AI contributions across the mathematics fields in the catalog, including algebra, analysis, probability, combinatorics, geometry, topology, number theory, algorithms, optimization, and formal proof.
In every arbitrary two-coloring of the Euclidean plane, each non-equilateral three-point set, including three distinct collinear points, has a monochromatic congruent copy. At most one equilateral side length can be missed, and an alternating-strip coloring shows that one exception can occur.
If a graph is not countably colorable, h_G(n)/n tends to infinity, where h_G(n) is the largest minimum number of edge deletions needed to make an n-vertex subgraph bipartite. Equivalently, a uniform linear deletion bound on all finite subgraphs forces countable colorability. The limit holds for the whole sequence, for graphs of any uncountable chromatic cardinality.
There are a finite field K of characteristic two, a finitely presented group G containing odd-prime torsion, and scalar a,b in K[G] with ab=1 but ba≠1. The manuscript gives a terminating prescription for the witness; it does not assert K=F₂.
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.
For every elliptic curve A/Q, every prime p and full p-power Selmer corank r in {0,1}, the manuscript claims analytic rank and Mordell–Weil rank r and finiteness of the whole Tate–Shafarevich group, without reduction, CM, residual-representation, torsion or isogeny restrictions.
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.
For two commuting measure-preserving maps T and S of a probability space, the bilinear averages N^-1 ∑_{n<N} f(T^n x)g(S^n x) converge almost everywhere when f∈Lp, g∈Lq, p,q∈[1,∞] and 1/p+1/q≤1. No mixing or invertibility assumption is imposed.
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 stationary integral m-varifold in an open subset of R^(m+n), with m,n≥1, the m-dimensional Hausdorff measure is zero for points lacking a neighborhood where the varifold is a positive integer multiple of one smooth embedded minimal m-submanifold.
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.
For the nearest-neighbor square-lattice O(4) model, the full Osterwalder–Schrader transfer gap, including rotation-invariant local-observable sectors and measured per original lattice time step, is asymptotic to 32 exp(π/4−1/2) √β exp(−πβ) as β tends to infinity.
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.
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.
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.
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.
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.
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.
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 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.
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.
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.
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.
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.
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.
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.
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 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.
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.
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 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.
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.
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.
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.
Claude agents devised an integer proof route for the s=1, index-two clauses of Theorem D in Jones’s Hadamard border note, replacing its rank and positive-semidefiniteness route. The formal development checks those Theorem D clauses.
For k≥3 and q≥k−1, let m=2kq+1. The numerical semigroup minimally generated by m,m+1,m+k,m+2k−1,m+2k has toric-ideal generator count 4k+2. The formal development also gives a nonredundant presentation.
For nonconstant g∈Q[t] of degree at most four, no u,v∈Q[t] satisfy u(t)v(t)(u(t)+v(t))=g(t)³+g(t)²+3g(t)−1. This excludes only those low-degree polynomial families.
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 H≤G of finite index n>1, Palcoux’s manuscript and formal development give a finite rational upper envelope for the number of intermediate subgroups and a strict bound c(p)n^((logₚn)/4), where p is the least prime divisor of n. Selected results also address prime-power and elementary-abelian sharpness.
For diag(1,1,1,−1,1,−1) and the local Hodge operator on coordinate plane 0123, fifteen orthogonal adjoint actions together with that operator generate sl(15,K) over every field of characteristic other than two. The same formal result gives dimension 224 and scalar rigidity for commuting endomorphisms, with determinant c²²⁴.
Birkbeck and Torzewski give uniform, strongly sheafy, nonnoetherian Tate domains that are not stably uniform, answering Kedlaya’s Question 7. One finite-jet example has a nonreduced rational localization; a weighted-parity example has a nonuniform localization that remains a domain.
For nonzero integers r,s with at least one odd, the word xʳyˢ is universal in Sₙ once n is sufficiently large and at least C log m(r,s), where m is the product of the distinct primes dividing rs. A separate construction shows the logarithmic order cannot generally be reduced.
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.
For every non-eventually-periodic sequence over a finite alphabet, the project constructs an infinite labelled population satisfying Alexander’s finite-root, finite-outdegree, finite real-birthdate-sublevel and incoming-label conditions but with no matching infinite path from any start. With Alexander’s earlier positive theorem, biological unavoidability is equivalent to eventual periodicity.
The proper hat-guessing number of K₈−e is 14. A PGL(2,13)-equivariant fourteen-colour strategy uses 990 normalized twin decisions and a 48,510-edge residual-orbit matching; exhaustive checking covers all 138,378,240 proper colourings, and a general upper bound gives equality.
For sufficiently large N, Sawhney bounds the size of A⊂[N] with ab+1 nonsquarefree for every a,b∈A by the size of the residue class 7 mod 25, and identifies the extremal 7 and possibly 18 mod 25 classes. The model-origin part is a squarefree-density sieve with prime exclusions depending on b and the progression difference.
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 subsets of {1,…,N} in which no element divides a nonempty subset sum of the others, Xeff proves the maximum size is N^(1/5+o(1)). The new upper bound matches the known lower exponent.
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.
The complement of a 243-vertex ternary Golay Cayley graph has domination number and one-guard-moves eternal domination number both 3, below its vertex clique-cover number.
One bounded continuous stable radial pair potential in three dimensions has |φ(r)|≤C r^(−3−1/32) for r≥1 and finite canonical free energy. At one common βc in [7/8,9/8], the free energy has a strict inverse-temperature derivative jump for every density in an open interval around 5p/3.
For a projective four-dimensional log canonical pair with rational boundary over an algebraically closed characteristic-zero field, every program permitted by Definition 1.1 has finitely many birational steps for every choice of negative extremal ray.
If a smooth connected rationally connected projective complex manifold has a specified holomorphic splitting TX=E₁⊕E₂ into positive-rank subbundles, both summands are integrable.
An explicit 2×2 disk-domain example meets the stated Lorist–Schwenninger Lemma 1 hypotheses with ‖T‖>1 and Re⟨E₁Tx,x⟩<0. It rules out a proposed one-step positivity shortcut.
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 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.
In ℤ/13ℤ, A={0,1,3,4,5,6,10} and B={0,1,4,6} have |A|+|B|=11≤12. Their restricted sumset is {1,3,4,5,6,7,9,10,11}, of size 9=|A|+|B|−2, but neither A nor B is an arithmetic progression. This contradicts the forward conclusion of Li–Liang Theorem A/3.1 in arXiv:2609.10148v2.
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.
Kimura reports exactly 16,453,493 imaginary quadratic fields with class number at most 2000; the largest absolute field discriminant on the list is 1,062,350,083.
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.
The initial negatively curved, Z2-coefficient counterexample to the generalized Lagrangian spectral-distance assertion, as identified in the AI disclosure; exclude the later human extensions.
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.
The original comparisons for three-element integer h-bases with exactly h summands: requiring a negative element gives a strictly smaller maximal covered interval for odd h or h=4, and the same maximum for even h other than 4. These are Aristotle’s original proofs, not Nathanson’s replacement arguments.
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.
Let f(n) count the iterations of Euler’s totient function needed to reach 1. The manuscript reports a constant α between 1/log 3 and 1/log 2 such that f(n)/log n tends to α on a set of natural density one. It also gives a quantitative mean-square bound, answering the first two parts of Erdős Problem 408.
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.
The model-origin deformation/Puiseux construction with prescribed residues used in the countable-representability argument, separating the authors’ general Lie-group extensions.
Kamijima, Nakajima, Noda and Sasaya report a nondegenerate Gaussian limit for N^(q/2−1)(log Z_N−E log Z_N) in the Gaussian q-body SYK model, for fixed even q≥4 and sufficiently small nonzero inverse temperature.
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.
Liu’s first release gives FC(4,9)=16 and a co-developed 3:1 core-petal charging argument yielding the nine-petal threshold and FC(4,n)=Θ(n²). A later version extends the asymptotic to FC(k,n)=Θₖ(n^(k−2)) for each fixed k≥2; that general proof is separate from the model-attributed first-release work.
For D(q)=q s q⁻¹ on n-cycles, Lefebvre-Naré’s October note claims depth p^(e−1) for odd prime powers p^e, depth 2^(e−1)−1 for powers of two, and the maximum prime-power depth for general n. The shorter proof uses a defect commutator and triangular Sylow-group filtration.
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.
The explicit conditional exceptional-set exponent g(theta)=theta(5-4theta)/(2(2-theta)), for the stated common Dirichlet-zero hypothesis theta in [1/2,2/3]. The separate 7/12 argument and earlier numerical chain are not the AI contribution.
Vu-Le reports that every compact connected planar shape movable through both left- and right-turning unit corridors from the same initial orientation has area at most 1.64495…; equality identifies Romik’s sofa up to rigid motion.
For every connected finite loopless multigraph of genus g≤6 and integers d,r≥0 with ρ=g−(r+1)(g−d+r)≥0, a degree-d divisor of Baker–Norine rank at least r exists. The new case is genus six, proved by descending special divisors from odd subdivisions to the original graph.
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.
Trevor Morris reports that Maker, moving first on an empty 15×15 square board, can force a rotated or reflected Snaky hexomino against any blocking play in the weak maker-breaker game.
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.
OpenAI’s manuscript reports that all Dirichlet L-functions and finite-order Hecke L-functions over Q(√−3) have no zeros with real part greater than 7/8.
For uniform real nearest-neighbor Lipschitz heights on triangular-lattice approximations of smooth simply connected domains, the October 6 manuscript claims a centered and normalized Dirichlet Gaussian free-field limit. With a tuned opposite two-arc boundary amplitude, the zero-height interface converges to chordal SLE₄ in uniform curve distance.
For every prime p, finite extension k/Q_p and smooth proper geometrically connected curve X/k of genus at least two, the manuscript claims a bijection between X(k) and sections of the full arithmetic étale fundamental-group sequence modulo conjugation by the full geometric group.
Every proper unit-distance colouring of the plane whose unit ball is the regular 14-gon uses at least five colours.
There are absolute constants A,C>0 such that, for every odd prime p, the least positive quadratic nonresidue n(p) is at most C(log p)^A; hence n(p)≪δ p^δ for each δ>0.
There is an explicit trace-preserving PPT channel Θ:M21(C)→M21(C) such that Θ∘Θ is not entanglement breaking.
For each integer d≥3 there are αd∈(0,1/3] and Cd<∞ such that every bounded infinity-harmonic function on the unit ball has gradient supremum and αd-Hölder seminorm on the half ball bounded by Cd times its oscillation on the unit ball.
For every deterministic one-tape machine and finite input, an effective smooth divergence-free mean-zero force on the flat three-torus gives a unique classical forced Navier–Stokes flow from rest, with zero pressure, whose fixed material particle enters one fixed open strip exactly when the machine halts. The force and velocity have bounded mixed derivatives and become one-periodic after initialization.
Every entire C² solution u:ℝ⁸→(−1,1) of Δu=u³−u with a strictly positive derivative in one fixed direction is a planar tanh heteroclinic, without an assumed limit at either end of that direction.
For the uniform measure on any compact convex body with nonempty interior in ℝᵈ, d≥2, and targets supported in one fixed compact set, the L² distance between their quadratic optimal maps is at most C times the one-third power of the targets’ 2-Wasserstein distance. No larger uniform exponent works in general, even for three-atom targets on a fixed cube.
There is a smooth compactly supported probability density on ℝ³, with smooth compactly supported square root, whose finite three-marginal Coulomb Kantorovich minimum is attained by a coupling but by no pair of measure-preserving Borel maps; nevertheless the Monge and Kantorovich infima are equal.
If a smooth almost complex structure on a closed connected smooth four-manifold is tamed by a symplectic form, then the same almost complex structure is compatible with some smooth symplectic form.
Every smooth normalized positive triple of closed two-forms on a closed connected oriented four-manifold deforms smoothly through positive closed triples to a hyperkähler triple while preserving each of its three cohomology classes.
OpenAI gives an adaptive-anchor and weighted-tree proof that every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point in the given norm; the manuscript also derives the commuting-family retract corollary from Bruck’s theorem.
For uniformly signed proper random 3-SAT clauses sampled independently with replacement, the finite positive limiting satisfiability threshold is a computable real: one deterministic machine takes a requested precision and returns a rational approximation within that error.
At the critical honeycomb self-avoiding-walk activity, the total weight of unrooted polygons of diameter at least r is r^{-2+o(1)}, their length-weighted tail is r^{-2/3+o(1)}, and the truncated second-length-moment mass through diameter r is at most r^{2/3+o(1)}.
For zero-field Sherrington–Kirkpatrick heat-bath dynamics at β=1, stationary and fair-spin quench autocorrelations converge jointly as random functions after multiplying correlations by n^{1/3} and scaling waiting times and lags by n^{2/3}; Gaussian and Rademacher couplings have the same limiting law, and the quench limit relaxes to the stationary limit as waiting time grows.
Assuming the scalar, annealed Cardy crossing formula for fair-colored critical Poisson–Voronoi percolation on every bounded Jordan quadrilateral, the jointly coupled near-critical crossing thresholds of rational polygonal quadrilaterals, normalized by each model’s expected number of unit-square color-pivotal sites, have conditional laws that converge in probability over Voronoi tessellations to the triangular-lattice limit.
For the Ising perceptron with independent Gaussian patterns, every bounded Borel real log-potential and each fixed positive temperature and pattern density have a limiting normalized free energy given by the manuscript’s explicit overlap-path variational formula, with convergence in expectation and probability.
