Family 211 Canonical conformal limits of subcritical FK planar maps
For every fixed `0 q 4`, we prove joint convergence of spherical Fortuin–Kasteleyn planar maps in their flag-triangle uniformization to the corresponding unit-area Liouville quantum gravity sphere decorated by an independent conformal loop ensemble. The convergence includes the area measure, deterministically rescaled graph distances between all vertex pairs, and the full nested interface collection, with interfaces converging uniformly up to reparameterization.
PQC lens (via Family 211): Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
View manuscript on GitHub → Open PDF (raw) →M379 · Family 211 The critical Liouville quantum sphere and geometric limits of FK maps at q=4
We construct the field and area law of the unit-area critical Liouville quantum sphere as a limit of ordinary subcritical quantum spheres, and equip it with its critical intrinsic metric. We then prove that spherical Fortuin–Kasteleyn planar maps at q = 4, embedded by their equilateral flag uniformizations, converge jointly to this sphere decorated by an independent nested conformal loop ensemble CLE4.
PQC lens (via Family 211): Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
View manuscript on GitHub → Open PDF (raw) →M380 · Family 211 Metric-measure limits of subcritical FK and spanning-tree planar maps
We resolve the finite spherical cases of Gwynne and Miller's graph-metric conjecture for critical Fortuin–Kasteleyn maps at each fixed `q (0,4)` and for uniform spanning-tree-decorated maps. After deterministic rescaling of graph distances, these maps, equipped with the vertex probability measure proportional to degree, converge in Gromov–Hausdorff–Prokhorov law to their ordinary unit-area Liouville quantum gravity spheres. Distances use every primal edge, and convergence holds through all positive integer edge counts.
PQC lens (via Family 211): Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
View manuscript on GitHub → Open PDF (raw) →M381 · Family 211 Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four
We prove the finite-volume continuum-random-tree prediction for critical Fortuin–Kasteleyn planar maps at every fixed q > 4. After rescaling graph distances by a constant times n−1/2, an n-edge map with normalized degree measure converges to the Brownian continuum random tree in the Gromov–Hausdorff–Prokhorov topology. The convergence holds through all positive integer sizes.
PQC lens (via Family 211): Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
View manuscript on GitHub → Open PDF (raw) →M382 · Family 212 No bigeodesics in planar first-passage percolation
We prove that planar first-passage percolation with independent identically distributed nonnegative nonatomic edge weights has almost surely no doubly infinite geodesic, provided the minimum of four independent weights has finite second moment. This resolves the planar no-bigeodesics conjecture under that moment assumption. The conclusion rules out all bigeodesics simultaneously, without any regularity assumption on the limit shape.
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View manuscript on GitHub → Open PDF (raw) →M383 · Family 212 Strict convexity and differentiability of the planar exponential first-passage limit shape
We prove that the limit shape of undirected nearest-neighbor first-passage percolation on ℤ2 with independent exponential edge weights is strictly convex and has a C1 boundary. This resolves the strict convexity and differentiability conjectures for the planar exponential model. More generally, we prove differentiability of the time-constant norm for every Gamma edge-weight law with positive shape and rate.
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View manuscript on GitHub → Open PDF (raw) →M384 · Family 213 Critical bond and site percolation on the cubic lattice
We prove that nearest-neighbor Bernoulli bond and site percolation on ℤ3 have no infinite cluster at their respective critical parameters. The proof combines a finite connection inequality for independent hyperedges with a finite-scale extension estimate and an adaptive exploration.
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View manuscript on GitHub → Open PDF (raw) →M385 · Family 213 No percolation at criticality on quasi-transitive graphs
We prove that critical Bernoulli bond percolation has no infinite cluster on any infinite connected locally finite quasi-transitive graph with critical probability less than one. This resolves the bond form of the criticality conjecture of Benjamini and Schramm.
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View manuscript on GitHub → Open PDF (raw) →M386 · Family 214 Nonuniqueness of percolation on nonamenable quasi-transitive graphs
We prove the bond-percolation nonuniqueness conjecture of Benjamini and Schramm: every infinite connected, locally finite, nonamenable quasi-transitive graph has a nonempty interval of parameters for which Bernoulli bond percolation almost surely has infinitely many infinite clusters. We also prove Hutchcroft's stronger operator-threshold conjecture, establishing `p_c p(2→2)≤ p_u`.
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View manuscript on GitHub → Open PDF (raw) →M387 · Family 215 The canonical massive continuum limit of the two-dimensional O(3) model
We construct a canonical interacting massive continuum limit of the two-dimensional nearest-neighbor `O(3)` model with unit-length spins, no external field, and no topological term. Normalized by susceptibility and second-moment correlation length, the limit exists as the bare coupling tends to infinity through all positive real values, without selecting subsequences. The limiting fields satisfy the Osterwalder–Schrader axioms and have a nonzero connected four-point correlation on separated time supports.
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View manuscript on GitHub → Open PDF (raw) →M388 · Family 215 An Isolated Particle Pole for the Two-Dimensional O(3) Spin Field
We consider the continuum spin field constructed from the nearest-neighbor two-dimensional `O(3)` model by the fixed prescription of the companion paper. We prove that its vector two-point spectral measure has a positive atom corresponding to the lowest mass, separated by a positive gap from all remaining mass support.
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View manuscript on GitHub → Open PDF (raw) →M389 · Family 215 Exact mass asymptotics for the two-dimensional O(4) lattice model
For the nearest-neighbor `O(4)` model on the square lattice at inverse temperature β, we prove the exact low-temperature asymptotic ` m(lat)(β) 32exp(π/4-1/2) sqrtβ exp(-πβ) (β→∞).` Here the mass is the gap of the…
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View manuscript on GitHub → Open PDF (raw) →M390 · Family 215 Sharp mass bounds for the two-dimensional O(4) model
We prove sharp mass bounds for the two-dimensional nearest-neighbor `O(4)` model. For all sufficiently large inverse couplings β, the full transfer gap, including rotation-invariant local-observable sectors, is bounded above and below by positive multiples of ` sqrtβ e⁻πβ`. We also prove that the model has a unique periodic local limit and a positive full gap at every finite β > 0, resolving the all-temperature lattice mass-generation conjecture for this periodic state.
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View manuscript on GitHub → Open PDF (raw) →M391 · Family 215 Exponential decay in two-dimensional classical O(n) models
We prove exponential decay of two-point correlations for the classical nearest-neighbor `O(n)` model on the square lattice, for every n ≥ 3 and every finite positive inverse temperature. The estimate is uniform over finite free-boundary subgraphs and bounded nonnegative edge strengths. This resolves positively the all-temperature exponential spin-decay conjecture for these models.
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View manuscript on GitHub → Open PDF (raw) →M392 · Family 216 The critical logarithmic correction for the planar XY model
We prove the critical logarithmic correction for the nearest-neighbor cosine XY model on the square lattice. Let `C(b_c)(r)` be the two-point correlation at critical inverse temperature, obtained by taking the free-box thermodynamic limit while the two sites remain r lattice steps apart along a coordinate axis. Then ` C(b_c)(r)=B(XY)r⁻¹/⁴(log r)¹/⁸(1+o(1)), B(XY) (0,∞),` as `r→∞`.
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View manuscript on GitHub → Open PDF (raw) →M393 · Family 216 Critical Center Magnetization in the Planar XY Model
Assuming the stated critical-height, local-renormalization, and spin-field inputs from the companion papers, we determine the center magnetization of the planar XY model in a square with aligned boundary spins at its mass-defined critical threshold. As `n→∞`, the magnetization is `A(XY)n⁻¹/⁸(log n)¹/¹⁶(1+o(1))`, where `A(XY)` is a finite, strictly positive model-specific constant.
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View manuscript on GitHub → Open PDF (raw) →M394 · Family 216 The Critical Spin Field of the Planar XY Model
Using the companion critical Bessel-height and pin-limit theorems, we prove that at the mass-defined critical inverse temperature of the nearest-neighbor XY model on the square lattice, the spin field on a square with boundary angles fixed to zero converges along the full sequence to the full-variance imaginary exponential of a zero-Dirichlet Gaussian free field with stiffness `2/π`.
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View manuscript on GitHub → Open PDF (raw) →M395 · Family 216 Essential Singularity of the Correlation Length in the Planar XY Model
We prove the Berezinskii–Kosterlitz–Thouless essential singularity for the correlation length of the nearest-neighbor cosine XY model on the square lattice. Let `m(b)` be the mass obtained by taking first the free-box thermodynamic limit and then the separation limit along a coordinate axis. As the inverse temperature b approaches bc from the massive side, ` √(b_c-b)log 1 m(b) A(XY),` where `A(XY)` is a finite, strictly positive model-specific constant.
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View manuscript on GitHub → Open PDF (raw) →M396 · Family 216 The critical correlation exponent of the planar XY model
For the ordinary nearest-neighbor XY model on the square lattice, we prove the Berezinskii–Kosterlitz–Thouless prediction that the critical spin-correlation exponent is 1/4. More precisely, at the mass-defined critical inverse temperature, the infinite-volume correlation is `n⁻¹/⁴⁺o(¹)`. The infinite-volume limit through free square boxes is taken before the separation limit.
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View manuscript on GitHub → Open PDF (raw) →M397 · Family 216 BKT universality for height and planar spin fields
We prove Gaussian scaling limits for the two-dimensional discrete Gaussian height model throughout its rough phase, including the physical roughening threshold, for every finite square-symmetric interaction set containing the nearest neighbors. After the natural lattice normalization, the critical effective temperature has the universal value `8π`.
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View manuscript on GitHub → Open PDF (raw) →M398 · Family 217 The low-temperature Sherrington–Kirkpatrick free-energy limiting law
For every fixed inverse temperature β > 1, we prove that the zero-field Gaussian Ising Sherrington–Kirkpatrick free energy, centered by its expectation and divided by its standard deviation, converges in distribution to a nondegenerate law as the system size tends to infinity through all integers. We also prove that its variance divided by n1/3 converges to a finite positive constant.
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View manuscript on GitHub → Open PDF (raw) →M399 · Family 217 The low-temperature Sherrington–Kirkpatrick fluctuation scale
For the zero-field Gaussian Sherrington–Kirkpatrick model at every fixed inverse temperature β > 1, we prove that the standard deviation of the log partition function is `n¹/⁶⁺o(¹)`. The same exponent describes its typical centered absolute fluctuations, establishing the predicted one-sixth exponent in this regime.
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View manuscript on GitHub → Open PDF (raw) →M400 · Family 218 Quenched SLE₃ limits for general weak random-bond Ising models
For the planar Ising model with bonds `J_e=1+ε xi_e`, where the ξe have any fixed bounded, nondegenerate, mean-zero iid law, we prove quenched chordal SLE3 convergence for sufficiently small ε > 0. The temperature is the spontaneous-magnetization threshold. Convergence holds in probability over environments for the full oriented curve law in deterministic Jordan-domain approximations.
PQC lens (via Family 218): Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
View manuscript on GitHub → Open PDF (raw) →M401 · Family 218 Quenched SLE₃ Universality for the Weak Random-Bond Ising Model
We prove quenched chordal SLE3 convergence for the planar Ising model with sufficiently weak independent symmetric two-valued ferromagnetic bonds. The temperature is the critical point defined by spontaneous magnetization. Convergence holds in probability over environments for the full oriented curve law in deterministic Jordan-domain approximations.
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View manuscript on GitHub → Open PDF (raw) →M402 · Family 218 Logarithmic Relative Fluctuations in the Weakly Disordered Planar Ising Model
Assuming the stated deterministic critical-reference estimates, we prove that the relative second moment of the quenched critical spin correlation in the square-lattice Ising model with independent fair bonds `1 pmε` grows as `(log r)¹/⁴⁺o(¹)` for each sufficiently small fixed ε > 0. The correlation is evaluated at the physical critical temperature, with the thermodynamic limit taken before the disorder moments and the large-distance limit.
PQC lens (via Family 218): Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
View manuscript on GitHub → Open PDF (raw) →M403 · Family 218 Conformal universality of bulk Ising correlations under weak interactions
We prove conformal universality of mixed bulk spin and energy correlations for the square-lattice Ising model with sufficiently small, square-symmetric, finite-range even multispin perturbations of either sign. One critical-temperature branch and two field normalizations apply in every bounded simply connected C2 Jordan domain with free, plus, or minus boundary conditions, and in the periodic thermodynamic plane state. Energies are centered in their actual states.
PQC lens (via Family 218): Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
View manuscript on GitHub → Open PDF (raw) →M404 · Family 218 SLE3 universality for weak finite-range Ising interactions
We prove that every sufficiently small, square-symmetric finite-range perturbation of the planar Ising contour energy has a domain-independent inverse temperature at which its spin interface converges to chordal SLE3. The result allows interactions of either sign, arbitrary admissible exterior contours, and uniformly approximated Jordan domains. Convergence holds for the full oriented curve law in uniform distance modulo increasing reparametrization.
PQC lens (via Family 218): Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
View manuscript on GitHub → Open PDF (raw) →M405 · Family 218 Buffered comparison and stopping-band resolution in critical Ising
We prove pointwise comparison and finite stopping-band approximation theorems for the critical nearest-neighbor Ising model on the square lattice. Arbitrary common pinned spins are allowed in the comparison: the likelihood ratio is controlled by a zero-field FK connection probability on the graph with those vertices deleted.
PQC lens (via Family 218): Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
View manuscript on GitHub → Open PDF (raw) →M406 · Family 219 Fixed-energy universality for weak Anderson disorder on random regular graphs
For every fixed degree d ≥ 3, we prove fixed-energy GOE universality for the adjacency matrix of a uniformly random simple labelled d-regular graph with independent uniform diagonal disorder. The disorder strength is positive, sufficiently small, and fixed as the graph grows. At each fixed energy in a compact subinterval of the clean spectral band, the full microscopic eigenvalue point process converges to the GOE bulk process after rescaling by the positive density of states of the corresponding infinite-tree operator.
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View manuscript on GitHub → Open PDF (raw) →M407 · Family 219 GOE bulk universality for fixed-degree random regular graphs
We prove the fixed-degree bulk-universality conjecture for random regular graphs. For every fixed integer d ≥ 3 and every fixed energy in the open Kesten–McKay bulk, the unfolded eigenvalue point process of the adjacency matrix of a uniform simple labelled d-regular graph converges to the GOE bulk process. The convergence holds along all admissible graph sizes, without additional conditioning.
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View manuscript on GitHub → Open PDF (raw) →M408 · Family 220 A directional zero–one law for finite-range-dependent random environments
We prove a directional zero–one law for uniformly elliptic nearest-neighbor random walks in stationary, ergodic, finite-range-dependent environments on ℤd, d ≥ 3. For every fixed nonzero real direction, the probability of escape in that direction, averaged over the environment, is either zero or one. Finite-range dependence is imposed on the full transition rows: collections of rows at distance greater than a fixed range are independent, including collections indexed by infinite deterministic sets.
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View manuscript on GitHub → Open PDF (raw) →M409 · Family 220 A directional zero–one law under strict ellipticity
We prove the directional zero–one conjecture for nearest-neighbor random walks in independent and identically distributed strictly elliptic environments on ℤd, d ≥ 3: the probability of escape in each fixed nonzero real direction is zero or one. Only strict positivity of the transition probabilities is required; no uniform lower bound or moment assumption is imposed.
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View manuscript on GitHub → Open PDF (raw) →M410 · Family 220 Directional transience implies ballisticity
We prove that almost-sure transience in a fixed direction implies a deterministic limiting velocity with positive projection in that direction for nearest-neighbor random walks in independent and identically distributed uniformly elliptic environments on ℤd, d ≥ 2. This resolves the ballisticity conjecture positively.
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View manuscript on GitHub → Open PDF (raw) →M411 · Family 221 The Mézard–Parisi formula for diluted spin glasses
We prove the Mézard–Parisi hierarchical cavity formula for diluted even-arity Ising models in the Panchenko–Talagrand class. This resolves the variational equality conjecture for that class: the limiting pressure equals the infimum of the trial functional over all finite hierarchy depths and trial laws. Only first moments of the interaction and external field are required.
PQC lens (via Family 221): Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
View manuscript on GitHub → Open PDF (raw) →M412 · Family 222 The free energy of the Ising random perceptron
We determine the limiting free energy of the Ising perceptron with independent Gaussian patterns for every bounded Borel log-potential, at every fixed positive temperature and pattern density. We give an explicit variational formula for the limit and prove convergence in expectation and probability.
PQC lens (via Family 222): Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
View manuscript on GitHub → Open PDF (raw) →M413 · Family 222 Microscopic jamming in the negative spherical perceptron
We prove a sharp feasibility threshold and limiting gap and force laws for the spherical perceptron with margin −1 and quadratic penalty. The limits are taken successively in system size, inverse temperature, and density approaching the threshold from above. They agree for Gaussian coordinates and for the equal mixture of centered Gaussian coordinates with variances `1-ε` and `1+ε`, for every sufficiently small fixed ε.
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View manuscript on GitHub → Open PDF (raw) →M414 · Family 222 The spherical perceptron with bi-orthogonally invariant disorder
We determine the limiting free energy of a spherical perceptron with a bounded continuous activation and a bi-orthogonally invariant disorder matrix. The singular values may have any compact limiting distribution, provided there are no outliers. We give an explicit variational formula for this limit.
PQC lens (via Family 222): Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
View manuscript on GitHub → Open PDF (raw) →M415 · Family 222 The free energy of the spherical random perceptron
We prove an exact variational formula for the limiting pressure of the spherical random perceptron with an arbitrary bounded continuous single-pattern potential. The formula holds at every fixed positive density and inverse temperature, with convergence in expectation and in probability.
PQC lens (via Family 222): Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
View manuscript on GitHub → Open PDF (raw) →M416 · Family 223 Square-lattice FK interfaces and nested loops for 1 <= q < 4
For every fixed `1≤ q 4`, critical square-lattice random-cluster Dobrushin interfaces converge as ordered curves to chordal ` SLE ( (q))`, where ` (q)=4π/ (- q/2)`, confirming the Rohde–Schramm prediction in this parameter…
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View manuscript on GitHub → Open PDF (raw) →M417 · Family 223 Self-dual random-cluster interfaces below one
For every fixed `0 q 1`, we prove that the Dobrushin interface of the square-lattice random-cluster model at its self-dual parameter converges to chordal SLEκ, where ` =4π/ (- q/2)…
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View manuscript on GitHub → Open PDF (raw) →M418 · Family 223 Quenched SLE Universality for Weakly Disordered FK–Ising Interfaces
We prove that critical FK–Ising interfaces with sufficiently weak, symmetric, independent two-valued bond disorder converge to chordal SLE16/3. The disorder strength is fixed as the mesh tends to zero, and convergence of the conditional curve laws holds in probability over the environment.
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View manuscript on GitHub → Open PDF (raw) →M419 · Family 223 Thermal FK–Ising interfaces and massive SLE
We prove convergence of thermal FK–Ising interfaces, for every fixed nonzero mass of either sign, on uniformly angle-bounded isoradial lattices in bounded simply connected domains. The limit is independent of the lattice and the admissible domain approximation. For positive mass it is the unique massive SLE `(16/3)` law with locally finite-energy drift prescribed by a massive boundary value problem; negative mass follows by duality and reversal. The theorem also allows Carathéodory approximations whose diameters diverge.
