Archive

A Cyclic Polytabloid Proof of Saxl's Conjecture

OpenAI

• Mathematics
• Theoretical Computer Science

For every m≥1, the tensor square of the staircase Specht representation of the symmetric group S_{m(m+1)/2} contains every irreducible complex representation.

Essential Singularity of the Correlation Length in the Planar XY Model

OpenAI

• Mathematics
• Theoretical Physics

For the nearest-neighbor cosine XY model on the square lattice, the manuscript claims sqrt(βc−β) log ξ(β) tends to a finite positive model-specific constant as β rises to βc from below. Here ξ is defined from the inverse axis mass after the infinite-volume and separation limits in that order.

GPT-6 Astra resolves Erdős Problem 74 on local bipartiteness

• Mathematics
• Theoretical Computer Science

There is a divergent integer-valued function f such that if every finite n-vertex subgraph of a graph becomes bipartite after at most f(n) edge deletions, then the graph has finite chromatic number. The formal proof gives three-colourability.

A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic

OpenAI

• Mathematics
• Theoretical Computer Science

For a finite field K of odd characteristic and a finitely generated group G with torsion, the construction gives a linear cellular automaton on all K^G configurations that is injective but not surjective.

A Counterexample to the Group-Ring Determinant Conjecture

OpenAI

• Mathematics
• Theoretical Computer Science

A finitely generated group G and matrix A over Z[G] are constructed with A invertible over Q[G] and Fuglede–Kadison determinant strictly between zero and one. This contradicts the unrestricted group-ring Determinant Conjecture.

Almost Affine Invariance Over Prime Fields: Green Problem 90

Jie Ma, Quanyu Tang, Max Wenqiang Xu

• Mathematics

Ma, Tang and Xu prove that half-density subsets of Fₚ can be almost invariant under every affine map with 1≤|a|≤K and |b|≤K exactly when K=o(log p). The AI-attributed new ingredient is the q-adic valuation formulation used for the sharp upper bound.

The Oddtown problem modulo a composite number

Boris Bukh, Ting-Wei Chao, Zeyu Zheng

• Mathematics

For every fixed modulus ℓ with at least two distinct prime factors, the AI-origin construction gives fℓ(n)≥ω(ℓ)n−o(n) through projective-hyperplane complements and low-rank correction blocks. This refutes fℓ(n)≤n outside prime powers.

A sharp 5/8 bound for an Erdős-Sós pairwise-sums problem

Ricky Cipollini

• Mathematics

Theorem 1.1 gives a constant C such that every sufficiently large A⊆{1,…,N} with |A|≥5N/8+C contains distinct a,b,c for which a+b, a+c and b+c also lie in A. An interval-union construction shows the 5/8 coefficient is sharp.

Quantitative lower bounds for trace reconstruction

OpenAI

• Mathematics
• Theoretical Computer Science

For every fixed deletion probability q∈(0,1), worst-case reconstruction of deterministic binary strings from independent deletion traces needs n^{Ω(log log n)} samples for any fixed positive success probability, even with unrestricted computation.

Claude counterexample refutes Erdős #619 diameter-four saving

• Mathematics

For every η>0, a family of connected triangle-free graphs needs at least (1−η)n new edges to become diameter at most four while staying triangle-free. The problem record states the stronger quantitative bound h₄(G)≥n−O(n^(8/9)(log n)^(2/9)).

Sharp Bounds on Ground State Energy of the SYK Model

Arpon Basu, Pravesh K. Kothari, Siddhant Midha

• Theoretical Physics

GPT-5.6 Pro suggested a twisted-boson moment identity and an isometric embedding and intertwining identity that transfer a bosonic witness into a lower-bound certificate. These steps support the authors’ E||H_SYK||=(1−o(1))√(2n)/k result for even k growing to infinity with k=o(√n), with a stated sparse extension.

Ten Advances in Mathematics and Theoretical Computer Science

OpenAI

• Mathematics

A connected bipartite two-degenerate graph H meeting the stated degree conditions has extremal number at least c n^(3/2+ε) for some c,ε>0. This contradicts the proposed exponent for that class.

Ten Advances in Mathematics and Theoretical Computer Science

OpenAI

• Mathematics
• Theoretical Computer Science

For every relative binary distance 0<δ<1/2, Chapter 2 gives an asymptotic rate bound strictly below the classical MRRW bound. Companion spherical-code bounds improve the localized linear-programming hierarchy for 0<s<1.

Ten Advances in Mathematics and Theoretical Computer Science

OpenAI

• Mathematics

A finite family of connected bipartite graphs, each containing a cycle, has joint extremal number O(n^(4/3−1/48)), while every member has extremal number Ω(n^(4/3)). This refutes the substantive cyclic version of the Erdős–Simonovits compactness claim.