For planar Ising bonds J_e=1+εξ_e with any fixed bounded, nondegenerate, mean-zero iid disorder law and sufficiently small ε>0, the full oriented interface law at the spontaneous-magnetization threshold converges in probability over environments to chordal SLE₃ in deterministic Jordan-domain approximations.
For the nearest-neighbor two-dimensional O(3) spin model with no external field or topological term, normalization by susceptibility and second-moment correlation length yields a unique continuum Schwinger hierarchy as the bare inverse coupling tends to infinity through all positive real values; the reconstructed field theory has a positive mass gap and a nonzero connected four-point function.
For critical spherical FK–Ising planar maps with n edges, stationary random walks using every incident half-edge converge jointly with the rescaled map metric and degree measure to Liouville Brownian motion on the unit-area √3-quantum sphere when time is accelerated by exactly n; the convergence holds for any fixed finite number of conditionally independent walks.
For every degree k≥2 there is c_k>0 such that, for each fixed intersective integer polynomial h of degree k with positive leading coefficient, every A⊆{1,…,N} avoiding nonzero differences h(m), m≥1, has |A|=O_h(N^{1−c_k}).
For every quartic Salem number β with 1<β<2, the unbiased Bernoulli convolution with contraction parameter λ=β⁻¹ is singular with respect to Lebesgue measure.
For a compact connected smooth manifold of dimension at least three with smooth boundary and any nonempty open boundary patch, equality of zero-frequency same-patch boundary measurements for two smooth metrics and smooth unitary connections on the trivial rank-two Hermitian bundle implies they differ only by a patch-fixing diffeomorphism and a patch-identity unitary gauge.
The three-dimensional periodic hard-sphere Boltzmann equation has one nonnegative initial density with bounded velocity support and finite mass, energy and absolute entropy that yields two distinct global entropy solutions, both satisfying exact local conservation of mass, momentum and kinetic energy and strong L¹ continuity.
For every integer n≥2 and real κ, a complete separable metric space is an n-dimensional Alexandrov space of curvature at least κ exactly when its Hausdorff-measure metric space has full support, is RCD((n−1)κ,n), and has Gigli distributional sectional curvature at least κ on the original global test classes.
Every connected closed Einstein four-manifold with positive Einstein constant and nonnegative sectional curvature that has even one zero-curvature two-plane has universal Riemannian cover isometric to a product of two equal-radius round two-spheres.
A uniformly discrete bounded-geometry coarse disjoint union of finite connected uniformly bounded-degree graphs has an infinite-order class in degree-one coarse K-homology that maps to zero under maximal coarse assembly, so that map is not rationally injective.
Every action of a countable discrete amenable group on a simple, separable, unital, infinite-dimensional, nuclear, stably finite, Jiang–Su-stable complex C∗-algebra absorbs the trivial Jiang–Su action up to cocycle conjugacy, with no condition on its action on traces.
For every unital expected inclusion N⊂M of von Neumann algebras with separable preduals, there is an expected unital amenable subalgebra P⊂N with P′∩c(M)=N′∩c(M), where c(M) is the continuous core.
For the zero-field Gaussian Sherrington–Kirkpatrick model, as system size tends to infinity first and QAOA depth then grows, the best limiting expected energy per spin approaches the Parisi ground-state value. For every accuracy, one finite depth and deterministic angles work independently of system size and disorder.
Every simple unitary strongly rational complex vertex operator algebra has polynomial energy bounds and CKLW strong locality; its smeared fields generate an irreducible conformal net.
An explicit entangled state on C¹⁰⊗C¹⁰ has zero distillable secret key against an eavesdropper holding a purification and the full public transcript, for completed local-instrument protocols with joint processing of copies and unlimited two-way public communication. Every admitted bit-output protocol remains at trace distance at least 1/10 from an ideal secret bit.
For any fixed nuclei at distinct positions with real charges at least one, total nuclear charge Z and M nuclei, the full two-spin nonrelativistic Coulomb Hamiltonian can strictly bind no more than Z+CM electrons, where C is universal and independent of the nuclear positions and charges.
Every nonzero one-homogeneous global minimizer of the one-phase Bernoulli energy in ℝᵈ is a flat half-space solution for each dimension 1≤d≤6.
There is a nonquadratic smooth entire function on ℝ¹⁰ with pointwise positive-definite Hessian that solves the classical affine maximal equation.
For each integer m≥1, there is a simply connected closed Kähler manifold of real dimension 22 with stable Morse number 80+1968m and a Hamiltonian diffeomorphism with exactly 80+1952m fixed points, all nondegenerate with contractible orbit loops.
Assuming the continuum hypothesis, there are infinite-dimensional real and complex Banach spaces with no bounded linear surjection onto any infinite-dimensional separable Banach space.
For every Grothendieck coherator in the Ara–Henry convention, every boundary-cellular infinity-groupoid model X, every n≥0 and every n-cell a of X, adjoining an (n+1)-disk along the source n-face of a gives a weak equivalence X→X⁺.
For separable nuclear complex C∗-algebras A and B, any isomorphism between their topological cones of all extended lower-semicontinuous tracial weights is realized by an isomorphism A⊗W⊗K ≅ B⊗W⊗K, where W is the Razak–Jacelon algebra and K the compact operators.
For every fixed positive half-integer spin and every nonnegative symmetric finite-range interaction generating ℤ³, the spontaneous magnetization deficit of the three-dimensional quantum Heisenberg ferromagnet is asymptotic to ζ(3/2)T^{3/2}/(8π^{3/2}√det D), where D is the positive quadratic one-magnon dispersion matrix.
For the fermionic V₁ Laughlin Hamiltonian at flux q=3(N−1), sufficiently weak projected bounded real scalar one-body potentials leave a unique ground state and a spectral gap bounded below uniformly for all sufficiently large N and all potential profiles of sup norm at most one.
For each hard-sphere radius a>0 and every sufficiently small fixed density ρ>0, there is a temperature T(a,ρ)>0 independent of volume at which the exact three-dimensional canonical Gibbs state has a positive condensate fraction in the constant orbital as N/L³ tends to ρ.
For every finite matrix size and 1/2<γ<3/2, a measurable positive semidefinite Hermitian matrix potential saturates the sharp one-dimensional Lieb–Thirring inequality exactly when one constant unitary basis makes it a direct sum of independent one-bound-state scalar sech² profiles and zero channels.
There is a finite connected aspherical complex with a linear combinatorial disk-filling bound whose word-hyperbolic fundamental group has no proper cocompact isometric action on any nonempty proper complete CAT(0) space.
For any group G and any finite system of m equations in n unknowns, with 1≤m≤n, if the exponent-sum matrix has row rank m over the rationals, then G embeds in the group formed by adjoining the unknowns and imposing the equations. Thus the system has a simultaneous solution in an overgroup.
For every fixed 0<κ<8, there is a deterministic Hausdorff gauge that almost surely gives positive finite measure to every nontrivial compact positive-time segment of ordinary chordal SLE_κ.
For the ferromagnetic four-state Potts broadcast model on every regular d-ary tree with d≥2 and every observed Poisson Galton–Watson tree of mean d>0, reconstruction occurs exactly when dλ²>1, including nonreconstruction at equality.
For every stationary ergodic, uniformly elliptic, finite-range-dependent nearest-neighbor environment on Z^d with d≥3, the annealed probability of escape in any fixed nonzero direction is either zero or one.
For every discrete group G, the Hattori–Stallings trace of every idempotent matrix over the complex Banach algebra ℓ¹(G) has finite support, contained in conjugacy classes of finite-order elements.
A finite nonempty lattice is the full congruence lattice of some finite algebra exactly when it admits the finite colored-graph witness specified in the manuscript, whose edge colors encode the required congruence relations.
There are fixed constants η,c>0 such that, for every sufficiently large n, n points can be placed in the unit square with every triangle of area at least c n^{-2+η}.
A real circulant Hadamard matrix of positive order n exists exactly when n=1 or n=4.
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 stable finite-range radial pair potentials in the stated regularity class, grand-canonical Newtonian gases started from spatially summable C¹ data with Gaussian density and spatial-gradient bounds have every fixed scaled factorial marginal converge in full phase-space L¹, uniformly on any finite interval where the corresponding nonlinear Boltzmann equation has a nonnegative classical solution with the required Gaussian bound.
For a finite pointed CW pair G ⊂ E and based map v:E→Sᵈ whose restriction to G becomes null after passage to complex cobordism, the restriction of a sufficiently large smash power v∧ℓ to the union where any fixed positive fraction of factors lie in G is stably null. The paper also derives a boundary-preserving cube compression theorem from a torus-slot bundle embedding of rank strictly below twice the source rank.
For an ICC group commensurable with a lattice in a finite product of the specified property-(T) algebraic groups over characteristic-zero local fields, any countably infinite ICC comparison group, and arbitrary normalized scalar cocycles at both ends, every bifinite correspondence between their twisted group factors is a closed bimodule summand of a finite sum of models from finite-index subgroup isomorphisms and finite-dimensional projective representations of the matched cocycle ratio. Conversely, every such model, sum, and closed summand is bifinite.
Under Assumption 2.1 for a four-dimensional local, unitary, positive-energy vacuum QFT—with a physical local stress tensor and dilatation current, discrete bounded-below finite-multiplicity scaling spectrum, finite scaling support of each original field, and compatible affiliated fields in a causally commuting bounded local net—the stress tensor admits a physical dimension-two scalar improvement that is symmetric, conserved, and traceless without changing translation or Lorentz charges. Its conformal currents obey unbroken local Ward identities on the physical field algebra.
For each fixed subextremal rotating Kerr bridge (M > 0 and 0 < |a| < M), smooth vacuum data in a sufficiently small weighted tenth-order neighborhood whose full maximal globally hyperbolic development admits a future continuous, nondegenerate Lorentzian extension with locally square-integrable weak connection form a meagre set. Hence inextendible data are residual in the weighted smooth topology.
Planar nearest-neighbor first-passage percolation with iid nonnegative nonatomic edge weights almost surely has no doubly infinite geodesic when the minimum of four independent weights has finite second moment.
For every integer b≥6 and finite-dimensional complex vector space V, there is a GL(V)-equivariant injection Sym⁶(Symᵇ V)→Symᵇ(Sym⁶ V).
A three-dimensional complete Noetherian normal local domain containing ℂ, with residue field ℂ, has no nonzero finitely generated module of depth three.
Under every flat local homomorphism of nonzero Noetherian local rings, Hilbert–Samuel multiplicity of the source does not exceed that of the target.
For every fixed r≥4, every K_r-free graph of sufficiently large maximum degree Δ has correspondence chromatic number O_r(Δ/log Δ).
Every graph can be partitioned into a universally bounded number of fixed edge pieces, each with ordinary random-graph containment threshold at most a universal constant times the original graph’s integral expectation threshold.
An ergodic invertible zero-entropy transformation of a standard nonatomic probability space is not measurably conjugate to any C∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold.
The unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized Liouville measure on the full phase space.
For every finite weakly reversible mass-action network with fixed positive rates, each positive stoichiometric compatibility class has one compact convex forward-invariant set entered eventually by every positive trajectory in that class, including when the class is unbounded.
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.
Every W^{1,p} homeomorphism between nonempty bounded connected open domains in R³, for 1≤p≤2, is a strong W^{1,p} limit of smooth diffeomorphisms onto the same target domain.
At every singular point of the first singular time for the mean-curvature flow of a smooth compact connected embedded surface in R³, the full fixed-center rescaling family converges locally smoothly on compact negative-time intervals to one multiplicity-one homothetic self-shrinker in the original ambient coordinates.
In sufficiently high dimension, a Ricci flow on a closed manifold reaches finite maximal time while scalar curvature remains uniformly bounded and full curvature diverges.
For every m≥2, a closed connected minimally immersed hypersurface of the unit round sphere S^{m+1} with non-totally-geodesic image has volume at least that of the smallest m-dimensional minimal Clifford product.
For every stationary integral m-varifold in an open Euclidean set, in every positive dimension and codimension, the singular set has Hausdorff dimension at most m−1.
For n≥4, every complete connected smooth n-manifold with the quadratic-form lower bound −4Δ+Scal≥1 maps continuously to a simplicial complex of dimension at most n−2 with every fiber’s original-metric diameter bounded only by n.
For every n≥3, a dimension-dependent constant aₙ>0 bounds the integral of the n/2 power of the negative part of scalar curvature below by aₙ times real simplicial volume, for every closed connected oriented smooth n-manifold and metric.
For every sufficiently large prime p, a connective p-complete spectrum of exact fp-type two lies outside the ordinary thick subcategory generated by the specified standard BP⟨2⟩ p-completion, even though its finite and telescopic localization comparisons hold.
For each n≥1 and prime p>n+1, the spectrum Lₙ₋₁L_{K(n)} of the p-completed sphere has an ordered 2ⁿ-stage filtration with the classical chromatic-splitting cofibers, whose first-stage map to the target is the canonical localization unit.
For every prime p, finite p-group G, subgroup H and height n≥0, the optimal chromatic fixed-point loss from H to G equals the shortest subnormal-chain length from H to G with cyclic quotients.
In the standard mod-three Adams Kervaire family, exactly the classes indexed 0, 2 and 3 survive to stable homotopy, in stems 10, 106 and 322, and each surviving detection coset contains an element of additive order three.
For every nonnegative height n, there is some prime p and a finite p-local spectrum whose Brown–Peterson homology is BP∗/(p,v₁,…,vₙ) with its canonical comodule structure and degree-zero generator.