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View manuscript on GitHub → Open PDF (raw) →M420 · Family 223 Natural Occupation Measures for Critical Square-Lattice FK Interfaces
We prove that the rescaled counting measure of a critical square-lattice Fortuin–Kasteleyn Dobrushin interface in the unit square converges to the Minkowski-content measure of its Schramm–Loewner limit, for every fixed cluster weight `1≤ q 4`. A single deterministic constant times the predicted power of the mesh gives the normalization. Convergence is joint with the ordered curve and includes the total mass, giving the scaling limit of the interface's total number of steps.
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View manuscript on GitHub → Open PDF (raw) →M421 · Family 223 Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces
For every fixed `1≤ q≤4`, critical square-lattice random-cluster Dobrushin interfaces converge to chordal ` SLE ( (q))`, where ` (q)=4π/ (- q/2)`, and the complete nested plane loop collections converge to whole-plane ` CLE (…
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View manuscript on GitHub → Open PDF (raw) →M422 · Family 224 From critical crossings to quenched near-critical universality in Voronoi percolation
Taking Cardy's crossing formula for critical Poisson–Voronoi percolation as an input, we prove a universal joint limit for the near-critical crossing thresholds of rational polygonal quadrilaterals under the monotone coupling. Conditional on the Poisson tessellation, the threshold law converges in probability over tessellations to the same law as on the triangular lattice. Each model is normalized by its own expected number of color-pivotal sites for a unit-square crossing.
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View manuscript on GitHub → Open PDF (raw) →M423 · Family 224 A Pivotal Amplitude for Voronoi Percolation from Cardy's Formula
Taking Cardy's conformal crossing formula for critical planar Poisson–Voronoi percolation in every bounded Jordan quadrilateral as an input, we prove that the expected number of color-pivotal cells for a unit-square crossing is asymptotic to a positive constant times `ε ⁻³/⁴`, where the point intensity is `ε ⁻²`. No rate of convergence in Cardy's formula is required.
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View manuscript on GitHub → Open PDF (raw) →M424 · Family 224 Cardy’s formula for critical Poisson–Voronoi percolation
We prove Cardy's formula for annealed crossing probabilities in critical planar Poisson–Voronoi percolation in every bounded Jordan quadrilateral. This proves the annealed crossing-probability form of the conformal-invariance conjecture for this model.
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View manuscript on GitHub → Open PDF (raw) →M425 · Family 225 The Gaussian free field limit of the balanced six-vertex model with variance multiplier 1/arcsin(c/2)
We prove that the height function of the square-lattice six-vertex model with weights `a=b=1` and `0 c≤2`, in the plane state obtained from balanced tori, converges to a multiple of the Gaussian free field. For unit height jumps and Green kernel `-(2π)⁻¹log|x-y|`, the squared multiplier is `1/ (c/2)`.
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View manuscript on GitHub → Open PDF (raw) →M426 · Family 226 The curve scaling limit of half-plane double dimers
We prove that the complete double-dimer loop ensemble for the Temperleyan square lattice in the upper half-plane converges to nested CLE4. The convergence holds along the full mesh limit and matches every macroscopic loop as an unparametrized curve. This resolves the half-plane Temperleyan form of the double-dimer CLE4 scaling-limit conjecture.
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View manuscript on GitHub → Open PDF (raw) →M427 · Family 227 Universality of critical quench autocorrelations in the Sherrington–Kirkpatrick model
We prove joint functional convergence of the stationary and quench autocorrelations of zero-field Sherrington–Kirkpatrick heat-bath dynamics at inverse temperature β = 1, with the same random limit for Gaussian and Rademacher couplings. Each site has a rate-one clock, mean spin autocorrelations are multiplied by n1/3, and waiting times and lags are measured in units n2/3. The quench starts from independent fair spins, and convergence is uniform on compact sets of positive waiting times and lags.
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View manuscript on GitHub → Open PDF (raw) →M428 · Family 227 Functional universality of critical SK autocorrelations
We prove that the stationary spin autocorrelation of the zero-field Sherrington–Kirkpatrick model at inverse temperature β = 1 has a common functional scaling limit for Gaussian and Rademacher couplings. With rate-one heat-bath clocks at every site, time is scaled by n2/3 and the mean spin autocorrelation is multiplied by n1/3. The functions converge in law uniformly on compact positive-time intervals.
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View manuscript on GitHub → Open PDF (raw) →M429 · Family 227 A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase
For every fixed inverse temperature `0 β 1`, we prove that the unscaled spectral gap of single-site heat-bath dynamics for the zero-field Gaussian Sherrington–Kirkpatrick model is bounded away from zero with probability tending to one over the disorder. Equivalently, the Gibbs law satisfies a dimension-free Poincaré inequality for all functions.
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View manuscript on GitHub → Open PDF (raw) →M430 · Family 227 Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase
We prove worst-case total-variation cutoff for the zero-field Gaussian Sherrington–Kirkpatrick heat-bath dynamics at every fixed inverse temperature `0≤β 1`. With rate-one refresh at each spin, the cutoff location is `log n/(2λ(β))` for a positive deterministic rate `λ(β)`. The location for uniformly chosen single-site update attempts is n times as large. Convergence is in probability over the disorder.
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View manuscript on GitHub → Open PDF (raw) →M431 · Family 227 Critical slowing down in the Sherrington–Kirkpatrick model
At the critical inverse temperature β = 1, we prove slow mixing for zero-field Gaussian Sherrington–Kirkpatrick heat-bath dynamics from typical equilibrium configurations held fixed as initial states. For every deterministic sequence `t_n=o(n²/³)` in rate-one-per-site time, the Gibbs mass of initial states whose time-tn total-variation distance from equilibrium exceeds 1/4 tends to one in probability over the disorder.
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View manuscript on GitHub → Open PDF (raw) →M432 · Family 227 Stretched-exponential barriers for typical SK initial states
At every fixed inverse temperature β > 1, we prove a stretched-exponential obstruction to mixing for the zero-field Sherrington–Kirkpatrick model from typical equilibrium configurations held fixed as initial states. The Gibbs mass of states whose total-variation distance from equilibrium at time `exp(n¹/¹⁰⁰⁰⁰)` exceeds 1/4 tends to one in probability over the disorder. This holds for rate-one-per-site heat-bath dynamics and after the same stated number of discrete update attempts.
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View manuscript on GitHub → Open PDF (raw) →M433 · Family 227 A typical-start upper bound for low-temperature SK Glauber dynamics
For every fixed inverse temperature β > 1, we prove subexponential mixing for rate-one-per-site heat-bath dynamics in the zero-field Gaussian Sherrington–Kirkpatrick model from a typical equilibrium configuration. If one Gibbs-sampled configuration is held fixed as the initial state, its total-variation distance from equilibrium is at most 1/4 by time `exp(n¹⁻¹/⁴⁰⁰⁰⁰⁰⁰⁰)`, with joint probability tending to one over the disorder and the sampled state.
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View manuscript on GitHub → Open PDF (raw) →M434 · Family 227 Critical mixing in the Sherrington–Kirkpatrick model
At the critical inverse temperature β = 1, we determine the worst-start total-variation mixing exponent for single-site heat-bath dynamics in the zero-field Gaussian Sherrington–Kirkpatrick model. The mixing time is `n²/³⁺o(¹)` when every site has rate one, or `n⁵/³⁺o(¹)` attempted uniform-site updates, in probability over the disorder.
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View manuscript on GitHub → Open PDF (raw) →M435 · Family 228 A continuum temperature singularity for a radial pair potential
We construct a stable radial pair potential in three dimensions whose canonical free energy has a strict downward derivative jump at one common finite positive inverse temperature throughout an open interval of positive densities. The potential has a divergent repulsive core, a nontrivial attractive interval, and an integrable tail satisfying `φ(r)=o(r⁻³)`. Its free-cube canonical free energy is finite at every positive inverse temperature and density.
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View manuscript on GitHub → Open PDF (raw) →M436 · Family 228 A radial continuum phase transition with algebraic decay
We construct a bounded, continuous, stable radial pair potential in three dimensions with `|φ(r)|≤ Cr⁻³⁻¹/³²` for r ≥ 1. Throughout an open interval of positive densities, its canonical thermodynamic free energy is finite at every positive inverse temperature and has a strict downward derivative jump at one common finite positive inverse temperature.
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View manuscript on GitHub → Open PDF (raw) →M437 · Family 229 The Reconstruction Threshold for the Ferromagnetic Four-State Potts Model
We establish the exact Kesten–Stigum reconstruction threshold for the ferromagnetic four-state Potts broadcast model on every regular d-ary tree with d ≥ 2 and every Poisson Galton–Watson tree of mean d > 0. Reconstruction occurs exactly when `dλ² 1`; we prove nonreconstruction at and below the threshold, including equality. In the Poisson model the whole tree is observed and the reconstruction advantage is averaged without conditioning on survival.
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View manuscript on GitHub → Open PDF (raw) →M438 · Family 229 A Capacity Criterion for Four-State Potts Reconstruction on Trees
For the ferromagnetic four-state broadcast model with `0 λ 1`, we prove that reconstruction on a bounded-degree deterministic rooted tree occurs exactly when its L3 capacity with edge resistances `λ⁻²|e|` is positive. This gives an exact criterion without regularity or growth-rate assumptions on the tree, including at the exponential critical boundary.
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View manuscript on GitHub → Open PDF (raw) →M439 · Family 229 The exact reconstruction threshold for the three-state symmetric channel
We determine the exact reconstruction threshold for the symmetric three-state broadcast process on every regular b-ary tree, b ≥ 2, and every observed Poisson Galton–Watson tree of mean d > 1. Reconstruction occurs exactly when `dλ² 1`, with d = b in the regular model; there is non-reconstruction at equality for either sign of the channel parameter. The Poisson advantage is averaged over trees and spins without conditioning on survival. This resolves the all-degree three-state regular-tree prediction.
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View manuscript on GitHub → Open PDF (raw) →M440 · Family 230 An exact Hausdorff gauge for SLE
For each `0 8`, we construct a deterministic Hausdorff gauge that almost surely assigns positive finite measure to every nontrivial compact positive-time segment of chordal Schramm–Loewner evolution. This answers Schramm's Hausdorff-measure existence problem in this parameter range. The entire trace has finite expected gauge measure in each bounded box.
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View manuscript on GitHub → Open PDF (raw) →M441 · Family 230 An explicit exact Hausdorff gauge for SLE
For each fixed `0 8`, let `d=1+ /8`. The gauge `h(r)=rd(loglog(¹/r))(²⁻d)/²` at sufficiently small radii almost surely gives positive finite Hausdorff measure to every nontrivial positive-time compact segment of chordal SLEκ. This gives an explicit solution to Schramm's Hausdorff-measure problem in this parameter range. The entire trace has finite expected measure in every bounded disk. The exponent-one iterated-logarithm gauge suggested by Schramm is not sigma-finite on any such segment.
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View manuscript on GitHub → Open PDF (raw) →M442 · Family 231 The free uniform spanning forest is a factor of IID
We prove that the free uniform spanning forest is a factor of IID on every infinite connected locally finite simple unweighted graph. One Borel rule works for all such graphs and uses no root, answering affirmatively the general factor question for unimodular random graphs. We also show that every translation-invariant strongly Rayleigh process indexed by a countable group is a factor of IID, including invariant determinantal processes with Hermitian positive-contraction kernels. This group-action conclusion requires no amenability.
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View manuscript on GitHub → Open PDF (raw) →M443 · Family 232 Gaussian free-field limits of weighted integer Lipschitz heights
We prove a Gaussian free field scaling limit for weighted integer Lipschitz heights on the triangular lattice. With zero boundary values and a factor x for each edge on which the height changes, the field converges after division by a positive constant depending only on x to the zero-Dirichlet Gaussian free field, for every fixed `x [1/ sqrt2,1]`. The convergence holds as a random distribution on every bounded C2 Jordan domain under inside lattice approximations with uniformly convergent boundary parametrizations.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M444 · Family 232 The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data
We prove that the centered uniform odd integer height function on triangular-lattice approximations of a smooth simply connected domain, with neighboring differences zero or two and boundary values +1 and −1 on two arcs, converges to a universal multiple of the Dirichlet Gaussian free field. This resolves the field part of Schramm's Problem 2.2. We give an absolutely convergent finite-volume formula for the normalization. The proof combines reflection positivity, a spectral sum rule, and boundary comparison with Gaussian moment identities.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M445 · Family 232 Uniform real Lipschitz surfaces on the triangular lattice
We prove the Gaussian free field and SLE `_4` scaling limits for uniformly sampled real nearest-neighbor Lipschitz heights on the triangular lattice, resolving Schramm's Problem 2.3. On approximations of smooth simply connected domains, the centered height field converges as a random distribution to a multiple of the Dirichlet Gaussian free field. At one tuned two-arc boundary amplitude, the zero-height interface converges in uniform curve distance to chordal SLE `_4`.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M446 · Family 233 The joint scaling limit of critical Ashkin-Teller currents
For each fixed point on the critical Ashkin–Teller line, including the four-state Potts endpoint, we prove the conjectured joint scaling limit of the height and both current-cluster collections in every bounded Jordan domain, for every admissible polygonal approximation. The height converges to a Gaussian free field with the predicted coupling constant, and the clusters converge to canonical recursive two-valued local sets of that same field. The joint limit includes all nesting depths and the distinguished wired boundary cluster.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M447 · Family 234 All-temperature pressure for orthogonally invariant Ising spin glasses
We determine the limiting pressure of an Ising spin glass with Haar orthogonal eigenvectors, a compact limiting spectral law, and no asymptotic outliers, at every fixed temperature. The pressure converges in expectation and almost surely to a variational formula. We also give the limiting pressure with deterministic external fields whose empirical laws converge in first-moment transport distance, and derive a formula for the zero-field ground-state energy by taking temperature to zero.
PQC lens (via Family 234): Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
View manuscript on GitHub → Open PDF (raw) →M448 · Family 235 A Limiting Satisfiability Threshold for Every Fixed Clause Size
For every fixed integer k ≥ 3, random k-SAT with independent uniformly signed clauses on distinct variables, sampled with replacement, has a finite positive limiting satisfiability threshold. We credit Gaia Carenini [[5]](https://eccc.weizmann.ac.il/report/2026/229/) with priority for resolving the satisfiability conjecture. This paper gives an alternative proof, using concentration of a capped last satisfiable index and a comparison between different system sizes.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M449 · Family 235 Linear Variance of the Random 3-SAT Hitting Time
For random 3-SAT on n Boolean variables, with independent uniformly signed clauses on three distinct variables sampled with replacement, we prove that the first unsatisfiable prefix has variance ` (n)`. The upper bound removes the logarithmic loss in the earlier variance estimate; the matching lower bound follows from Wilson's transition-width theorem.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M450 · Family 235 Variance of the Random k-SAT Hitting Time
Let Hn be the index of the first unsatisfiable prefix in random k-SAT on n variables, with independent uniformly signed clauses using k distinct variables and sampled with replacement. For every fixed k ≥ 4, we prove ` Var (H_n)= Theta_k(n)`. For k = 3, the variance is bounded below by a positive multiple of n and above by a constant multiple of `nlog n`; the companion paper on random 3-SAT sharpens this to ` (n)`.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M451 · Family 235 Computing the Random 3-SAT Threshold
The limiting satisfiability threshold of uniform random 3-SAT is a computable real. We credit Gaia Carenini [[4]](https://eccc.weizmann.ac.il/report/2026/229/) with priority for resolving the satisfiability conjecture, which establishes the threshold's existence. We prove that one finite deterministic machine can approximate it to any prescribed accuracy. A deletion estimate gives explicit lower certificates, while a finite hierarchical approximation of the soft pressure gives upper certificates at every larger rational density.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M452 · Family 236 The sharp factor-of-IID threshold for the free Ising model on regular trees
For every integer d ≥ 3 and inverse temperature β ≥ 0, the free zero-field ferromagnetic Ising measure on the infinite d-regular tree is a factor of IID exactly when `tanhβ≤(d-1)⁻¹/²`. We prove the positive implication, including equality, resolving the conjecture of Nam, Sly and Zhang; the strict converse is known. The spin factor can be chosen to commute with every tree automorphism on every label input.
PQC lens (via Family 236): Quantum many-body and statistical-physics results inform error correction, randomness, and hardware noise models.
View manuscript on GitHub → Open PDF (raw) →M453 · Family 237 Radial transfer estimates and polygon length laws for honeycomb walks
We prove critical diameter-tail exponents −2 for unrooted honeycomb polygons and −2/3 for length-weighted polygons, together with a truncated second-length-moment bound of exponent 2/3. Consequently, polygons conditioned to have length at least n have diameter `n³/⁴⁺o(¹)` in probability under either weight.
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View manuscript on GitHub → Open PDF (raw) →M454 · Family 237 Critical honeycomb chords with prescribed boundary endpoints
Critical self-avoiding walks between prescribed, macroscopically separated boundary ports of a regular honeycomb hexagon have length `R⁴/³⁺o(¹)` in probability, where R is the scale of the hexagon. We prove the corresponding statements for half-plane arches, parallel cuts and nonparallel pure cuts. The half-plane law also has mean length `R⁴/³⁺o(¹)`. A separate strip argument gives endpoint mean laws on one density-one set of heights and in an aligned, critically weighted mixture of all even heights in a macroscopic interval.
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View manuscript on GitHub → Open PDF (raw) →M455 · Family 237 Cylinder loop weights and planar nesting
We determine the growth exponent, at every fixed positive loop fugacity, of the critical honeycomb partition function for disjoint polygons separating two prescribed markers on a balanced cylinder. At fugacity two the cylinder exponent is 1/6; the corresponding planar nesting exponent and middle-strip nesting exponent are 1/12.
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View manuscript on GitHub → Open PDF (raw) →M456 · Family 237 Mass and covering exponents for fixed-length honeycomb walks
For uniform self-avoiding walks of every sufficiently large integer length on the honeycomb lattice, we prove diameter exponent 3/4 and simultaneous local-mass and covering exponent 4/3, with arbitrary positive exponent slack and arbitrary polynomial failure probability.
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View manuscript on GitHub → Open PDF (raw) →M457 · Family 237 Signed cylinder propagation and marked polygons on the honeycomb lattice
At the critical activity and loop fugacity two, we prove a 1/6 partition exponent for disjoint honeycomb polygons separating opposite marks on a balanced infinite cylinder, and a 1/12 planar nesting exponent. For ordered first-exit chords in a regular hexagon of side R, with both boundary ports summed and diameter at least `R/10`, the finite critical partition is `R³/⁴⁺o(¹)` and the normalized mean length is `R⁴/³⁺o(¹)`.
