AI contributions across algorithms and computation, logic and formal proof, and optimization in the catalog, including their connections to mathematics.
The manuscript gives converse bounds that completely characterize the storage/repair-bandwidth tradeoffs of exact-repair regenerating codes for (n,k,d)=(6,5,5) and (7,6,6), without assuming linear codes. Previously known layered codes with space sharing attain the bounds.
For every finite two-player entangled game G with entangled value below one, the value of winning all n independent repetitions decays exponentially in n; the Chapter 6 bound depends on the gap and answer alphabet sizes.
An explicit fixed rational cubic in 14 variables gives a quartically regularized polynomial whose exact minimum shift is nonnegative but not a sum of polynomial squares for every positive regularization parameter. A second explicit cubic has a unique global minimizer with positive-definite Hessian there and retains a strict SOS gap for every positive parameter.
The archived preprint proves Erdős–Kac normal laws for the number of prime factors, with or without multiplicity, of base-b palindromes and reversed base-b primes for every b≥2. The reversed-prime theorem also allows a fixed leading digit.
For every fixed k≥3, random k-SAT with independent uniformly signed proper clauses sampled with replacement has a positive finite limiting density αₖ: satisfiability probability tends to one below αₖ and zero above it. The claim makes no assertion at αₖ.
An OpenAI manuscript claims a real homogeneous hyperbolic polynomial whose closed cone has no exact finite affine semidefinite lift, even with arbitrary real coefficients and any finite number of auxiliary variables.
The authors prove λ₁(G)²+λ₂(G)²≤2(1−1/ω(G))|E(G)| for every finite simple noncomplete graph G with at least two vertices. A stronger weighted theorem bounds the two largest positive eigenvalues of a symmetric nonnegative zero-diagonal matrix supported on G by (1−1/ω(G)) times its squared Frobenius norm.
OpenAI’s Theorem A constructs a C∞ positive-definite metric g on (-1,1)^2 such that no open neighborhood of the origin admits a C∞ isometric immersion into R³. The paper also says g can agree with the Euclidean metric to every order at the origin. The Lean endpoint directly states the smooth positive metric and no-local-immersion claim; the flat-jet refinement is not part of that endpoint.
For every k≥2, a nonempty pool of ranked blocks on pairwise coprime square-free moduli, with ranks strictly decreasing along nonzero k-th-power-residue difference arcs, gives α(k,P)≤liminf log D_k(N)/log N and eventually D_k(N)≥N^ρ for every ρ<α(k,P). Two separate nine-block instances give α(4,P₄)=0.912145042700…>0.9121 and α(6,P₆)=0.950825559086…>0.9508; each exceeds the specified six-residue Ruzsa-transfer comparator.
For every simple graph G on seven vertices, either G or its complement has an injective unit-distance representation in Euclidean R³, where every edge has length one and nonedges are unconstrained. This proves the d=3 case of Frankl–Kupavskii–Swanepoel Problem 2; their general 2d+1-vertex question remains open. Combining with the published lower bound gives f_D(7)=3, although the lower bound is outside the formal target.
For d,l≥1, a two-fold Langford sequence of order l and defect d exists exactly when 3l≥2d−1. Each number p in [d,d+l−1] occupies endpoints of two disjoint pairs at distance p among 4l positions; the proof uses sixteen small cells, concatenation and signed-permutation families.
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.
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.
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.
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.
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.
For every n>0 and every set of n² lattice points, the induced square-grid graph has at most as many spanning trees as the n×n square. Equality holds exactly for translates of that square.
For every nonnegative rational n×n matrix, Anari’s formal development constructs one rational-output algorithm and an absolute constant c<√2 with a two-sided cⁿ approximation guarantee for the permanent. It also proves deterministic polynomial time in the ordinary binary input size.
A released AI-generated sum-of-squares and Lean candidate encodes Φ(A)≤1226/625−||A−U₅||²_F/625 for nonnegative 5×5 matrices whose entries sum to five. If its encoding matches the original problem, it implies unique equality at U₅.
Protti’s planar acyclic four-terminal instance needs upper overload at least 335 when the largest demand is 294 if cost cannot increase. A rational family gives a lower bound (299−41√41)/32≈1.139747 for the unrestricted planar constant.
A connected 13-vertex graph formed by joining an independent seven-vertex set to two disjoint triangles has independence number 7, largest induced bipartite subgraph size 9 and Havel–Hakimi residue 2. These values violate Conjecture 109’s proposed bound of 6.
For every finite uniformly dense rank-three matroid M with |E(M)|=3k and k>0, the version-2 Lean theorem constructs a cyclic ordering in which each consecutive triple is a basis. It includes non-simple and non-paving matroids.
For k≥4, the ordered three-summand count Fₖ(t) modulo 2ᵏ−1, with total binary weight below k, equals (k+23)3^(k−4) when the k-bit target has exactly two nonadjacent zero digits. The paper proves balancing only through carry mass five and does not classify global maximizers.
For the paper’s one-layer, attention-only transformer on Boolean inputs with a dedicated query token and linear threshold readout, the autonomous core results characterize head complexity by sums of admissible linear-fractional functions, bound it below by threshold degree, and determine it exactly for symmetric Boolean functions by the number of sign changes. In particular, n-bit parity requires exactly n heads.
For every finite family of independent Bernoulli variables, increasing each success probability from pᵢ to rᵢ with 0 ≤ pᵢ ≤ rᵢ ≤ 1/2 cannot decrease either the Rényi or Tsallis entropy of their sum, for any fixed entropy order 0 ≤ q ≤ 2. The statement includes zero and fair probabilities, support entropy at q=0 and Shannon entropy at q=1.
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.
Anthropic reports recovering a reproducible lrand48-and-timestamp recipe for all five NIST prime-curve seeds, five additional ANSI prime-curve seeds and four NIST binary-curve seeds. The reported timestamps and candidate indices regenerate each 160-bit public seed. The NIST B-163 seed remains unexplained.
The AI-origin improvement in Appendix B gives non-expansive line embeddings with O(log h) average distortion for the stated vertex-weighted metrics of connected graphs excluding the one-subdivision of K_h as an induced minor. The paper derives balanced separators of size O(h log h √m) for the corresponding m-edge graphs.
The support-criterion project reports that any symmetric real probability law with 0<a≤|X|≤b almost surely and b/a≤√3+√2 satisfies E exp(tX)≤exp(t² E[X²]/2). The inequality is strict for nonzero t, the variance proxy is optimal, and every larger universal support ratio admits a symmetric four-point counterexample.
For any real law with X² in [A,B], A=3−√6 and B=3+√6, let δ=3(E[X²])²−E[X⁴] and let X₀ be the four-node Gaussian quadrature law identified by zero deficit. For every feasible 0≤δ≤30+12√6, W1(L(X²),L(X₀²))≤√(δ/3), using ordinary infimum-over-couplings Wasserstein distance. The bound is attained at every feasible deficit and does not require symmetry of X.
For every symmetric law with X² in [A,B], A=3−√6 and B=3+√6, set v=E[X²] and h=E[(X²−A)(B−X²)]. For every real t, exp(vt²/2)−E exp(tX)≥h t⁴/24+t⁸/25200. Both coefficients are optimal for this two-term bound. A positive coefficient on the intrinsic fourth deficit 3v²−E[X⁴] cannot be paired with the optimal eighth-order coefficient.
Gu, Kemeny, Yelin and Lukin report lower Toffoli and peak-qubit counts in seven specified circuit benchmarks, plus a new prime-field modular-inversion architecture. For fixed-point Gaussian evaluation, their stated benchmark falls from about 24,000 Toffolis and 362 qubits to 1,583 Toffolis and 139 qubits; the reported worst-case error is 9.99×10⁻⁸.
A Gemini agent pipeline discovered and formally proved closed systems of Collatz-like acceleration rules for eleven specified six-state, two-symbol Turing machines. Each system summarizes exact execution from its fixed blank-tape starting configuration; the machines’ eventual halting behavior remains unresolved.
Möbius invariance and the stated corollaries for finite distinct spherical Ramsey configurations, relative to the credited OpenAI classification and classical identification of the Möbius action.
Lau reports an explicit refinement of Mazur’s positive-density Collatz result: writing L=2^(2^(2^87640)), for every X≥2^L at least X/L positive integers n<X reach 1 within floor(10.46 log n) ordinary Collatz steps. The formal development also gives an odd-starting-value version with accelerated-step bounds and target-reaching variants for prescribed positive targets not divisible by three.
For every s≥3, allowed base q and k≥1, the explicit N=q^(sk) digital net has covering radius ≤√s N^(−1/s); when q≥5a*_s+4 its point separation is ≥(1−a*_s/q)N^(−1/s), with the stated bound on a*_s.
Every compact connected planar set that can be moved continuously through both a right-turning and left-turning unit-width hallway from the same starting orientation has area at most M=1+4Y²+arctan(Y), where Y>0 solves 4Y³+3Y=1. Equality holds exactly for rotated and translated copies of Romik’s sofa, and the sofa attains M.
For any finite-Kraus 1→2 qubit channel whose mean fidelity over the six Pauli eigenstates is at least 5/6−ε, the Kraus operators have total squared Frobenius distance at most 6ε from the span of the two Bužek–Hillery Kraus operators. For every input matrix ρ, the channel output is within 200√ε‖ρ‖F of the Bužek–Hillery channel in Frobenius norm.