For every positive order delta, the three-dimensional Bochner–Riesz multiplier is bounded on L3(R³). Interpolation and duality yield the conjectured strict Lp boundedness range for all 1 <= p <= infinity.
A smooth Kähler metric in the Fubini–Study class of complex projective ten-space has a maximal Calabi flow with finite existence time and scalar curvature diverging like a positive constant times (T−t)⁻¹ᐟ² at a point.
Every smooth Riemannian metric on the standard sphere Sⁿ, for n≥2, has infinitely many prime closed geodesics with pairwise distinct images.
The real Banach space ℓ₁ has metric Markov cotype two in the Mendel–Naor Cesàro formulation.
There are closed connected aspherical topological four-manifolds with word-hyperbolic common fundamental group that are homotopy equivalent but not homeomorphic.
In every positive degree, the mod-two stable Hurewicz image of the sphere is spanned by the images of η, ν, σ and the Kervaire-invariant-one classes that exist.
For every prime p, height n≥1 and integer degree t, the homotopy group π_t of the K(n)-local p-completed sphere is a finitely generated p-adic-integer module.
For every prime p and height n≥1, the dualizable K(n)-local stable-homotopy category has exactly n+2 thick tensor ideals, forming one strict chain; its Balmer spectrum has n+1 points.
A compact oriented smooth four-manifold contains finitely many immersed disk maps with framed algebraic dual spheres whose embedded boundary circles bound no pairwise disjoint proper locally flat disks, even without prescribed output homotopy classes or framings.
For every maximal abelian subalgebra of a complex von Neumann algebra, every self-adjoint operator is a strong-operator limit of self-adjoint approximants of norm at most three times its norm, each admitting norm paving to relative error ε with at most Cε⁻⁶ projections for a universal C.
A finitely generated torsion-free group has an infinite-order degree-zero K-homology class in the kernel of the coefficient-free reduced Baum–Connes assembly map.
A finite three-electron molecule with two equal positive integer nuclear charges has a normalizable interacting Coulomb ground state whose spin-summed density is not the density of any noninteracting ground-state ensemble for a single real spin-independent local potential in L^(3/2)(R³)+L∞(R³).
For each fixed dimension d and positive rational epsilon, every epsilon-log-canonical Fano contraction of a normal integral variety over an algebraically closed characteristic-zero field admits, near each closed base point, a klt complement whose Cartier index is bounded only by d and epsilon.
For a surjective connected-fiber holomorphic map from a smooth compact connected Fujiki-class-C manifold with rational simple-normal-crossing boundary to a normal compact irreducible complex space, the total log Kodaira dimension is at least the sum of the very general fiber log Kodaira dimension and Campana’s orbifold-base Kodaira dimension, including boundary coefficient one.
For every smooth connected complex algebraic curve, every rank r, and prescribed quasi-unipotent boundary conjugacy classes, including nonsemisimple classes, one finite extension of the number field generated by their eigenvalues makes ambient integral points Zariski dense in the exact-boundary locus of the SL_r character variety and in every irreducible component.
At every fixed point on the critical Ashkin–Teller line, including the four-state Potts endpoint, the height and both current-cluster collections converge jointly in bounded Jordan domains: the height to a Gaussian free field and the clusters to canonical recursive two-valued local sets of that field, including all nesting depths and the wired boundary cluster.
For each fixed edge-change weight x∈[1/√2,1], zero-boundary integer Lipschitz heights on triangular-lattice inside approximations of a bounded C² Jordan domain converge, after division by a positive x-dependent constant, to the zero-Dirichlet Gaussian free field as a random distribution.
The complete loop ensemble of the Temperleyan square-lattice double-dimer model in the upper half-plane converges along the full mesh limit to nested CLE₄, matching every macroscopic loop as an unparametrized curve.
For the balanced-torus plane state of the square-lattice six-vertex model with a=b=1 and 0<c≤2, its unit-jump height field converges modulo constants to a Gaussian free field whose squared multiplier is 1/arcsin(c/2) under the Green-kernel normalization −(2π)^(-1)log|x−y|.
For each fixed 1≤q<4, the critical square-lattice random-cluster Dobrushin interface converges as an ordered curve in uniformly approximated marked bounded Jordan domains to chordal SLE with κ(q)=4π/arccos(−√q/2).
For every fixed degree d≥3, sufficiently weak fixed positive uniform diagonal disorder on a uniformly random simple d-regular graph yields the GOE bulk microscopic eigenvalue point process at each fixed energy separated from the clean spectral edges, after rescaling by the corresponding tree density of states.
For the nearest-neighbor cosine XY model on the square lattice, the free-box thermodynamic two-point correlation at critical inverse temperature obeys C(r)=B_XY r^(-1/4)(log r)^(1/8)(1+o(1)) along a coordinate axis, with 0<B_XY<∞.
There is a two-dimensional ramified regular local ring A of mixed characteristic (0,5) for which K₅(A)→K₅(Frac A) has a nonzero kernel.
Every finite symmetric tensor category over an algebraically closed field of positive characteristic has an exact faithful strong symmetric monoidal functor to some finite higher Verlinde category Ver_{p^n}(k).
For every algebraically closed field K of characteristic p>0 and every fixed bound on defect-group order, blocks of finite groups represent only finitely many K-linear Morita-equivalence classes.
For a smooth strictly convex planar billiard with positive boundary curvature, a continuous foliation of a full near-grazing phase annulus by individually invariant essential curves forces an analytic boundary and a jointly analytic collar of smooth strictly convex caustics.
An explicit homogeneous hyperbolic polynomial of degree 20 in 23 real variables has a closed hyperbolicity cone that cannot be represented as the positive-semidefinite region of any finite real symmetric linear matrix pencil in the original variables.
For every eta in (0,1), there is a compact subset of [0,1] of measure greater than 1−eta that contains no affine copy of {2^(−n):n>=1} under any translation or nonzero real dilation of either sign.
For arbitrary complex F,G in L3(R²), the supremum of the triangular Hilbert transform over both finite hard-truncation endpoints is bounded in L^(3/2)(R²) by C times the two L3 norms.
For every eta > 0 and every sufficiently large integer N, there is a Littlewood polynomial with N consecutive coefficients in {−1,1} whose maximum modulus on the unit circle is at most (1+eta) sqrt(N).
Every nonempty connected bounded semialgebraic subset that is open in a complex affine algebraic variety is smooth and biholomorphic to a bounded symmetric domain if a discrete group of biholomorphisms acts properly with compact quotient.
For function fields of transcendence degree at least two over algebraically closed fields of characteristic different from a fixed prime ell, two perfect-closure field isomorphisms preserving the constant fields that induce the same pro-ell abelian-by-central Galois data up to ell-adic unit scaling agree up to Frobenius.
For every n≥2 and real K, each regular point of a noncollapsed RCD(K,n) space has an ambient-open neighborhood bi-Lipschitz to an open subset of Euclidean n-space, with distortion bounded only by n.
Every connected closed smooth Riemannian manifold of positive dimension whose global injectivity radius equals its diameter is, up to scale and isometry, a standard compact rank-one symmetric space.
For some sufficiently large even N, the cotangent bundle of S⁹×Sᴺ⁻¹ contains a closed exact smoothly embedded Lagrangian diffeomorphic to the base that is not Hamiltonian isotopic to the zero section.
For every closed connected smooth Riemannian manifold of dimension at least three with strictly negative sectional curvature, normalized Liouville measure has maximal entropy for the geodesic flow if and only if the metric is locally symmetric.
A finite connected three-dimensional CW complex satisfies the D(2) homological finiteness conditions but is not homotopy equivalent to any finite complex of dimension at most two.
For every finite group G and prime p with trivial largest normal p-subgroup, the poset of nontrivial elementary abelian p-subgroups of G has nonzero augmented reduced rational homology.
For any nontrivial smooth knot in the three-sphere, Dehn surgeries along distinct slopes never produce orientation-preservingly homeomorphic filled three-manifolds.
For every finite N≥2, the undeformed SU(N) BFSS matrix quantum-mechanics Hamiltonian after removal of center-of-mass coordinates has exactly one normalizable zero-energy state.
For finite-range bounded interactions of fixed local dimension on any finite induced subgraph of the square lattice, a unique ground state separated by a uniform global spectral gap has entanglement entropy at most C times the crossing-edge boundary size for every set of sites.
A finitely generated group has two essentially free probability-measure-preserving actions with different costs: its Bernoulli action has cost strictly greater than one, while a suitable finite-height extension has cost closer to one.
Every finitely generated one-relator group containing no subgroup isomorphic to a Baumslag–Solitar group BS(m,n), with m and n nonzero, is word-hyperbolic.
For a symmetric n×n matrix whose entries on and above the diagonal are independent uniform signs, the probability of singularity is (1/2+o(1))^n.
For G=Spin(2n), n≥2, if three dominant integral weights sum to the root lattice, then a nonzero invariant in their tensor product after any positive integral dilation implies a nonzero invariant at the original weights.
For every finite group, every prime p, and every p-block B, the number of irreducible Brauer characters in B equals the number of conjugacy classes of B-weights.
A countable characteristic-zero division ring is algebraic over its center, generated by two elements as an algebra over that center, and infinite-dimensional over the center.
Over every characteristic-zero field, for a homogeneous ideal containing a regular sequence of any positive length with degrees at least two, a corresponding lex-plus-powers ideal has the same Hilbert function and at least as large a graded Betti number in every homological and internal degree.
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).
The two-color graph Ramsey number of the n-dimensional binary cube is at most C·2^n for one absolute constant C.
For every fixed dimension d≥3, a constant c_d>0 makes every n-point set of distinct points in ℝᵈ determine at least c_d n^(2/d) distinct distances.
For every finite coloring of the positive integers and every k≥1, some k-element set has all its nonempty subset sums and all its nonempty subset products in one color class.
For every invertible mixing probability-preserving transformation and every fixed finite tuple of bounded measurable functions, consecutive multiple ergodic averages converge almost everywhere, along all positive averaging lengths, to the product of the functions’ integrals.
In each dimension, every convex body has isotropic constant no greater than that of a simplex, with equality only for simplices.
For every connected simple complex algebraic group G, smooth projective connected complex curve X, and complex level c outside Q, the full twisted D-module categories on Bun_G(X) and Bun_G-dual(X) are equivalent at dual shifted levels c and -1/(r c), retaining the given global forms and all connected components.
For every smooth connected projective complex variety whose anticanonical line bundle has a smooth Hermitian metric with semipositive curvature, some positive anticanonical tensor power has a nonzero global section.
If a smooth projective complex base satisfies all ordinary descendant Virasoro constraints, then the projectivization of every algebraic vector bundle of rank at least two over it satisfies them in every genus and individual integral curve class, with all cohomology insertions.
The ordinary discrete fundamental group of every connected smooth special compact Kähler manifold has an abelian subgroup of finite index.
Every smooth complex quintic threefold admits a numerical Bridgeland stability condition with the support property whose hyperplane-tensor and structure-sheaf spherical-twist symmetry shifts phase by 2/5, with the normalized central charge of Toda’s quintic Gepner conjecture.
At genus 10^60 with 3 times binomial(10^60,3) markings, an explicitly defined formal tautological class maps to zero in the Chow ring and rational cohomology of the compactified pointed-curve moduli space but lies outside the span of Pixton’s original graph-formula relations.
Every complex K3 surface has the convex approximation property: for every positive integer m, every holomorphic map from a neighborhood of a compact convex subset of C^m into the surface can be uniformly approximated there by entire maps C^m into it.
On a smooth connected compact Kähler manifold, if B is an effective rational simple-normal-crossing divisor with coefficients below one, M is a nef rational line bundle, K_X+B is pseudo-effective, and K_X+B+M is nef, then its real Bott–Chern first Chern class is represented by a semiample rational line bundle.
Assuming logarithmic Iitaka subadditivity for the specified smooth projective reduced-boundary fibrations, every analytically nef Q-Cartier log canonical divisor K_X+Delta on a normal irreducible compact Kähler space with effective rational boundary is semiample.
For a smooth proper hyperbolic curve over an algebraic closure of a p-adic local field and finitely many disjoint sufficiently small open disks, there is one connected finite étale cover with a component mapping isomorphically onto the entire disk exterior and every component above each disk having degree divisible by p.
Assuming the stated finite-level Ramanujan–Arthur decomposition, every occurring cuspidal excursion parameter for a split connected semisimple group over a global function field at a specified full finite level has one algebraic SL2 and commuting Weil centralizer map recovering the parameter on the entire Weil group, including inertia.
Every continuous odd absolutely irreducible two-dimensional 2-adic representation of the absolute Galois group of Q that ramifies at only finitely many finite primes occurs in the full completed varying-weight Hecke algebra at some odd tame level, possibly with auxiliary primes.
For arbitrarily large odd nonlinear powers, some Sobolev index k>8 admits a nonempty open set of Hᵏ initial data on the twelve-dimensional torus whose defocusing nonlinear Schrödinger solutions undergo classical finite-time self-similar blowup.
A complete smooth nonnegatively Ricci-curved metric on Euclidean space in dimension 16 has more harmonic functions of pointwise polynomial growth at most integer degree 50000 than Euclidean space has harmonic polynomials of that degree.
There is a contractible complex three-dimensional domain with a complete Kähler metric whose real sectional curvatures lie between two negative constants, yet it has no bounded holomorphic map to complex three-space with nowhere-vanishing Jacobian and is not biholomorphic to a bounded domain.
There is a closed connected orientable smooth three-manifold that carries a smooth metric without conjugate points but carries no smooth metric of everywhere nonpositive sectional curvature.