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View manuscript on GitHub → Open PDF (raw) →M458 · Family 237 Cylinder amplitudes and logarithmic bridge-length windows on the honeycomb lattice
We prove a logarithmic window of critical honeycomb bridge lengths: summed through height `2h(log h)¹/⁶⁴`, bridges of lengths between `h⁴/³(log h)⁻¹/⁸` and `2h⁴/³(log h)¹/²` have mass at least `h³/⁴(log h)⁻C`.
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View manuscript on GitHub → Open PDF (raw) →M459 · Family 237 Marked polygon correlations and one-arc bounds
At the critical honeycomb vertex activity, the squared-length mass of simple polygons of diameter at most H, counted modulo translations, is at most `H²/³⁺o(¹)`. We also prove a quantitative two-mark cylinder estimate and a polynomial one-arc bound uniform even for arbitrarily unequal marked intervals.
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View manuscript on GitHub → Open PDF (raw) →M460 · Family 237 Disk transfer representations and confined bridge mass
We prove that critical honeycomb bridges crossing a strip of width R have mean length `R⁴/³⁺o(¹)`, with the same exponent after confinement to a fixed multiple of the strip width. The initial boundary port is fixed, the terminal port is summed, and all finite lengths receive their critical weights. The bridge mass is comparable to R−1/4; the half-plane arch kernel at endpoint separation m is comparable to m−5/4.
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View manuscript on GitHub → Open PDF (raw) →M461 · Family 237 Polynomial vacuum representations and bridge mass for honeycomb walks
We prove that, under the all-length critical measure, a honeycomb bridge crossing a strip of h layers, with its initial port fixed and its terminal port free, has mean length `h⁴/³⁺o(¹)`. We also obtain central visit probability `R⁻²/³⁺o(¹)` and mean length `R⁴/³⁺o(¹)` for macroscopic free-boundary chords in a regular hexagon.
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View manuscript on GitHub → Open PDF (raw) →M462 · Family 237 Renewal and changes of law for critical honeycomb walks
For critical honeycomb self-avoiding walk, we prove spatial exponent 3/4 for the infinite irreducible-bridge law and for bridges of every sufficiently large compatible even length. We establish the corresponding thermal laws at every large discount scale, including all positive length and spatial moments. For unrestricted uniform walks, the endpoint lower law holds on a common set of lengths of natural density one.
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View manuscript on GitHub → Open PDF (raw) →M463 · Family 237 Critical strip-crossing mass on the honeycomb lattice
At the critical weight of the regular honeycomb lattice, the total weight of self-avoiding paths crossing a strip of height N is comparable to N−1/4. The first horizontal-displacement moment of return paths is comparable to N3/4.
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View manuscript on GitHub → Open PDF (raw) →M464 · Family 237 Uniform marked-polygon estimates and sharp finite bridge moments
We prove a uniform bound for critical honeycomb polygons through two axial marks on a periodic staircase. The bound retains an explicit power of the ratio between the period and the marked separation. We also prove sharp finite strip estimates: bridge mass of order h−1/4, first-length mass at most `Ch¹³/¹²`, and mass at least `ch⁻¹/⁴` on bridges with length at least `ch⁴/³`.
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View manuscript on GitHub → Open PDF (raw) →M465 · Family 237 Cap-selected amplitudes and triangle chords for honeycomb walks
At the critical honeycomb fugacity, self-avoiding port-to-port chords in an equilateral lattice triangle of side R, summed over both boundary endpoints and restricted to diameter at least `R/100`, have partition sum comparable to R3/4 and mean length `R⁴/³⁺o(¹)`. We also determine the amplitude selected by two vacuum caps on a cylinder of circumference N and prove that it grows as `N¹/⁶⁺o(¹)`.
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View manuscript on GitHub → Open PDF (raw) →M466 · Family 238 Optimal-order mixing of the Thorp shuffle
We prove that the Thorp shuffle on `2d` cards mixes in ` (d)` complete shuffles. The full permutation law after `1600d` shuffles converges to uniform in total variation as `d→∞`, uniformly over the initial deck, while a support count gives a lower bound of `2d-O(1)`.
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View manuscript on GitHub → Open PDF (raw) →M467 · Family 238 Random coordinate frames and partial permutation laws
For the Thorp shuffle on `2d` cards, we prove that the worst-start total-variation distance after `32800d` physical shuffles tends to zero as `d→∞`. Combining our fixed-list estimate with the companion Fourier transfer improves this bound to `512d` shuffles. These results follow from bounds on partially observed permutation laws in random coordinate frames.
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View manuscript on GitHub → Open PDF (raw) →M468 · Family 238 From partial permutation information to Fourier bounds
We show how information about partial permutations controls full permutation laws. Let n tend to infinity through multiples of eight. If the images of a uniformly chosen `7n/8` labels approach the uniform injection law in average total variation, and the sign mean tends to zero, then the product of two independent permutations with the given law converges to uniform on Sn.
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View manuscript on GitHub → Open PDF (raw) →M469 · Family 238 Conditional information under deterministic coordinate sweeps
We bound the information remaining after paths of specified cards in the Thorp shuffle on `n=2d` cards have been observed. After a fixed number of deterministic coordinate sweeps, the joint endpoint law of further cards is close to uniform on the available positions, on average over the observed paths, provided a fixed positive fraction of labels lies outside both lists. Combined with a Fourier transfer, these bounds give full-deck mixing after `2048d` physical shuffles.
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View manuscript on GitHub → Open PDF (raw) →M470 · Family 238 Conditional permutations in a revealed switching environment
Fix half the labels in a Thorp shuffle on `2d` positions and reveal their complete trajectories. We prove that, after an explicit absolute number of coordinate sweeps, the conditional permutation of the remaining labels approaches uniform in expected total variation as `d→∞`, uniformly in the initial layout. The resulting constructions give full-deck mixing in a constant multiple of d physical shuffles.
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View manuscript on GitHub → Open PDF (raw) →M471 · Family 238 Routing densities and representation contraction for Thorp sweeps
For `N=2d` cards, we prove that an absolute number of coordinate sweeps of the Thorp shuffle brings the full permutation law to total-variation distance tending to zero from uniform. The mixing time therefore has optimal order ` (log N)` in physical shuffles.
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View manuscript on GitHub → Open PDF (raw) →M472 · Family 238 Row–column symmetry and contraction of coordinate sweeps
We prove that the Thorp shuffle on `N=2d` labeled cards has full-permutation total-variation mixing time ` (log N)` in physical shuffles. An absolute number of coordinate sweeps suffices for the upper bound.
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View manuscript on GitHub → Open PDF (raw) →M473 · Family 238 Random-subspace tests and trace smoothing for coordinate sweeps
We prove that an absolute number of coordinate sweeps of the Thorp shuffle on `n=2d` positions brings the full permutation to total-variation distance at most ` frac12n⁻⁵` from uniform, uniformly over the initial deck. Thus `O(d)` physical shuffles suffice, which is optimal in order.
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View manuscript on GitHub → Open PDF (raw) →M474 · Family 238 Compatibility entropy and the spectrum of a Thorp sweep
We prove that the Thorp shuffle on `n=2d` cards mixes in ` (d)` physical shuffles. After a sufficiently large fixed number of coordinate sweeps, the total-variation distance of the full permutation from uniform tends to zero as `d→∞`.
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View manuscript on GitHub → Open PDF (raw) →M475 · Family 238 Signed tensor densities and diagram budgets for the Thorp shuffle
We prove that a fixed number of coordinate sweeps of the Thorp shuffle on `2d` cards makes the squared L2 distance of the full permutation density from uniform tend to zero as `d→∞`. In particular, the total-variation mixing time is ` (d)` physical shuffles.
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View manuscript on GitHub → Open PDF (raw) →M476 · Family 238 Conditional coordinate sweeps and analytic transfer
For coordinate sweeps with near-uniform line laws and line sizes among powers of two in a suitable fixed interval, we prove a Schatten-moment bound after conditioning on any feasible collection of prescribed card trajectories. An analytic transfer to binary sweeps then shows that a fixed number of sweeps mixes the Thorp shuffle on `2d` cards in `O(d)` physical steps, matching the order of the support lower bound.
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View manuscript on GitHub → Open PDF (raw) →M477 · Family 238 A strict four-row permanent inequality and permutation moments
We prove a permanent inequality with exponent strictly below two for laws on S4 sufficiently close to uniform and with exactly uniform coordinate marginals. Applied to a four-row recursion, it shows that an absolute number of coordinate sweeps brings the full permutation of the Thorp shuffle on `N=2d` cards to total-variation distance tending to zero from uniform as `N→∞`. Thus the mixing time has the optimal order `O(log N)` in physical shuffles.
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View manuscript on GitHub → Open PDF (raw) →M478 · Family 239 The sharp exponential rate of singularity for symmetric Bernoulli matrices
Let An be a symmetric `n× n` matrix whose entries on and above the diagonal are independent uniform signs. We prove ` (det A_n=0)=≤ft( frac12+o(1) )n.`
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M479 · Family 239 The sharp singularity rate for biased symmetric sign matrices
Let An be a symmetric random matrix whose entries on and above the diagonal are independent signs, equal to 1 with fixed probability `p (0,1) 1/2 `. We prove that ` P(det A_n=0)=(p²⁺(¹⁻p)²⁺o(¹)n`. The rate is attained by the event that two rows agree.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M480 · Family 240 A CH obstruction to a prescribed categoricity threshold
Assuming the continuum hypothesis, we construct an abstract elementary class with Löwenheim–Skolem number ℵ0 that is categorical in every sufficiently large cardinal but has at least two nonisomorphic models of cardinality ` (ω_2)`. Thus categoricity does not transfer down to the proposed bound ` ((2 aleph_⁰)^⁺`, which equals ` (ω_2)` under CH. Consequently, if ZFC is consistent, the prescribed-threshold form of Shelah's categoricity conjecture is not provable in ZFC.
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View manuscript on GitHub → Open PDF (raw) →M481 · Family 240 Eventual categoricity for abstract elementary classes
We prove Shelah's eventual categoricity conjecture for abstract elementary classes in ZFC. For each infinite bound on the Löwenheim–Skolem number there is a uniform threshold such that categoricity in any one cardinal at or above that threshold implies categoricity in every cardinal at or above the same threshold.
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View manuscript on GitHub → Open PDF (raw) →M482 · Family 241 Rigidity of the Turing degrees
We prove that every order automorphism of the full partial order of Turing degrees is the identity, resolving the rigidity conjecture for the Turing degrees positively.
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View manuscript on GitHub → Open PDF (raw) →M483 · Family 242 Single-fold Diophantine representations
Every recursively enumerable set of natural-number tuples has a polynomial Diophantine representation with exactly one complete auxiliary tuple for each member. This proves the single-fold conjecture and, consequently, the finite-fold conjecture.
PQC lens (via Family 242): Changes hardness or approximation barriers that underpin cryptographic reductions and parameter selection.
View manuscript on GitHub → Open PDF (raw) →M484 · Family 243 Choiceless polynomial time with counting does not capture polynomial time
We prove that choiceless polynomial time with counting does not capture polynomial time on unordered finite structures, confirming the noncapture conjecture of Blass, Gurevich and Shelah. A linear-consistency query over 𝔽3 in a fixed binary vocabulary is decidable in polynomial time but not in the full counting formalism.
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View manuscript on GitHub → Open PDF (raw) →M485 · Family 243 Witnessed symmetric choice is strictly stronger than choiceless polynomial time with counting
We prove that witnessed symmetric choice strictly increases the expressive power of choiceless polynomial time with counting. A fixed sentence with one witnessed-choice occurrence defines a Boolean query on every finite input that is not definable in the original counting formalism.
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View manuscript on GitHub → Open PDF (raw) →M486 · Family 244 The Partition Principle does not imply Choice
We prove that the Partition Principle does not imply the Axiom of Choice: if ZF is consistent, then so is ZF with the Partition Principle, Choice for ordinal-indexed families, and the negation of the Axiom of Choice. Separately, over every countable transitive model of ZFC, we construct a transitive symmetric model of this theory with the same ordinals and no new countable sequences of ground elements.
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View manuscript on GitHub → Open PDF (raw) →M487 · Family 245 Weak and strong normalization in pure type systems
We prove that every weakly β-normalizing pure type system is strongly β-normalizing. Both properties quantify over all legal expressions in all valid contexts, and reduction acts inside type annotations. No functionality hypothesis is required. This resolves the β-Barendregt–Geuvers–Klop conjecture.
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View manuscript on GitHub → Open PDF (raw) →M488 · Family 246 A Modulus Proof of Cannon’s Conjecture
We prove that every hyperbolic group whose boundary is homeomorphic to the two-sphere admits a proper cocompact isometric action on hyperbolic three-space with finite kernel. This resolves Cannon's conjecture positively.
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View manuscript on GitHub → Open PDF (raw) →M489 · Family 247 An infinite finitely presented residually finite 2-group
We prove that the infinite, ordinarily finitely presented periodic Steinberg group ` = St (12)(R)` of a companion paper is residually finite. Its finite-index subgroup `G= ( → St (12)( F_2))` is infinite, ordinarily finitely presented, and residually finite, and every element of G has finite 2-power order. The orders of its elements are unbounded. Phases of long words in the companion's graded algebra allow finite degree truncations to detect all elements of Γ, including the central kernel.
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View manuscript on GitHub → Open PDF (raw) →M490 · Family 247 An infinite finitely presented periodic group
We construct an infinite group with an ordinary finite presentation in which every element has finite order, answering the finitely presented Burnside question negatively. We also construct an infinite-dimensional finitely presented nonunital nil associative algebra over 𝔽2 that is Jacobson radical but not nilpotent. Its unitization is finitely presented, algebraic, and infinite-dimensional.
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View manuscript on GitHub → Open PDF (raw) →M491 · Family 248 Thompson's group F is nonamenable
We prove that Thompson's group F is nonamenable. This confirms Geoghegan's conjecture and resolves the amenability problem for F.
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View manuscript on GitHub → Open PDF (raw) →M492 · Family 249 A finitely generated counterexample to the Eilenberg–Ganea conjecture
We construct a finitely generated residually finite group of integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture.
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View manuscript on GitHub → Open PDF (raw) →M493 · Family 250 Finite algebraic envelopes and the Boone–Higman conjecture
We prove the Boone–Higman conjecture. A finitely generated group has decidable word problem if and only if it embeds in a finitely presented simple group.
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View manuscript on GitHub → Open PDF (raw) →M494 · Family 250 Simple F∞ overgroups of groups with decidable word problem
Every finitely generated group with decidable word problem embeds in a simple group of type F∞. This resolves the higher-finiteness strengthening of the Boone–Higman conjecture.
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View manuscript on GitHub → Open PDF (raw) →M495 · Family 250 A universal group of type F∞
We construct a single group of type F∞ containing every finitely presented group. Its finitely generated subgroups, up to isomorphism, are exactly the finitely generated recursively presented groups. This answers the F∞ form of the higher-dimensional Higman embedding question.
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View manuscript on GitHub → Open PDF (raw) →M496 · Family 251 Unitarizability implies amenability for discrete groups
We prove that a discrete group is amenable if and only if every uniformly bounded representation on a complex Hilbert space is similar to a unitary representation. This resolves Dixmier's unitarizability problem affirmatively for discrete groups.
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View manuscript on GitHub → Open PDF (raw) →M497 · Family 251 Strong Ulam Stability Characterizes Amenability
A countable discrete group is amenable if and only if it is strongly Ulam stable: every sufficiently accurate unitary almost representation, on any complex Hilbert space, is uniformly close in operator norm to a genuine representation on the same space. We prove the converse to Kazhdan's amenable stability theorem, answering the question of Burger, Ozawa, and Thom. The inclusion of infinite-dimensional Hilbert spaces is essential.
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View manuscript on GitHub → Open PDF (raw) →M498 · Family 252 A torsion-free hyperbolic group that is not residually finite
We construct a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question for hyperbolic groups negatively.
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View manuscript on GitHub → Open PDF (raw) →M499 · Family 253 An infinite finitely presented simple amenable group
We construct an infinite finitely presented simple amenable group, giving a positive answer to the finite-presentation existence question for simple amenable groups.
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View manuscript on GitHub → Open PDF (raw) →M500 · Family 254 Harmonic heights and the Artin K(pi,1) conjecture
We prove that the standard Salvetti complex of every Artin group with finitely many standard generators is aspherical. This resolves the Artin `K(π,1)` conjecture in finite rank.
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View manuscript on GitHub → Open PDF (raw) →M501 · Family 254 An Artin group with no geometric CAT(0) action
We construct an Artin group on 116 generators that admits no proper, cocompact isometric action on a nonempty proper CAT(0) space. This refutes the CAT(0) conjecture for Artin groups.
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View manuscript on GitHub → Open PDF (raw) →M502 · Family 254 Parabolic intersections in Artin groups
We prove that every intersection of parabolic subgroups of any finite-rank Artin group, with arbitrary finite or infinite Coxeter labels, is parabolic. This resolves the Parabolic Intersection Conjecture affirmatively.
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View manuscript on GitHub → Open PDF (raw) →M503 · Family 255 Quasi-isometric recognition of virtually polycyclic groups
We prove that every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic. This resolves the lattice-recognition conjecture of Eskin, Fisher and Whyte, which allows the ambient solvable Lie group in the conclusion to differ from the one in the hypothesis.
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View manuscript on GitHub → Open PDF (raw) →M504 · Family 256 Nonsingular systems of equations over arbitrary groups
Every finite nonsingular system of equations over an arbitrary group has a simultaneous solution in an overgroup. This proves Howie's conjecture on nonsingular systems.
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View manuscript on GitHub → Open PDF (raw) →M505 · Family 256 The Kervaire theorem for groups
We prove that no free product of a nontrivial group with an infinite cyclic group is normally generated by one element. This resolves the Kervaire conjecture positively.
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View manuscript on GitHub → Open PDF (raw) →M506 · Family 257 A hyperbolic group with no geometric CAT(0) action
We construct a hyperbolic group with a finite classifying space that admits no geometric action on a proper complete CAT(0) space. Consequently, a finite aspherical simplicial complex with a linear combinatorial disk-filling inequality need not have a finite locally CAT(0) homotopy model, and hence need not have a finite locally CAT(−1) model. Here metric models carry geodesic length metrics inducing the given complex topology.
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View manuscript on GitHub → Open PDF (raw) →M507 · Family 258 Baumslag-Solitar-free one-relator groups are hyperbolic
We prove Gersten's conjecture: every finitely generated one-relator group containing no subgroup isomorphic to a Baumslag–Solitar group ` BS (m,n)`, for nonzero integers m, n, is word-hyperbolic.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M508 · Family 258 Virtual compact specialness of hyperbolic one-relator groups
We prove that every word-hyperbolic one-relator group is virtually compact special. Combined with the work of Kielak–Linton, this resolves Wise's virtual free-by-cyclic conjecture for hyperbolic one-relator groups: every such group is virtually free-by-cyclic, with the free kernel allowed to have infinite rank.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M509 · Family 259 A group without fixed price
We construct a finitely generated group with two essentially free probability-measure-preserving actions of different costs. This gives a negative answer to Gaboriau's general fixed-price problem. The group is an amalgam of a free group of rank 100 with a direct product. Its Bernoulli action has cost bounded away from one, whereas finite height extensions have costs tending to one.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M510 · Family 260 The spacetime Penrose inequality with charge and original-data rigidity
Under the stated energy, decay, and trapping hypotheses, we prove the sharp charged spacetime Penrose upper-area inequality for one-ended three-dimensional initial data with source-free electric and magnetic fields. The theorem allows arbitrary second fundamental form, nonzero ADM momentum, and disconnected boundary. Writing m for invariant ADM mass, Q for total charge magnitude, and rA for the minimum-enclosing-area radius, the bound is m ≥ Q and `r_A≤ m+√(m²⁻Q²)`.