For every natural k≥0, the convergent series Σₙ≥1 σₖ(n)/n! is irrational, where σₖ(n) is the sum of kth powers of positive divisors of n. The k>0 cases answer Erdős Problem 252; the Lean theorem also handles k=0.
For every N≥0 and h≥0, constant-step gradient descent xᵢ₊₁=xᵢ−(h/L)∇f(xᵢ) on a differentiable convex function with L-Lipschitz gradient in a real Hilbert space satisfies ‖∇f(xN)‖≤L‖x0−x*‖max{1/(1+Nh),|1−h|^N}. One-dimensional Huber and quadratic examples attain the two branches. The compared Lean contract also classifies sampled equality trajectories and proves the same sharp factor for the terminal residual of relaxed proximal minimization with actual minimizers.
Cureton and Thorner’s version-two manuscript reports that for every number field K, its Dedekind zeta function has no zeros in Re(s)>31/32, excluding its pole at s=1. A corollary gives the same half-plane for finite-order Hecke L-functions wherever they are holomorphic, including s=1 for nonprincipal characters.
The specified seven-vertex induced Paley tournament has transitive number 4 and capacity between √18 and 5; every tournament on at most six vertices has capacity equal to its transitive number, so seven is minimal.
For a finite open chain with positive strong-bond lower bounds βᵢ, boundary parameter s≥0 and weak-to-strong ratios qᵢ∈[ℓᵢ,ρᵢ], the minimum scalar squared gap minq λmin(DβQ(q)ᵀQ(q)Dβ) is positive and equals the minimum over the ordered one-switch endpoint vectors q⁽ᵏ⁾, with lower endpoints before k and upper endpoints after k. The same value sharply bounds the squared nonzero gap for arbitrary finite-dimensional complex matrix channels satisfying the stated singular-value constraints.
On a connected open subset of the Euclidean plane, if the canonical local geodesic midpoint maps of the dual torsion-free connections D+KC and D−KC are complementary near every diagonal point, then their totally symmetric cubic tensor field C is constant. The formal proof constructs the short geodesic branches and also proves a differential obstruction criterion and a five-direction test.
The-Anh Vu-Le’s project reports that Gerver’s sofa is the unique area-maximizing moving sofa up to rigid motion, and that sofas with area deficit ε are within C√ε after translation in its Hausdorff-distance model, with exponent 1/2 optimal.
For a balanced probability law ν on the 24 permutations of four labels at total-variation distance at most r≤1/24 from uniform, and nonnegative rows fᵢ normalized by the uniform mean of fᵢ² equal to one, 1−Eν∏ᵢfᵢ(σ(i))≥(1−24r)/9 ΣᵢVar(fᵢ). The coefficient is sharp at each radius and 1/24 is the exact admissible radius.
Deterministic integer 3SUM in O(n^1.99896) and APSP in O(n^2.99791) in the pinned upstream word RAM model, for polynomially bounded input numbers.
Proudfoot and Perrault report that the exact tensor-rank matrix-multiplication exponent is strictly below 2.3710449 over every commutative ring. Their formal theorem uses the exact rational bound 23710449/10000000; the accompanying paper rounds this to 2.371045.
For F(q)=Σₙ≥₁1/(qⁿ−1), a degree criterion excludes polynomial relations of degree at most e over Q(q) when eλ log M(q)<log q, where λ=(42963+428220/π²)/202140. It gives nonquadraticity for every integer q≥2 and irrationality for rational q=a/b>1 when log b/log a<0.5728; a certified refinement reaches 0.573290.
For a minimally four-generated numerical semigroup S=⟨n₀<n₁<n₂<n₃⟩, the minimal number of generators of its toric ideal is at most 2W over any field, where W=(n₃−n₀)/gcd(n₁−n₀,n₂−n₀,n₃−n₀). Consequently it is at most twice the ordinary width n₃−n₀.
Every set of six distinct points on a circle satisfies the FieldCriterion tensor condition of the OpenAI Euclidean Ramsey classification. More generally, every spherical finite set of at most m+4 points, where m is its affine-span dimension, satisfies that criterion in its own coordinate field; every spherical set with algebraic coordinates also satisfies it.
Navot’s AI-assisted Lean developments extend OpenAI’s matrix multiplication exponent bound ω ≤ 9/4 from complex numbers to every field, including finite fields and positive characteristic, and subsequently to every associative unital ring, including noncommutative rings. The ring development establishes exact integer coefficient tensor-rank identities and, for every ε > 0, correct division-free arithmetic programs with cost at most C n^(9/4 + ε) at every positive matrix size, for one C independent of n. Palomar registered the field development on 8 October and the ring development on 11 October 2026.
Conditional on R₄(3)≤61, the sixth Schur number satisfies S(6)≤1801; the general Ramsey-to-Schur bound and the stated frontier restrictions are proved with the Ramsey assertion left as an explicit premise.
Every regular language over a finite alphabet has a generalized regular expression over that same alphabet whose Kleene-star nesting height is at most thirteen, with complement taken relative to the same free monoid.
For each k≥4, an explicit construction maps a finite compatible-choice system to two equal-size simple uncolored graphs that are k-dimensional Weisfeiler–Leman equivalent exactly when the system has no successful choice; a successful choice is detected after two joint-refinement rounds or one separate-refinement round.
Every exact affine semidefinite lift of the perfect-matching polytope of the complete graph on n vertices has matrix size at least 2^{c n} for some absolute c>0 and all sufficiently large even n.
For every binary string of known length n and known rational deletion retention probability p, one randomized algorithm reconstructs the string with probability at least 2/3 using an explicit sample budget and bit time polynomial in that budget and the binary input length; for each fixed p, both are quasipolynomial in n.
Given a prime p in binary and positive n,s in unary, one deterministic polynomial-bit-time machine outputs a tuple of matrices over F_p of dimension O(n³s⁶) that detects every nonzero division-free noncommutative formula of size at most s in n variables over every field of characteristic p.
For every fixed r≥3, every vertex-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)) for all 0<p<1, without requiring monotonicity.
For every fixed C>1, approximating Uniform Sparsest Cut within factor C is NP-hard on graphs with nonnegative rational capacities and unit demand between every pair of distinct vertices.
With exact rank-value oracles, the number of common integer bases of two equal-rank polymatroids can be approximated by an FPRAS whose oracle calls and non-oracle bit work are polynomial on every run in ground-set size, binary input length, inverse relative error, and log inverse failure probability.
For every n, one quantum oracle circuit generated in polynomial time from n using H, T, T† and CNOT approximates the channel of every n-qubit unitary to full diamond-norm error at most 1/2 after a suitable Boolean oracle is chosen; circuit size, qubits, queries and query length are polynomial in n.
The truth of existential–universal sentences over the reals is decidable in one fixed level of the counting hierarchy, even when their integer polynomials are specified by arithmetic circuits; in particular, the existential theory of the reals lies in the counting hierarchy.
There are total Boolean functions whose bounded-error randomized query complexity cannot obey any universal bound of the form O((1+Q(f))^α) with α<4, where Q is bounded-error quantum query complexity.
For every fixed d≥3 and every sufficiently large even n, a deterministic algorithm outputs in polynomial bit time a simple connected nonbipartite d-regular n-vertex graph whose nonconstant adjacency eigenvalues lie strictly between ±2√(d−1).
There are fixed positive rational ε and δ and a deterministic reduction mapping an End-of-Line instance of binary length N to a generalized circuit of length N(log N)^O(1). A solution to the original instance can be recovered in polynomial time from any polynomial-length rational assignment ε-satisfying all but a δ fraction of gates.
For every fixed A>0 and sufficiently large dimension d, a one-pass learner retaining at most Ad² bits between independent exact Gaussian linear measurements needs at least c_A d log(1/ε) measurements to estimate a uniformly random unit vector to angular error ε≤1/10 with success probability at least 2/3.
On positive integral metrics of polynomially bounded diameter, a supplied anchor with distinct accurate proxies for all but logarithmically many comparison clusters and sufficiently small total proxy error supports a polynomial-time k-median recovery algorithm opening at most k facilities and attaining a (1+2/e+ε) factor against those comparison centers with any requested confidence.
A polynomial-time uniform family of quantum circuits over one fixed finite bounded-arity gate set outputs the complete prime factorization of every integer N≥2 with probability exactly one, using polynomially many gates and qubits in the input bit length.
For every finite two-player game with finite-dimensional entangled value v<1 and arbitrary correlated questions, the maximum probability of winning at least a fraction v+δ of k repetitions decays at least as exp(−κδ¹³k/(1+log(|A||B|))) for one universal κ>0 and 0<δ<1−v.
Approximating the electronic Coulomb spectral infimum in three-dimensional continuum space is QMA-hard when positive integer nuclear charges use binary encoding, nuclei have distinct rational positions and electron count is unary; a deterministic polynomial-time reduction produces a promise gap of at least one.
For any pure type system, if every legal expression in every valid context has a terminating β-reduction sequence, then every β-reduction sequence from every such expression terminates, including reductions inside type annotations.
An explicit isomorphism-invariant query on all finite structures in one fixed vocabulary of eight relations is decidable in deterministic polynomial time but is not definable in choiceless polynomial time with counting.
For every C>0, some nonconstant Boolean function on a finite sign cube has the sum of its linear Fourier coefficients greater than C√deg(f), where degree is that of its real multilinear representation.
A fixed deterministic machine with one writable tape can be simulated through a supplied time cap T≥2 in O(T^(2/5) polylog(T+2)) work-space bits while preserving its finite-control and halting outcome.