A separable reflexive real Banach space has an asymptotically midpoint uniformly convex given norm with averaged midpoint modulus at least √(1+t²/12)−1 for t>0, but admits no equivalent asymptotically uniformly convex norm; depth-k countably branching diamonds have distortion at least √(1+k/12) in it.
For a nonzero real Banach space with the ordinary approximation property, K-convexity holds if and only if both the space and its dual have finite Rademacher cotype, with possibly different exponents.
For every irrational rotation angle θ in (0,1), there is a continuous nonnegative circle weight f with one zero and integral log f equal to negative infinity such that the weighted rotation U f(V) has no nontrivial invariant projection in its associated hyperfinite type II₁ factor.
An explicit separable unital full group C∗-algebra admits no unital embedding into the norm ultrapower B^ω of any nonzero unital nuclear C∗-algebra B for any free ultrafilter ω on the natural numbers.
For the Thorp shuffle on 2^d cards, the full permutation law after 1600d complete shuffles tends to uniform in total variation as d grows, uniformly over initial decks; a support bound requires at least 2d−O(1) shuffles, giving mixing time Θ(d).
For Ising interactions with Haar orthogonal eigenvectors, empirical eigenvalues converging to a compactly supported law, and extreme eigenvalues converging to the support edges, the normalized pressure converges in expectation and almost surely at every fixed positive temperature to the paper’s overlap-hierarchy variational functional.
One stable radial pair potential in three dimensions with a divergent repulsive core and integrable o(r⁻³) tail has canonical free energy at all positive temperatures and densities, with a strict finite jump in its inverse-temperature derivative at one common positive temperature over a nonempty open interval of densities.
For diluted even-arity Ising models in the Panchenko–Talagrand class, finite-system pressure converges to the infimum of the hierarchical cavity trial functional over every finite hierarchy depth and trial law.
A finite Euclidean configuration is Ramsey for monochromatic congruent copies at its original scale exactly when its coordinate field admits the paper’s rational-tensor matrix satisfying the pointwise evaluation identities and identity-matrix product identities; the criterion covers every affine dimension.
For every fixed ε>0, all but o(n) points of any n-point planar set determine at least n^(1−ε) distinct nonzero distances as n grows, uniformly over the set.
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.
There is a countable uniformly discrete metric space whose real Lipschitz-free space has the approximation property but fails every bounded approximation property.
For every proposed constants a,b≥1, some origin-symmetric convex body K in a finite dimension violates Pietsch’s proposed dimension-free covering-entropy comparison with the cube: log N(K,B∞)>b log N(B∞°,a⁻¹K°).
Every real Banach space with Markov type p>1 for some p is superreflexive; with the established converse, nontrivial Markov type characterizes superreflexivity.
There are separable real Banach spaces X and Y with a global bi-Lipschitz bijection but no bounded linear isomorphism between them; X contains an isometric copy of c₀(ℓ₂), while Y contains no bounded linear copy of that space.
Every complex C∗-algebra in which each positive element is properly infinite is strongly purely infinite, satisfying the full positive-element diagonalization property.
There is a bounded self-adjoint tuple X in a von Neumann algebra with faithful normal tracial state for which both microstates and nonmicrostates free entropy are finite yet −∞<χ(X)≤χ*(X)−1/2<∞.
For every qubit generalized amplitude-damping channel with both parameters in [0,1], unassisted classical capacity equals its one-use Holevo information and an attained one-variable maximum; Holevo information is additive over all repeated uses.
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.
For every d≥3 and β≥0, the free zero-field ferromagnetic Ising law on the infinite d-regular tree is a factor of IID exactly when tanh β≤1/√(d−1), including equality.
On every infinite connected locally finite simple unweighted graph, the free uniform spanning forest is a factor of independent identically distributed vertex labels under one root-free Borel equivariant rule.
For every positive integer n outside {2,4,9}, some irreducible complex representation of Sₙ has a tensor square containing every irreducible complex representation of Sₙ.
A finite-dimensional algebra over F₂(q,H₁,H₂) has a finite-dimensional nonprojective Gorenstein-projective module Z with Extⁱ(Z,Z)=Extⁱ(Z,Λ)=0 for every i>0.
One explicit 66×66 binary pattern has no polynomial ordered matrix-removal bound: for every c,C>0 there are binary matrices far from avoiding it but with fewer than c ε^C n^132 induced ordered copies.
For random triangle removal from Kₙ, the number of edges left at termination divided by n^(3/2) converges in L² to 1/(2√2).
For every integer d≥3, there are arbitrarily large finite pure connected d-dimensional simplicial complexes with uniformly bounded vertex degree and uniform F₂ coboundary expansion in every degree below d.
For every finite simple graph H with h≥1 edges and at most n vertices, the threshold for G(n,p) to contain an ordinary copy of H is at most C p_E(n,H)(1+log₂ h) for a universal C.
For every natural unit interval graph, its chromatic quasisymmetric function has nonnegative coefficients in the elementary basis over ℕ[q].
A noninvertible C¹ self-map of a compact smooth manifold without boundary has zero topological entropy while its induced action on second real homology has eigenvalue 2.
A C∞ diffeomorphism of the three-torus preserves standard volume and has simple Lebesgue Koopman spectrum on its entire mean-zero L² space.
The trilinear Hilbert transform with fixed slopes 1, 2, and 3 maps three complex L3(R) inputs to L1(R) with a uniform norm bound.
Every smooth connected complex projective Fano variety of dimension six with nef tangent bundle is rational homogeneous.
Every holomorphic one-parameter family of isolated hypersurface singularities in C^3 with constant Milnor number is topologically right-trivial, by a homeomorphism preserving the parameter, origin section, and defining function.
Every smooth connected complex projective contact Fano manifold of complex dimension at least three is contact-isomorphic, with its specified contact distribution, to the adjoint variety of a simple complex Lie algebra.
Every sequence of generalized log canonical flips on a normal globally Weil Q-factorial compact Kähler fourfold terminates when each flip is a projective small diagram with the stated opposite ample signs, rational boundary, and one fixed nef Q-Cartier b-divisor on a projective modification.
For every sufficiently large admissible Hassett discriminant, a very general smooth cubic fourfold in that divisor is irrational although its Kuznetsov component is equivalent to the ordinary derived category of a projective K3 surface.
A smooth projective complex fourfold has large fundamental group but a non-Stein universal cover with no positive-dimensional compact complex-analytic subvarieties.
For every finite quiver, including loops and multiple arrows, and every regular stability character with stable semistable locus and free gauge action, the equivariant cohomology algebra of its Nakajima variety is canonically isomorphic to the coordinate ring of the scheme-theoretic stability-cocharacter fixed locus on the flavor-deformed Coulomb branch.
For a smooth projective complex curve of genus at least two, the perverse and weight filtrations agree on the full rational cohomology of fixed-determinant, trace-free Higgs and character moduli spaces in every composite rank coprime to the degree.
For a projective log canonical threefold pair over an algebraically closed field of characteristic p > 3, if the rational log canonical divisor K_X+B is Q-Cartier, nef, and has numerical dimension one, then it is semiample.
Every continuous open homomorphism between the Galois groups of solvably closed Galois extensions of number fields is induced by a unique equivariant field embedding in the opposite direction, without a kernel restriction.
For every epsilon > 0, infinitely many positive integers n have more than n^(1-epsilon) preimages under Euler’s totient function.
For all integers m≥n≥3 except (m,n)=(3,3), R(C_m,K_n)=(m−1)(n−1)+1; the exceptional value is R(C₃,K₃)=6.
In ZFC, two self-dual partitional matroids on a countably infinite common ground set admit neither a packing/covering partition nor an intersection witness.
Some absolute ε>0 and C bound the number of unordered halving pairs in every sufficiently large even n-point planar set with no three collinear by Cn^(4/3−ε).
For every sufficiently large prime q, a finite intersecting (q+1)-partite (q+1)-uniform hypergraph has covering number q+1 and exactly q+1 nonisolated vertices in each part.
For every finite real self-similar probability measure with positive weights and nonzero contraction ratios of absolute value below one, dim_H μ=min(1,h_RW/χ), even with exact overlaps and unequal signed ratios.
For C² potentials with known minimizer at zero and I≤∇²V≤2I, exact value-and-gradient oracle sampling to total-variation error at most 1/10 uses at most Cε d^ε queries for every ε>0, while every algorithm needs at least c log d queries eventually.
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.
A regular tetrahedron has a finite cylinder cover with compact triangular perpendicular bases whose total area is strictly less than half its minimum orthogonal projection area.
On any bounded connected smooth domain in three-dimensional Euclidean space, equality of full static displacement-to-traction maps for two smooth real Lamé-modulus pairs with μ>0 and 3λ+2μ>0 on the closure forces both moduli to agree throughout the domain.
For n≥2, p,q>0 and real weights A,B satisfying (n+A)/(p+1)+(n+B)/(q+1)>n−2, the Hénon–Lane–Emden system has no strictly positive entire solution continuous at the origin and C² away from it.
On every compact connected smooth Riemannian manifold of dimension at least two satisfying weak Ma–Trudinger–Wang curvature, each open tangent injectivity domain is convex.
In the unit cubic flat three-torus, every volume-minimizing region is a ball, a circular tube about a shortest closed geodesic, a coordinate slab or the complement of one of these, in the predicted volume ranges, including both types at the transition volumes 4π/81 and 1/π.
For n≥3, finitely many closed standard symplectic 2n-balls of positive capacities Rᵢ embed disjointly into the interior of a ball of capacity R exactly when the sum of Rᵢⁿ is below Rⁿ and every pair satisfies Rᵢ+Rⱼ<R.
Every surjective isometry between the unit spheres of nonzero real Banach spaces extends uniquely to a surjective real-linear isometry of the full spaces.
For every nonzero unital separable complex C∗-algebra A and every free ultrafilter ω on the natural numbers, its norm central-sequence algebra has no characters if and only if A absorbs the Jiang–Su algebra under minimal tensor product.
Every type II₁ factor with separable predual is generated as a von Neumann algebra by one bounded operator, equivalently two self-adjoint operators.
For every complex von Neumann algebra M and every integer k≥2, each bounded M-valued Hochschild k-cocycle has a bounded M-valued primitive, so Hᵏ_b(M,M)=0.
For every ε>0 there is a universal δ>0 such that any two unital von Neumann algebras on the same complex Hilbert space at Kadison–Kastler distance below δ are conjugate by a unitary u with norm distance from the identity below ε.
Every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is itself virtually polycyclic.
There exists an infinite group that is finitely presented, simple and amenable.
Every centered log-concave probability measure with a Lebesgue density in R^n and linear a-subgaussian marginals satisfies a logarithmic Sobolev inequality with constant at most C a^2, uniformly in dimension.
For initial data in H^(1/3)(R^2), the planar Schrödinger evolution converges almost everywhere to the data as positive time tends to zero, with the pointwise evolution defined by a Gaussian regularization limit first.
For the half-wave evolution in three spatial dimensions, the spacetime L3 norm on times [1,2] is bounded by the input L3 Sobolev norm with every positive derivative loss epsilon. The estimate yields the conjectured strict Euclidean local-smoothing range for all 2 < p < infinity.
The Fourier extension operator for the two-sphere maps bounded complex-valued data in L-infinity(S^2) to Lp(R^3) for every p > 3, with a constant depending only on p.
Every positive-dimensional smooth complex Fano manifold of dimension n, Picard number rho, and pseudoindex iota satisfies rho(iota - 1) <= n; equality holds exactly for a product of rho copies of projective space of dimension iota - 1.
Every connected minimal compact complex surface of class VII with positive second Betti number contains a global spherical shell.
Two reduced weighted-homogeneous complex hypersurface germs with isolated singularities are ambiently homeomorphic but have multiplicities two and three. They lie in one finite complex ambient dimension divisible by eight.
Every positive-dimensional compact connected Kähler manifold with no nonconstant holomorphic map from the complex line has ample canonical bundle and is projective.
There is a singular normal integral affine surface over the complex numbers whose tangent module is free of rank two on the whole surface.
For every smooth connected projective complex surface with geometric genus pg = 0, the Albanese map on integral degree-zero zero-cycles is an isomorphism. In the remaining general-type case pg = q = 0, this says the degree map CH0(S) to Z is an isomorphism.
For each fixed smooth closed connected Riemannian surface, the nodal set of every nonzero real Laplace eigenfunction of eigenvalue λ>0 has length at most C√λ, with C depending on the surface.
For every positive finite n and n=ω, small strict globular n-categories admit a proper combinatorial Thomason model structure Quillen equivalent to simplicial sets, with weak equivalences detected by the Street nerve.
In ZFC there exists a unital infinite-dimensional simple complex C∗-algebra with a faithful tracial state whose nonzero irreducible representations are all unitarily equivalent, yet it is not an algebra of compact operators.
There is a separable unital complex C∗-algebra with normalized 2-quasitraces but no tracial state; its quasitraces fail additivity on a fixed pair of positive contractions.
For two independent finite-mean-energy bosonic inputs with any finite positive number of modes, beam-splitter output satisfies N(ρC)≥ηN(ρA)+(1−η)N(ρB) for 0≤η≤1, where N(ρ)=g⁻¹(S(ρ)/n) is entropy photon number per mode.
The standard Salvetti complex of every Artin group with finitely many standard generators is aspherical, equivalently its universal cover is contractible.
There exists a torsion-free word-hyperbolic group that is not residually finite.