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View manuscript on GitHub → Open PDF (raw) →M511 · Family 260 A Charged Reduction of the Spacetime Penrose Inequality in Spatial Dimensions at Least Four
Using the companion neutral spacetime Penrose theorem exactly at its stated strong-decay, future-trapped, positive-area, and future-timelike scope, we prove the sharp purely electric upper-area inequality in every spatial dimension n ≥ 4 for one-ended charged data satisfying the charged dominant energy condition and ` div _g E=0`, with the specified decay, integrability, and finite-flux assumptions. The data may have arbitrary interior topology and ADM momentum, with a future or past trapping sign chosen independently on each boundary component.
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View manuscript on GitHub → Open PDF (raw) →M512 · Family 260 Spacetime Penrose inequalities: enclosing area, charge, and rigidity
We prove sharp spacetime Penrose inequalities for invariant ADM mass and minimum enclosing area in three and four spatial dimensions. Under their respective designated-end conventions, the neutral numerical results allow arbitrary second fundamental form, nonzero momentum, disconnected weakly future trapped boundary, and finitely many ends.
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View manuscript on GitHub → Open PDF (raw) →M513 · Family 260 The Kerr–Newman Penrose Inequality for Axisymmetric Electrovacuum Exteriors
We prove the sharp Kerr–Newman Penrose inequality ` m²≥ A 16π + Q² 2 + π(Q⁴⁺⁴J²) A.` Here `Q²⁼Q_e²⁺Q_b²`, and J is the conserved total angular momentum, including its electromagnetic contribution. The result applies to smooth axisymmetric electrovacuum exteriors in the stated one-ended topological and decay class, with a connected outermost, outer-area-minimizing future marginally outer trapped boundary, Coulomb electromagnetic asymptotics, zero ADM momentum, and the explicitly assumed physical-area condition `A≥4π√(Q⁴⁺⁴J²)`.
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View manuscript on GitHub → Open PDF (raw) →M514 · Family 260 Electromagnetic tails and the Kerr–Newman Penrose inequality
We construct smooth axisymmetric electrovacuum exteriors that violate the Kerr–Newman Penrose inequality when J is the bare gravitational ADM angular momentum and the electromagnetic fields have only `O(r⁻²)` decay, allowing angularly varying leading tails. The examples have zero total electric and magnetic charge even though both electromagnetic fields are nonzero. They have a connected outermost, outer-area-minimizing future marginally outer trapped boundary and lie strictly on the physical area branch.
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View manuscript on GitHub → Open PDF (raw) →M515 · Family 260 The nonmaximal anti-de Sitter Penrose Inequality and original-data rigidity
We prove the spacetime Penrose Inequality for smooth three-dimensional initial-data exteriors with one spherical asymptotically anti-de Sitter end, satisfying the stated decay and integrability assumptions, the dominant energy condition, and a compact weakly future outer trapped boundary. We assume that the hyperbolic metric four-flux is future timelike; its Lorentz norm is the mass, and the area is the infimum over full enclosing cuts.
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View manuscript on GitHub → Open PDF (raw) →M516 · Family 260 The Penrose inequality for maximal asymptotically hyperbolic initial data
We prove the sharp Penrose inequality for three-dimensional maximal asymptotically hyperbolic initial data with spherical conformal infinity. Under the dominant energy condition, the stated decay and integrability assumptions, and a future-timelike mass covector, the invariant mass is bounded below by the Schwarzschild–anti-de Sitter mass associated with the minimum enclosing area of a weakly future outer-trapped boundary. The boundary may be disconnected, and no restriction is imposed on the compact topology.
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View manuscript on GitHub → Open PDF (raw) →M517 · Family 260 A local Penrose inequality for conformal perturbations of Schwarzschild–anti-de Sitter data
We prove the asymptotically hyperbolic Penrose inequality for sufficiently small maximal vacuum conformal perturbations of a positive-mass Schwarzschild–anti-de Sitter exterior, for every fixed decaying transverse-traceless seed and every solution branch satisfying the stated decay and mass assumptions. The bound uses the area of the marginally outer trapped boundary itself; it requires neither outermostness nor outer area-minimization, and no bulk-versus-boundary domination condition.
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View manuscript on GitHub → Open PDF (raw) →M518 · Family 260 The spacetime Penrose inequality and enclosing area
We prove the sharp spacetime Penrose inequality for smooth one-ended initial-data exteriors in every spatial dimension n ≥ 3, under the dominant energy condition, weak future trapping, the stated differentiated decay and positive enclosing area. These hypotheses force the ADM energy-momentum to be future timelike. Area is the infimum over full enclosing cuts in the original metric.
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View manuscript on GitHub → Open PDF (raw) →M519 · Family 260 Conformal flow and the Riemannian Penrose inequality with minimizing frontiers
We give a detailed conformal-flow proof of the numerical Riemannian Penrose inequality in every dimension n ≥ 3 for complete asymptotically flat exteriors with nonnegative scalar curvature and full compact frontiers that are outer minimizing and locally perimeter minimizing. The metric extends smoothly through the possibly singular frontier; neither spin nor frontier connectedness is assumed. The proof follows the conformal-flow approach of Bray, Bray–Lee, and Bi–Zhu.
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View manuscript on GitHub → Open PDF (raw) →M520 · Family 260 Equality and rigidity in the spacetime Penrose inequality
Equality in the spacetime Penrose inequality identifies the original initial data as a spacelike slice of a Schwarzschild–Tangherlini exterior, smoothly attached to its horizon, under the stated outermostness and decay hypotheses. We prove this in every spatial dimension n ≥ 3 for a strong-decay exterior class and for weaker decay in dimensions three and four. In dimension three the finite-component theorem forces a single spherical horizon and gives qualitative strictness for disconnected boundaries.
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View manuscript on GitHub → Open PDF (raw) →M521 · Family 260 Boundary graph deformations for the spacetime Penrose inequality
We construct graph and conformal deformations of trapped initial-data exteriors in three and four spatial dimensions. The resulting metrics have nonnegative scalar curvature, strictly negative inner mean curvature, a lower bound protecting every enclosing cut, and an arbitrarily small upper error in ADM energy. We also give a direct four-dimensional maximal vacuum construction that retains a noncompact decaying second fundamental form. These constructions give boundary routes to the corresponding numerical Penrose inequalities.
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View manuscript on GitHub → Open PDF (raw) →M522 · Family 260 Area-controlled end replacement and the Bondi Penrose inequality in the CKS class
We replace a three-dimensional Cha–Khuri–Sakovich hyperboloidal end by an asymptotically flat end while preserving the dominant energy condition and each fixed compact interior. For strictly future-timelike CKS initial-data charge, the replacements lose asymptotically no enclosing area and their ADM masses tend to the invariant Bondi mass. Combining this construction with the companion numerical spacetime Penrose theorem yields the sharp Bondi bound for possibly disconnected weakly future trapped boundaries in this class.
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View manuscript on GitHub → Open PDF (raw) →M523 · Family 261 Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three
For each fixed dimension d ≥ 3 and each fixed sufficiently small positive disorder strength, we prove that the Anderson operator on ℤd with independent uniform site potentials almost surely has purely absolutely continuous spectrum with nonzero weight on an open energy interval. The interval may depend on d but is independent of the disorder strength. This settles the purely absolutely continuous energy-range question in Simon's Problem 1 for this model.
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View manuscript on GitHub → Open PDF (raw) →M524 · Family 261 Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder
For each fixed positive disorder strength, we prove that the nearest-neighbor Anderson operator on the square lattice with independent uniform site potentials almost surely has pure-point spectral type throughout its spectrum. This resolves the pure-point assertion of the two-dimensional Anderson localization conjecture for the uniform single-site law.
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View manuscript on GitHub → Open PDF (raw) →M525 · Family 262 Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality
We classify all equality cases in the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and `1/2 γ 3/2`. For measurable Hermitian positive semidefinite potentials W with `∫_ R tr (Wγ⁺¹/²) ∞`, equality holds precisely for direct sums, in one constant unitary basis, of scalar one-bound-state solitons and zero channels. The nonzero solitons may have independent scales and centers.
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View manuscript on GitHub → Open PDF (raw) →M526 · Family 262 Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials
We prove the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and every exponent `1/2 γ 3/2`. The optimal constant is the scalar one-bound-state constant, independently of the matrix size. The inequality bounds the full sum of negative eigenvalue moments for every measurable Hermitian positive semidefinite potential W satisfying `∫_ R tr (Wγ⁺¹/²) ∞`. The matrices may have arbitrary rank and need not commute at different points.
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View manuscript on GitHub → Open PDF (raw) →M527 · Family 262 Sharp one-dimensional Lieb–Thirring constants
We resolve affirmatively the remaining cases of the scalar one-dimensional Lieb–Thirring conjecture: for every ` frac12 γ frac32`, the optimal constant is the one-bound-state constant. The estimate holds for every nonnegative potential in `Lγ⁺¹/²( R)`, with all negative eigenvalues included.
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View manuscript on GitHub → Open PDF (raw) →M528 · Family 263 Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms
We prove that a molecule with M fixed nuclei of real charges at least one and total nuclear charge Z strictly binds at most `Z+CM` electrons, where C is universal. The result holds for the full nonrelativistic Coulomb Hamiltonian with two spin states and arbitrary distinct nuclear positions. For neutral atoms with integer nuclear charge Z ≥ 1, the first ionization energy and, for every ground state, the radius outside which one half of an electron remains are each bounded above and below by positive universal constants.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M529 · Family 263 Generalized ionization energies for full Coulomb atoms
We prove the energy part of the generalized ionization conjecture for full nonrelativistic Coulomb atoms with two electron spin states. The energy needed to remove m electrons is asymptotic to `a(TF)m⁷/³` whenever `m→∞` and `Z/m→∞`, with the Thomas–Fermi constant in atomic units. We also obtain both conjectured iterated limits.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M530 · Family 263 Generalized outer-electron radii of neutral Coulomb atoms
We prove the radius part of the generalized ionization conjecture for neutral nonrelativistic Coulomb atoms with two spin states. For every choice of ground states, the upper and lower large-nuclear-charge limits of the radius defined by an expected exterior electron mass m are both asymptotic to `(81π²/2)¹/³m⁻¹/³` as m tends to infinity.
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View manuscript on GitHub → Open PDF (raw) →M531 · Family 264 Generic Future Inextendibility with Square-Integrable Connection Near a Fixed Kerr Spacetime
For each fixed rotating subextremal Kerr background with mass M > 0 and rotation `0 | a| M`, we prove that the smooth vacuum data near its complete two-ended bridge whose full maximal globally hyperbolic development admits a future `C⁰ W¹,²(loc)` extension form a meagre set in the weighted smooth topology. The extension metric is continuous and nondegenerate, and its weak connection is locally square-integrable. The neighborhood requires smallness of only the tenth weighted seminorm, while the topology tests every finite order.
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View manuscript on GitHub → Open PDF (raw) →M532 · Family 264 Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes
For each fixed Kerr spacetime with mass M > 0 and rotation `0 a M`, we prove that a dense Gδ set of nearby smooth, complete two-ended vacuum data has a maximal globally hyperbolic development with no future C1 extension. The neighborhood and genericity are defined in a weighted smooth topology, and ambient extensions may be nonvacuum.
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View manuscript on GitHub → Open PDF (raw) →M533 · Family 264 Quantitative Near-Kerr Evolution and Generic C2 Future Inextendibility
For every fixed rotating subextremal Kerr bridge, we prove that smooth vacuum data admitting a C2 future extension of their full maximal globally hyperbolic development form a meagre set in a neighborhood defined by one finite-order seminorm. The topology allows arbitrary symbol-bounded asymptotically flat tails and imposes no symmetry.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M534 · Family 265 A two-dimensional area law from a global spectral gap
We prove an entropy area law for the unique ground state of a finite-range Hamiltonian on any finite induced subgraph of the square lattice. A lower bound on the spectral gap of the full Hamiltonian and fixed bounds on the local dimension, interaction range, and interaction strength suffice. For every set of sites, its entanglement entropy is bounded by a constant times the number of edges crossing its boundary, independently of the size and shape of the domain.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M535 · Family 265 Polynomial PEPS approximation of gapped square-grid ground states
We prove that the unique ground state of a uniformly gapped nearest-neighbor Hamiltonian on an `L× L` square lattice admits a projected entangled-pair state approximation with bond dimension polynomial in L and global vector error at most L−1 after normalization. Only the gap of the full Hamiltonian is assumed. The result is an existence theorem, with constants uniform over Hamiltonians of fixed local dimension, interaction strength, and gap.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M536 · Family 266 The maximum number of mutually unbiased bases in dimension six
We prove that the maximum number of mutually unbiased orthonormal bases in ℂ6 is three, resolving Zauner's dimension-six MUB conjecture. The upper bound is computer-assisted: under the stated binary64 arithmetic and compiler conditions, a complete execution of the documented verification pipeline excludes four arbitrary complex bases.
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View manuscript on GitHub → Open PDF (raw) →M537 · Family 266 Exact Fourier certificates for complex Hadamard matrices of order six
We prove the Fourier-vanishing conjecture of Matolcsi, Ruzsa, and Weiner: every complex Hadamard matrix of order six outside Tao's cubic equivalence class has a vanishing character sum at every permutation of `(1,1,1,-1,-1,-1)`. We also give an exact certificate excluding seven mutually unbiased bases in ℂ6; by Weiner's completion theorem, this yields an upper bound of five. Both certificates use only integer and rational arithmetic, and the complete verifier is included.
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View manuscript on GitHub → Open PDF (raw) →M538 · Family 267 Bose–Einstein condensation at positive temperature in the dilute hard-sphere gas
We prove Bose–Einstein condensation at positive temperature in the three-dimensional dilute hard-sphere gas. For each fixed exclusion distance and every sufficiently small fixed density, there is a strictly positive temperature, independent of the volume, at which the exact canonical Gibbs state has a positive condensate fraction in the thermodynamic limit. The condensate occupies the constant orbital.
PQC lens (via Family 267): Directly shapes quantum threat models, algorithm timelines, or query-complexity assumptions used in security proofs.
View manuscript on GitHub → Open PDF (raw) →M539 · Family 267 Quantum Depletion and Momentum Distribution in the Dilute Hard-Sphere Bose Gas
We prove the full scaled Bogoliubov momentum distribution for the depleted particles in the three-dimensional hard-sphere Bose gas at zero temperature. The thermodynamic limit is taken at each fixed density before the dilute limit, uniformly over all pure and mixed ground states. All thermodynamic accumulation values have the same dilute asymptotic: the limiting nonzero-momentum occupation measure has total mass `8/(3 sqrtπ)`, giving the depletion fraction ` 8 3 sqrtπ √(ρ a³)⁺o(√(ρ a³)` for density ρ and hard-sphere exclusion distance a.
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View manuscript on GitHub → Open PDF (raw) →M540 · Family 267 Quantum Depletion for Fixed Bounded Repulsive Potentials
We prove the Bogoliubov quantum-depletion asymptotic for the ground state of a three-dimensional Bose gas with a fixed bounded, nonnegative, radial interaction of finite range and positive scattering length a. The thermodynamic limit is taken at each fixed density ρ before the dilute limit. Every thermodynamic accumulation value of the fraction outside the constant mode is ` 8 3 sqrtπ √(ρ a³)⁺o(√(ρ a³)` as `ρ downarrow0`.
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View manuscript on GitHub → Open PDF (raw) →M541 · Family 267 A density-uniform condensate bound for dilute Bose gases
For each fixed bounded measurable, nonnegative, radial interaction of finite range in three dimensions that is not zero almost everywhere, we prove a positive lower bound on the constant-orbital condensate fraction that is uniform over all sufficiently small densities and all temperatures between zero and the square of the density. The interaction and density remain fixed in the thermodynamic limit.
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View manuscript on GitHub → Open PDF (raw) →M542 · Family 267 Ground-state condensation in the dilute hard-sphere gas
We prove Bose–Einstein condensation in every ground state of a dilute three-dimensional hard-sphere Bose gas. At every sufficiently small fixed gas parameter, the constant orbital contains a positive fraction of the particles in the thermodynamic limit. The fraction can be chosen independently of the gas parameter, and the conclusion holds for arbitrary complex ground states.
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View manuscript on GitHub → Open PDF (raw) →M543 · Family 268 The periodic spin-one Haldane gap
We prove a uniform positive spectral gap for the pure antiferromagnetic spin-one Heisenberg chain on even periodic rings, establishing the positive even-periodic formulation of the spin-one Haldane conjecture.
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View manuscript on GitHub → Open PDF (raw) →M544 · Family 268 A boundary-field gap for the spin-one Heisenberg chain
We prove a uniform spectral gap for odd open spin-one antiferromagnetic Heisenberg chains with the same fixed magnetic field at both endpoints. The field `h=3/5` gives a gap greater than `log(10)/392` on every chain of `2L+1` sites with L ≥ 960. Tasaki's index theorem then gives index −1 for every subsequential local limit of these boundary-selected ground states.
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View manuscript on GitHub → Open PDF (raw) →M545 · Family 269 Uniform Stability of the Spherical Laughlin Gap
We prove that the fermionic Laughlin V1 Hamiltonian at filling 1/3 on expanding round spheres retains a unique ground state and a uniform spectral gap under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. With coefficient one for each pair projector and Laughlin flux `q=3(N-1)`, the gap and perturbation threshold are uniform for all sufficiently large particle numbers and all potential profiles of supremum norm at most one.
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View manuscript on GitHub → Open PDF (raw) →M546 · Family 269 A Fock-space inequality and the Laughlin spectral gap
We prove the spherical fermionic Laughlin spectral-gap conjecture for the full V1 interaction. With coefficient one for each pair projector, the gap above the Laughlin state at filling 1/3 on the round sphere is at least 1/25 for all sufficiently large systems. More generally, `H_Q²≥γ H_Q` for some fixed `γ 1/25` on the entire lowest-Landau-level Fock space for all sufficiently large flux Q, independently of particle number.
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View manuscript on GitHub → Open PDF (raw) →M547 · Family 270 The unique threshold bound state of the SU(N) BFSS model
For every finite N ≥ 2, we prove that undeformed ` SU (N)` BFSS matrix quantum mechanics has exactly one normalizable zero-energy state after removing the center of mass. This establishes the threshold-bound-state conjecture.