Over every characteristic-zero field, for all sufficiently large n every syntactically homogeneous depth-five ΣΠΣΠΣ circuit computing the (1,1) entry of n independent n×n matrix products has at least n^(√n/400) gates.
For every n≥4, an explicit n-state two-way nondeterministic finite automaton has a complement requiring at least ½·2^⌊(n−4)/127⌋−1 states in any two-way nondeterministic finite automaton.
Every degree-at-most-d real polynomial threshold function on the uniform n-dimensional Boolean cube has average sensitivity at most 8d√n for 1≤d≤n.
For any nonnegative row and column margins with equal totals, an exact uniform sampler for integer contingency tables terminates almost surely and has expected bit complexity polynomial in both dimensions and the binary input length.
For every fixed depth, polynomial qubit bound and ε>0, sufficiently large measured-output quantum circuits using arbitrary one-qubit and unbounded-arity Toffoli gates have some input on which parity succeeds with probability below 1/2+ε.
Every recursively enumerable subset of natural-number n-tuples, for n≥1, has an integer-polynomial Diophantine representation with exactly one complete natural-number auxiliary tuple for each member and none for nonmembers.
There are total Boolean functions with unbounded block sensitivity divided by sensitivity squared; in particular, a fixed exponent α>2 holds along a sequence with unbounded block sensitivity.
For a binary channel on a uniform discrete cube under independent symmetric noise, a noisy coordinate maximizes retained information at fixed initial information; for Boolean functions this gives I(f(X);Y) ≤ 1 − h₂(ε) at crossover probability 0≤ε≤1/2.
One fixed polynomial-time Boolean language has exact depth-three OR–AND–OR circuit size greater than 2^(A√n) for every fixed A>0 at all sufficiently large input lengths n.
For every finite matroid, one independent sample per element supports an online rule with expected reward at least 2^-310 times the expected offline optimum, even when the arrival-order adversary sees all samples, online values, and the rule’s seed.
For every graphical degree sequence on n≥4 vertices, the lazy simple-undirected switch chain that proposes switches uniformly on four vertices has total-variation mixing time at distance 1/4 at most 2n⁸.
A deterministic polynomial-time algorithm finds a minimum-makespan schedule for unit-length jobs with arbitrary acyclic precedence constraints on three identical parallel machines.
A randomized algorithm approximates the number of perfect matchings in every finite simple undirected graph to relative error ε with failure probability at most δ in worst-case bit time polynomial in the input length, ε⁻¹, and log δ⁻¹; it returns zero with certainty when there is no perfect matching.
The complex border determinantal complexity of the m×m permanent is Ω(m³), allowing affine-linear determinant representations and coefficientwise limits.
For every fixed 0<δ<1/3, it is NP-hard to distinguish three-colorable finite simple graphs from graphs whose largest independent set has fewer than δ times the vertices.
For every fixed rational 0<δ<1, it is NP-hard to distinguish satisfiable 2-to-1 games from unweighted games of value at most δ, with an alphabet depending only on δ.
For some absolute constant C, every finite simple undirected graph on n vertices has an edge partition into at most Cn simple cycles and single edges.
OpenAI reports that every finite simple cubic bipartite planar graph that is 3-vertex-connected has a Hamiltonian cycle.
The manuscript reports one deterministic algorithm computing the exact complex discrete Fourier transform at every length n in O(n(log n)^(1−10^−13)) operations, including scalar preparation, scheduling and indexing, in its specified arithmetic model.
OpenAI reports that no algorithm can decide whether an integer-coefficient polynomial has a rational solution when the number of variables is part of the input.
OpenAI reports a deterministic polynomial-time reduction from 3SAT to fixed-alphabet Unique Games with completeness at least 1−ε and soundness at most δ for each fixed ε,δ in (0,1/2).
OpenAI reports a uniform deterministic logarithmic-space simulation of polynomial-time randomized logarithmic-space decision procedures with bounded error, yielding L=RL=BPL.
OpenAI reports randomized k-server competitive ratio O(log²(k+1)) on every metric against oblivious requests, matching the stated worst-case lower-bound order; distinct initial server positions need no additive term.
OpenAI reports uniform deterministic complete factorization of a densely represented polynomial over a binary-specified prime field in bit complexity polynomial in (degree+1) log p, including irreducible factors and multiplicities.
OpenAI reports that every order automorphism of the full Turing-degree partial order is the identity in ZFC, without a definability or regularity restriction on the automorphism.
OpenAI reports that consistency of ZF implies consistency of ZF plus the Partition Principle and Choice for ordinal-indexed families but without the full Axiom of Choice.
In Arnosti’s random bipartite matching model with independent uniform job neighbourhoods of prescribed sizes and a common positive integer bin capacity, RANDOM-VERTEX stochastically dominates RANKING in the number of accepted jobs.
The manuscript reports a deterministic quasipolynomial algorithm for the nonnegative-value vertices of finite turn-based stochastic mean-payoff games.
Claude’s original sparse-output thin-matrix multiplication algorithm, as presented in Section 2 and related to the certified 3SUM/APSP theorems. Human-optimized numerical parameters, new reductions, data structures and Hinted OMv extensions retain human credit.
The manuscript reports a randomized classical Subset Sum algorithm in O(2^(0.49n)) time for polynomial-bit inputs.
For every ε>0, the OpenAI construction multiplies n×n complex matrices with Oε(n^{9/4+ε}) division-free arithmetic operations; consequently ω(C)≤9/4.
In an exact partition of any finite or infinite solvable group into at least two left cosets, two of the corresponding subgroups have the same index. The cardinality form of Erdős Problem 274 follows.
The model-origin optimality arguments for five through seven freely rotated unit squares in a disk, separating known small cases and later human commentary.
The manuscript reports that the complex matrix-multiplication exponent is at most 9/4.
For positive-integer two-size Gasoline instances with deliveries in {1,K}, the position-first iterative-rounding algorithm under the doubly stochastic assignment-LP interpretation has capacity at most OPT+K−2. For K≥3, the original-index tie rule has matching instances with OPT=K and returned capacity 2K−2, giving worst-case ratio 2−2/K.
At least one of the eight classical Riemann zeta values ζ(7),ζ(9),…,ζ(21) is irrational; the theorem names no particular value.
The project-defined Volkenborn-normalized 2-adic value zeta2(7)=J(6)/768 is outside the image of Q in Q₂, with convergence of the defining Riemann sums proved in Lean.
The artifact reports (3,4,5,20,22,36) schedulable but (3,4,5,20,22,32) unschedulable, refuting the proposed six-task kernel cap.
For p=3 and every prime q≠2, no finite or infinite group G satisfies Glauberman–Norton Hypothesis (B): it would need an embedding of H=(additive F_{3^q})⋊(norm-one units), a finite abelian subgroup Q of order prime to 3 and y∈Q, with the embedded prime-field line normalizing Q and its y-conjugate normalizing the embedded U.
On every separable infinite-dimensional complex Hilbert space, the paper constructs a nonzero bounded norm-quasinilpotent operator S whose commutant has no nonzero proper closed invariant subspace. The operator S itself has ordinary invariant subspaces.
On every separable infinite-dimensional complex Hilbert space, the commutant of the constructed operator is a proper unital complex algebra, closed in the strong operator topology and transitive.
A separable real Banach space Z has no closed linear copy of c₀ yet is globally bi-Lipschitz equivalent to Z×c₀. The construction also yields bi-Lipschitz universality for separable metric spaces.
L(4)=16 and L(5)≥49 for the Schur degree of block sums. The two results refute Eliahou–Revuelta Conjecture 5.6 at n=4 and n=5.
There exists a countable domain R that is left PCI and left V but neither right Ore, right PCI nor right V. Thus right PCI does not imply left PCI, and left V-domain does not imply right V-domain.
For joint-update k-dimensional Weisfeiler–Leman equivalence, with k≥2 binary encoded as part of the input and graphs given by explicit adjacency matrices, the test is EXPTIME-complete. The hardness also holds for connected equal-order simple uncolored graphs of maximum degree three.
Theorem 1.1 claims that for every fixed integer r ≥ 4, each finite K_r-free graph on n vertices with average degree d ≥ 2 has an independent set of size at least c_r n log(d)/d, with c_r > 0 independent of n and d. It removes a log-log denominator in the fixed-clique-size average-degree bound; the paper does not optimize c_r or give an efficient algorithm.
Every regular language over a finite alphabet has a generalized regular expression over that same alphabet with at most three nested Kleene stars, allowing union, concatenation and complement.
For every nontrivial increasing graph property on n≥2 vertices invariant under relabeling and 0<ε<1/2, the probability width from ε to 1−ε is at most 2¹⁹ log(1/(2ε))/(log n)².
For each h≥2, an h+3-state one-way nondeterministic automaton recognizes liveness of products of binary relations, while every equivalent two-way deterministic automaton with s states satisfies 2^⌊(h−2)/31⌋≤4(s+2)². The finite alphabet grows with h.
For an ordinary mean-payoff game of complete binary input length L with arbitrary signed binary edge weights, the September 25 deterministic manuscript reports a 2^O((log(L+2))²)-bit-operation algorithm for the exact winning set, values and optimal positional strategies.
Project-defined Volkenborn-normalized zeta2(7) and zeta2(9) are not both rational in Q₂, and every defining J(s) limit exists.
At least one of project-defined Volkenborn-normalized zeta2(7), zeta2(9), zeta2(11) is not rational in Q₂, and every defining J(s) limit exists.