Ten Advances in Mathematics and Theoretical Computer Science

OpenAI

• Mathematics

A recursive k-edge-coloring of a complete graph without a monochromatic triangle gives Rₖ(3)≥(c k^(1/3)/log k)^k for all sufficiently large k. Thus log Rₖ(3)=Θ(k log k).

Ten Advances in Mathematics and Theoretical Computer Science

OpenAI

• Mathematics
• Theoretical Computer Science

For the n×n permanent over the complex numbers and n≥32, Chapter 5 gives at least n⁴/(128 log₂ n) variable leaves for division-free arithmetic formulas and n⁴/(192 log₂ n) for valid rational formulas. A separate circuit lower bound has different scope.

Ten Advances in Mathematics and Theoretical Computer Science

OpenAI

• Mathematics
• Theoretical Computer Science

A direct 3SAT reduction proves NP-hardness of approximating Euclidean closest vector within n^(1/400), under the dimension and promise conventions of Chapter 7. The chapter gives separate consequences for binary nearest codeword and finite p norms.

Ten Advances in Mathematics and Theoretical Computer Science

OpenAI

• Mathematics
• Theoretical Computer Science

For the optimal Cohn–Elkies linear-programming packing bound LP_d, OpenAI gives the exponential root limit √(e/(2π)). This yields the sphere-packing upper bound (√(e/(2π))+o(1))ᵈ and corresponding asymptotic Fourier sign-uncertainty radii.

Ten Advances in Mathematics and Theoretical Computer Science

OpenAI

• Mathematics
• Theoretical Computer Science

OpenAI gives a finitely presented group that lacks finite permutation approximations, establishing the existence of non-sofic groups. Its construction combines a Kun–Thom criterion with an expander-matching argument involving binary Leavitt algebra and Thompson’s group V.

Two Möbius groups suffice: Euclidean Ramsey sets on spheres of every dimension, after the tensor classification

Will Strinz

• Mathematics
• Theoretical Computer Science

Strinz reports that every finite set of distinct points on a circle is Euclidean Ramsey when all but at most three points have coordinates algebraic over Q. The circle’s center and radius may be arbitrary real numbers.

Ten Advances in Mathematics and Theoretical Computer Science

OpenAI

• Mathematics
• Theoretical Computer Science

For every positive dimension n, a compact convex body with nonempty interior, barycenter 0 and no interior lattice point other than 0 has volume at most (n+1)ⁿ/n!. A centered simplex attains equality. This is the inequality, separate from the later equality classification.

Ten Advances in Mathematics and Theoretical Computer Science

OpenAI

• Mathematics
• Theoretical Computer Science

OpenAI constructs infinitely many pairwise nonisomorphic finitely generated ICC property-(T) groups whose tracial group von Neumann factors are mutually isomorphic. The family refutes Connes’s rigidity conjecture and a finite-to-one strengthening.

A counterexample to the periodicity conjecture for finite-dimensional algebras

Haruhisa Enomoto

• Mathematics

Enomoto constructs a 36-dimensional algebra Λ(q) over an algebraically closed field with an element q of infinite multiplicative order. Every simple right module has period four and the algebra is twisted periodic, yet a two-dimensional module has pairwise nonisomorphic even syzygies, so the algebra is not periodic.

The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture

OpenAI

• Mathematics
• Theoretical Computer Science

For every torsion-free discrete group G and commutative characteristic-zero domain R, the OpenAI Bass manuscript claims each scalar idempotent in R[G] is 0 or 1. The separate trace-rank assertion concerns algebraic K₀(C[G]).

Determinant Divisibility of Centered Latin Squares: Binary Criteria, Universal Counterexamples, and p-Adic Bounds

Oleksiy Babanskyy

• Mathematics

For even-order Latin squares, Babanskyy gives binary-rank and adjugate criteria for an extra factor of two in the centered determinant. An explicit family has an odd normalized determinant quotient for every n≡2 mod 4 with n≥6.

Cycles of length divisible by five in graphs of minimum degree five - The k=5 case of Dean's conjecture

Elias Botsford

• Mathematics
• Theoretical Computer Science

Botsford’s revised theorem says every finite simple graph with minimum degree at least five has a simple cycle whose length is divisible by five. The proof reduces a minimal counterexample to 5-weak graphs and treats bipartite, triangular and triangle-free cases.

Depth-Two Smith Profiles and Carry Geometry for Affine Hjelmslev–Radon Incidence

Oleksiy Babanskyy

• Mathematics

Babanskyy determines the complete p-primary Smith profile of the integer affine point–line incidence matrix over Z/p²Z for every prime p, including the nonunit factors p, p² and p³ and their multiplicities. The argument also gives a nonsplit transfer extension and a two-chart carry presentation.