PQC lens (via Family 270): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M548 · Family 270 Positive eigenvalues of the relative SU(2) BFSS Hamiltonian
We prove that the relative ` SU (2)` BFSS Hamiltonian has infinitely many positive eigenvalues tending to infinity, with square-integrable eigenvectors. The operator is defined by closing the gauge-invariant supercharge form. This refutes, at N = 2, the exclusion of normalizable positive-energy states stated in the original BFSS paper.
PQC lens (via Family 270): Touches integer factorization, elliptic-curve, or analytic-number-theory assumptions behind classical public-key cryptography.
View manuscript on GitHub → Open PDF (raw) →M549 · Family 271 Bloch's Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions
We prove Bloch's T3/2 law for the spontaneous magnetization of three-dimensional quantum Heisenberg ferromagnets at every fixed positive quantum spin. The result holds for every nonnegative symmetric finite-range interaction whose support generates ℤ3, including spatially anisotropic couplings. The leading magnetization deficit has the exact coefficient determined by the determinant of the quadratic one-magnon dispersion.
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View manuscript on GitHub → Open PDF (raw) →M550 · Family 271 The first lattice correction to Bloch's law
We prove the first lattice correction to Bloch's law for the three-dimensional nearest-neighbor quantum Heisenberg ferromagnet at every fixed spin `S= tfrac12,1, tfrac32, `. The spontaneous-magnetization deficit agrees with the full ideal-magnon density up to `o(β⁻⁵/²)`. In addition to the leading Bloch term, this gives the correction `3ζ(5/2)(β S)⁻⁵/²/(128π³/²)`. The magnetization is the right derivative at zero field of the thermodynamic pressure; the volume limit precedes the field derivative, and the low-temperature limit is taken last.
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View manuscript on GitHub → Open PDF (raw) →M551 · Family 271 The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet
We prove the spherical magnetization law for the three-dimensional nearest-neighbor isotropic quantum Heisenberg ferromagnet at every fixed positive quantum spin and every sufficiently low fixed positive temperature. As the even periodic cubes grow, the symmetric zero-field magnetization converges in moments to a uniform direction with a deterministic positive magnitude. This magnitude equals the right derivative at zero field of the infinite-volume pressure.
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View manuscript on GitHub → Open PDF (raw) →M552 · Family 271 Spontaneous magnetization in the quantum Heisenberg ferromagnet
For every dimension d ≥ 3 and every spin `S frac12,1, frac32, `, we prove that the nearest-neighbor isotropic quantum Heisenberg ferromagnet has a translation-invariant, spontaneously magnetized equilibrium state at every sufficiently low positive temperature. The same state satisfies the KMS condition for the zero-field dynamics and has magnetization at least `S/4`. This resolves the low-temperature ordering problem in the spontaneous-magnetization formulation.
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View manuscript on GitHub → Open PDF (raw) →M553 · Family 272 Entanglement with zero distillable secret key in local dimension ten
We construct an entangled state on ` C¹⁰ C¹⁰` with zero distillable secret key for the local-instrument protocols specified here. They allow joint processing of all copies and unlimited two-way public communication, with the input as the only shared private resource and an eavesdropper holding a purification and the complete public record. The construction also disproves the unrestricted two-map PPT-composition conjecture. A separate trace-preserving PPT channel on `M(21)( C)` has a square that is not entanglement breaking.
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View manuscript on GitHub → Open PDF (raw) →M554 · Family 273 The entropy photon-number inequality
We prove the entropy photon-number inequality for two independent bosonic inputs with finite mean energy, for every finite number of modes. This resolves the entropy photon-number conjecture in this finite-energy setting while allowing arbitrary entanglement among the modes within either input. As a consequence, we determine the exact minimum output entropy at fixed input entropy for every finite tensor power of an identical thermal attenuator, over finite-energy inputs that may be entangled across modes.
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View manuscript on GitHub → Open PDF (raw) →M555 · Family 274 Product-projection localization and the QAC0 parity lower bound
We prove that constant-depth quantum circuits with arbitrary one-qubit and unbounded-arity Toffoli gates cannot compute parity with any fixed positive worst-case advantage using polynomially many qubits. Ancillas start in zero, only one output qubit is measured, and all final garbage is unrestricted. This resolves Moore's parity conjecture in the measured-output model.
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View manuscript on GitHub → Open PDF (raw) →M556 · Family 274 Regular trajectories, pruning and quantum parity
We prove that constant-depth quantum circuits with arbitrary one-qubit gates and unbounded-arity Toffoli gates cannot compute parity with any fixed positive worst-case advantage using polynomially many total qubits. Ancillary qubits are initialized to `|0 `, one output qubit is measured, and all other final registers may be discarded without restriction. This resolves Moore's parity conjecture in the measured-output model.
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View manuscript on GitHub → Open PDF (raw) →M557 · Family 275 Continuum Coulomb hardness with binary nuclear charges
We prove that approximating the electronic Coulomb spectral infimum in three-dimensional space is QMA-hard when positive integer nuclear charges are encoded in binary. The nuclei have distinct rational positions, the electron number is unary, and the energy is minimized over all antisymmetric continuum states and spin sectors. A deterministic classical polynomial-time reduction produces instances with threshold separation at least one. The nuclear charges may be exponentially large, but every output has polynomial bit length.
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View manuscript on GitHub → Open PDF (raw) →M558 · Family 275 QMA-hardness of continuum Coulomb energy with unit nuclear charges
We prove that approximating the electronic ground-energy infimum for clamped unit-charge nuclei is QMA-hard on the full spinful fermionic continuum space. A deterministic classical polynomial-time reduction produces polynomially many nuclei at distinct rational positions and polynomially many electrons, with polynomial rational bit lengths and threshold separation at least one. No orbital basis, magnetic field, or additional external potential is supplied, and no binding assumption is imposed.
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View manuscript on GitHub → Open PDF (raw) →M559 · Family 276 Classical capacity and entropy inequalities for generalized amplitude damping
We determine the unassisted classical capacity of every qubit generalized amplitude-damping channel, for all damping strengths and thermal occupations. It equals the one-shot Holevo capacity and is given by a one-variable maximum. A binary pure-state ensemble attains the one-use optimum, and its independent products attain the optimum at every block length.
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View manuscript on GitHub → Open PDF (raw) →M560 · Family 277 Threshold parallel repetition for finite-dimensional entangled games
For every finite two-player game with entangled value v < 1, we prove exponential decay for the probability of winning at least a `v+δ` fraction of k independent repetitions, uniformly over all finite-dimensional joint strategies. The result allows arbitrary correlated question distributions and holds for every `0 δ 1-v` and k ≥ 1. Its universal rate is proportional to `δ⁵/(1+log(| A|| B|))`, where ` A, B` are the answer alphabets. For each fixed question distribution, we also obtain an explicit cubic rate in δ.
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View manuscript on GitHub → Open PDF (raw) →M561 · Family 278 A Coulomb ground-state density without Kohn-Sham ensemble representation
We construct a finite three-electron Coulomb molecule whose spin-summed ground-state density cannot be reproduced by any ground-state ensemble of noninteracting electrons in a single real, spin-independent local potential in `L³/²( R³)⁺L^∞( R³)`. The molecule has two equal positive integer nuclear charges, specified by an exact finite nonnumerical formula.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M562 · Family 279 Exact quantum factoring over a fixed finite gate set
We give a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer N ≥ 2 with probability one. A fixed finite set of bounded-arity gates suffices, and both the gate count and the number of qubits have polynomial worst-case bounds in the input length.
PQC lens (via Family 279): Directly shapes quantum threat models, algorithm timelines, or query-complexity assumptions used in security proofs.
View manuscript on GitHub → Open PDF (raw) →M563 · Family 280 Strongly rational unitary vertex operator algebras and conformal nets
Every simple unitary strongly rational vertex operator algebra generates a completely rational conformal net. All its simple grading-restricted modules are unitarizable, its canonical fusion forms are positive, and the Carpi–Weiner–Xu functor gives a braided unitary tensor equivalence from its grading-restricted finite-length module category onto the finite-index sectors of the net.
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View manuscript on GitHub → Open PDF (raw) →M564 · Family 281 QAOA attains the SK ground-state energy in the thermodynamic-first limit
We prove that the Quantum Approximate Optimization Algorithm (QAOA) approaches the ground-state energy per spin of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity first and circuit depth then increases. For every accuracy, some finite depth and deterministic angles, independent of system size and disorder, achieve that accuracy in the limiting expected energy per spin using the standard cost Hamiltonian and transverse-field mixer.
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View manuscript on GitHub → Open PDF (raw) →M565 · Family 281 Full support of the zero-temperature Sherrington-Kirkpatrick order parameter
We prove that every admissible integrable minimizer of the zero-temperature Parisi functional for the pure, zero-field Sherrington–Kirkpatrick model has full relative Stieltjes support on `[0,1)`. Thus its support has no gaps at any overlap scale below one. We use the covariance normalization ` (t)=t²/2`.
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View manuscript on GitHub → Open PDF (raw) →M566 · Family 282 Scale and conformal symmetry in four-dimensional operational quantum field theory
We prove that scale symmetry implies local conformal symmetry for a class of four-dimensional unitary, positive-energy quantum field theories with a Poincaré- and scale-invariant vacuum. The class has a discrete, bounded-below spectrum of scaling dimensions with finite multiplicities, and each original field has finite scaling support.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M567 · Family 283 Polynomial-Time Unitary Synthesis from a Boolean Oracle
We give a positive answer to the constant-error formulation of the Aaronson–Kuperberg unitary synthesis problem. For every n, a quantum oracle circuit generated in polynomial time from n alone can approximate the channel of every n-qubit unitary to full diamond-norm error at most 1/2, after a suitable Boolean oracle is chosen. Using the fixed gates `H,T,T^,CNOT`, the circuit has polynomially many qubits, elementary gates, and oracle calls, and its oracle queries have polynomial length. The oracle may depend on the target unitary;
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M568 · Family 284 A Nearly Quartic Separation Between Randomized and Quantum Query Complexity
We construct total Boolean functions with a nearly quartic separation between bounded-error randomized and quantum query complexity. Writing these complexities as ` R(f)` and ` Q(f)`, the examples rule out every universal bound ` R(f)=O((1+ Q(f))^α)` with α < 4. Thus the known quartic upper bound has the optimal exponent, disproving the conjectured cubic bound. Both complexities count worst-case bit queries with error at most 1/3 on every input.
PQC lens (via Family 284): Directly shapes quantum threat models, algorithm timelines, or query-complexity assumptions used in security proofs.
View manuscript on GitHub → Open PDF (raw) →M569 · Family 285 A torsion-free counterexample to reduced Baum–Connes injectivity
We disprove rational injectivity of the coefficient-free reduced Baum–Connes assembly map for torsion-free groups. Specifically, we construct a finitely generated torsion-free discrete group whose degree-zero assembly map has a kernel class of infinite order.
PQC lens (via Family 285): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M570 · Family 285 A torsion-free counterexample to the Kadison–Kaplansky projection conjecture
We construct a finitely generated torsion-free discrete group whose reduced group C∗-algebra contains a projection other than zero and the identity. This disproves the Kadison–Kaplansky projection conjecture.
PQC lens (via Family 285): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M571 · Family 285 An irrational-trace counterexample to reduced Baum–Connes
We construct a finitely generated discrete group whose reduced group C∗-algebra contains a projection of irrational canonical trace. Its K0-class lies outside the image of the coefficient-free reduced Baum–Connes assembly map. This disproves the coefficient-free reduced Baum–Connes conjecture for countable discrete groups.
PQC lens (via Family 285): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M572 · Family 286 Arithmeticity of twisted finite correspondences for lattices over local fields
We prove arithmetic exhaustion for bifinite correspondences between arbitrarily scalar-twisted group factors of ICC groups commensurable with property- `(T)` lattices over characteristic-zero local fields and twisted group factors of arbitrary countable ICC groups. Every such correspondence is a closed summand of a finite direct sum of models obtained from actual finite-index subgroup isomorphisms and finite-dimensional projective representations of the matched cocycle ratio.
PQC lens (via Family 286): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M573 · Family 286 The arithmetic category and stable recovery of lattice factors
We prove stable canonical recovery for scalar-twisted group factors of ICC groups commensurable with property- `(T)` lattices over characteristic-zero local fields. Every specified stable isomorphism to a scalar-twisted factor of any countably infinite ICC group recovers the group, the cocycle up to a normalized cochain, and equal amplification scales, with an implementing partial isometry in the given matrix corners.
PQC lens (via Family 286): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M574 · Family 287 An isomorphism of the free group factors
We solve the free group factor isomorphism problem affirmatively by proving that `L( F_2)` and `L( F_3)` are isomorphic as tracial von Neumann algebras. The classical free group factor alternative then implies that all interpolated free group factors, including `L( F_∞)`, are isomorphic and have fundamental group ` R( 0)`.
PQC lens (via Family 287): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M575 · Family 288 Kadison's similarity theorem through uniform derivation estimates
We prove that every bounded complex-linear unital algebra homomorphism from a unital complex C∗-algebra into the bounded operators on an arbitrary Hilbert space is similar to a `*` -homomorphism. This resolves Kadison's similarity conjecture positively. We also obtain one universal hyperreflexivity constant for all unital von Neumann algebras on arbitrary complex Hilbert spaces.
PQC lens (via Family 288): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M576 · Family 289 Universal strong Kadison–Kastler stability
We prove that sufficiently close unital von Neumann algebras are conjugate by a unitary arbitrarily close to the identity. The tolerance depends only on the prescribed distance of that unitary from the identity, uniformly over all algebras, representations, and Hilbert spaces. This resolves the strong Kadison–Kastler conjecture.
PQC lens (via Family 289): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M577 · Family 289 Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings
We give a negative answer to the unrestricted small-spatial-embedding problem for one-sided near inclusions of von Neumann algebras. On separable complex Hilbert spaces, we construct pairs of unital von Neumann algebras with common identity whose one-sided gaps tend to zero. Spatial embeddings of the source into the target exist, but every implementing unitary stays a fixed positive distance from the identity. The obstruction therefore concerns small implementing unitaries, rather than the existence of spatial embeddings.
PQC lens (via Family 289): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M578 · Family 289 Close Separable C*-Algebras Without Spatial Conjugacy
We disprove the separable C∗-algebraic spatial form of the Kadison–Kastler conjecture. For every ε > 0, we construct unital, norm-separable C∗-algebras on a common separable complex Hilbert space, with the same identity and Kadison–Kastler distance less than ε, that are not conjugate by any unitary. The two algebras have the same von Neumann closure.
PQC lens (via Family 289): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M579 · Family 290 Expected amenable subalgebras preserving core commutants
We prove that every inclusion `N M` of von Neumann algebras with separable preduals and a faithful normal conditional expectation contains an expected amenable subalgebra `P N` such that `P' c(M)=N' c(M)`, where `c(M)` denotes the continuous core of M. This resolves the relative bicentralizer conjecture in this setting.
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View manuscript on GitHub → Open PDF (raw) →M580 · Family 290 Bounded recovery for modular spectral averages
We prove a bounded spectral recovery theorem for a von Neumann algebra with a faithful normal state whose centralizer consists only of scalars. A positive averaged squared norm for an arbitrary bounded Hilbert-space operator on shrinking modular spectral bands can be recovered on uniformly bounded algebra elements with shrinking spectral support. We apply this theorem to spectral intertwining rigidity and obtain an alternative proof of bicentralizer triviality for type III1 factors with separable predual.
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View manuscript on GitHub → Open PDF (raw) →M581 · Family 291 Equivariant Jiang–Su Stability for Amenable Actions in the Unital Stably Finite Case
Every action of a countable discrete amenable group on a simple, separable, unital, infinite-dimensional, nuclear, stably finite complex C∗-algebra that is already Jiang–Su stable absorbs the trivial action on the Jiang–Su algebra up to cocycle conjugacy. No restriction is imposed on the action on the tracial-state simplex. This proves the unital, stably finite case of Szabó's Conjecture A on automatic equivariant Jiang–Su stability.
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View manuscript on GitHub → Open PDF (raw) →M582 · Family 291 Cuntz comparison and Jiang–Su absorption
We prove that strict comparison in the extended-functional sense implies Jiang–Su absorption for separable simple nuclear non-elementary C∗-algebras. This resolves the corresponding implication of the Toms–Winter regularity problem, including nonunital algebras and allowing unbounded traces. More generally, every separable nuclear C∗-algebra whose Cuntz semigroup is almost unperforated and fully almost divisible absorbs the Jiang–Su algebra.
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View manuscript on GitHub → Open PDF (raw) →M583 · Family 291 Nuclear dimension and Jiang–Su stability without elementary subquotients
For separable nuclear C∗-algebras with no nonzero elementary ideal subquotients, finite nuclear dimension is equivalent to Jiang–Su stability. More generally, `dim(nuc)(A_0 Z)≤1` for every separable nuclear A0. This proves Robert and Tikuisis's Conjecture (C1) and the nuclear-dimension equivalence in the nonsimple Toms–Winter regularity question.
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View manuscript on GitHub → Open PDF (raw) →M584 · Family 291 Tracial projection methods and uniform property Gamma
For a simple, separable, unital, infinite-dimensional, nuclear, stably finite C∗-algebra with traces, real rank zero of the uniform tracial ultrapower of its uniform tracial completion implies uniform property Γ. This answers Problem XXI of Schafhauser, Tikuisis and White affirmatively. Independently, strict comparison implies Jiang–Su absorption for simple, separable, unital, infinite-dimensional nuclear algebras, resolving the unital Toms–Winter conjecture.
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View manuscript on GitHub → Open PDF (raw) →M585 · Family 292 An explicit obstruction to nuclear norm-ultrapower embeddings
We give a negative answer to Kirchberg's norm-ultrapower embedding problem. We construct an explicit separable unital full group C∗-algebra that admits no unital embedding into Bω, for any nonzero unital nuclear C∗-algebra B and any free ultrafilter ω on ℕ. In particular, it does not embed unitally into ` O_2^ω`.
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View manuscript on GitHub → Open PDF (raw) →M586 · Family 293 Invariant-projection counterexamples for every irrational rotation
For every irrational angle, we construct a continuous nonnegative circle weight with exactly one zero and logarithmic integral `-∞` whose weighted rotation has no nontrivial invariant projection in the associated hyperfinite type II1 factor. This answers negatively the question of Zhu, Fang, and Shi, with the angle prescribed in advance. The resulting operator is nonzero and norm-quasinilpotent.
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View manuscript on GitHub → Open PDF (raw) →M587 · Family 293 Backward intertwiners and a transitive commutant
We give a negative answer to the hyperinvariant-subspace problem by constructing, on every infinite-dimensional separable complex Hilbert space, a nonzero bounded norm-quasinilpotent operator with no nonzero proper closed hyperinvariant subspace. Its commutant is a proper strongly closed unital transitive complex operator algebra.
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View manuscript on GitHub → Open PDF (raw) →M588 · Family 294 A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness
We refute Kaplansky's quasitrace conjecture by constructing a separable unital complex C∗-algebra that admits normalized 2-quasitraces but has no tracial state. As a consequence, we show that the minimal tensor product of two unital simple stably finite complex C∗-algebras can be properly infinite, even when one factor is the reduced free-group algebra `C_r*( F_2)`.