For each fixed integer s≥6, the OpenAI manuscript reports r(s,t)=t^(s−1)/(log t)^(s−2+o(1)) as t grows through all integers. It determines the logarithmic exponent, not a constant-factor asymptotic.
The electronic Coulomb ground-energy promise problem is QMA-hard for distinct rational positions of unit-charge nuclei, unary electron count and rational thresholds separated by at least an inverse polynomial, with minimization over the full three-dimensional spinful fermionic continuum.
For every positive pattern density and inverse temperature, and every bounded continuous single-pattern potential, the spherical random-perceptron pressure converges in expectation and probability to the manuscript’s variational formula.
For every Boolean function on the uniform discrete cube, including biased outputs, and every product-noise correlation ρ∈[−1,1], the manuscript claims H(Ef)−E H(Tρf)≤1−H(ρ), where H(t)=sqrt(1−t²). Signed coordinates attain equality.
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.
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.
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]).
For every fixed ε>0, a deterministic polynomial-time algorithm on finite rational metric k-median instances with specified candidate facilities opens at most k facilities and attains factor 1+2/e+ε. Together with known P≠NP hardness, this gives infimum approximation factor 1+2/e.
For rational-coefficient affine linear matrix pencils, a universal nonadaptive polynomial-size list of rational matrix evaluations detects nonsingularity over the free skew field. The paper also gives a rank-detecting list for rectangular rational pencils, with polynomial construction and output bounds in the dimensions and variable count.
For the zero-field Gaussian Sherrington–Kirkpatrick model at every fixed 0<β<1, rate-one-per-site heat-bath dynamics satisfy a dimension-independent Poincaré inequality for all observables with disorder probability tending to one. Uniformly chosen single-site updates have gap at least 1/(Cβ n).
There are absolute constants c>0 and C such that every A⊆{1,…,N} with no nonzero square difference satisfies |A|≤CN^(1−c) for all N≥1. The source gives no useful numerical value for c.
For the fermionic Laughlin state at filling 1/3 on the round sphere and flux 3(N−1), the unperturbed V1 Hamiltonian has a uniform gap for all sufficiently large particle numbers N. The released Lean scope gives a 1/25 Fock-space distance estimate and a 1/100 finite spherical gap.
For the nearest-neighbor isotropic quantum Heisenberg ferromagnet on ℤᵈ with d≥3 and every positive half-integer spin S, a translation-invariant zero-field KMS equilibrium state exists at sufficiently low positive temperature with magnetization at least S/4.
For each fixed q>4, critical finite Fortuin–Kasteleyn planar maps with n edges, distances scaled by a q-dependent constant times n⁻¹ᐟ², and normalized degree measure converge in distribution to the Brownian continuum random tree in Gromov–Hausdorff–Prokhorov topology.
OpenAI claims that every infinite Löwenheim–Skolem bound admits a uniform threshold above which one categorical cardinal forces categoricity throughout the remaining tail, in ZFC.
The manuscript reports arbitrarily large additive configuration-LP gaps and additive-approximation hardness for rational item sizes greater than 1/6.
The manuscript reports a randomized algorithm taking (n+m)^(1+o(1)) time and returning a maximum matching with probability at least 2/3.
The manuscript reports a randomized (1+ε)-approximation in N^(1+o(1)) worst-case expected time for each fixed rational ε in (0,1).
The manuscript reports a deterministic polynomial-time algorithm producing a common superstring of length at most twice optimum.
Every finitely generated group with decidable word problem embeds injectively in a nontrivial simple group of type F∞, meaning it has a classifying CW complex with finitely many cells in each dimension.
For every sufficiently large prime s≡2 mod 3 and sufficiently large odd n depending on s, the September 23 manuscript constructs an intersecting (s^n+1)-partite uniform hypergraph with matching number one and covering number s^n+1, exceeding Ryser’s proposed bound s^n.
Over F₂(q,H₁,H₂), with three algebraically independent parameters, the construction gives a finite-dimensional symmetric algebra A and a finite-dimensional nonprojective A-module M with Ext_A^i(M,M)=0 for every i>0.
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 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.
For every smooth integral complex projective surface S with ample L, the manuscript claims that all sufficiently large r have maximal multipoint Seshadri constant ε(S,L;p)=√(L²/r) at very general distinct r-tuples, with a nef boundary divisor on the blowup.
The manuscript claims an absolute integer C such that every finite nonempty simple graph G has list chromatic number at most C times its largest clique-minor order h(G). The constant is not optimized.
The manuscript constructs two distinct uniformly positive bounded measurable scalar conductivities on the ball B(0,3) in R³. Both equal one near the boundary, differ on a set of positive measure and have the same full weak Dirichlet-to-Neumann operator.
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 a central projection halving every specified trace on every coefficient. This yields uniform property Gamma for each specified free ultrafilter.
There are constants C>0 and 1≤β<4/3 such that every finite n-point planar set has at most Cn^β unordered unit-distance pairs.
A real radial Schwartz function on the plane has nonnegative Fourier transform, is nonpositive beyond unit distance, and has normalized origin values making the Cohn–Elkies linear-programming bound attain the triangular-packing density π/(2√3).
For a loopless multigraph on 2m vertices with a perfect matching and feasible edge marginals x, the maximum entropy H(x) satisfies F(x)−(2−2/m)B(x)≤H(x)≤F(x), where F and B sum the binary-coordinate entropy terms. Optimizing gives the linear-error estimate proposed by Anari, Oveis Gharan and Vinzant.
For all positive integers m and n, the ordinary crossing number of the complete bipartite graph Kₘ,ₙ equals ⌊m/2⌋⌊(m−1)/2⌋⌊n/2⌋⌊(n−1)/2⌋ for plane drawings with continuous simple edge paths.
In every dimension d≥2, an IID uniformly elliptic nearest-neighbor random walk environment on Zᵈ with almost-sure transience in a fixed direction has an almost-sure annealed velocity that is deterministic and has positive projection in that direction.
An Artin group on 116 generators, with off-diagonal labels only 2, 3 or infinity, has no proper cocompact isometric action on any nonempty proper CAT(0) metric space, in any dimension.
For every integer n≥3 and finite β>0, spin two-point correlations in finite free-boundary nearest-neighbor square-lattice O(n) models with edge strengths in [0,β] obey a uniform A exp(−m|x−y|) bound. Free-boundary limits inherit the decay.
The September 23 manuscript claims that the intersection of any two conjugates of standard parabolic subgroups is parabolic in every finite-rank Artin group, including nonspherical types and infinite Coxeter labels.
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.
The manuscript reports deterministic multiplication in O(n (log n)^(1−κ)) time with κ = 2^−182 on a fixed finite-tape Turing machine.
For every origin-symmetric compact convex body K with nonempty interior in Rⁿ, n≥2, the Gromov width of int(K)×int(K°) is 4. Every symplectic ball of capacity c<4 embeds.
Kegel, Li and Ren give an effective family of finite presentations with two generators and ten relators for which deciding triviality is impossible. The count is an achieved presentation size, not a minimum.
Kegel, Li and Ren give an effective family of finite presentations with four generators and eleven relators for which deciding triviality is impossible. These are fixed presentation slots, not a minimality result.
Kegel, Li and Ren give a finite presentation with three generators and nine relators whose word problem is unsolvable. The count is an achieved presentation size, not a minimum.
Alexander Temerev reports that every planar search path for an unknown straight shoreline has competitive ratio at least 13.8111351794611…, matching the logarithmic spiral. The proof permits paths with decreasing radius or reversing polar angle.
For the centered Hardy–Littlewood maximal operator over axis-parallel cubes in R², the optimal weak-type (1,1) constant c₂ is at least Φ=1.6855099933…, so c₂>1.685.
The model-extracted generic polynomial-calculus/Res(parity) sufficient criterion in Theorem 9.1, rather than attributing the complete bit-PHP lower-bound architecture to AI.
For weighted k-matroid intersection with k≥2, the natural LP has integrality gap at most k−1+1/k. A deterministic LP-relative algorithm achieves that factor; the paper separately extends the bound to p-matchoids.
A ten-word code in F₃⁴ and an eight-degree lift give square-difference-free polynomial sets of size 810ᵉ below degree 8e and 27·810ᵉ below degree 8e+4. They refute Naslund’s Conjecture 13 at q=3, k=2 for every n divisible by four with n≥8, and imply a growth-exponent lower bound of 16/21.
For each fixed acyclic 0–1 matrix pattern P, Gishboliner and Li prove Ex(n,P)≤n^(1+O_P(1/log log n)), establishing the weak n^(1+o(1)) Pach–Tardos statement.
Every finite-alphabet configuration c:Z²→A with P_c(m,n)≤mn for some positive m,n has a nonzero global period.
Coester, Koutsoupias and Zbysiński report that the work function algorithm is k-competitive on every metric space.
Lean proofs establish Dimitrov’s Conjectures 1 and 3, the stated rational lower bounds involving Euler’s totient φ(n), Dedekind’s ψ(n) and σ(n), for every integer k≥1 and n≥2. Equality in each bound holds exactly when n is prime.
A group built as EL₃(LC(X,F₂)⋊ℤ), with X a binary Toeplitz subshift, is infinite and simple, has Kazhdan property (T), and is locally embeddable into finite groups. It is therefore sofic and hyperlinear. The Lean project compares two existential statements using this same witness.
Every four-partite four-uniform hypergraph with matching number two has vertex-cover number at most six.