A discrete group is amenable if and only if every uniformly bounded representation of it on a complex Hilbert space is similar to a unitary representation.
A finitely generated group has decidable word problem if and only if it embeds in a finitely presented simple group.
There is a finitely generated residually finite group of integral cohomological dimension two and geometric dimension three.
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.
For every infinite connected locally finite nonamenable quasi-transitive graph under Bernoulli bond percolation, p_c<p_{2→2}≤p_u, yielding a nonempty parameter interval with infinitely many infinite clusters almost surely.
There is a universal C such that every finite loopless k-edge-connected multigraph on at least two vertices has a spanning tree meeting each nontrivial cut in at most C/k times the number of original cut edges.
Every order isomorphism between Bruhat intervals in arbitrary Coxeter systems preserves the corresponding equal-parameter Kazhdan–Lusztig polynomial.
For every n≥3, the ordinary crossing number of Kₙ equals ¼⌊n/2⌋⌊(n−1)/2⌋⌊(n−2)/2⌋⌊(n−3)/2⌋.
There is an absolute K such that W(r,k)>k^(k⌊log₂ r⌋/100000) for all k≥K and r≥2, so W(r,k)^(1/k) diverges for each fixed r≥2.
A finite tile in ℤ³ admits a translational tiling but no fully periodic tiling; its unit-cube thickening has the same property in ℝ³ even with arbitrary real translations.
The standard sine map of the two-dimensional torus has positive metric entropy with respect to normalized area for every sufficiently large positive parameter.
Every invertible mixing probability-preserving transformation is mixing of every finite order: for any k≥3 measurable sets, their translated intersection tends to the product of their measures as successive time gaps diverge.
A subset of real Hilbert space has doubling constant at most 76,800 yet admits no bi-Lipschitz embedding into any finite-dimensional Euclidean space at any finite distortion. Moreover, every infinite-dimensional real Banach space contains a compact doubling subset with the analogous obstruction and a universal doubling bound.
Every prescribed-order finite sequence of vectors in the Euclidean unit ball of R^d has one choice of signs whose every signed prefix has norm at most C sqrt(d), with C independent of sequence length. Consequently every indexed zero-sum family admits an ordering with every unsigned prefix bounded by C sqrt(d).
For each fixed real 1 < p < infinity with p != 2 and distortion D > 1, every n-point subset of a real Lp space has a nonlinear embedding into finite-dimensional l_p with dimension at most exp(C(p,D) (log n)^gamma), where gamma = 2-p for p < 2 and 1-2/p for p > 2.
For all origin-symmetric convex bodies K and L in every dimension and every interpolation parameter t in [0,1], the volume of their logarithmic Wulff combination is at least |K|^(1−t)|L|^t.
The density-one triangular lattice minimizes lower energy per particle among all locally finite planar configurations of centered-disk density one for every nonnegative completely monotone function of squared distance, including infinite-energy comparisons.
For every convex body K in Rⁿ with n≥4, its normalized projection-body volume is at least the ellipsoid value, with equality exactly for ellipsoids.
For every real f in W¹,¹(R²), the centered-disk maximal function has a globally integrable weak gradient and satisfies ||∇Mf||₁ ≤ C||∇f||₁ for an absolute constant C.
For a planar Lipschitz unit vector field with Lipschitz constant K>0, short Hilbert transforms with outer scale at most 1/(10⁶K) have a universal L² bound independent of the inner hard truncation, including fields depending on both coordinates.
For d≥4 and 2≤n≤d−2, an n-Ahlfors–David regular measure in Rᵈ with uniformly L²-bounded positive hard truncations of its n-dimensional Riesz transform is uniformly n-rectifiable.
For every complex-valued function on the circle with finite integral of |f| log(2+|f|), its ordinary symmetric Fourier partial sums converge almost everywhere to f along the full sequence.
If the tangent bundle of a compact connected Kähler manifold splits into two positive-rank integrable holomorphic subbundles, its ordinary universal cover is a product whose tangent factors lift the specified summands.
On P¹ × P¹, coordinatewise power pullbacks of an explicit rank-two bundle, tensored with O(1,1), are ample for every positive power yet admit no smooth strictly Griffiths-positive Hermitian metric for all sufficiently large powers.
An explicit degree-five polynomial in four complex variables is not a coordinate there but becomes a coordinate after one new variable; each fiber is affine three-space without an ambient rectification.
There is an explicit integral complex affine fourfold X such that X × A¹ is isomorphic to A⁵ but X is not isomorphic to A⁴.
For each fixed prime degree at least five, volume-weighted packets of totally real fields equidistribute to Haar measure as their multiplier-order discriminants grow, with no escape of mass, arbitrary local homothety type, and varying fields allowed.
The normal projective base of any projective Lagrangian fibration from a compact irreducible holomorphic symplectic Kähler manifold is projective space.
A singular n-dimensional complex algebraic klt germ with zero boundary has normalized volume at most 2(n−1)^n, with equality exactly at an analytic ordinary double point.
For every fixed C>0, a positive lower proportion of consecutive prime gaps pₙ₊₁−pₙ exceed C log pₙ.
If V(x) counts distinct values of Euler’s totient function up to x, then V(cx)/V(x) tends to c for every fixed real c>0.
Over all primary Eisenstein primes of norm at most X, the sum of normalized cubic Gauss sums has the sharp-cutoff asymptotic (6/5)c₍*₎X^(5/6)/log X, where c₍*₎=(2π)^(2/3)/(3Γ(2/3)).
Every irreducible integer quartic with no fixed prime-square divisor takes squarefree values with the predicted positive Euler-product density.
For an absolutely almost simple simply connected algebraic group over any global function field, including characteristic two, every noncentral abstract normal subgroup of its rational points is the inverse image of an open normal subgroup of the finite product of its anisotropic local groups.
For every irreducible subvariety X of an abelian variety over a number field, there are only finitely many maximal atypical subvarieties of X relative to its smallest containing torsion coset.
For every fixed 0<a,b<1, the natural density of integers n with P⁺(n)≤nᵃ and P⁺(n+1)≤nᵇ is ρ(1/a)ρ(1/b), where P⁺ is the largest prime factor and ρ is the Dickman function.
In the initially empty infinite square-board weak Maker–Breaker game, Maker can complete a translated, rotated or reflected Snaky hexomino within 21 legal Maker claims against every legal Breaker play.
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.
For every convex body K in R^n, n≥1, the manuscript reports volume product at its Santaló point at least (n+1)^(n+1)/(n!)², with equality exactly for simplices.
The manuscript reports arbitrarily large finite simple graphs with independence number at most two and chromatic number greater than their largest clique-minor order. It also claims violation of the ordinary fractional-coloring weakening.
OpenAI reports that rational Hodge classes on abelian varieties with good reduction pair rationally with complementary divisor products on the reduction, compatibly across prime-to-p and crystalline realizations, including residue characteristic two.
OpenAI reports the complete Birch–Swinnerton-Dyer leading-term formula for elliptic curves over the rationals whose full q-power Selmer group has corank zero or one at some prime q, including all local prime factors.
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 that, for each elliptic curve over the rationals, its quadratic twists of analytic rank zero and one each have density one half among signed squarefree parameters ordered by absolute value.
OpenAI reports cancellation with a power-of-logarithm saving for ordinary two-point Liouville averages along fixed nonproportional affine forms, and a binary corrected Elliott result under uniform nonpretentiousness.
OpenAI reports that the rational Grothendieck–Teichmüller Lie algebra is free on one generator in each odd weight starting at three, with the corresponding weight-completed isomorphism.
OpenAI reports that no set differing from the positive primes at finitely many integers is a sumset A+B of two nonnegative-integer sets that each contain at least two elements.
OpenAI reports that divergence of the totient-weighted approximation sum gives infinitely many successful shifted approximations for almost every real number, for each fixed real shift and arbitrary finite-valued tolerances.
OpenAI reports at least c_a x/(log x)² primes in each sufficiently large interval (x,2x) for which a fixed integer a is a primitive root, whenever a is neither minus one nor a square.
OpenAI reports modularity of every elliptic curve over every imaginary quadratic number field, with an automorphic representation matching the local parameters at all places.
OpenAI reports that K_X+mL is globally generated on every smooth complex projective n-dimensional variety for every ample line bundle L and every integer m at least n+1.
OpenAI reports that, at r at least ten very general complex plane points, the total multiplicity of every nonzero effective degree-d curve is strictly less than d times the square root of r.
OpenAI reports area-integrability of the s-th power of the derivative of every conformal bijection from a simply connected planar domain to the unit disk whenever 4/3<s<4, together with the stated inverse-square integral-means exponent.
OpenAI reports that every compact subset of Euclidean d-space, d at least two, with Hausdorff dimension strictly greater than d/2 has a distance set of positive Lebesgue measure.
OpenAI reports the three-dimensional Kakeya maximal L³ estimate with tube-radius loss at most C_ε δ^(-ε) for every positive ε, uniformly over unit tubes of radius δ.
OpenAI reports that every origin-symmetric convex body in n-dimensional real space has volume product at least 4^n/n!, with equality precisely for invertible linear images of Hanner polytopes.
OpenAI reports that the sum of squared Gaussian first-moment lengths of the cells of every finite measurable Euclidean partition is at most 9/(8π), with the three-sector planar construction attaining the bound in the stated dimensions.
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 positive-integer set with divergent reciprocal sum contains arithmetic progressions of every finite length, through fixed-length bounds r_k(N)≤C_k N exp(−c_k(log N)^ε_k).
OpenAI reports a bipartite graph with 35 vertices and 66 edges whose homomorphism density in a finite simple host is below the predicted edge-density bound; it also reports failure of the associated forcing assertion.
OpenAI reports that every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance exactly two as at directed distance one.
OpenAI reports a finitely presented torsion-free group with a finite two-dimensional classifying space whose group algebra over the two-element field contains nonzero elements with product zero.
OpenAI reports a finitely presented torsion-free group with a finite two-dimensional classifying complex and elements a,b in its characteristic-two group algebra satisfying ab=1 but ba≠1.
OpenAI reports one fixed finite-dimensional complex algebra with finite-dimensional modules of unbounded finite projective dimension, refuting universal finiteness of little finitistic dimension.
OpenAI reports a nondegenerate limiting law for centered and variance-normalized total free energy in the zero-field Gaussian Ising Sherrington–Kirkpatrick model at each fixed β>1, with variance asymptotic to a positive constant times n^(1/3).
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.
OpenAI reports an infinite discrete group with an ordinary finite presentation that is residually finite and has finite two-power order for every element, with unbounded element orders.
OpenAI reports that Thompson’s group F, viewed as a discrete group, is nonamenable.
OpenAI reports purely absolutely continuous spectrum of nonzero weight on an open energy interval for the lattice Anderson model with independent uniform site potentials at each fixed dimension d≥3 and sufficiently small positive disorder.
OpenAI reports that the maximum number of mutually unbiased orthonormal bases in complex dimension six is three, using an exhaustive computer-assisted exclusion of four bases under specified arithmetic and compiler conditions.
OpenAI reports a uniform positive spectral gap above the ground energy for the pure antiferromagnetic spin-one Heisenberg chain on even periodic rings.
OpenAI reports that every bounded complex-linear unital homomorphism from a unital complex C*-algebra to operators on a Hilbert space is conjugate by a bounded invertible operator to a *-homomorphism.
OpenAI reports vanishing of universal-cover L²-Betti numbers outside degree two for closed connected aspherical topological four-manifolds, and more generally finite aspherical integral Poincaré complexes of formal dimension four.
OpenAI reports the generalized Cartan–Hadamard isoperimetric inequality in every dimension for complete simply connected smooth manifolds with sectional curvature at most κ≤0, comparing finite-volume finite-perimeter sets with equal-volume model balls.
OpenAI reports that every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to complex Euclidean space.
OpenAI reports that every nonzero first-positive Neumann eigenfunction on a smooth bounded simply connected planar domain has nonvanishing interior gradient, placing all global extrema on the boundary even for a multiple eigenvalue.
Let K be a field of characteristic two with |K|=2ⁿ, gcd(k,n)=1 and dₖ=4ᵏ−2ᵏ+1. If Δ is the image of b↦(b+1)^dₖ+b^dₖ+1, then for distinct nonzero v₁,v₂∈K exactly 2^(2n−3) triples (x,y,z)∈Δ³ satisfy v₁x+v₂y+(v₁+v₂)z=0.
OpenAI claims that rational Hodge classes on any complex CM abelian variety admit algebraic-cycle representatives with rational coefficients, in all codimensions.
For the manuscript’s smooth one-ended three-dimensional Einstein–Maxwell exterior data satisfying charged dominant energy and weak future trapping assumptions, invariant ADM mass obeys m≥Q and √(A/4π)≤m+√(m²−Q²), where A is minimum enclosing area in the original metric.
For every countable discrete group, the manuscript claims amenability is equivalent to strong Ulam stability: sufficiently accurate unitary almost representations on any complex Hilbert space are uniformly close in operator norm to genuine representations on the same space. It gives a fixed-distance obstruction for nonamenable groups.
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.
For the relative SU(2) BFSS Hamiltonian defined by closing its supercharge quadratic form, the manuscript claims an orthonormal sequence of square-integrable gauge-invariant eigenvectors with positive eigenvalues tending to infinity.
For exact pure and mixed ground states of a three-dimensional hard-sphere Bose gas, the nonzero-momentum occupation measure approaches the scaled Bogoliubov profile: first take the thermodynamic limit at fixed density, then the dilute limit. Its mass gives depletion 8/(3√π)√(ρa³)+o(√(ρa³)), uniformly over thermodynamic accumulation values.