PQC lens (via Family 294): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M589 · Family 295 Vanishing of higher bounded Hochschild cohomology
We prove that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with values in the algebra itself, has a bounded primitive. Together with the established degree-one inner-derivation theorem, this resolves the Kadison–Ringrose cohomology conjecture positively.
PQC lens (via Family 295): Advances algebraic structures (factors, lattices, traces) that appear in advanced crypto constructions and side-channel theory.
View manuscript on GitHub → Open PDF (raw) →M590 · Family 296 Relative generation and the generator problem for finite factors
We prove that every type II1 factor with separable predual is generated by one operator, equivalently by two self-adjoint operators. Combined with the established direct-integral reduction to this case, this gives an affirmative solution of the generator problem for von Neumann algebras with separable predual.
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View manuscript on GitHub → Open PDF (raw) →M591 · Family 297 A counterexample to Naimark's problem in ZFC
We construct in ZFC a unital infinite-dimensional simple complex C∗-algebra with a faithful tracial state whose nonzero irreducible representations are all unitarily equivalent. This gives a negative answer to Naimark's problem without additional set-theoretic assumptions.
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View manuscript on GitHub → Open PDF (raw) →M592 · Family 298 A finite-entropy separation of microstates and nonmicrostates free entropy
We answer the finite-entropy equality question for microstates and nonmicrostates free entropy negatively. We construct a bounded self-adjoint tuple X in a von Neumann algebra with faithful normal tracial state such that `-∞ (X)≤ *(X)- tfrac12 ∞`. Here χ is the original microstates entropy with an operator-norm cutoff and a limsup over matrix sizes. The counterexample uses a large but fixed number of variables.
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View manuscript on GitHub → Open PDF (raw) →M593 · Family 299 The Kirchberg–Rørdam character criterion
We prove the Kirchberg–Rørdam character criterion: for every free ultrafilter on ℕ, a nonzero unital separable complex C∗-algebra absorbs the Jiang–Su algebra if and only if its norm central-sequence algebra has no characters. We also prove that the infinite minimal tensor power of every nonzero unital separable complex C∗-algebra without characters is ` Z` -stable and contains a unital copy of ` Z`.
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View manuscript on GitHub → Open PDF (raw) →M594 · Family 300 Approximation Paving over Arbitrary Maximal Abelian Subalgebras
We prove the approximation-paving conjecture of Popa and Vaes for every maximal abelian subalgebra of a complex von Neumann algebra. For each `0 ε 1`, every self-adjoint operator is a strong limit of self-adjoint operators of norm at most three times its norm, each admitting a norm paving with error at most ε times the approximant's own norm. The number of projections is at most `Cε⁻⁶` for a universal constant C. No separability or conditional-expectation hypothesis is required.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M595 · Family 300 Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras
We prove the quadratic strong-operator paving conjecture of Popa and Vaes. For every `0 ε 1`, every self-adjoint element of a von Neumann algebra admits strong-operator paving over each maximal abelian subalgebra with at most `5×10⁸ε ⁻²` projections. The bound is uniform over representations and requires no separability or conditional-expectation assumption.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M596 · Family 301 The trace cone classifies Razak–Jacelon stabilizations
We prove that the canonical topological cone of all extended lower-semicontinuous tracial weights determines a separable nuclear C∗-algebra after tensoring with the Razak–Jacelon algebra and the compact operators, answering Robert's trace-cone classification question positively. The isomorphism realizes the prescribed cone map, with arbitrary ideal structure and without a density assumption on the finite domains of the weights.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M597 · Family 302 Filtered products and boundary-preserving compression in complex cobordism
We prove that, in sufficiently large smash powers, an `MU` -null restriction to a finite pointed subcomplex becomes stably null on the entire union of products with a fixed positive proportion of restricted factors. This coherent vanishing theorem yields boundary-preserving compression of cube-valued maps on every compact metrizable input space.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M598 · Family 302 Radius of comparison equals half the mean dimension
We prove the integer-action case of the Phillips–Toms conjecture: for every minimal homeomorphism of an infinite compact metrizable space, the radius of comparison of its crossed product equals one half of its mean dimension, including equality at infinity. For these systems, zero mean dimension and the small boundary property are equivalent to Jiang–Su stability and to finite nuclear dimension of the crossed product; in this case its nuclear dimension is at most one.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M599 · Family 303 Weak pure infiniteness and O-infinity absorption
We prove that every complex C∗-algebra in which each positive element is properly infinite is strongly purely infinite. This answers the ordinary-to-strong part of Kirchberg–Rørdam's comparison question. For exact algebras, we also prove that proper infiniteness of one fixed finite amplification of every positive element implies proper infiniteness of each positive element. Consequently, every separable nuclear algebra with this fixed-amplification property absorbs ` O_∞`, without assumptions of unitality or simplicity.
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View manuscript on GitHub → Open PDF (raw) →M600 · Family 304 The Hilbert–Smith conjecture in every finite dimension
We prove the Hilbert–Smith conjecture in every finite dimension: every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected Hausdorff second-countable finite-dimensional topological manifold without boundary is a Lie group.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M601 · Family 305 A boundary-only obstruction to four-dimensional disk embedding
We disprove the four-dimensional disc embedding conjecture without a fundamental-group hypothesis, even when no homotopy classes or output framings are prescribed. We construct a compact oriented smooth four-manifold containing finitely many disc maps with framed algebraic dual spheres whose boundary circles bound no disjoint locally flat discs. Consequently, the free group on two generators is not good in the sense of Freedman–Quinn, and neither is any group containing it as a subgroup.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M602 · Family 305 A marked tensor obstruction to four-dimensional disk embedding
We disprove the unrestricted four-dimensional disk-embedding conjecture. We construct immersed disks in a compact oriented smooth four-manifold with framed algebraic dual spheres satisfying the usual equivariant intersection and reduced self-intersection conditions, but with no pairwise disjoint locally flat replacements that preserve the boundary maps and induced normal framings. The obstruction holds even when the replacement disks' relative homotopy classes are not prescribed.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M603 · Family 305 A PD4 group without an aspherical manifold model
We construct a finitely presented integral Poincaré duality group of dimension four that has a finite classifying space but is not the fundamental group of any closed aspherical topological four-manifold. This gives a negative answer to Wall's manifold-realization question in dimension four.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M604 · Family 306 Purely cosmetic surgery on knots in the three-sphere
We prove the purely cosmetic surgery conjecture for knots in the three-sphere. Distinct Dehn surgery slopes on a nontrivial smooth knot never give orientation-preservingly homeomorphic manifolds.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M605 · Family 307 Failure of rational injectivity for maximal coarse assembly
We construct a uniformly discrete bounded-geometry space whose maximal coarse assembly map has an infinite-order element in its kernel. The space is a coarse disjoint union of finite connected graphs of uniformly bounded degree, so maximal coarse assembly need not be rationally injective even for such graph unions.
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View manuscript on GitHub → Open PDF (raw) →M606 · Family 307 A counterexample to the coarse Novikov conjecture
We disprove the coarse Novikov conjecture: ordinary coarse assembly need not be rationally injective for uniformly discrete spaces of bounded geometry. We construct a coarse disjoint union of finite graphs of uniformly bounded degree and an infinite-order class in its degree-one coarse K-homology whose image under ordinary coarse assembly in the K-theory of the reduced, locally compact Roe algebra vanishes.
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View manuscript on GitHub → Open PDF (raw) →M607 · Family 308 Finite Smith–Toda Complexes at Varying Primes
For every nonnegative integer n, we construct a Smith–Toda complex `V(n)` at some prime p depending on n. It is a finite p-local spectrum whose Brown–Peterson homology is `BP_*/(p,v_1,v_n)` with its canonical comodule structure and generator in degree zero. Each listed generator is killed to its first power.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M608 · Family 308 A finite Smith–Toda complex V(4) at the prime 1009
We construct a Smith–Toda complex `V(4)` at the prime 1009. It is an ordinary finite 1009-local spectrum whose Brown–Peterson homology is `BP_*/(1009,v_1,v_2,v_3,v_4)`, with its canonical comodule structure and generator in degree zero.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M609 · Family 309 The Kervaire invariant problem at the prime three
We solve the Kervaire invariant problem at the prime three for the standard Kervaire classes in the mod-three Adams spectral sequence. These classes survive precisely at indices 0, 2, and 3, and each surviving detection coset contains an element of additive order three. In particular, there is an order-three Kervaire element in stem 322.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M610 · Family 310 Rational homology and Quillen's conjecture
We prove Quillen's conjecture for all finite groups and all primes. More precisely, if a finite group G has trivial largest normal p-subgroup `O_p(G)`, then the poset of nontrivial elementary abelian p-subgroups of G has nonzero augmented reduced rational homology. This establishes the stronger rational-homology form of the conjecture.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M611 · Family 311 Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra
We classify the prime and radical ideals of the Lubin–Tate deformation ring invariant under an open Morava stabilizer subgroup, at every prime and positive height. This proves Chai's invariant-ideal conjecture in the standard finite-residue-field formulation. It also proves the Hovey–Strickland conjecture: the dualizable `K(n)` -local category has exactly `n+2` thick tensor ideals, and its Balmer spectrum is a chain of `n+1` points.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M612 · Family 312 The Grothendieck homotopy hypothesis via elementary expansions
We prove the Grothendieck homotopy hypothesis for every Grothendieck coherator in the Ara–Henry convention: its weak globular infinity-groupoids recover the homotopy theory of spaces. We also resolve Henry's pushout conjecture, showing that elementary expansions preserve components and all homotopy groups of cellular infinity-groupoids.
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View manuscript on GitHub → Open PDF (raw) →M613 · Family 313 Finite generation for the K(n)-local sphere
We prove that the homotopy groups of the `K(n)` -local sphere are finitely generated ℤp-modules in every integer degree, for every prime p and positive height n. Equivalently, the same conclusion holds after `K(n)` -localizing any finite p-local spectrum. This answers the degreewise finiteness question of Hovey and Hovey–Strickland.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M614 · Family 314 Cyclic length and chromatic fixed-point loss
For a finite p-group G and a subgroup H, we prove that the optimal chromatic fixed-point loss equals the shortest length of a subnormal chain from H to G with cyclic quotients. The equality holds at every prime and every nonnegative height, resolving positively the equality proposed by Kuhn and Lloyd.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M615 · Family 315 The Singer conjecture in dimension four
We prove the four-dimensional Singer conjecture: the L2-Betti numbers of the universal cover of a closed connected aspherical topological four-manifold vanish outside degree two. More generally, the same vanishing holds for finite connected aspherical integral Poincaré complexes of formal dimension four, including nonorientable ones.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M616 · Family 316 The Stable Hurewicz Image of the Sphere at Two
We prove Curtis's conjecture: in every positive degree, the mod-two stable Hurewicz image of the sphere is spanned by the images of η, ν, σ and the Kervaire-invariant-one classes that exist. Consequently, Eccles's conjecture holds for every sphere Sn with n > 0.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M617 · Family 317 Thomason Model Structures in Every Strict Higher Dimension
We prove the higher-dimensional Thomason model-structure conjecture of Ara and Maltsiniotis. For every `1≤ n≤∞`, the category of small strict globular n-categories admits a proper combinatorial model structure that is Quillen equivalent to simplicial sets. Its weak equivalences and fibrations are detected by the twice-extended Street nerve `Ex²N_n`, and the Quillen equivalence is given by `c_nSd² dashvEx²N_n`. Thus strict higher categories model the homotopy theory of spaces in every positive finite dimension and in dimension ω.
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View manuscript on GitHub → Open PDF (raw) →M618 · Family 318 Filtered chromatic splitting at generic primes
For every n ≥ 1 and prime `p n+1`, we construct a `2n` -stage ordered filtration of `L(n-1)L(K(n))S_p^ ` with the classical chromatic-splitting cofibers. The map from the first stage to the target is the canonical localization unit.
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View manuscript on GitHub → Open PDF (raw) →M619 · Family 318 The height-three chromatic overlap: an explicit filtration and its attachments
For every prime p ≥ 5, we construct an explicit eight-stage filtration of `L_2L(K(3)) S_p^ ` by the local-sphere layers in the height-three chromatic-splitting pattern. The map from its first stage to the overlap is the canonical unit, and two signed fracture formulas identify all attachments for the chosen local maps and compatibility homotopy. A companion canonical-map theorem further implies that the first height-one attachment is nonzero.
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View manuscript on GitHub → Open PDF (raw) →M620 · Family 318 A rational obstruction to strong chromatic splitting at height three
For every prime p ≥ 5, the canonical map `L_0L(K(3))S→ L_0L(K(2))L(K(3))S` for the sphere spectrum S is nonzero on π−3. Consequently, the height-three strong chromatic splitting formula is false in this range, even as an equivalence of underlying `E(2)` -local spectra without specified summand maps.
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View manuscript on GitHub → Open PDF (raw) →M621 · Family 318 Failure of finite assembly for a chromatic overlap at the prime three
At the prime three and height three, the chromatic overlap cannot be constructed from the rational, height-one, and height-two local spheres by finitely many sums, shifts, cofibers, and retracts in the category of `E(2)` -local modules over the derived 3-complete sphere. This gives a negative answer to the ordinary finite-assembly question for chromatic overlaps.
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View manuscript on GitHub → Open PDF (raw) →M622 · Family 318 Counterexamples to weak chromatic splitting: sphere kernels and descent exponents
For the derived p-completion of the sphere, the canonical weak chromatic splitting map has no homotopy retraction at height p for every prime p ≥ 5, and at height `p+1` for every prime p ≥ 7. The classical sphere product `β_1(p⁻¹)^²` gives a nonzero kernel class in both ranges.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M623 · Family 319 Counterexamples to the Hahn-Wilson conjecture at height two
We disprove the Hahn–Wilson conjecture at height two. For every sufficiently large prime p, we construct a connective p-complete spectrum X of exact fp-type two outside the ordinary thick subcategory generated by the specified standard form of `BP langle2 rangle_p^ `. The same spectrum satisfies both localization comparisons `L_2fX L_2X` and `L(T(2))X L(K(2))X`.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M624 · Family 320 Nonhomeomorphic closed aspherical four-manifolds with the same homotopy type
We construct closed connected aspherical topological four-manifolds that are homotopy equivalent but not homeomorphic, disproving the homeomorphism-existence formulation of the Borel conjecture in dimension four. Their common fundamental group is word-hyperbolic. One of the manifolds also has a self-homotopy equivalence not homotopic to a homeomorphism.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M625 · Family 321 A Counterexample to Wall's D(2) Problem
We give a negative answer to Wall's finite `D(2)` problem. We construct a finite connected three-dimensional CW complex satisfying the `D(2)` finiteness condition but not homotopy equivalent to any finite CW complex of dimension at most two. The example has infinite fundamental group.
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View manuscript on GitHub → Open PDF (raw) →M626 · Family 322 A positive solution to Tingley’s problem
Every surjective isometry between the unit spheres of real Banach spaces extends uniquely to a surjective real-linear isometry, giving an affirmative solution to Tingley's problem. If distances between different radii are not preserved, we realize the positive maximal defect in a possibly enlarged pair of Banach spaces. We then align extremal chords using common supports and Darbo's fixed-point theorem and obtain a contradiction from support and convexity estimates.
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View manuscript on GitHub → Open PDF (raw) →M627 · Family 323 Relative independence of the separable quotient problem
The separable quotient problem asks whether every infinite-dimensional Banach space has a separable infinite-dimensional quotient. We prove that this statement is independent of ZFC, relative to the consistency of ZFC with a measurable cardinal. This holds over both the real and complex fields. For spaces of norm density ℵ1, we obtain independence relative to the consistency of ZFC alone.
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View manuscript on GitHub → Open PDF (raw) →M628 · Family 324 Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic
There are separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic. This gives a negative answer to the separable Banach-space Lipschitz-isomorphism problem.
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View manuscript on GitHub → Open PDF (raw) →M629 · Family 324 Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0
We construct a separable real Banach space Z that contains no linear copy of c0, yet is bi-Lipschitz equivalent to `Z oplus_∞ c_0`. The same space contains a bi-Lipschitz image of every separable metric space.
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View manuscript on GitHub → Open PDF (raw) →M630 · Family 325 A direct proof of the complete Crouzeix inequality
We give a direct proof of the sharp constant-two numerical-range inequality for matrix-valued polynomials in all finite base and coefficient dimensions. This resolves the complete Crouzeix conjecture in its matrix formulation, including matrices whose numerical ranges are points or line segments.
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View manuscript on GitHub → Open PDF (raw) →M631 · Family 325 The complete Crouzeix theorem: optimal similarity and a common positive boundary representation
We resolve the complete Crouzeix conjecture by proving the sharp constant-two numerical-range inequality for every bounded operator on a complex Hilbert space and every matrix-valued polynomial. No separability assumption is needed. The closure of the numerical range is a complete 2-spectral set, and the bound extends to finite matrix-valued functions holomorphic near that closure.
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View manuscript on GitHub → Open PDF (raw) →M632 · Family 326 The cotype–cotype conjecture under the approximation property
We prove the cotype–cotype conjecture under the ordinary approximation property. A nonzero real Banach space with this property is K-convex if and only if both the space and its dual have finite Rademacher cotype, possibly with different exponents.
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View manuscript on GitHub → Open PDF (raw) →M633 · Family 327 Nontrivial Markov Type Forces Superreflexivity
We prove that every real Banach space with Markov type p > 1 is superreflexive. Together with the known converse, this characterizes superreflexivity by nontrivial Markov type. This answers Naor's question: every real Banach space with nontrivial Markov type admits an equivalent uniformly smooth norm.
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View manuscript on GitHub → Open PDF (raw) →M634 · Family 328 Fixed Points of Nonexpansive Maps in Reflexive Banach Spaces
Every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. This resolves the reflexive-space fixed point problem for the given norm.
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View manuscript on GitHub → Open PDF (raw) →M635 · Family 329 Counterexamples to the duality conjecture for metric entropy
We disprove Pietsch's dimension-free duality conjecture for metric entropy. For every proposed pair of universal constants, we construct origin-symmetric convex bodies that violate the corresponding covering-entropy inequality, already when the covering body is a cube.
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View manuscript on GitHub → Open PDF (raw) →M636 · Family 330 A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces
We construct a countable uniformly discrete metric space whose real Lipschitz-free space has the approximation property but fails the bounded approximation property, answering Kalton's question negatively. The identity can be approximated on every compact set by finite-rank operators, but no uniform bound on their norms is possible.
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View manuscript on GitHub → Open PDF (raw) →M637 · Family 331 Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space
We construct a separable reflexive real Banach space whose given norm is asymptotically midpoint uniformly convex but which admits no asymptotically uniformly convex equivalent norm. Its averaged midpoint modulus is at least `√(1+t²/12)-1`, and the countably branching diamond of depth k has distortion at least `√(1+k/12)` in this space. This gives a negative answer to the reflexive diamond converse for asymptotic uniform convexifiability.