For m≥2 labeled supplies on [0,1], independent uniform demands and independent uniform replacement after each online match, the project proves a policy whose one-period root-mean-square cost, from every fixed initial inventory, has limsup at most (72000/log 2)·log(m+1)/m. After a scheduled m-period replacement phase, expected average cost including that phase obeys the same bound for every horizon N≥2m².
Almeida and Knudstorp report that Medvedev’s logic of finite problems is undecidable, by reducing periodic plane tiling to non-theoremhood.
Every finite loopless Eulerian digraph with no repeated ordered arc, including digraphs with opposite arcs, contains every orientation of every t-edge tree when it has n vertices and more than (t−1)n arcs. Complete bidirected graph unions show the bound is sharp.
In Bucić’s alternative proof that the uniform Turán density of the three-uniform tetrahedron K₄³ is 1/2, ChatGPT 6 Pro found the finite sum-of-positive-quadratic-forms identity and proposed the tensor projection inequality used in the proof.
Becker, Greger and Peters report that every approval-based multiwinner election has a core committee and that one can be found in polynomial time.
An explicit independent set in the 213rd strong power of the 11-cycle gives Shannon capacity C₁₁ ≥ 5.295526013632343.
Moffat’s Theorem 3.1 claims to bound any single-letter entropy certificate whose classes contain product laws by c<=1-h(1/sqrt(2))/sqrt(2)=0.383099... . Theorem 3.4 gives a refined c<=0.382885260... ceiling when the iid protocol has a singleton class and other classes admit component hiding. These are limits of specified certificate methods, not bounds on the true union-closed constant. The paper's c>=0.38284 construction remains conditional on numerical hypotheses and is separate.
For every n≥1, the integers 1 through n can be partitioned into nonempty blocks of at most three distinct integers, with each block summing to a power of three.
Guo, Fang and Lu prove the strict bound K_G^C>1.35584631827168 using a Gaussian Hermite multiplier, exact rational parameters and a weighted operator-norm criterion.
Chernikov constructs a complete simple theory with a definable group and a global type whose left-translation orbit has size at most 2^ℵ₀, although the group has no left-invariant Keisler measure. This refutes the bounded-orbit implication to definable amenability outside NIP.
For every ε>0, arbitrarily large n admit an n-element sum-distinct set of positive integers with maximum element N≤ε·2^n. The formal statement excludes N=0 and gives no effective n as a function of ε.
Let f(n) be the minimum number of distinct prime divisors among pairwise sums of any n-element subset of positive integers. The compared theorem proves f(n)/log n→∞. Separate Lean resolutions give stronger polynomial lower bounds, including a square-root bound.
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.
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.
For all sufficiently large X, the largest consecutive prime gap below X is at least a constant times log X (log₂ X)² log₄ X/(log₃ X)². The argument uses a short-translates proposition and a weighted sieve.
Astra proved in Lean that a complex polynomial p exists with p(0)=0 and p′(0)=1 such that |p(c)/c|>1 at every critical point c. Thus the universal K=1 mean-value inequality fails at z=0.
For n≥k+1, every simple graph with at least (k−1)n/2+1 edges contains every tree on k+1 vertices. A separate counting lemma also reaches the classical strict threshold when (k−1)n is odd.
Dong and Jain report that, for a nonnegative n-by-n matrix whose bipartite support graph has girth at least even g>=4, Theorem 1.3 proves Bethe(A)<=per(A)<=2^(2n/g)Bethe(A); disjoint unions of g-cycles attain the upper bound when g divides 2n. The model-developed intermediate proof had a weaker exp(n Theta(log g/g)) factor; the exact model-origin step retained in the final sharp proof is not identified.
Independent random one-qubit complex Cliffords, or exact one-qubit 2-designs, form a depth-one n-qubit tensor-product ensemble with entangled anticoncentration error (2/3)^n and negligible distinctness for polynomially many queries.
Bostanci and coauthors construct a unitary quantum oracle U for which QMA(2)^U differs from QMA^U. Their oracle problem has a one-query QMA(2) protocol with linear-size unentangled proofs, while QMA needs exponential resources in query count or proof size.
Freitag and Mutchnik give a stable theory of an infinite-dimensional vector space over the division ring of fractions of the quantum graph algebra of the random graph, refuting stable forking.
For every base b≥2 and sufficiently large digit sum q congruent to a square modulo b−1 (automatic for b=2), the authors construct at least exp(c_b√q) squares n² of bounded size with gcd(n,b)=1 and base-b digit sum q. The explicit Lean statement gives q≥2304b⁴, n²≤b^(3q), and at least 2^(⌊√q/(36b)⌋) examples for b≥3, with a separate binary case.
For k≥2 in any positive integer dimension, the magnitude of the kth hard-sphere Mayer coefficient, multiplied by k!, equals a sum of no-broken-circuit region volumes over spanning trees. In dimension three, each tree-region volume is at most (17/32)^ν(T)κ^(k−1), where κ is unit-ball volume and ν(T) is the largest packing of disjoint ordered forks a<b<c with tree edges ab and ac.
For every finite metric voting profile, Shah’s v2 random-size stable-lottery rule has distortion at most 107200742653/50000000000 = 2.14401485306. This supersedes the v1 mixed-rule bound of 2.3282.
The free Heyting algebra F₂ on two generators cannot be the lattice of subterminal objects of an elementary topos. The obstruction extends to any Heyting algebra surjecting onto F₂, including free F_N for N≥2.
For PR[S] on square-free integers S⊂[2,n], with edges where gcd exceeds one, the n=51 graph has an exact integer Rayleigh witness showing its largest adjacency eigenvalue exceeds its number of distinct degree values.
For each δ∈(0,1), sufficiently large maximum degree Δ and finite simple n-vertex graphs of girth at least 5, Theorem 1 gives irreducible proper-q-coloring Glauber dynamics with spectral gap Ωδ(1/n) and mixing time Oδ(n² log q+n log(1/ε)) when q≥(1+δ)Δ. Theorem 3 extends the method to antiferromagnetic Potts models under its stated β and q conditions; Theorem 5 gives a local spectral-contraction criterion for general multi-spin systems. The result is rapid mixing, not the conjectured exact q≥Δ+2 threshold or uniformly optimal O(n log n) bound.
For Borel-definable global Keisler measures over a model of a discrete first-order theory, the authors prove the reverse implications: self-averaging gives a definable measure whose canonical random extension is generically stable, which in turn gives a frequency interpretation measure. Together with earlier forward implications, these yield equivalence.
For iid Bernoulli(p) bits with 0<p≤1/2 and a full-row-rank k×n binary linear encoder H, its optimal bitwise-MAP distortion D(H) obeys k/n≥h(p)(1−D(H)/p). A nonasymptotic entropy bound for every affine subspace drives the converse.
For a fixed n×d real matrix U with UᵀU=I, d≥1, 0<ε≤1 and 0<δ≤1, the fully independent signed-hash SparseStack construction gives failure probability at most δ with column sparsity below 1356 max(1,⌈log₂(d/δ)⌉)/ε and row count below 306456(d+max(1,⌈log₂(d/δ)⌉))/ε².
Alpöge and Buckmaster construct smooth compactly supported initial velocity and forcing smooth through blowup time for 3D incompressible Euler on R³. A finite-energy classical solution develops unbounded vorticity in finite time, with divergent time-integrated vorticity sup norm and uniqueness in the stated preterminal class.
For Z={α>1: some nonzero real ξ makes every floor ⌊ξαⁿ⌋ even for n≥1}, Stephan constructs a witness showing √3∈Z and proves that among integer square roots √m with m≥2, only √2 lies outside Z.
For positive coprime unequal integers a and b, an AlphaProof Nexus Lean proof shows that Sharipov’s degree-eight cuboid polynomial Pₐ,ᵦ(t) is irreducible over the integers. Its route uses quartic factor analysis and an infinite descent for x⁴+34x²y²+y⁴=z².
Huang, Liu and Liu give a finite two-receiver discrete memoryless broadcast channel T for which Marton’s two-use sum-rate quantity M(T⊗2) exceeds twice its one-use value M(T), with a certified gap above 1.8821×10⁻⁶ nats. Its one-letter Marton region is therefore strictly smaller than capacity.
A ZFC model has the property that every family (Aₓ) of subsets of ℝ with outer measure below one admits an infinite X such that x∉Aᵧ for distinct x,y∈X. Combined with Hechler’s earlier CH counterexample, this gives relative ZFC independence for the first question of Erdős 501.
For four colors, with x=y allowed, the reported proof gives R(c)=40c+41 for every integer c≥2 in x+y+c=z. It reduces the universal statement to R(2)=121 and two unsatisfiable spoke templates.
A deterministic exact algorithm counts linear extensions of any n-element poset in O*(1.89^n) time.
Dupont and collaborators report ω<2.371177 by improving the combination-loss optimization and refining it with AlphaEvolve.
The Paley graph P(17) and each join P(17)∨Kₜ satisfy χ(H)≤ω(H)(ω(H)+1)/2 for every induced subgraph H, yet are not perfectly divisible.
Xu and Wang exhibit a strongly convex rational quadratic program with A=B=I₂ and identity third block whose direct three-block ADMM has a bounded non-KKT orbit of minimal period 66. A separate Kimi search gives a locally attracting period-23 orbit in dimension three that can be converted to all-identity blocks.
An explicit 8×8 nearly decomposable support has a unique global permanent minimizer on its full doubly stochastic face. At a specified zero, the permanental cofactor exceeds the permanent although that zero is not a tie point, refuting the Foregger–Sinkhorn implication.