For globally generic cuspidal complex automorphic representations of split connected adjoint absolutely simple groups of types G₂, F₄, E₆, E₇ or E₈ over global function fields, every local component is tempered, including arbitrarily ramified places and without a characteristic restriction.
For every epsilon>0, the OpenAI-authored manuscript constructs unital norm-separable C*-algebras on a common separable Hilbert space, with common identity and Kadison-Kastler distance below epsilon, which share their von Neumann closure but are not conjugate by any unitary. It does not assert abstract nonisomorphism or a von Neumann counterexample.
The OpenAI-authored manuscript constructs unital von Neumann algebra pairs on separable Hilbert spaces with one-sided near-inclusion gaps tending to zero, yet every unitary embedding the source into the target stays a fixed distance from the identity. Spatial embeddings do exist; this does not refute two-sided Kadison-Kastler stability.
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.
Every family of 14-subsets of a 24-point set that covers every 4-subset has at least 20 blocks, so C(24,14,4)≥20. The Schönheim point-derivation bound then gives C(25,15,5)≥34.
For every repetition threshold α with (8/3)^+≤α<3, the lengths of α-free uniform binary morphisms are exactly the positive integers other than 3 and 6; each admitted length has a complementary witness. At the strict threshold 8/3, length 5 is impossible.
For Thue–Morse digit-sum parity t, maximum monochromatic progression length A(d), and earliest maximizing start i(d): i(2^n+1)=3·2^(2n)−2^n−1 for n≥2; i(2^(2n)−1)=3·2^(4n)−2^(2n)+1 for n≥1; and i(2^(2n+1)−1)=2^(2n+1)−1 for n≥0.
For every linear code of length n and encoded coordinate i, the expected uniform-with-replacement recovery reads satisfy a_i(C)+a_i(C⊥)=n under the time-zero convention; therefore C is recovery balanced exactly when its dual is.
The manuscript reports a randomized classical Subset Sum algorithm in O(2^(0.49n)) time for polynomial-bit inputs.
Bizeul, Klartag and Lehec report a universal upper bound for the Poincaré constant of every isotropic log-concave probability measure in every dimension, resolving the KLS conjecture in its functional formulation.
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.
For five, six and seven freely rotated, interior-disjoint unit squares in a closed disk, the project reports exact least radii √(5/2), √q* and √13/2, respectively, where q* is its specified quartic root. It classifies all optimal packings up to rotation and relabelling: the five- and six-square optima are unique, while seven squares admit a three-parameter family.
The model-origin optimality arguments for five through seven freely rotated unit squares in a disk, separating known small cases and later human commentary.
Hegarty, Ognissanti and Wedin give convergent heterogeneous Hegselmann–Krause trajectories in the synchronized bounded-confidence and bounded-influence models whose proximity digraphs never stabilize, plus constant-digraph trajectories that fail eventual pseudostability. These counterexamples refute Mirtabatabaei and Bullo Conjectures 2.2 and 2.3 and Theorem 6.4(iv).
The manuscript reports that the complex matrix-multiplication exponent is at most 9/4.
Oren Ben-Bassat reports that every polystable representation of a finite quiver over any complete non-Archimedean field admits a split HKKP harmonic norm, without spherical-completeness or value-group restrictions.
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.
For n≥3, Proposition 1.2 bounds the spatial L¹ norm of the gradient of the Dirichlet Green’s function for the linearized Monge–Ampère operator by C h^β, uniformly in its pole on a section of height h. The convex C³ potential has Hessian determinant between positive constants; the nested sections satisfy the paper’s compact-containment hypotheses.
For every ordinal ω<α<ω₁, Almeida’s compact-range Vietoris power K(α,ord), with its range-plus-finite-coordinate topology, is second-countable and Lindelöf but neither σ-compact nor Menger. The Lean theorem ordinal_full covers this interval by a delayed diagonal.
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 four positive masses in the planar Newtonian problem, each cyclic order has exactly one strictly convex central configuration up to similarity. The v2 theorem also gives Q≥K/4 at every convex configuration, nondegenerate local minima and analytic dependence.
For the derived p-completed sphere X, the canonical map L_{n−1}X→L_{n−1}L_{K(n)}X is claimed to have no homotopy retraction at n=p for primes p≥5 and at n=p+1 for primes p≥7. The proposed kernel is the image of β₁^((p−1)²); at p=5 it lies in π₆₀₈L₄X.
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.
The September manuscript proves that every admissible integrable zero-temperature Parisi minimizer for the pure zero-field Sherrington–Kirkpatrick model, with covariance ξ(t)=t²/2, has full relative Stieltjes support on [0,1). It uses a rational coercive identity and strict curvature on constant intervals.
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.
OpenAI reports the sharp invariant-mass lower bound in terms of minimum enclosing area for one-ended spacetime initial-data exteriors in every dimension n≥3, under the paper’s energy, trapping and decay assumptions.
The manuscript reports matching bounds exp(Θ(√(log d · log log d))) for the least ℓ₁ distortion of unit-cost edit distance on strings of length at most d.
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.
For each fixed 0<κ<8, let d=1+κ/8. The September 26 manuscript claims that the explicit small-radius gauge r^d(log log(1/r))^((2−d)/2) gives positive finite Hausdorff measure to every compact positive-time segment of ordinary chordal SLEκ almost surely, with finite expected measure of the full trace inside bounded disks.
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.
OpenAI reports the sharp constant-two numerical-range inequality for matrix-valued polynomials, uniformly over all finite base and coefficient dimensions.
The manuscript reports a uniform finite bound on component size for every fixed step limit in the Gaussian-prime graph.
For each r≥4, the r-extension of the four-cycle has product exponent X(F)=r but admits Ω(n^(r−1/2)) F-free configurations, exceeding the conjectured exponent r−1. The AI-origin contribution is identifying this counterexample family.
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 the three-state symmetric channel Kλ(i,i)=(1+2λ)/3 and Kλ(i,j)=(1−λ)/3 for i≠j, with −1/2≤λ≤1, reconstruction from depth-ℓ spins and the observed tree on every regular b-ary tree (integer b≥2) and observed Poisson(d) tree (d>1) occurs exactly when dλ²>1 (d=b in the regular case). At admissible equality there is nonreconstruction. Poisson advantage averages over the tree without conditioning on survival.
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.
For each dimension d and t>0, the manuscript claims a bound N(d,t) on [k_S:k] for a boundary prime S with coefficient at least t on an ordinary projective log-canonical Q-pair (X,B) over a characteristic-zero field, where X is normal integral, H⁰(X,O_X)=k, B is effective and K_X+B is Q-linearly trivial.
The OpenAI manuscript claims a domain in some sufficiently large fixed finite complex dimension, diffeomorphic to real Euclidean space, with a complete Kähler metric of real sectional curvature at most −1 and only constant bounded holomorphic functions. The curvature is not bounded below.
For every minimal homeomorphism h of an infinite compact metrizable space X, the manuscript claims rc(C(X)⋊ₕZ)=mdim(X,h)/2, including infinite values. The new part is the lower bound for every finite open cover.
Every word-hyperbolic one-relator group is virtually compact special: a finite-index subgroup is the fundamental group of a finite nonpositively curved cube complex locally isometric to a Salvetti complex. With Kielak–Linton this yields virtual free-by-cyclic structure, allowing an infinite-rank free kernel.
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.
For the Thue–Morse target in Alexander’s binary avoiding population, the project gives an attained exact maximum matching-path length L(v) for every natural starting vertex v, including zero, through a seven-case formula based on the factorization of ⌊v/2⌋+1 and adjacent digit-parity bits. It also proves a finite digit recurrence.
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.
The manuscript reports that every a/b with 1 ≤ a < b and sufficiently large b has an expansion with O(log log b) distinct positive unit fractions.
The manuscript reports h(k) ≪ k²/(log log(3k))², answering the quadratic-bound question in its stated formulation.
For each smooth projective complex threefold X, ample integral H and rational B₀, the manuscript gives k≥0 and R*>0 depending only on (X,H,B₀) such that every finite-slope, zero-tilt-slope semistable E satisfies ch₃^(B₀+bH)(E)≤(a²/6+k)H²ch₁^(B₀+bH)(E) for all a>0 and real b. The k correction vanishes uniformly in b for a>R*.
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.
Three OpenAI manuscripts jointly claim Zilber–Pink for Hodge-generic curves in the Siegel threefold A₂ over algebraic number fields without a boundary hypothesis. The combined result uses separate finiteness theorems for non-CM elliptic squares, E×CM cases and quaternionic-division points.
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.
OpenAI claims that every smooth complete intersection in complex projective space satisfies ordinary unreduced descendant Virasoro constraints coefficientwise in every genus and curve class, with arbitrary cohomology insertions including primitive and odd classes.
For normalized univalent disk maps, OpenAI proves M_{−1}[f′](r)≤C(1−r)^(−1/4+ε) for some ε>0 and all 1/2≤r<1. Thus the bounded-image universal spectrum B_b(−1)<1/4.
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.
An explicit polynomial F in four complex variables has zero-fibre coordinate ring isomorphic to C[3], while its gradient vanishes at a point, so F is not a coordinate. Adding variables extends this counterexample to every ambient dimension at least four.
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 order-six complex Hadamard matrix outside Tao’s cubic equivalence class, the normalized character sum vanishes at every permutation of (1,1,1,−1,−1,−1), allowing row and column permutations and arbitrary unit phases.
For x′=y−F(x), y′=−x and every real polynomial F of degree at most five, there are at most two geometrically distinct isolated periodic orbits, including degenerate cycles. Some such system has exactly two.
For projective generalized log-canonical Q-pairs over an algebraically closed characteristic-zero field with a fixed nef rational b-divisor, the manuscript claims a birational model that extracts no divisors and preserves discrepancy inequalities, with nef adjoint when pseudo-effective, or a relative-Picard-number-one Mori fibre space otherwise.
For fixed local dimension q, interaction bound J and global spectral gap Δ>0, the manuscript claims a PEPS with bond dimension at most CLᶜ and normalized vector error at most L⁻¹ for the unique ground state of any on-site plus nearest-neighbour Hamiltonian on an open L×L square grid.
For every discrete group G, OpenAI claims that the Hattori–Stallings trace of every class in K₀(C G) vanishes on each infinite-order conjugacy class. The claim concerns the complex group ring; an integral identity-only consequence needs a separate known finite-order input.
For compact connected smooth manifolds of dimension at least three, the manuscript claims that equal density-valued energy Dirichlet-to-Neumann forms measured on any nonempty proper relatively open boundary patch determine two smooth metrics up to a smooth diffeomorphism fixing that patch.
For primes p≤x with equal weight, the normalized logarithms of the ordered prime factors of p−1, counted with multiplicity, converge in every fixed finite joint distribution to Poisson–Dirichlet(1). The manuscript also derives a fixed-threshold Dickman law for the largest factor.
For split connected semisimple groups over global function fields and every effective divisor, including arbitrary multiplicity, rational and ℓ-adic cuspidal-function spaces split into direct sums indexed by dual-group nilpotent orbits. The same orbit controls all unramified Hecke places.
An OpenAI manuscript claims a finitely presented integral Poincaré-duality group of dimension four with a finite oriented aspherical Poincaré complex, but no closed aspherical topological four-manifold model.
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.
Assuming the four stated characteristic conditions on the split connected semisimple group G over a finite field, every canonical unramified Arthur-filtration subspace on the full space of finitely supported automorphic functions is defined over Q and is independent of ℓ, including noncuspidal functions.
The OpenAI manuscript claims that, for dimensions 3 through 9, every smooth locally uniformly convex affine-maximal graph complete in its induced Euclidean metric is an elliptic paraboloid. The manuscript also derives the affine-metric-complete immersed-hypersurface corollary. The companion smooth nonquadratic entire example in dimension 10 is a separate contribution and does not supply affine-metric completeness.
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.
A connected normal complex projective variety has a universal cover biholomorphic to a semialgebraic open subset of a projective variety exactly when that cover is D×C^m×F, with D a bounded symmetric domain and F simply connected, normal and projective.
For the spherical perceptron at margin −1 with quadratic penalty, after successive limits in system size, inverse temperature and density approaching the feasibility threshold from above, the gap and force laws have exponents γ=1/(2+θ), with 0.4126930<γ<0.4126934 and 0.4231063<θ<0.4231088. The same limits are claimed for a sufficiently small fixed Gaussian scale mixture.
Every subset of R⁴ containing a unit segment in every direction has Hausdorff dimension four, without a measurability, compactness or stickiness condition.
For every compact smooth positively curved surface Σ in R³, including smooth boundary, and every 3<p<∞, the Fourier extension operator maps Lᵖ(Σ) to Lᵖ(R³).
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).
For a connected reductive group on a smooth projective connected curve over an algebraically closed field of characteristic p>0, the restricted geometric Langlands functor reaches every spectral component under either finite-field descent or, with p very good and p not dividing the Weyl-group order, an arbitrary algebraically closed base. Both regimes require a nondegenerate invariant Lie-algebra form (also on Levi centers), Chevalley restriction for every Levi, semisimple centralizers that are Levi, and nilpotents lying in parabolic unipotent radicals over every field extension.
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 disappearance of infinite bond-percolation clusters at criticality on every infinite connected locally finite quasi-transitive graph whose critical threshold is below one.
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.
OpenAI claims a finite degree-dependent upper bound for the total number of isolated periodic trajectories of any planar polynomial vector field.