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View manuscript on GitHub → Open PDF (raw) →M638 · Family 331 Midpoint lenses in segment spaces
For a real segment-forest dual with unbounded finite component heights and the real infinite-height coordinate predual, we bound the tail of an arbitrary displacement in a symmetric lens by `2√(R²⁻ |x |²)`, where R is the lens radius and x is its finitely supported center. Both given norms are asymptotically midpoint uniformly convex, although neither space admits an equivalent asymptotically uniformly convex norm. The finite-height forest dual is reflexive.
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View manuscript on GitHub → Open PDF (raw) →M639 · Family 331 Distortion of countably branching diamonds from midpoint and tree energies
We derive quantitative distortion bounds for countably branching diamond graphs from midpoint estimates and direct tree energies. In the dual of a finite-height segment forest, every distortion-D embedding of the depth-k diamond satisfies `D²≥1+k/4`. The same bound holds in the infinite-height coordinate predual. We also obtain power-type distortion bounds for path-cost and recursive tree norms.
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View manuscript on GitHub → Open PDF (raw) →M640 · Family 331 Exact asymptotic moduli in a Daugavet subspace of L1
For an infinite-dimensional real subspace of L1 whose unit ball is totally bounded in measure and whose norm has the Daugavet property, we compute the averaged midpoint and one-sided asymptotic moduli at every unit center. They are `max t/2,t-1 ` and `max 0,t-2 `, respectively. The same weak-neighborhood geometry excludes every equivalent asymptotically uniformly convex norm. A quantitative realization of the Kadets–Werner construction supplies a space with both hypotheses.
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View manuscript on GitHub → Open PDF (raw) →M641 · Family 331 Midpoint convexity from bounded tree potentials and path costs
Four real Banach spaces defined by bounded tree potentials satisfy the averaged asymptotic midpoint bound ` widehatδ(t)≥√(1+t²/4)-1` for `0 t 1`, while none admits an asymptotically uniformly convex (AUC) renorming. On finite-height trees, the globally constrained norm equals the least additive cost of a Hilbert vector and root paths. The quadratic path and segment-start outer Hilbert sums are reflexive and asymptotically midpoint uniformly convex, and admit no equivalent AUC norm.
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View manuscript on GitHub → Open PDF (raw) →M642 · Family 331 Independent products in real L1: asymptotic midpoint convexity without AUC renormings
For a countably branching tree, the closed real L1 spans of products of independent exponential or Gaussian-square multipliers along its paths have positive averaged asymptotic midpoint moduli and admit no equivalent asymptotically uniformly convex norm. The renorming obstruction holds for every positive nonconstant mean-one multiplier with finite second moment.
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View manuscript on GitHub → Open PDF (raw) →M643 · Family 331 Midpoint convexity from two recursive potentials
We study tree norms computed by two least nonnegative fields whose difference is the vector. For Euclidean child aggregation, two root-sum spaces have an averaged asymptotic midpoint modulus of at least `t³/128` for `0 t 1`, including a reflexive joining-root space. A reflexive construction with a fixed zero root also satisfies a homogeneous cubic estimate. These spaces admit no equivalent asymptotically uniformly convex norm. We also obtain sixth-power and cubic estimates when the aggregation exponent depends on the height of a finite component.
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View manuscript on GitHub → Open PDF (raw) →M644 · Family 332 Metric Markov Cotype Two of ℓ1
We prove that the real Banach space ℓ1 has metric Markov cotype two, answering a question of Mendel and Naor. As a consequence, every Lipschitz map from an arbitrary subset of a real Hilbert space into ℓ1 extends to the whole Hilbert space with a universal multiplicative loss in its Lipschitz constant, resolving Ball's extension problem for this target.
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View manuscript on GitHub → Open PDF (raw) →M645 · Family 333 Smooth isometric immersions of closed surfaces into Euclidean four-space
Every closed smooth Riemannian surface admits a smooth isometric immersion into Euclidean four-space, without an orientability assumption. This resolves the closed-surface form of the classical four-dimensional isometric-immersion problem.
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View manuscript on GitHub → Open PDF (raw) →M646 · Family 334 A Smooth Metric with No Local Isometric Immersion into Three-Space
We construct a smooth positive-definite Riemannian metric on `(-1,1)²` that agrees with the Euclidean metric to every order at the origin, yet no neighborhood of the origin admits a smooth isometric immersion into Euclidean…
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View manuscript on GitHub → Open PDF (raw) →M647 · Family 335 An integral scalar curvature bound for real simplicial volume
We prove the integral scalar-curvature inequality proposed by Gromov. For every dimension n ≥ 3, there is a constant `a_n 0`, depending only on n, such that ` ∫_M( Scal _g⁻)n/² dV_g≥ a_n |M |` for every closed connected oriented smooth n-manifold M and every smooth Riemannian metric g. Here ` Scal _g⁻:=max 0,- Scal _g `, and ` |M |` is real simplicial volume. The proof uses the nonnegative-scalar-curvature vanishing theorem of the companion paper on rational inessentiality.
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View manuscript on GitHub → Open PDF (raw) →M648 · Family 335 Positive scalar curvature forces rational inessentiality
We prove that every closed connected oriented smooth manifold admitting strictly positive scalar curvature is rationally inessential: its rational fundamental class maps to zero under the classifying map. No spin or fundamental-group hypothesis is needed, and there is no upper dimension bound. This proves the Gromov–Lawson aspherical conjecture; moreover, every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat.
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View manuscript on GitHub → Open PDF (raw) →M649 · Family 336 Spectral scalar curvature and uniform Urysohn width
For every n ≥ 4, a complete connected smooth Riemannian n-manifold without boundary satisfying `-4 + Scal ≥1` as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most `n-2` whose entire fibers have diameter bounded only in terms of n. The bound is measured in the original metric. This extends the uniform Urysohn width theorem from a pointwise scalar-curvature lower bound to a spectral lower bound.
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View manuscript on GitHub → Open PDF (raw) →M650 · Family 336 Spectral scalar curvature and Urysohn width in dimension three
Every connected complete smooth Riemannian three-manifold without boundary satisfying `-4 + Scal ≥λ 0` as a quadratic-form inequality admits a continuous map to a graph whose entire fibers have diameter at most `500/ sqrtλ` in the original metric. No orientability, spin, compactness, or bounded-geometry assumption is required.
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View manuscript on GitHub → Open PDF (raw) →M651 · Family 336 Positive scalar curvature and uniform codimension-two width
We prove the quantitative continuous form of Gromov's scalar-curvature conjecture in every dimension n ≥ 4. Every complete connected smooth boundaryless n-manifold with scalar curvature at least one admits a continuous map to a simplicial complex of dimension at most `n-2` whose entire fibers have diameter bounded only in terms of n. We also obtain the continuous macroscopic-dimension conclusion for universal covers of closed manifolds with positive scalar curvature in every dimension n ≥ 2.
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View manuscript on GitHub → Open PDF (raw) →M652 · Family 337 Generalized Cartan–Hadamard isoperimetry and Euclidean equality rigidity
We resolve the generalized Cartan–Hadamard isoperimetric conjecture in every dimension. In a complete simply connected smooth manifold with sectional curvature at most κ ≤ 0, every finite-volume set of finite ambient perimeter has perimeter at least that of the equal-volume ball in curvature κ. For bounded positive-volume sets, equality in the Euclidean comparison holds precisely when the set agrees up to null sets with an open region isometric, with its induced metric, to a round Euclidean ball.
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View manuscript on GitHub → Open PDF (raw) →M653 · Family 338 Uniformization of complete Kähler manifolds with positive bisectional curvature
We prove that every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to complex Euclidean space. This resolves Yau's uniformization conjecture positively in every complex dimension.
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View manuscript on GitHub → Open PDF (raw) →M654 · Family 339 Entropy equality and local symmetry in negative curvature
For every closed connected smooth Riemannian manifold of dimension at least three with strictly negative sectional curvature, we prove that normalized Liouville measure has maximal entropy for the unit-speed geodesic flow if and only if the metric is locally symmetric. This resolves Katok's entropy rigidity conjecture positively in these dimensions, including all rank-one symmetric types at arbitrary scale.
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View manuscript on GitHub → Open PDF (raw) →M655 · Family 340 A counterexample to the nearby Lagrangian conjecture
We disprove the unrestricted nearby Lagrangian conjecture. For some sufficiently large even integer N, we construct a closed exact smoothly embedded Lagrangian in `T*(S⁹× SN⁻¹)` that is diffeomorphic to the base but is not Hamiltonian isotopic to the zero section.
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View manuscript on GitHub → Open PDF (raw) →M656 · Family 341 Deforming hypersymplectic four-manifolds to hyperkähler triples
Every smooth normalized positive triple of closed two-forms on a closed connected oriented four-manifold admits a smooth deformation, with each cohomology class fixed, to a hyperkähler triple. This resolves Donaldson's hypersymplectic deformation conjecture.
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View manuscript on GitHub → Open PDF (raw) →M657 · Family 342 Taming implies compatibility on four-manifolds
We prove that every smooth almost complex structure on a closed four-manifold which is tamed by a symplectic form is compatible with a symplectic form. This gives a positive solution to Donaldson's tamed-to-compatible conjecture.
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View manuscript on GitHub → Open PDF (raw) →M658 · Family 343 Symplectic Ball Packings in Higher Dimensions
We prove that, for all integers n ≥ 3 and k ≥ 1 and all positive real capacities `R_1,R_k`, the closed standard symplectic `2n` -balls of these capacities embed disjointly into the interior of a ball of capacity R > 0 if and only if ` ∑(i=1)k R_in Rn, R_i+R_j R (i≠ j).` Capacity is π times the squared Euclidean radius, and each embedding is defined on a neighborhood of its closed source ball. This proves Siegel and Yao's Conjecture A.
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View manuscript on GitHub → Open PDF (raw) →M659 · Family 344 The metric Blaschke theorem
We prove the metric Blaschke conjecture: every connected closed smooth Riemannian manifold of positive dimension whose global injectivity radius equals its diameter is, up to scale, a standard compact rank-one symmetric space.
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View manuscript on GitHub → Open PDF (raw) →M660 · Family 345 Infinitely many closed geodesic images on every Riemannian sphere
We resolve the sphere case of the closed-geodesic infinitude problem: every smooth Riemannian metric on the standard sphere Sn, n ≥ 2, has infinitely many prime closed geodesics with pairwise distinct images. The two-sphere case is classical; the result in higher dimensions includes degenerate metrics. Using finite covers and theorems of Perelman and Rademacher–Taimanov, we also obtain the same conclusion for every smooth Riemannian metric on a nonempty closed smooth three-manifold.
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View manuscript on GitHub → Open PDF (raw) →M661 · Family 346 A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold
The singular set of every stationary integral m-varifold in an open Euclidean set has Hausdorff dimension at most `m-1`, in every positive dimension and codimension. This sharp bound proves the Euclidean singular-set conjecture recorded by Brena, Decio, and De Lellis.
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View manuscript on GitHub → Open PDF (raw) →M662 · Family 346 Almost-everywhere regularity of stationary integral varifolds
We resolve the almost-everywhere regularity conjecture of Brena, Decio, and De Lellis for stationary integral varifolds of arbitrary positive dimension and codimension in Euclidean open sets. The singular set has zero measure in the dimension of the varifold: near almost every support point, the varifold is a constant positive integer multiple of a smooth embedded minimal submanifold. The corresponding statement also holds on round spheres.
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View manuscript on GitHub → Open PDF (raw) →M663 · Family 347 Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two
We disprove the stable Morse lower bound for nondegenerate Hamiltonian fixed points with examples in fixed real dimension twenty-two. For every integer m ≥ 1, we construct a simply connected closed Kähler manifold with stable Morse number `80+1968m` and a Hamiltonian diffeomorphism with exactly `80+1952m` fixed points, all nondegenerate and with contractible orbit loops. The deficit is therefore unbounded, and the fixed-point count is at most 127/128 of the stable Morse number.
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View manuscript on GitHub → Open PDF (raw) →M664 · Family 347 Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number
We construct a simply connected closed Kähler manifold of real dimension 3332 and minimal Chern number one whose Hamiltonian fixed-point count attains the cyclic integral Floer bound while falling below the stable Morse number. The Hamiltonian diffeomorphism has exactly `1 872 232` fixed points, all nondegenerate and with contractible orbit loops, whereas the stable Morse number is `1 872 264`. Thus the stronger stable Morse bound fails by exactly 32, even when the cyclic integral bound is sharp.
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View manuscript on GitHub → Open PDF (raw) →M665 · Family 347 Hamiltonian Fixed Points Below the Stable Morse Number
We disprove the stable Morse lower bound for nondegenerate Hamiltonian fixed points by constructing a simply connected closed Kähler manifold with sixteen fewer fixed points than its stable Morse number. All the corresponding periodic orbits are contractible. The construction combines a Hamiltonian involution with integral homology torsion at two different primes; its proof uses finite-dimensional Morse theory and complex blowups.
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View manuscript on GitHub → Open PDF (raw) →M666 · Family 347 A nondegenerate counterexample to the Morse-number Arnold bound
We construct a smooth one-periodic Hamiltonian on a closed symplectic twelve-manifold whose time-one map has fewer fixed points than the manifold's ordinary Morse number. Every fixed point is nondegenerate and has a contractible Hamiltonian trajectory. This disproves the Morse-number form of the Arnold conjecture. The deficit can be arbitrarily large among twelve-dimensional examples.
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View manuscript on GitHub → Open PDF (raw) →M667 · Family 347 Three fixed points on the symplectic quadric threefold
We construct a smooth Hamiltonian diffeomorphism of the complex quadric threefold with exactly three fixed points, at least one of which is degenerate. The critical number and the unit-inclusive rational cup length of this manifold are both four. Thus the example disproves the unrestricted critical-number and rational cup-length forms of the Arnold conjecture.
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View manuscript on GitHub → Open PDF (raw) →M668 · Family 348 Zero-Plane Rigidity for Einstein Four-Manifolds
We prove that a closed Einstein four-manifold with positive Einstein constant, nonnegative sectional curvature, and a zero-curvature plane has universal Riemannian cover isometric to a product of two round two-spheres. Under the normalization ` Ric =3g`, both spheres have radius `1/ sqrt3`. The proof extends coupled estimates for the two Weyl curvature blocks to the boundary of the sectional-curvature cone and determines their equality case.
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View manuscript on GitHub → Open PDF (raw) →M669 · Family 348 An L² Einstein Gap for Nonnegatively Curved Four-Manifolds
We prove a universal, scale-invariant L2 gap for the trace-free Ricci tensor on simply connected closed four-manifolds with nonnegative sectional curvature. If the trace-free Ricci energy is sufficiently small, the manifold is diffeomorphic to S4, `CP²`, or `S²× S²`. The proof uses the classification of positive-Einstein, nonnegatively curved four-manifolds supplied by the companion zero-plane rigidity theorem, stated explicitly as the classification premise of our result.
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View manuscript on GitHub → Open PDF (raw) →M670 · Family 348 Positively curved Einstein four-manifolds
We prove the classification conjecture for connected smooth closed Einstein four-manifolds with strictly positive sectional curvature. Up to positive scaling and isometry, every such manifold is the round four-sphere, the complex projective plane with its Fubini–Study metric, or real projective four-space with its round metric. No orientability assumption is needed.
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View manuscript on GitHub → Open PDF (raw) →M671 · Family 349 The Solomon–Yau least-volume theorem
We prove the Solomon–Yau least-volume conjecture for minimal hypersurfaces of unit round spheres: in every dimension m ≥ 2, a closed connected minimal immersion with non-totally-geodesic image has volume at least the smallest m-dimensional minimal Clifford product. Volume is measured on the domain, so covering multiplicities are included.
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View manuscript on GitHub → Open PDF (raw) →M672 · Family 350 Sharp nodal length on smooth surfaces
We prove that a nonzero real Laplace eigenfunction with eigenvalue λ > 0 on a fixed smooth closed connected Riemannian surface has nodal length at most `C sqrtλ`. Together with the known lower bound, this proves Yau's conjecture in this setting.
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View manuscript on GitHub → Open PDF (raw) →M673 · Family 350 Smooth counterexamples to Yau's nodal upper bound in dimensions three and four
We construct a smooth metric on the three-sphere, arbitrarily close to the round metric in the smooth topology, and a smooth metric on `S²× T²` for which sequences of exact real Laplace eigenfunctions have unbounded nodal measure divided by the square root of the eigenvalue. Each sequence belongs to one fixed metric. Thus the upper-bound part of Yau's nodal conjecture fails for smooth metrics in dimensions three and four.
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View manuscript on GitHub → Open PDF (raw) →M674 · Family 350 Power-law violations of Yau's nodal upper bound
We construct a smooth Riemannian metric on `S⁴× S¹` and a sequence of real Laplace eigenfunctions whose nodal four-volume grows faster than `λ¹/²⁺ε_⁰` for one fixed `ε_0 0`. This disproves the smooth upper-bound assertion in Yau's nodal-set conjecture and the proposed bound with an arbitrarily small positive power loss.
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View manuscript on GitHub → Open PDF (raw) →M675 · Family 351 A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup
We disprove the scalar-curvature extension conjecture in its unrestricted all-dimensions form. In sufficiently high dimension, we construct a Ricci flow on a closed manifold whose scalar curvature remains uniformly bounded while full curvature diverges at a finite maximal time. In one fixed sufficiently high dimension, the examples have two-sided power-law curvature blowup with arbitrarily large exponents.
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View manuscript on GitHub → Open PDF (raw) →M676 · Family 351 Bounded scalar curvature and smooth extension of four-dimensional Ricci flow
We prove that a smooth Ricci flow on a closed real four-manifold extends on the same manifold through every finite time at which its scalar curvature remains uniformly bounded. This resolves the scalar-curvature extension problem in dimension four.