For every complete first-order theory, SOP₂ implies SOP₃. The reverse implication was already known, so the two dividing lines coincide.
Liehr, Taylor and Yu prove that every countable Bessel sequence of bound 1 in a complex Hilbert space has its full Gram matrix, including diagonal entries, represented by correlations of complex functions on [0,1] bounded in modulus by 1.
For every fixed r,s≥1, both restricted-permutation count sequences a_(r,s)(n), forbidding π_(i+r)−π_i=s, and b_(r,s)(n), forbidding the corresponding absolute difference, are claimed to be holonomic. The argument uses an interleaving bijection with finitely many seams and D-finite closure.
For the Geode G defined by S−1=(t₂+⋯+t₆)G, a coefficient-collapse identity reduces its diagonal coefficient to a one-dimensional integral and coefficient extraction. It enables an exact 8,367-digit computation of G[1000^(5)] using modular arithmetic and bounded Chinese remainder reconstruction.
The 10×42 Chomp bar has exactly three winning first bites, at (5,36), (7,30) and (8,26), by exhaustive exact game-state computation.
For every p between sqrt(2+sqrt(3)) and 2, every horizon T and every predetermined nonnegative gradient-descent stepsize schedule, Ma and Chen construct a smooth convex problem in dimension at most T+1 with last-iterate error at least c_p L R²/(T+1)^p.
In the paper’s small-ε regime, a randomized adaptive method using one independent fresh sample per query needs Ω(ΔL/ε² + ΔLσ²/ε⁴) queries to reach expected gradient norm at most ε for smooth nonconvex objectives with gap Δ, L-Lipschitz gradient and uniformly bounded, unbiased fixed-law gradient noise.
For n independent standard Gaussian points in R^d, a positive semidefinite S with x_iᵀSx_i=d for every i exists with high probability when limsup n/d²<1/4 and fails to exist with high probability when liminf n/d²>1/4.
Under the paper’s pth-order smoothness and oracle assumptions, the final Halpern-based method for smooth monotone variational inequalities attains residual rate Õ(T^(−p)), or Õ(ε^(−1/p)) oracle complexity.
Fredslund-Hansen, Mozes and Weimann report that, for a connected undirected unweighted planar graph with a designated face whose facial walk has k edges, the number of distinct facial distance-difference patterns is O(k^2), matching the known Omega(k^2) lower bound. The model supplied the forbidden-configuration and fixed-cardinality insight and a short prefix-trie proof; the paper's separate diameter algorithm is human-developed downstream work.
For planar graphs at sufficiently small fixed hard-core activity, including λ≤1/1443, Guo and Zhang give a randomized polynomial-time approximation scheme for the partition function without a maximum-degree bound.
For any number of indivisible goods and agents with nonnegative additive valuations, an allocation exists with bundle sizes differing by at most one, envy-free up to one good, and no balanced fractional reallocation that Pareto improves it.
For every complex polynomial of degree at least two with all zeros in the closed unit disk, each zero lies within distance one of a zero of its derivative, including repeated and boundary zeros.
For the single-neuron modular-addition setting, Appendix A extends Memana’s human counterexample to selection-independent Clarke trajectories and dead-neuron examples under the specified ReLU, dead-zone smoothing and fixed-step full-batch gradient-descent dynamics.
Lalonde gives a bipartite inner-product game with four and three inputs and six outputs. It has a perfect strategy using a nonmaximally entangled state, but no perfect finite-dimensional strategy using a maximally entangled state.
For every deterministic fractional online vertex-cover algorithm and every ε>0, a finite bipartite edge-arrival instance forces a competitive ratio of at least 2−ε. The paper extends the lower bound to randomized integral or fractional algorithms against an oblivious adversary.
For the released fixed four-terminal planar acyclic gadget with coordinatewise positive cost-difference scenarios, the normalized additive upper arc-deviation supremum is (299−41√41)/32 with one scenario, 17/8 with two, 3 with three, and 4 with at least four. The two-scenario value is not attained under scenario-wise cost non-increase.
In the Boolean-output quantum random oracle model, every finite-round perfectly complete key-agreement protocol with quantum local operations, independent initial private states and classical communication admits a passive eavesdropper that recovers the key with certainty using O((q_A+q_B)^5) classical oracle queries. The eavesdropper may use unbounded computation.
For every finite connected simple graph G=(V,E), a parallel chip-firing game with 2|E|−|V|<|σ|<2|E| has eventual minimal period two and activity one half.
For every fixed positive diameter k, the maximum number nₖ(d) of vertices in a graph of maximum degree at most d and diameter at most k satisfies nₖ(d)/dᵏ→1 as d grows. Halved-flag graphs supply the construction.
Xu constructs a fixed finite forbidden graph family with chromatic parameter p=2 and extremal surplus greater than c n^(3/2) for all sufficiently large n, yet with an extremal graph at each such order whose complement is connected.
Assuming the randomized rational-weighted exact-volume Small-Set Expansion Hypothesis, no randomized polynomial-time algorithm can guarantee both additive ε accuracy in rational sparse least squares and support O(k κₛ₊ₖ^(1−γ)) for any fixed γ∈(0,1]. The hard matrices have full column rank and at most two nonzeros per row.
Feng, Fu and Guo give a fully polynomial randomized approximation scheme for s-to-t reachability probability in general directed graphs with independent edge-opening probabilities. An existing reduction extends it to undirected graphs.
There are full-dimensional 0/1-polytopes of dimension 4n+1 whose graph edge expansion decreases exponentially with dimension.
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.
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.
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.
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.
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.
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.
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.
For every subDL theorem A → B, GPT-5.6 Sol’s terminal-hierarchy argument claims that A and B share a variable with the same sign at the same relevant-arrow depth.
Meiburg constructs a total Boolean function on 2,017,584 variables with block sensitivity at least 14,011 and spectral sensitivity at most 89.0162. Composition yields an exponent of at least 2.127, refuting a proposed quadratic upper bound.
For an unknown d-dimensional state and m adaptively chosen observables 0≤A≤I, Chen, O’Donnell, Pelecanos and Wright give online expectation estimates within ε with success probability at least 9/10 using O(log(m)√log(d)/ε³) or O(√m/ε²) copies, respectively.
Buys, Polak and Zuiddam give Lean-formalized Shannon-capacity lower bounds for C7, C11, C13, C15, C19, C21 and C23 using seven explicit base tuples and Gao’s product lemma. Their C7 bound is 3.258805369885… and their C11 bound is 5.294502522149….
A specified connected graph with 33 vertices and 36 edges has domination polynomial divisible by x+4. Its exact factorization includes x¹¹(x+2)(x+4)(x²+3x+1), so −4 is an integer domination root.
For positive rational u=c/d and r=a/b in lowest terms, with d even, d dividing lcm(b,2), and u≤r, the integer trace of the fractional Gaussian binomial coefficient at u dominates that at r coefficientwise for every k≥1. This proves the Han–Xiong comparison for every r≥1/2 and reduces the remaining parameters to 1/(2m).
Domina and Ceriotti give noncongruent atom-centered environments with identical complete correlations through seven-neighbor order, including three-dimensional lifts. For every finite correlation order and angular cutoff, they also construct indistinguishable noncongruent environments.
The Paley tournament on 43 vertices has predictability 181/301, above the five-voter threshold 3/5, yet cannot be induced by five voters. The proof uses a co-backing constraint for cyclic triangles and an exact finite shell-and-screen enumeration.
For the least possible largest distinct factor f(n) in n! = a₁⋯aₖ with n<a₁<⋯<aₖ, Wang’s manuscript claims that (f(n)−2n)log(n)/n tends to 4029639598/25970038185. Its new step is a matching upper-bound construction.
Allen, Dhawan and Noel prove that for every fixed girth g≥4 there are d-degenerate graphs of girth at least g with fractional chromatic number at least (1−o(1))d/log d as d grows. ChatGPT solved the authors’ optimization problem for the fractional clique function used in this lower-bound proof.
KEMENY SCORE is NP-complete for exactly three unweighted complete rankings. Peters reduces from SIMPLE MAX CUT with long even edges to a unit-margin majority tournament realizable by three voters.
Temerev and Doria report that the minimum escape-path length for the golden gnomon, an isosceles triangle with unit equal sides and 36-degree base angles, is 1.282676025459… . A symmetric seven-piece path attains the value, and their lower bound covers every rectifiable competitor.
For any centered Gaussian vector, including singular covariance, and positive real exponents αᵢ, the expected product of |Xᵢ|^αᵢ is at least the product of the corresponding marginal expectations.
Theorem 1.2 gives a marked profinite presentation of the absolute Galois group of Q₂ with four generators, a tame relation, an explicit wild-word relation, and a pro-2 condition on the normal subgroup generated by the wild generators.
For every fixed power k≥1, sufficiently many variables give a lattice monomial in the Newton polytope of the k-th Vandermonde power with zero coefficient. The new step is an explicit even-power seed for k≥4; earlier cases and propagation complete the result.
GPT-5.5 Pro’s Proposition 2 expresses the rank-two partial-trace quadratic form q⁽²⁾₋₁/₂(C) as projection terms. Song and Chen then use a separate human-developed lemma to prove nonnegativity and two-copy nondistillability at the Werner threshold.
Conditional on Conjecture 1.1 about large prime divisors, the authors construct an S*_m-decoding polynomial with the optimal k+1 nonzero terms for every fixed k, for suitable k-prime m and finite field. The construction gives the stated fixed-server PIR tradeoff; finite prime checks make it unconditional through 15 servers.