AI-built Lean proof reports Conway’s omnific refinement property

• Mathematics
• Theoretical Computer Science

For cut-defined omnific integers a,b,c,d represented as numeric games, a Lean theorem says ab=cd implies omnific e,f,g,h with a=ef, b=gh, c=eg and d=fh, including zero cases. The theorem is stated at each Lean universe, rather than putting all surreal numbers into one type.

Depth-Three Smith Profiles and Newton-Minor Geometry for Affine Hjelmslev–Radon Incidence

Oleksiy Babanskyy

• Mathematics

Babanskyy reports every nonzero Smith factor of the integer affine point–line incidence matrix over Z/p³Z for every prime p. The profile reaches exponent p⁵ and has six multiplicities, with polynomial formulas for odd p and explicit p=2,3 rows. The new step resolves a q-Pascal common image through q-Newton minors and a cross-chart carry calculation.

Counterexamples to two conjectures about matroids

Matt Larson

• Mathematics
• Theoretical Computer Science

A graphic matroid formed from four paths between two vertices, three of length 26 and one of length 1, has flat counts W₇₄²=967,813,250,625 < W₇₃W₇₅=982,359,393,275. This refutes Mason’s plain log-concavity conjecture for matroid flats.

AI-centered Lean proof classifies base-three partition counts

• Mathematics

For partitions of {1,…,n} into nonempty blocks of at most three elements whose sums are powers of three, the count is exactly one on N_u={1,2,3,4}∪{3ᵗ+d:t≥2, d∈{−4,−2,−1,0,1,2,3,5}}; exactly two on E₂={13}∪{3ᵗ−3:t≥2}∪{3ᵗ−6:t≥3}; and at least three otherwise.

Short proofs in combinatorics, probability and number theory II

Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke, Gregory Valiant

• Mathematics

Only finitely many integers n have n−k² prime for every integer k≥1 with gcd(k,n)=1 and k²<n. This is the original a=1 Erdős Problem 1141 statement.

Polynomial Hitting Lists for Noncommutative Rational Formulas

OpenAI

• Mathematics
• Theoretical Computer Science

Given unary n and a size bound s, a deterministic machine constructs in polynomial bit time a polynomial-size list of rational matrix tuples of polynomial total output length and common polynomial dimension. Every admissible nonzero noncommutative rational formula over Q of size at most s has a defined, invertible evaluation on a listed tuple.

Star Fleet reports Erdős Problem 796 second-order constant

• Mathematics

For the corrected Erdős Problem 796 formulation, Star Fleet reports that (g₃(n)−n log log n/log n)/(n/log n) converges to an explicit constant written Mertens.M plus a variational limit.

Uniform exclusion of Landau–Siegel zeros

OpenAI

• Mathematics

There is an absolute c>0 such that every real zero 0<β<1 of the L-function of a primitive nonprincipal real Dirichlet character with conductor q≥3 satisfies 1−β≥c/log q.

Integral tail signatures and the Hilbert–Smith conjecture

A. Dabrowski's AI agents

• Mathematics

The selected formal development claims that a continuous additive p-adic-integer action is excluded on a connected, Hausdorff, second-countable, finite-dimensional manifold without boundary. It also claims that a second-countable locally compact Hausdorff group acting continuously and faithfully on such a manifold has a compatible smooth Lie-group structure.

The geometric case of the Erdős similarity conjecture

OpenAI

• Mathematics

For each fixed q∈(0,1) and η∈(0,1), there is a compact set E in [0,1] of measure greater than 1−η containing no translated copy of {qⁿ:n≥1} under any nonzero real dilation of either sign. E may depend on q.

Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras

OpenAI

• Mathematics

For every 0<ε<1, every self-adjoint x in any von Neumann algebra M admits strong-operator paving over any maximal abelian subalgebra A with at most 5×10⁸ε⁻² projections. The count is fixed before the inclusion, operator and strong neighborhood, without separability or conditional-expectation assumptions.

Short proofs in combinatorics, probability and number theory II

Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke, Gregory Valiant

• Mathematics

A single real sequence has uniformly bounded partial exponential sums A_k= sup_N |Σ_{n<N} exp(2πikx_n)|=O(√(k log(2k))) for every positive integer k. This implies the weaker limsup Ã_k=o(k) sought in Erdős Problem 987.