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View manuscript on GitHub → Open PDF (raw) →M677 · Family 351 Path selection and an elliptic inequality on degenerating Ricci-flat trees
We prove a sequential elliptic inequality for the renormalized Einstein–Hilbert functional on a fixed finite tree of four-dimensional Ricci-flat spaces. As the joining lengths diverge and the scale-neutral weighted Ricci error M tends to zero, the functional satisfies `E=o(M)`. The root end has decay exponent greater than one, and every joined quotient group is nontrivial. We also prove the multivariable path-selection theorem for asymptotic expansions used to obtain this estimate.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M678 · Family 352 A finite-time singularity of Calabi flow on projective space
We construct a smooth Kähler metric in the Fubini–Study class of ` CP ¹⁰` whose Calabi flow develops unbounded scalar curvature in finite time. This disproves Chen's smooth long-time existence conjecture, even in a class containing a constant-scalar-curvature metric.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M679 · Family 353 A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten
We construct a smooth nonquadratic entire graph in dimension ten that solves the classical affine maximal equation and has positive-definite Hessian everywhere. This gives a smooth counterexample to the entire-graph affine Bernstein assertion in dimension ten. Hessian positivity is pointwise; no global uniform lower bound or completeness of the Berwald–Blaschke metric is asserted.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M680 · Family 353 The affine Bernstein theorem in dimensions three through nine
We prove the Euclidean-complete affine Bernstein conjecture in dimensions three through nine: a smooth locally uniformly convex affine maximal graph is an elliptic paraboloid whenever its induced Euclidean metric is complete. The same conclusion holds, without an initial graph assumption, for connected smooth open (noncompact and without boundary) affine-complete locally uniformly convex immersed hypersurfaces that are classically affine maximal in these dimensions.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M681 · Family 354 The Isoperimetric Conjecture for the Cubic Flat Three-Torus
We prove the isoperimetric conjecture for the cubic flat three-torus and classify all minimizers, including the equality cases at the transition volumes `4π/81` and `1/π`. The minimizing regions are balls, circular tubes about shortest closed geodesics, coordinate slabs, and their complements.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M682 · Family 355 Uniqueness of tangent flows at the first singular time of embedded surface mean-curvature flow
We prove that, at each singular point of the first singular time of the mean-curvature flow of a smooth compact connected embedded surface without boundary in ℝ3, all fixed-center rescalings converge locally smoothly on compact negative-time intervals to one multiplicity-one homothetic self-shrinker flow. The limit is unique in the original ambient coordinates, including its position and axes. No mean-convexity assumption or prescribed tangent model is required.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M683 · Family 356 Gigli’s distributional curvature characterization of Alexandrov spaces
For every integer n ≥ 2 and κ ∈ ℝ, we prove that a complete separable metric space is an n-dimensional Alexandrov space of curvature at least κ if and only if, with reference measure ` Hn`, it is a full-support `RCD((n-1),n)` space whose distributional sectional curvature is at least κ in Gigli's original global test classes. This resolves Gigli's characterization conjecture in dimensions at least two.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M684 · Family 356 Weak Hessian bounds along every geodesic in RCD spaces
On a full-support `RCD(K,N)` space with `1 N ∞`, we prove that a bounded globally Lipschitz function whose distributional Hessian is bounded above by a bounded continuous function satisfies the corresponding second-derivative inequality along every minimizing geodesic.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M685 · Family 357 Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces
We resolve the regular-point bi-Lipschitz conjecture in the noncollapsed setting. For every integer n ≥ 2 and every real K, every regular point of a noncollapsed `RCD(K,n)` space has an open neighborhood bi-Lipschitz homeomorphic to an open subset of ℝn. The bi-Lipschitz constant depends only on n, and the neighborhood uses the restricted ambient distance.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M686 · Family 358 A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric
We construct a closed connected orientable smooth three-manifold that admits a smooth Riemannian metric without conjugate points but admits no smooth Riemannian metric of nonpositive sectional curvature.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M687 · Family 359 A negatively pinched Kähler threefold without bounded holomorphic coordinates
We construct a contractible domain in complex dimension three with a complete Kähler metric whose real sectional curvatures lie between two finite negative constants. It admits no bounded holomorphic map to ℂ3 with nowhere-vanishing Jacobian and is therefore not biholomorphic to a bounded domain. This gives a negative answer to the negatively pinched Kähler uniformization question.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M688 · Family 359 One-sided negative sectional curvature and the holomorphic Liouville property
We construct, in some sufficiently large fixed finite complex dimension m, a domain in ℂm diffeomorphic to ` R²m` that admits a complete Kähler metric with real sectional curvature at most −1 and has only constant bounded holomorphic functions. Its sectional curvatures are unbounded below. The construction gives a negative answer to the one-sided bounded-holomorphic-function question.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M689 · Family 360 Global Support and Convex Injectivity Domains under Weak MTW
We prove that weak Ma–Trudinger–Wang curvature on a smooth, connected, compact Riemannian manifold of dimension at least two without boundary implies convexity of every tangent injectivity domain, resolving Villani's conjecture in this setting. Conjugate cut points are allowed. More generally, every ordinary subgradient of a squared-distance cost potential is a minimizing velocity with a global supporting mountain. No density hypothesis is used.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M690 · Family 360 Uniform Bi-Holder Transport from Weak MTW
We prove that the weak Ma–Trudinger–Wang condition on a fixed smooth connected compact boundaryless Riemannian manifold of dimension at least two implies a common Hölder estimate for optimal transport maps and their inverses over the entire class of probability densities with fixed positive upper and lower bounds. The maps have homeomorphic representatives, conjugate cut points are allowed, and no density regularity is assumed.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M691 · Family 361 A counterexample to integer-degree harmonic dimension comparison
For some even n ≥ 8 and integer k ≥ 2, we construct a complete smooth metric on ℝn with nonnegative Ricci curvature whose space of real harmonic functions of pointwise polynomial growth at most k has dimension larger than the Euclidean harmonic-polynomial dimension. The metric is Euclidean near the origin, has asymptotic volume ratio strictly between zero and one, and has nonunique tangent cones at infinity. This answers the integer-degree form of Yau's dimension comparison question in the negative.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M692 · Family 361 A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison
For every `4/9 v 1` and `1 c 9v/4`, all sufficiently large integers k admit a complete smooth metric on ℝ3 with nonnegative Ricci curvature, asymptotic volume ratio v, and at least `c(k+1)²` linearly independent real harmonic functions of pointwise growth at most k. This answers Yau's integer-degree dimension comparison question negatively in dimension three, with a fixed-factor excess over the Euclidean count.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M693 · Family 362 Global classical solutions of the three-dimensional relativistic Vlasov–Maxwell system
We prove global existence and uniqueness for arbitrary smooth admissible initial data in the three-dimensional, one-species relativistic Vlasov–Maxwell system. The particle density is initially compactly supported, and the electromagnetic fields have finite energy and bounded derivatives of all orders. The solution remains smooth on every finite time interval. This resolves the large-data global classical regularity problem for this model, without size or symmetry restrictions on the data.
PQC lens (via Family 362): Directly shapes quantum threat models, algorithm timelines, or query-complexity assumptions used in security proofs.
View manuscript on GitHub → Open PDF (raw) →M694 · Family 363 Nonuniqueness with local conservation for the hard-sphere Boltzmann equation
We prove nonuniqueness for the three-dimensional periodic hard-sphere Boltzmann equation among global entropy solutions satisfying exact local conservation of mass, momentum, and kinetic energy. We construct one nonnegative initial density with bounded velocity support and finite mass, energy, and absolute entropy that gives rise to two distinct such solutions. Both are strongly continuous in L1, and their collision gain and loss terms are integrable with every polynomial velocity weight on every bounded time interval.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M695 · Family 363 Nonuniqueness for the periodic hard-sphere Boltzmann equation
We prove nonuniqueness for the periodic hard-sphere Boltzmann equation by constructing two distinct global renormalized solutions with the same nonnegative initial density on ` T³× R³`. This density has bounded velocity support and finite mass, energy, and absolute entropy. Both solutions conserve local mass and total momentum and satisfy the global energy and entropy-dissipation inequalities. On a common initial interval, they are strongly continuous in L1, and their collision gains and losses are integrable.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M696 · Family 364 The Boltzmann–Grad limit for stable radial potentials on regular kinetic intervals
We derive the nonlinear Boltzmann equation from a grand-canonical Newtonian gas with initial pair exclusion. We treat stable, finite-range radial potentials that are C2 except for an allowed repulsive singularity at the origin, and C1 initial probability densities with spatially summable Gaussian bounds on the density and its spatial gradient.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M697 · Family 364 Hard-sphere fluctuations on the regular Boltzmann lifespan
We prove a finite-dimensional central limit theorem away from equilibrium for a deterministic grand-canonical hard-sphere gas in three dimensions. The initial probability density is smooth, with spatially summable Gaussian velocity bounds on the density and its spatial gradient. The limit holds on every finite interval on which the classical Boltzmann solution has uniform Gaussian velocity decay. The empirical measure is centered by its exact microscopic expectation.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M698 · Family 365 Determination of a metric and a unitary connection from one boundary patch
We prove that zero-frequency boundary measurements on any nonempty open boundary patch determine both a smooth Riemannian metric and a smooth unitary connection on the trivial Hermitian rank-two bundle over a compact connected smooth manifold of dimension at least three with smooth boundary. Both inputs and observations are restricted to the same patch. The metric and connection are determined up to a diffeomorphism and a unitary gauge that restrict to the identity on the measured patch.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M699 · Family 365 Smooth anisotropic uniqueness in the Calderón problem from one boundary patch
We resolve the smooth anisotropic Calderón uniqueness problem with arbitrary same-patch measurements. A smooth Riemannian metric on a compact connected manifold of dimension at least three is determined, up to a diffeomorphism fixing the measured patch, by its zero-frequency Dirichlet-to-Neumann energy form with both input and observation on any nonempty open boundary patch.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M700 · Family 365 Nonuniqueness for bounded measurable scalar conductivities in three dimensions
We construct two distinct uniformly positive bounded measurable scalar conductivities on a ball in ℝ3 with the same full Dirichlet-to-Neumann operator. Both conductivities equal one near the boundary. This gives nonuniqueness in the scalar Calderón problem at bounded measurable regularity.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M701 · Family 366 Interior regularity of planar Mumford–Shah minimizers
We prove the interior regularity assertion of the planar Mumford–Shah conjecture for reduced absolute minimizers with bounded fidelity data. Every interior point of the closed discontinuity set has a neighborhood consisting of a `C¹,α` arc, an arc ending at that point, or three such arcs meeting at 120 degrees. Only finitely many global connected components meet any relatively compact open set.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M702 · Family 367 The critical dimension for one-phase Bernoulli minimizers
We prove that seven is the critical dimension for the one-phase Bernoulli problem: every nonzero one-homogeneous global minimizer in dimensions at most six is flat, while a nonflat one-homogeneous global minimizer exists in dimension seven. It follows that the interior free boundary of a local minimizer is smooth in dimensions at most six. In dimension n ≥ 7, its singular set has Hausdorff dimension at most `n-7`, and this bound is sharp. In dimension seven, the singular set is locally finite.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M703 · Family 368 Strong diffeomorphic approximation in three dimensions for 1≤p≤2
We resolve the three-dimensional Ball–Evans approximation problem for `1≤ p≤2`. Every `W¹,p` homeomorphism between arbitrary bounded domains in ℝ3 is a strong `W¹,p` limit of smooth diffeomorphisms onto the same target.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M704 · Family 368 Strong diffeomorphic approximation in three dimensions for p>2
We resolve the three-dimensional Ball–Evans approximation problem for every finite p > 2. Every `W¹,p` homeomorphism between arbitrary bounded domains in ℝ3 can be approximated strongly in `W¹,p` by smooth diffeomorphisms onto the same target.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M705 · Family 369 Strict hot spots and absence of interior critical points on smooth simply connected planar domains
We prove the strict hot spots conjecture for smooth bounded simply connected planar domains. More precisely, every nonzero eigenfunction for the first positive Neumann eigenvalue has nonvanishing gradient in the interior, so all its global maxima and minima lie on the boundary. This holds even when the eigenvalue is multiple.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M706 · Family 370 The Subcritical Hénon–Lane–Emden Conjecture
We prove the subcritical Hénon–Lane–Emden conjecture: for every dimension n ≥ 2, positive powers p, q, and real weights A, B, the system has no strictly positive entire solution when `(n+A)/(p+1)+(n+B)/(q+1) n-2`. Solutions need only be continuous at the origin and classical elsewhere, without a condition at infinity.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M707 · Family 371 Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus
We prove stable self-similar finite-time blowup for a supercritical defocusing nonlinear Schrödinger equation on the twelve-dimensional torus. For a sufficiently large odd power, the blowup initial data contain a nonempty open set in a high Sobolev space. As a consequence, Gaussian Fourier initial data of arbitrarily high Sobolev regularity can have positive probability of finite-time blowup.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M708 · Family 372 Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem
We prove that the full static displacement-to-traction map uniquely determines both real smooth Lamé moduli on every bounded connected smooth domain in ℝ3, provided μ > 0 and `3λ+2μ 0` on the closure. This resolves the smooth three-dimensional isotropic elastic Calderón uniqueness problem under these positivity assumptions.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M709 · Family 373 A counterexample to the Monge ansatz for the three-marginal Coulomb cost
We construct a smooth compactly supported probability density on ℝ3, with a smooth compactly supported square root, for which the three-marginal Coulomb transport minimum is not attained by any pair of measure-preserving Borel maps. Nevertheless, the Monge and Kantorovich infima agree: we construct preserving maps whose costs approach the Kantorovich minimum. For every d ≥ 2 and s >
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M710 · Family 374 Sharp One-Third Stability of Brenier Maps
For the uniform probability measure ρ on a compact convex body in ℝd, d ≥ 2, quadratic optimal transport maps are one-third Hölder continuous in `L²(ρ)` with respect to the target's 2-Wasserstein distance. The constant is uniform over all targets supported in a fixed compact set, and the exponent is optimal. A three-atom family on a fixed cube disproves Letrouit's conjectured uniform square-root estimate.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M711 · Family 375 A positive resolution of De Giorgi's conjecture in dimension eight
We resolve De Giorgi's conjecture positively in dimension eight.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →M712 · Family 376 Finite Instructions and Solenoidal Shear Flows
We realize finite reciprocal affine instruction maps by smooth incompressible shear flows on a flat three-torus. Applied to a reversible one-head recorder with a finite transition table, this gives a complete machine-to-fluid construction: a fixed particle enters a fixed open strip exactly when a given machine halts, with zero initial velocity, zero pressure, and a solenoidal mean-zero force that is periodic after initialization. The construction acts on full closed rectangles and controls every intermediate trajectory.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M713 · Family 376 A Fixed Particle Test for Computation in a Forced Viscous Flow
We construct smooth external forces for incompressible Navier–Stokes flow on a fixed flat three-torus at any fixed positive computable viscosity, from zero initial velocity, such that a fixed particle enters a fixed open set exactly when a prescribed Turing machine halts. Every mixed derivative of the force and velocity is bounded and square-integrable in time in spatial supremum norm. The construction uses the machine's ordinary instructions, records their history in a third coordinate, and compensates for finer spatial gates by longer time steps.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M714 · Family 376 Universal Computation with Eventually Stationary Navier–Stokes Forcing
At every fixed positive computable viscosity, we construct smooth mean-zero forces on the flat three-torus that become stationary after time one and make a fixed particle, initially in a fluid at rest, enter a fixed open set exactly when a prescribed Turing machine halts. The force has bounded derivatives of every order and a finite effective description. A reversible recording table is realized on whole planar rectangles by Hamiltonian motions, then driven by a mean-zero spatial clock.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M715 · Family 376 Geometric Programs for Solenoidal Forcing
We construct smooth solenoidal mean-zero forces, periodic from time zero, for which a fixed particle on a flat three-torus reaches a fixed open strip exactly when a given machine halts. The initial fluid velocity and the pressure are zero. We also realize positive diagonal maps on coding sheets by incompressible shears, with explicit normal compensation for changes of planar area, and reciprocal maps on whole boxes of positive thickness.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M716 · Family 376 Prefix Instructions and Incompressible Flows
Finite prefix instructions may change area and erase information. We give explicit history processors and smooth incompressible motions that retain that information and realize every instruction on its full domain. A first application assigns each machine and finite input a smooth mean-zero force on the flat unit three-torus, at any fixed positive computable viscosity. The solution starts from rest; a fixed particle enters a fixed open strip exactly when the machine halts. The force repeats with period one after an initial loading interval.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M717 · Family 376 Computation under Rapidly Vanishing Navier–Stokes Forcing
At any fixed positive computable viscosity, smooth forces can make a fixed fluid particle detect the halting of an arbitrary machine, starting from rest, while every mixed derivative of the force and velocity decreases faster than every inverse power of time. We give three complete memory constructions: compact moving curls, alternating fractional coordinates, and a periodic lattice on the flat three-torus. Each machine step takes one unit of physical time.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M718 · Family 376 Velocity-Field Detection of Computation in Forced Navier–Stokes Flows
We construct smooth forces for three-dimensional incompressible Navier–Stokes flow, from rest at any fixed positive computable viscosity, whose velocity field detects whether a prescribed machine halts. On the unit flat torus, a pointwise test of the third velocity component uses successively shorter stirring intervals in a fixed spatial region. On ` R²× T`, an integral test uses an expanding array of translation regions and a force with globally bounded mixed derivatives.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M719 · Family 376 Scalar Potentials and Slow Clocks for Forced Fluid Computation
We construct smooth forces of fixed compact spatial support for three-dimensional incompressible Navier–Stokes flow from rest whose particle at the origin detects halting by entering a fixed half-space. Every mixed derivative of the force and velocity decays at rate `O((1+t)⁻¹⁻j)` for time order j. Three scalar potentials lift a coded rectangle, perform its instruction, and lower its image; an unbounded logarithmic clock supplies the decay.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M720 · Family 376 Incompressible Box Transport and Finite Computation
We realize finite positive diagonal affine maps of determinant one by effective smooth incompressible flows on neighborhoods of entire closed rational solid boxes. The source and target families are each disjoint, but may overlap each other. The construction uses localized curls, evacuation to storage and obstacle detours. A balanced three-stack recorder then assigns every machine and finite input a smooth Navier–Stokes force with one compact spatial support, periodic after a loading interval, at any fixed positive computable viscosity.
Catalog note: Formal proof artifacts improve assurance for crypto implementations and protocol verification pipelines.
View manuscript on GitHub → Open PDF (raw) →M721 · Family 377 Uniform Interior `C¹,α` Estimates for Infinity-Harmonic Functions
We prove a uniform interior `C¹,α_d` estimate for bounded infinity-harmonic functions in every dimension d ≥ 3. For some `α_d (0,1/3]` depending only on the dimension, the gradient's supremum norm and αd-Hölder seminorm on B1/2 are bounded by a dimension-dependent constant times the oscillation on B1. The exponent is not explicit, and no endpoint or boundary regularity claim is made.
Catalog note: General mathematics result in the OpenAI catalog. QTL does not classify this family as directly relevant to post-quantum migration planning.
View manuscript on GitHub → Open PDF (raw) →
Verification note: OpenAI states that not all manuscripts are formalized in Lean and that some unformalized results may contain errors.
Manuscript rows mirror every preprints/*.pdf link in CONTENTS.md (721 listed; OpenAI’s header cites 722 total). Family rows cover all 372 families.
QTL recommendations
- Separate “claimed” from “deployed.” Treat catalog entries as early-warning intelligence. Production decisions should still anchor on NIST FIPS standards (ML-KEM, ML-DSA, SLH-DSA) and your CBOM—not on preprints from any AI lab.
- Add AI-mathematics to threat-intake. Assign ownership for monitoring large model releases the same way you track NIST drafts and IETF PQ TLS work.
- Invest in agility, not just parameters. If discovery accelerates across number theory and lattice analysis, the winning architecture is swappable algorithms, monitored dependencies, and rehearsed incident playbooks—not a one-time RSA sunset project.
- Pair with readiness tooling. Use QTL’s PQC Readiness assessment to translate external research shocks into inventory priorities and executive timelines.
What we did not do
This page does not assert that OpenAI has broken modern cryptography. It maps where their released mathematics intersects security-relevant fields so CISOs, cryptographers, and program managers can prioritize review time. For primary sources, start with OpenAI’s overview PDF and CONTENTS.md manuscript map.