Ananth, Ragavan and Sahai give efficient, perfectly correct, one-time private-key encryption of one bit using a random nonidentity phase-free n-qubit Pauli and a random eigenstate. After key disclosure, both separated receivers recover the bit with probability at most 1/2+O(2^(−n/2)) against information-theoretic attacks.
For synchronizing one-cluster automata of positive level, Zhu proves the relative extending-word property and reset bound (m−1)(n−1)+mℓ≤(n−1)^2. Binary examples attain his refined 3n−5 bound.
For finite-dimensional unitaries satisfying approximate order-two and orbit-commutator relations through 2M, with M polynomial in the target tolerance, Dogon and Vidick prove nearby exact lamplighter relations. Their argument gives polynomial upper bounds on Hilbert–Schmidt stability rate and radius.
For every n≥2, a two-colouring of the edges of the n-cube in which antipodal edges have opposite colours contains a monochromatic path from some vertex to its antipode.
The Laurent polynomial f(x,z)=(1−z⁻¹)((1−x)+xz) has zero pure product-integral moments for every positive n but marked moment (−1)^(n−1)/(n+1). It refutes xz(1,1), with padding for all mixed k,l≥1. Its SU(2) lift F=(1+c)(ad+b), G=−c has the same pure and marked Haar moments and refutes the SU(2) Mathieu conjecture.
Testing whether a supplied family of meet-irreducibles is the complete canonical family of a finite implication closure system is coNP-complete, even with acyclic implications of premise size at most three and a genuine irredundant supplied family. The hard direction reduces 3-UNSAT to a missing member.
The rooted terminal-only Manhattan cost-radius spanning-tree decision problem is NP-complete under polynomial-time many-one reductions through a weak reduction from positive-integer Partition.
Liu and Fitzsimmons give an explicit planar homogeneous cubic polynomial vector field with integer coefficients whose origin is globally asymptotically stable and has a positive radially unbounded C¹ homogeneous Lyapunov function, but has no positive definite homogeneous polynomial with nonpositive Lie derivative and no local real-analytic weak Lyapunov function.
Assadi, Jiang and Xiang construct optimal blueprints showing that no deterministic or randomized single-pass semi-streaming algorithm using O(n polylog n) memory and constant success probability can guarantee better than a one-half approximation for maximum bipartite matching. Greedy attains one half.
Star Fleet's construction shows that for every odd n at least 201, every integer k at least n·2^n·ceil(49n/100)^n+2 is the count of a tiling of the unit n-cube by positive homothetic cubes. This gives c(n)=o(n^n) along odd dimensions and refutes the proposed universal lower bound c(n)≫n^n; it does not determine the best bounds for c(n) in all dimensions.
For each composition α=(α₁,…,α_k) of n, the number of alternating permutations of 2n with record composition α is ∏_j binom(2s_j−1,2α_j−1)E_(2α_j−1), where s_j=α₁+…+α_j and E_m is an Euler number. The paper also realizes these counts in a noncommutative symmetric-function lift.
Every finite connected Eulerian multigraph of minimum degree at least four has, for a fixed Euler tour, an edge partition into connected 2-regular circuits compatible with that tour’s transitions.
Kerger reports Ω(d²/log(d+1)) exact-function-value queries at accuracy Θ(d^−1/2) for deterministic convex Lipschitz minimization over a d-dimensional ball.
Jumagulov certifies that the dimension-36 Cohn–Elkies two-point LP optimum is at least 146.1036734821, more than 32.91 times the Kschischang–Pasupathy packing’s center density.
For every ε>0 and all sufficiently large even n, the complete graph K_n contains at least (1−ε)n edge-disjoint perfect matchings such that every pair forms a Hamilton cycle. This is the model-origin construction of Theorem 1.3.
For exact double covers of {0,1,2}^d by proper subboxes, Patrick White’s model-origin modular and line-rigidity argument gives f(4)≥19, f(5)≥33 and the unconditional bound f(6)≥60.
Under the stated balanced stochastic-block-model signal and community-growth assumptions, the authors give a high-probability polynomial-time pure node-private estimator attaining the prior exponential-time method’s minimax exact-recovery risk, with privacy parameter of order log n.
A polynomial-time algorithm finds a planar embedding of minimum edge-outerplanarity for every finite loopless planar graph, including graphs with parallel edges. It reduces edge-peeling depth to minimum face-depth of an auxiliary planar graph.
For every odd integer base b>1, the fair {0,2} base-b Cantor measure admits no Fourier frame. The middle-third case answers Strichartz’s question.
A 39-vertex dumbbell graph violates Graffiti Conjecture 143, which bounds the variance of positive adjacency eigenvalues by graph size divided by average distance. The witness works under both standard average-distance conventions.
For the integer-normalized trigonometric permanents at prime p=29, s₂₉=1,053,859 is positive and s′₂₉=−4,806,838,304 is negative. These exact values contradict both sign predictions in Conjecture 4.6(ii).
A bipartite graph on 13 vertices has 22 edges and a unique minimum dominating set of size 4, exceeding the proposed bound m(13,4)=21 in Koch–Narayan Conjecture 1.
Lollipop graphs with 118 or 120 vertices violate Graffiti Conjecture 154 when “deviation” means the standard deviation of adjacency eigenvalues. A 144-vertex graph violates it under both standard average-distance conventions.
For every n≥1, the number of nonnesting permutations of {1,1,…,n,n} avoiding both 1132 and 3312, under the published containment convention, equals 3ⁿ−3·2ⁿ⁻¹+1.
Theorem 6.1 gives an explicit quantitative stable-phase-retrieval bound for the closed real L² span of independent centred normalized random variables under indexed two-sided L¹ bounds on all but at most one coordinate. Its proof uses a head–tail concentration versus diffusion argument and Sperner-type anticoncentration.
AxiomProver proved in Lean that an integral x_n forces the squarefree kernel K_n to divide 1+m² with |m|≥√(K_n−1), that large exceptional x_n lie near a π/2 lattice point, and that the exceptional set has O(log N) members through N. Ono combines these with his asymptotic lower bound for K_n to show nonintegrality for almost all n.
For every nonempty inclusion-free hypergraph on n vertices, every d≥1 and arbitrary real edge offsets, Faber and Harris prove that at least n times the sum of j^(n−1) for j=0,…,d−1 assignments in [d]^n isolate a unique minimum edge. The n-singleton hypergraph attains equality.
For every fixed-in-advance sequence of T Euclidean unit vectors in dimension d, Aden-Ali’s Gaussian triplet walk assigns signs in O(dT) time. With probability at least 1−δ, every prefix has ℓ∞ discrepancy at most sqrt(18 log(2T²/δ)).
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.
Nikolenko constructs a 17-terminal common-point instance proving the universal cost-preserving unsplittable-flow constant is at least 1.282494797984843521. A 3/2 congestion ceiling applies only to common-point cells of complement mass two.
Nikolenko gives a six-terminal planar instance requiring congestion factor at least 58676765987259/50000000000000, or 1.17353531974518, under cost nonincrease. He also identifies exact local envelopes E(2)=1, E(3)=9/8 and E(4)=(299−41√41)/32.
For every depth p and even ring size N with 2p+2≤N, the optimal QAOA approximation ratio for MaxCut on the ring of disagrees is exactly (2p+1)/(2p+2).
Under OVH or SETH, Saha, Xu and Ye prove n^(2−o(1)) lower bounds for exact Furthest Pair, Bichromatic Closest Pair, Maximum Inner Product and Hopcroft’s problem in every efficiently constructible superconstant dimension.
For every feasible set of row and column margins on an m×n binary matrix, with m,n≥2, the lazy 2×2 swap chain has spectral gap at least 1/[binom(m,2)binom(n,2)].
A five-element coverage function with marginals (3/10, 7/20, 3/10, 7/20, 7/20) has ordinary concave closure 4 and pairwise-independent closure at most 479/160. Their ratio is at least 640/479, above the conjectured universal 4/3 bound.
For all sufficiently large k, the least n>2k for which (n−k)⋯(n−1) has no prime factor in (k,2k) exceeds exp((log k)²/(20 log log k)).
For n=3, K=2 and d=4, Peng gives positive definite symmetric matrices arbitrarily close to the identity with ||W_SS|| > ||W_RS||, refuting the SS–RS half. For all n≥2, K≥1 and d≥1, the RS–GD half holds when (1−1/(4n²+1))I ≤ Aᵢ ≤ I.
For every M≥2 with N=4M−5, a nonempty open set of complex N×M measurement matrices has a noninjective phase-retrieval map. Hence an IID complex Gaussian matrix is noninjective with positive probability, so p_M<1.
A biconnected graph with nine vertices and ten edges, with s=1 and t=3, gives a longest path of six under MaxSTN and seven under MinSTN, refuting the proposed inequality for these algorithms.
AlphaEvolve found 51- and 151-unit-disk configurations whose odd-covered areas are estimated by Monte Carlo calculations at about 2.73 and 2.51, respectively.
A connection from relaxed thin perfect matching to a single-tree cut sparsifier helps yield an O(log n log log n)-thin perfect matching relative to any fractional perfect matching, allowing edges outside its support. The separate O(n log n) bound requires support-constrained edges.