AI-assisted Hadamard constructions cover eight orders

• Mathematics

Using periodic T-matrix quadruples of orders 103 and 163 and symmetric-circulant Williamson quadruples of orders 7, 11, 13, 17, 19, 23 and 29, a formalized Cooper–Wallis construction gives Hadamard matrices at orders 2884, 4532, 7004, 7172, 7828, 8476, 9476 and 11948. A separate parity result rules out Miyamoto’s displayed C2-matrix ingredient at positive block orders not congruent to 1 mod 4.

AI-found residue sets improve a polynomial-free lower bound

• Mathematics

For positive-integer sets avoiding {x,x+y,x+y²} for every nonzero integer y, Claude-found residue sets modulo 145 of sizes 10 and 32 satisfy Younis’s two-set conditions. They give lower-bound exponent 1/2+log(10)/(3log(145))+log(32)/(6log(145)), between 0.7702 and 0.77029.

AI-built Lean program triangulates simple polygons in linear time

• Mathematics
• Theoretical Computer Science

For every closed simple polygon with n vertices, including collinear vertices and equal heights, an explicit exact-real-RAM program returns a conforming triangulation with n−2 triangles in at most C(n+1) steps and natural words bounded by K(n+1). It is a variant of Chazelle’s algorithm that completes the visibility map directly below n=2³⁰+2.

AI-derived determinant restriction for finite-order matrices

• Mathematics

For m>0 and N≥1, the formalized refinement says GL(N,ℤ) contains an element of order m with determinant −1 exactly when m is even and ψ(m)+1≤N, where ψ sums φ(p^k) over exact prime-power factors except 2. The same criterion holds over ℚ, and the least integral dimension is ψ(m)+1.

AI-assisted proof gives sharp graph-matrix norm orders

• Mathematics

For each fixed finite simple graph shape in the globally injective dense independent-sign model and fixed p>0, the pth norm moment has matching upper and lower orders n^((h+i₀−s)/2)log(2n)^(b*/2), with h the role count, i₀ isolated interior roles, s the minimum boundary-separator size and b* the largest attached floating-component count over minimum separators.

AI-assisted Walsh embedding reaches linear row scaling

• Mathematics
• Theoretical Computer Science

For n=2^k, a fixed real orthonormal n×d frame, 0<ε<1 and 0<δ<1/2, two independent sign rerandomizations of the normalized Walsh transform followed by uniform row sampling preserve all squared norms within ε with probability at least 1−δ. The stated row count is min(n,⌈8192(d+27⌈log₄(4d/δ)⌉³)/ε²⌉), which is O(d/ε²) for fixed δ.

AI-assisted sharp bounds for Hansen polytopes of cographs

• Mathematics

For every nonempty cograph G, its Hansen polytope in dimension d=|V(G)|+1 has normalized Mahler volume at least one, at least 3^d nonempty faces and at least 2^d d! full flags. Each equality holds exactly for threshold graphs.

AI-assisted inverse theorem for short zero sums in C₃³

• Mathematics

For a length-24 positional sequence of nonzero elements of C₃³, every pair of nonempty zero-sum subsequences of length at most three has zero intersection sum exactly when its positions can be permuted into three copies of eight distinct elements with no short nonempty zero sum.

Three Conjectures on Binary Channels for the Doubly Symmetric Binary Source

Georg Pichler

• Mathematics
• Theoretical Computer Science

For a doubly symmetric binary source with crossover p∈[0,1], the convex hull of rate triples from arbitrary binary channels equals that from binary symmetric channels. At p=0 and parameters a,d∈(0,1), no binary channel pair with rates no larger than the stated Z/S pair has greater I(U;V), and no pair with rates at least those of the stated Z/Z pair has smaller I(U;V). These are value bounds, not uniqueness claims.

AI proof of Gallardo’s all-ones polynomial equivalence

• Mathematics

For every n≥2 over GF(2), the polynomial 1+X+⋯+X^n is irreducible exactly when its composition with X²+X+1 is irreducible. The formalization also checks the corresponding OEIS set statement and excludes n=1.

AI-derived bounds for Mazur’s greedy drawing construction

• Mathematics
• Theoretical Computer Science

For a well-formed Mazur edge, cycle, series and join scheme with n≥2 vertices and j nontrivial joins, the span satisfies T+1≤2^j(n−2j), with equality at (n,j)=(2,0) or 2j+3≤n. Its worst case is T=2^(n/2)−1 for even n and T=3·2^((n−3)/2)−1 for odd n. Companion theorems give interior dart mass ≤aₛbₛT, Dirichlet row norm ≤2(n−1)²T and |det M|≤(2(n−1)²T)^n, bound the candidate products T²|det M| and T³|det M|, and prove z=adj(M)u with Mz=Δu and boundary zₛ=Δhₛ(T−hₛ) when hₛ≤T, where Δ=det M.