The manuscript reports a proof that Catalan’s constant is irrational.
The manuscript reports that the irrationality exponent of π equals 2.
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.
The manuscript reports R(5,t) = t⁴/(log t)^(3+o(1)) as t tends to infinity.
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 the manuscript’s unit-background planar Coulomb renormalized-energy class and admissible square-cutoff families, every admissible field has energy at least that of the density-one triangular lattice. A prior equivalence then identifies the linear term of optimal spherical logarithmic energy.
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.
For every sufficiently large prime s≡2 mod 3 and sufficiently large odd n depending on s, with r=s^n+1, the manuscript constructs an r-edge-colored complete graph requiring exactly r monochromatic trees to cover its vertices, even when trees may overlap and singletons count.
Every holomorphic line bundle with nonzero nef isotropic first Chern class on a compact irreducible holomorphic symplectic Kähler manifold is semiample, according to the OpenAI manuscript. The resulting map has a normal projective base of half the complex dimension and Lagrangian fibers.
For every finite nonempty ground set and nonempty proper increasing family F, OpenAI proves q_f(F)≤25·512^4 q(F), comparing fractional and integral expectation thresholds with the same covering budget of one half and unrestricted fractional support.
For every finite ground set, p∈(0,1) and family D of p-product measure at least 1−2^(−75), OpenAI claims the sets not contained in unions of 2^75 members of D form a p-small family.
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 sufficiently large dimension n, an OpenAI manuscript constructs a centrally symmetric convex body whose unrestricted translative covering density exceeds c n log n for an absolute c>0. With Rogers’ O(n log n) upper bound, this fixes the worst-case unrestricted order and rules out a universal O(n) bound.
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 claims arbitrarily large finite graphs G with independence number at most two and μ(G)+1<|V(G)|/2≤χ(G), contradicting the proposed inequality χ(G)≤μ(G)+1.
The manuscript claims that every connected smooth closed Einstein four-manifold with strictly positive sectional curvature is, up to positive scaling, round S⁴, Fubini–Study CP², or round RP⁴, without assuming orientability.
For simple separable unital infinite-dimensional nuclear C*-algebras, the companion manuscripts claim strict comparison implies Jiang–Su absorption, completing the unital Toms–Winter equivalence with finite nuclear dimension. A separate extended-functional theorem also treats nonunital algebras under its own assumptions.
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.
For separable nuclear complex C*-algebras with no nonzero elementary ideal subquotients, the manuscript claims finite nuclear dimension, Jiang–Su stability and nuclear dimension at most one are equivalent. It separately claims dim_nuc(A₀⊗Z)≤1 for every separable nuclear A₀.
For every fixed degree d≥3 and energy strictly inside the Kesten–McKay bulk, the unfolded adjacency-eigenvalue point process of a uniform simple labelled d-regular graph converges at that fixed energy to the GOE bulk process along all admissible graph sizes, without extra conditioning.
Every closed connected oriented smooth manifold of dimension at least two with positive scalar curvature is rationally inessential, without spin or fundamental-group restrictions. The aspherical obstruction also covers nonorientable manifolds through the orientation double cover.
The OpenAI manuscript claims that the nearest-neighbor Anderson operator on ℓ²(Z²), with independent site potentials uniform on [−h,h], has almost surely pure-point spectral type and a complete eigenbasis for every fixed h>0. It also identifies the spectrum as [−4−h,4+h].
For the stated deterministic grand-canonical three-dimensional hard-sphere gas with smooth spatially summable Gaussian-bounded initial density and gradient, exactly centered empirical fluctuations converge at every fixed finite set of times and tests to a Gaussian linear fluctuating Boltzmann field on each prescribed finite interval where the classical Boltzmann solution has uniform Gaussian velocity decay.
OpenAI claims a group of type F∞ containing every finitely presented group. Its finitely generated subgroups, up to isomorphism, are exactly the finitely generated recursively presented groups.
The September OpenAI manuscript claims an infinite-dimensional finitely presented nonunital nil associative algebra over F₂ that is Jacobson radical but not nilpotent. Its unitization is finitely presented, algebraic and infinite-dimensional.
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.
A finitely generated torsion-free discrete group has a scalar projection e in its reduced group C*-algebra whose canonical trace lies strictly between zero and one-half.
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 an absolutely almost simple simply connected group G over a number field k, every noncentral abstract normal subgroup of G(k) is the inverse image of an open normal subgroup of the product of G(kᵥ) over anisotropic nonarchimedean places.
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.
OpenAI claims globally smooth unique evolution for arbitrary admissible large initial data in the three-dimensional one-species relativistic Vlasov–Maxwell model, under its support and field assumptions.
OpenAI claims that every compact boundaryless smooth Riemannian surface has a smooth length-preserving immersion into R⁴, including nonorientable surfaces.
OpenAI claims Lie-group structure for any locally compact second-countable Hausdorff group with a faithful jointly continuous action on a connected finite-dimensional manifold under the stated topological conventions.
OpenAI claims a normal trace-preserving *-isomorphism between the free group factors L(F₂) and L(F₃), giving the affirmative branch of the classical isomorphism alternative.
OpenAI claims that a hyperbolic group with a two-sphere boundary has a proper cocompact action by isometries on hyperbolic three-space, allowing a finite kernel.
OpenAI claims strict positivity of the Tor Euler characteristic for nonzero finite modules of complementary dimension over a regular local ring, when their tensor product has finite length.
OpenAI claims a conformal circle-domain model for an arbitrary planar domain, allowing uncountably many complementary components.
The manuscript reports an L₁ embedding with a universal distortion bound for every finite connected planar graph with positive real edge lengths.
The manuscript reports density at most C n log n for a single-lattice covering by translates of every convex body in dimension n ≥ 2.
The manuscript reports deterministic multiplication in O(n (log n)^(1−κ)) time with κ = 2^−182 on a fixed finite-tape Turing machine.
The manuscript reports a nine-dimensional counterexample using rank-one orthogonal projectors on R⁴ with the Frobenius metric.
The manuscript reports that five colors cannot suffice, implying that the plane’s chromatic number is six or seven if the result holds.
For each c>4, Colbrook’s polynomial ODE on ℝ⁵ has uniformly negative divergence and a compact global attractor. Its finite-time local Lyapunov and asymptotic Kaplan–Yorke dimension maxima are 4+4/c on an irrational-flow invariant two-torus, while equilibrium and periodic-point maxima are 3.
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.
For a reduced planar absolute Mumford–Shah minimizer, each interior discontinuity point has a local C¹ arc, endpoint or 120-degree triple junction, with endpoints and junctions locally finite.
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 every increasing antipodal function h on the Boolean cube and every nonnegative allocation of each nonempty Fourier coefficient’s squared mass among its coordinates, the induced linear function flows upward to h₊². This is Kahn’s flow conjecture.
For every transcendental radius r>2, a seven-point set on a circle of radius r, including r=π, has a finite coloring of Euclidean space with no monochromatic congruent copy in any dimension.
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 sufficiently large n=2ᵏ, Wu’s Walsh–Hadamard norm and identity map Uₙ:Xₙ→ℓ∞ⁿ make C₂ʳ(Uₙ)/max(C₂ᵍ(Uₙ),‖Uₙ‖(2,1)) grow at least as c√log(n+1). Thus the proposed universal operator-cotype bound fails already at q=2.
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 even integer N>2 is a sum of positive integers a and b for which the Liouville function satisfies λ(a)=λ(b)=−1.
Tsiokos reports a simply connected rational polyhedral tile that tiles R³ by congruent copies, reflections allowed, while every tiling has no infinite-order symmetry. Each tiling has a unique nested supertile hierarchy and a finite symmetry group of order at most 24.
For an absolute c>0, every family of subsets of [n] whose pairwise intersections contain a nontrivial three-term arithmetic progression has size at most (1/2−c)2^n. Keevash also proves this for 3-uniform hypergraphs with bounded pair-codegree.
Flake, Gruber and Heidersdorf report that the universal rigid monoidal category on one object has an abelian envelope without the quotient property.
Sauers constructs a separable unital C*-algebra with compact nonempty trace space whose uniform tracial completion has a trace discontinuous in the uniform 2-norm, and a factorial tracially complete pair with the same failure.
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.
The decimal Copeland–Erdős constant formed by concatenating the primes is not simply strongly normal in base 10, and therefore is not strongly normal in base 10.
For every real p>4, Xiong constructs an equilateral set of 8m points in ℓₚ^(8m−2) for some power-of-two m, exceeding the proposed dimension-plus-one maximum.
Every four-partite four-uniform hypergraph with matching number two has vertex-cover number at most six.
For every nonzero real ξ, the fractional parts of ξ(3/2)ⁿ cannot all lie in the closed interval [2/7,5/7]. Equivalently, ||ξ(3/2)ⁿ||<2/7 infinitely often. The certificate has depth 15 and 166 blocks across 32 rank strata.
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.
For integers b,c≥2, the Stoneham number α(b,c)=Σ_{k≥1}1/(c^k b^{c^k}) is normal in base b exactly when some prime divisor of c does not divide b. In particular α(6,4) is not 6-normal despite neither parameter dividing the other.
Almeida and Knudstorp report that Medvedev’s logic of finite problems is undecidable, by reducing periodic plane tiling to non-theoremhood.
For integers b,c,d≥2 with gcd(b,c)=gcd(b,d)=1, the sum of the two Stoneham constants α(b,c)+α(b,d) is normal in base b, including when c=d or the two parameters share prime factors.
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.
The constant ξ=Σ_{a,c≥0}1/(2^a3^c·2^{2^a3^c}) is normal in base two: every finite binary word has its expected limiting frequency when overlapping occurrences are counted.
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.
Lorist and van Neerven construct two explicit 2^n-dimensional Banach-space families at p=2. One has Hilbert-transform constant of order n and UMD constant of order √n; the other reverses them, making both classical quadratic comparison exponents sharp over real and complex spaces.
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 positive viscosity, the manuscript reports a three-dimensional incompressible Navier–Stokes flow starting from rest with smooth compactly supported forcing, bounded kinetic energy, and unbounded velocity in finite time. It claims Clay alternatives C on R³ and D on the periodic torus.
The manuscript reports smooth compactly supported divergence-free initial data on R³ whose unforced three-dimensional incompressible Euler solution has finite maximal lifespan, with unbounded velocity gradient and divergent time integral of maximum vorticity.
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.
For n≥r≥2, t≥1 and any n-vertex r-uniform hypergraph H containing no tight r-tree T with t edges, |E(H)|≤(t−1)|∂H|/r≤(t−1)binom(n,r−1)/r, where ∂H is the (r−1)-shadow. The bound is attained for infinitely many n.
GPT-6 Astra constructed a strictly stationary, centered φ-mixing sequence with finite second moments and divergent partial-sum variance. Along a subsequence, its variance-normalized sums converge to zero in probability rather than to a standard normal law.
For every positive dimension n, the Euclidean kissing number τₙ is an attained maximum and satisfies τₙ≤(√6)ⁿ. A planar hexagon gives τ₂=6, so √6 is the least base in a bound τₙ≤αⁿ valid in every dimension without a prefactor.
The proper hat-guessing number of K7−e is 12. Either two disjoint S(5,6,12) completion designs or a sharply five-transitive orbit-map construction meets the twin-completion criterion; the general upper bound gives equality.
The proper hat-guessing number of K6−e is 10. Two order-sensitive twin rules from a sharply four-transitive action on 11 points satisfy the coordinate-line criterion; Hall completion gives a winning ten-color strategy, matching the general upper bound.
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.
GPT-6 Astra constructed a nil algebra over the algebraic closure of F₂ with a nonnilpotent 2×2 matrix. In its unitization, the algebra is a nil two-sided ideal, refuting Krempa’s matrix form of Köthe’s conjecture.
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.
For every rational r with 1 ≤ r < 2, there is a finite bipartite graph H whose extremal number ex(n,H) grows as Θ(n^r). Here ex(n,H) is the largest number of edges in an n-vertex graph containing no copy of H; Θ means matching upper and lower bounds up to constant factors.
The proper hat-guessing number of K5−e is 8. An F₂³ construction and Hall-completion argument give a winning eight-color strategy, matching the prior general upper bound.
For every real Hilbert space H, including nonseparable spaces, the one-dimensional Besicovitch rectifiability threshold σ₁(H) is at most 0.6934. In particular, a planar Borel set of finite H¹ measure is countably one-rectifiable when its lower one-density exceeds any fixed threshold greater than 0.6934 almost everywhere.
For every convex P in the Boolean lattice Q_n and all k≥0, w(P×Q_k)≥w(Q_{n+k})|P|/2^n. The k=0 case proves the Daykin–Frankl relative-width conjecture.
Qiu and Yan prove interior curvature bounds in every dimension n≥3 for smooth admissible graphs on B₃⁄₂ solving σ₂(κ[u])=1 with bounded height and gradient. The model-suggested compression inequality in Lemma 3.4 controls a two-surface Jacobi inequality; the model also supplied the auxiliary η in equation 62.
Curry constructs a nonnegative smooth, arbitrarily C²-small perturbation of a strongly pseudoconvex CR 3-structure with a nowhere-zero closed canonical-bundle section, but every local C¹ CR function has zero differential at the origin. It is therefore not locally embeddable.
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.
For the three-dimensional space-homogeneous Boltzmann equation with collision kernel B=|v−v*|^γ/(4π), 0≤γ≤1, Silvestre constructs smooth positive radial two-Maxwellian initial densities f_R=(1−p)M₁+pM_R. For 0<p<p_γ and sufficiently large R, the entropy production D initially increases: dD(f_R)/dt>0 at t=0, with a positive leading term proportional to R^γ log R. The range includes constant-angle Maxwell molecules and hard spheres.