Anari’s Theorem 4 reports that the flip–repair walk associated with any real skew-symmetric n×n matrix has spectral gap at least 2/n. In the flat 0/1 determinant case it also gives an entropy bound and nearly linear mixing; this feeds a near-uniform sampler for strongly connected directed Eulerian multigraphs with m arcs in O(m^(3/2) polylog(m/ε)) time.
For the Lipschitz-function growth constant c(Q_d) of the d-dimensional hypercube, log c(Q_d)≤(3/4+o(1))(log d)/d. The argument compares graph homomorphisms with K_(d,d).
The AlphaProof Nexus report says its agent ruled out monochromatic quantum graphs for the diagonal four-, six- and ten-particle cases N=d=4, 6 and 10 in the Krenn–Gu perfect-matching model.
For every finite simple connected graph G, the maximum number L_S(G) of leaves in a spanning tree is at least 2(ℓ(G)−1), where ℓ(G) is the average local independence number.
For every n-element Sidon set A of natural numbers, AlphaProof Nexus proves 16 I(A+A)+100n+16≥n², where I counts sums whose neighboring integers are both absent from A+A. Thus the minimum isolated-sum count grows at least quadratically, answering the finite-set form of Erdős Problem 152.
A counterexample on Z/3Z disproves the intended complex-valued equality of two quadratically structured function spaces in Ben Green’s Problem 57.
For an n×n Hermitian positive semidefinite matrix A, an entropy-corrected concave Gram-factor relaxation P̂ satisfies e^(−γn)P̂≤per(A)≤P̂ and gives a deterministic polynomial-time e^((γ+ε)n)-approximation for every ε>0. The exponent is optimal up to lower-order terms under the cited NP-hardness, assuming P≠NP.
AxiomProver generated Lean proofs of Ballantine–Beck–Feigon–Maurischat Conjectures 2, 5, 7, 8, 9 and the intended corrected reading of 10, covering coprimality, divisibility, special values and a recurrence. It also found an n=4 counterexample to a printed binary log-concavity claim.
Some computably enumerable truth-table degree contains no computably enumerable irreducible many-one degree. Cintioli derives the example using Degtev’s c.e. semirecursive set with rigid complement and the fact that its simple-set representatives cannot be irreducible cylinders.
An infinite-chromatic graph has the property that, for every nonnegative function f(m) tending to infinity, every m-vertex finite subgraph contains an independent set of size at least m/2−f(m). A stronger construction bounds local odd-cycle-transversal number by any prescribed unbounded integer profile.
For every primitive set A contained in [x,∞), x≥2, the sum f(A)=Σₐ∈A 1/(a log a) is at most 1+O(1/log x).
For every odd m≥3, the directed Cayley graph of (Z/mZ)^5 with the five positive coordinate generators decomposes into five directed Hamilton cycles.
For semifree noncommutative differential graded algebras over a nontrivial computable unital commutative ring, stable tame isomorphism, quasi-isomorphism and derived Morita equivalence are undecidable.
A sparse set B that is not an asymptotic additive basis gives, for every A of Schnirelmann density 0<α<1 and every cutoff N, a shift b∈B for which A∪(A+b) covers at least (α+f(α))N of [1,N], with f(α)>0.
For asymptotically strictly decreasing sequences, and for every index set Q with at least two members, the possible global product-intersection sets are exactly the subsets of positive integers containing 1. The paper also specifies the Q=0 and Q=1 cases.
For a feasible d-dimensional Max-Cut SDP solution strengthened by triangle inequalities, a randomized polynomial-time rounding algorithm attains expected cut value at least (α_GW+2^(−O(d))) times the SDP value.
For every m≥1, an explicit K₄-free graph on 20m+31 vertices is 4-chromatic and vertex-critical, while every cycle has at most ten chords. This refutes the proposed unbounded forced chord count under the stated local 3-colorability condition.
A polynomial-time algorithm finds an optimal size-k committee for any Thiele rule on voter-interval approval profiles, including voter-specific scoring sequences.
For smooth convex-concave objectives with a monotone K-Lipschitz gradient operator and a saddle point, the schedule αₜ=1/(K√(t+γ)), βₜ=γ/(t+γ), γ≥2, gives squared-gradient residual O(1/t) at the last iterate of anchored gradient descent-ascent.
AlphaEvolve-generated graph searches report R(3,13)≥61, R(3,18)≥100, R(4,13)≥139, R(4,14)≥148, R(4,15)≥159, R(4,16)≥174, R(4,18)≥209, R(4,19)≥219 and R(4,20)≥237.
In the IID request model, the stationary expected access cost of Transposition is at most OPT+1, where OPT orders items by decreasing request probability. GPT-5 Pro proposed the central inequality sⱼ≤pⱼ and a coefficient-nonnegativity route; Christian Coester supplied the final proof.
For the specified row-stochastic 4×4 matrices supported on a 4-cycle, every nonreal eigenvalue a+ib has 0≤a≤1, a+|b|≤1 and (b²+a²+a)²+2a²−b²≥0. Conversely, upper-half-plane points with b>0, 0≤a≤1, a+b<1 and the polynomial nonnegative occur; the stated boundary arcs are attained. This includes Ran and Teng’s Conjecture 20.
For every numerical semigroup S and integer p≥0, AxiomProver’s Lean proof establishes Fel’s formula for the normalized alternating syzygy quantity Kₚ(S) in terms of universal symmetric polynomials, gap power sums and a final polynomial term.
For odd k≥3 and integers n with gcd(n,k)=gcd(n+1,k)=1, AxiomProver found a Jacobi-symbol reformulation and the floor-sum identity N_k(n)=F_k(n+1)−F_k(n). The authors combine that identity with classical Jacobi facts to establish the parity formula N_k(n)≡⌊(k+1)/4⌋ mod 2 and make the earlier genus-zero and genus-one k-differential formulas unconditional.
For increasing positive integers aₙ with liminf aₙ^(1/2ⁿ)>1, Aletheia’s accepted proof shows that Σₙ 1/(aₙaₙ₊₁) is irrational. This is the original Erdős #1051 statement.
The original clarified Erdős #728 factorial-divisibility solution obtained by GPT-5.2 Pro and Aristotle; later sharpness and strengthened corollaries in v5 are separate human/model follow-ups.
DeepMind reports Lean proofs for IMO 2024 problems 1, 2, and 6, and publishes the specific proof artifacts.
For a differentiable convex function with a minimizer, trajectories of the r=3 Nesterov ODE converge to a minimizer. Jang and Ryu adapt this argument to show that both iterates xₖ and yₖ of standard finite-dimensional Nesterov accelerated gradient converge to one minimizer for an L-smooth convex function under the stated growing tₖ schedule.
Harmonic reports that Aristotle generated machine-verified solutions for 2025 IMO Problems 1–5 from human-formalized problem statements, with the Lean proofs published.
The selected contribution is GPT-5-Thinking’s original rational-trace strategy for the completeness barrier; the current published proof uses Phillip Harris’s determinant instead.
An advanced Gemini Deep Think model produced natural-language solutions to five of the six 2025 International Mathematical Olympiad problems, earning 35 of 42 points under the official coordinators’ grading.
The AlphaEvolve report gives a rank-48 decomposition of the 4×4 matrix-multiplication tensor over the complex numbers, yielding a bilinear algorithm using 48 scalar multiplications.
AlphaGeometry solved 25/30 specialized IMO-AG-30 classical Euclidean geometry translations using learned auxiliary-point constructions plus a symbolic deduction and algebra engine; the paper does not claim all natural-language IMO geometry conditions are represented.
The authors report packing heuristics that outperform tested baselines on their online one-dimensional bin-packing experiments.
DeepMind reports AI-discovered assembly sorting routines translated to C++ and incorporated in LLVM libc++. Its tests report up to 70% improvement for length-five inputs and about 1.7% for inputs exceeding 250,000 elements.
The authors report an algorithm multiplying 4×4 matrices over the two-element field using 47 scalar multiplications, compared with 49 from two levels of Strassen’s algorithm.
AI contributions across algorithms and computation, logic and formal proof, and optimization in the catalog, including their connections to mathematics.
Following this subject includes its nested topics and problems. Contributions that belong to several fields are counted once within the subject.
Background review: 2026-10-14. Selected sources:
Publication dates and editorial changes.
Exact Storage-Repair-Bandwidth Tradeoffs for (6,5,5) and (7,6,6) Exact-Repair Regenerating Codes
Formally Verified Autoresearch for Analyzing the Expressivity of Multi-Head Attention
At the Limits of Computability: Steps Towards Autonomous AI Discovery on the Busy Beaver Frontier
Revisiting Separators in String Graphs, Region Intersection Graphs, and Induced-Minor-Free Graphs
AI-led proof gives a sharp global squared-law transport bound
OpenAI reports bi-Lipschitz c₀ absorption without a linear c₀ copy
AI-assisted criterion yields Erdős–Kac laws for palindromes and reversed primes
OpenAI reports simple F∞ overgroups for decidable-word-problem groups
AI-assisted proof refutes the Foregger–Sinkhorn tie-point conjecture
OpenAI formalizes an alternative proof of random k-SAT thresholds
OpenAI reports a hyperbolicity cone with no finite semidefinite lift
AI-assisted positive model makes Erdős 501 independent of ZFC
Codex-derived rule lowers randomized metric distortion to 2.1441
AI-assisted Lean proof settles the Bollobás–Nikiforov inequality
OpenAI gives a smooth metric with no local immersion in three-space
AI agents extend a difference-free construction to all powers
GPT-found cubic field refutes a Lyapunov converse conjecture
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