For a quadratic real root α>1 of X²−aX−b whose conjugate has modulus below one, the Route A exponent condition A(α)<1 gives an open interval missed by every orbit (ξαⁿ mod 1) with ξ in the paper’s class C(α), so none is uniformly distributed. The criterion is equivalent to (log₂α−1)(log₂(α/|b|)−1)>1 and yields an explicit gap for α=2+√5.
For even N=2m≥2, pairwise distinct real numbers x₁,…,x₂ₘ and odd D=2r+1≥N−1, Ma proves that the matrix [(xⱼ−xᵢ)^D] is nonsingular. After sorting the nodes, the proof gives the stronger sign (−1)^(m choose 2) Pf[((xⱼ−xᵢ)^D)]>0.
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.
GPT 6 Astra’s manuscript claims lim inf of consecutive prime gaps is at most 186, by proving DHL[40,2] and applying it to an explicit admissible 40-tuple of diameter 186.
For n ≥ 2 and 1 ≤ v ≤ π(n), the count C_{n,v} of minimal generators of the truncation ideal I_n with least support index v equals Legendre's sifting function Φ(n,p_v).
Yoshito Ishiki constructs real Banach spaces X and Y and a bijection U from X to Y that preserves distances between every pair of points in each closed ball of radius 1/4, but is not a global isometry. This answers the unrestricted Scottish Book Problem 155 negatively.
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.
From a property-(T) group Γ with an injective non-surjective endomorphism, Eckhardt forms its ascending HNN extension G and generalized wreath product W=(⊕_(G/Γ) Z/2Z)⋊G. His Theorem 4.3 gives a specified nonidentity element killed by every homomorphism of W to a matrix-ultraproduct quotient, showing W is not MF.
For the two-qubit state ρ=(|Φ+⟩⟨Φ+|+|01⟩⟨01|)/2, where |Φ+⟩=(|00⟩+|11⟩)/√2, Holevo and Shirokov show that entanglement of formation has no global supporting affine functional. Along a perturbation toward |10⟩, it decreases on a square-root scale.
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 every sufficiently large genus g, Cai constructs a closed hyperbolic surface with systole at least log g−12 log log g. The lower limit of the largest possible systole divided by log g is therefore at least one.
An eleven-block construction yields α∞≤liminf log D(N)/log N for the largest square-difference-free subset D(N) of {1,…,N}. Every exponent ρ<α∞ holds eventually, and the formal proof certifies α∞>0.7537.
The tilted residue-class sieve gives Y(X)≫X log X/log₃ X for covering and maximal prime gaps G(T)≫log T log₂ T/log₄ T for all sufficiently large T.
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.
Tang et al.’s Theorem 1.2 constructs a real meromorphic F on C with all 0-, 1- and ∞-points on the real axis, each infinite, and for every other spherical value a, the a-point divisor in each half-plane fails the Blaschke condition. Consequently F is not of bounded type in either half-plane despite omitting three prescribed values there; Möbius postcomposition and affine precomposition extend the example to any triple and Euclidean half-plane.
The AI-associated dyadic randomization step underlies Theorem A: for 1≤r,s<∞ with 1/r+1/s>1, a bilinear form on compactly supported bounded scalar functions with an (r,s)-sparse bound has a convex-body sparse bound for its coordinate-wise C^n extension, with stated n^(1+1/r+1/s) and η^−1 dependence. Theorem 2.6 proves a dyadic version; the non-dyadic extension and final theorem were completed by the human authors.
Over any characteristic-zero field, Wang and Zhang prove Fröberg’s predicted generic Hilbert series for equal-degree forms in four variables when the degree is 5 or 7 and the generator count is any r≥1. Exact rank certificates cover the previously open middle ranges r=6–11 and r=6–21, respectively; propagation covers the remaining counts.
For n>2, Schlagenhauf constructs a bounded W¹,ⁿ map from the n-ball to Rⁿ⁺² that is discontinuous at the origin yet solves the critical n-Laplace system with an antisymmetric Lⁿ potential in distributions. The potential can lie in every L⁽ⁿ,ᑫ⁾ for q>2.
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 positive ξ, the fractional parts of ξ(3/2)ⁿ cannot all lie in [1/6,13/24), a specific single interval of length 3/8. A finite block certificate and a general soundness argument establish this exclusion.
For each integer a ≥ 1, the affine Erdős Problem 270 constants C(a,0), …, C(a,a−1), C(a,a+1) are algebraically independent; every other C(a,b) with b ≥ 1−a is a nonconstant rational affine combination of them and is transcendental.
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.
Station v2 Section 6.13 S1: a conference graph with parameters (q,(q-1)/2,(q-5)/4,(q-1)/4) gives R(B_q,B_(q+1))=4q+3. S2: every prime power Q>3 with Q=3 mod 8 gives R(B_((Q-1)/2),B_((Q+1)/2))=2Q+1. Exclude the third Yamada–Pott family requiring human expert synthesis.
The three exact model-generated 604-point kissing configurations in dimension 11 in Station v2 Section 6.3. Two are reported as new isometry classes; the overlapping concurrent construction does not establish sole AI priority.
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.
The Alpöge–Claude manuscript claims a compact complex threefold, obtained by completing a modular family of complex two-tori, whose underlying smooth manifold is diffeomorphic to S⁶. Philip Engel supplies a separate geometric exposition and self-contained proof of the construction.
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.
ChatGPT 5.6 suggested the two-way trace graph TG_{d,h} of Remark 2.2 on projective points over F_(2^h). Ihringer and Mattheus bound its clique size by binom(d−2,ceil((d−2)/2)) and independence number by (d^h+1)². Their stronger one-way ordered graph and Theorem 1.3 are human-developed refinements.
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 the Hopf threefold X=(C³\{0})/⟨z↦e^-1 z⟩, Xia and Zhang’s Theorem 1.1 constructs a smooth Hermitian form ω and smooth potentials φ_j with ω+dd^cφ_j>0 yet ∫_X(ω+dd^cφ_j)^3→∞. Thus this compact complex manifold lacks the bounded mass property. The paper’s proof uses an elliptic fibration, an exact mass identity and fiberwise heat-kernel potentials.
Among real quadratic Pisot numbers α, the golden ratio is the only one in Dubickas’s set S: every other α admits a nonzero ξ for which all floors ⌊ξαⁿ⌋ are even for n≥1. For α=1+√2, ξ=(2−√2)/4 gives the exact Pell identity ⌊ξ(1+√2)⁽ⁿ⁺¹⁾⌋=Pₙ−[n odd].
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.
For the explicit iid block-refreshed smooth divergence-free velocity field U on T³ and the specified divergence-free initial magnetic mode, Theorem 1.1 gives, for each κ in [0, log(π/e)/(4π²)], an almost surely finite κ-dependent L^κ with ||B_t^κ||₂≥(L^κ)^−1 e^(t/2) for all t≥0 and a uniform positive moment bound on L^κ. Null sets can depend on κ; no single deterministic field working simultaneously for every κ is proved.
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.
For the Kasami function F(x)=x^(4^k−2^k+1) over F_(2^n), with gcd(k,n)=1, Nagy and Vajda prove the conjectured ordered triple count 2^(2n−3) for all distinct nonzero coefficients when k mod n is 1, 2, n−2 or n−1. They also check admissible n≤13 by exhaustive computation.
Liu constructs a smooth polarized complex projective fivefold that is K-polystable for all normal ample algebraic test configurations of every positive exponent, yet has no constant-scalar-curvature Kähler metric.
Every finite planar point set of at least 10^11055931 points has four collinear points or six pairwise visible points.
Jung constructs a 31-vertex simple plane triangulation with maximum induced forest 15, then simple maximal planar graphs Mₖ with 31k vertices, minimum degree five and maximum induced forest 15k for every k≥2. The ratio 15/31 refutes the Albertson–Berman half-forest conjecture.
Dupont and collaborators report ω<2.371177 by improving the combination-loss optimization and refining it with AlphaEvolve.
In each of dimensions four and five, for every sufficiently small nonzero ε, an explicit smooth odd function on S^(d−1) cannot agree with a linear functional on any rotated normalized A_d root system. Through Kuperberg’s equivalence, these give counterexamples to Makeev’s universal-cover conjecture in those dimensions.
On the Boolean cube {−1,1}^n with uniform measure, for every fixed 0<a<1, n≥1 and u>1, convolution by the biased-coin product measure μ_a satisfies sup_{f≥0, ||f||₁=1} u·λ({T_{μ_a}f≥u})≤C_a/√(log u), with an a-dependent constant. The bound is dimension free.
GPT-5.6 Sol proposed the Hodge-obstruction and rank-improvement strategy behind the bound Vol B_R(p)≤C(n)R^(n−2) on complete connected n-manifolds with Ric≥0 and Scal≥1.
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.
Poliakova uses ChatGPT-found finite obstruction subgraphs to show dim W(H₄)≥5>4 and dim W(E₆)≥7>6 for weak orders of Coxeter groups. These region posets of simplicial arrangements disprove the proposed equality of order dimension and rank.
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.
Waldron proves a cone theorem for projective Q-factorial log-canonical fourfold pairs over eligible positive- and mixed-characteristic bases p>5, assuming log resolutions for the relevant pairs and numerical equivalence K_X+Δ≡_U M≥0.
For an integral domain k, the commutative ring R=k[X₁,X₂,…]/(Xᵢ(1−Xⱼ):i<j) has exactly the pure ideals 0, I=(x₁,x₂,…), and R. Thus 0 is purely-prime but not purely-maximal.
Le, Sun and Tran construct smooth three-dimensional uniformly elliptic nondivergence equations with I≤Aₘ≤2⁸¹I, common boundary data and bounded solutions whose gradient L¹ norms diverge. They also obtain a uniformly convergent approximation limit that is not locally of bounded variation.
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 odd n≥3, Lu and Yang obtain a global Lipschitz family of exact hyperplane-section isometries after a positive-degree bundle pullback. Combined with their degree argument, this proves that a real Banach space whose n-dimensional subspaces are all linearly isometric is Hilbert when 2≤n<dim X.
For every complete first-order theory, SOP₂ implies SOP₃. The reverse implication was already known, so the two dividing lines coincide.
Alpöge and Furman report an unconditional lower bound of 0.6725 for the proportion of nontrivial zeta zeros that are simple and lie on the critical line.
For every integer b≥1, the dihedral and cyclic Ramsey numbers of the alternating 3-path versus K_b both equal 2b−1.
Every oriented graph on n=2δ+2 vertices with minimum outdegree δ has a vertex whose exact second outneighborhood is at least as large as its first. With Fisher’s tournament theorem, this covers n≤2δ+2, without proving Seymour’s conjecture for all oriented graphs.
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 the three-dimensional liquid-drop energy given by De Giorgi perimeter plus half the Coulomb self-interaction, balls uniquely minimize at every fixed volume 0<V≤5(2−2^(2/3))/(2^(2/3)−1)≈3.51, up to translation and null sets. Above that threshold, no minimizer exists.
Niebel constructs a real scalar diffusion coefficient on a bounded interval that is bounded, uniformly elliptic, one-half Hölder in time with spatial L∞ values, and arbitrarily close to the heat coefficient, yet its zero-data variational solution with continuous L² forcing has a time derivative outside space-time L². Extensions give real symmetric isotropic counterexamples in all dimensions and bounded domains.
For the two-layer solid standard Young-tableau shape [[n,n],[n,1]], Padhi gives an explicit two-term hypergeometric formula for the count g(n) and proves the conjectured second-order degree-12 recurrence for every n≥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 three-row tableaux with runs of length at least two, G(n)~C₁8ⁿ/n⁴; for odd runs, H(n)~C₂(7+5√2)ⁿ/n⁴. The constants are positive finite boundary sums, with numerical estimates about 0.52128605909 and 0.6389278129.
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.
There are noncongruent simply connected strictly convex Euclidean plane domains with real-analytic boundaries and identical Steklov spectra, including multiplicities. Such pairs can approach a disk smoothly.
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.
For fixed matrix dimension, whether the mean of i.i.d. binary random matrices is diagonalizable over R or C can be predicted with only finitely many errors almost surely. The argument uses eventual predictability of semialgebraic properties of bounded means.
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.
Kehoe’s working preprint recasts global cut conditions in the general-region half-collinear single-minus graviton recursion as positive directed-tree flows and open-root-cone membership. Retarded vertices become weighted sums over feasible spanning trees; directed Matrix-Tree determinants apply in chambers with a full arborescence family. It classifies four- and five-block chambers and gives one non-decay five-graviton identity.
For odd n=2k+1>1 and doubly stochastic A, Lavi proves max per(I−A)=3·2^(k−2). Every maximizer is a simultaneous permutation conjugate of the half-weight triangle block plus k−1 swap blocks.
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.
At canonical point C of the DiVincenzo family, an explicit Schmidt-rank-two tight-frame witness proves one-copy undistillability but two-copy distillability for every local dimension d≥3.
At inverse temperature β=1, the quenched law of N^(1/3) times the two-replica overlap converges to an explicit random Airy₁-based measure for both Ising and spherical Sherrington–Kirkpatrick models. Its second moment gives the limit of N^(2/3) E⟨overlap²⟩, answering Talagrand’s Conjecture 11.7.5.
For any field extension K⊂L, associative L-algebra A and A-module M, Greenfeld’s manuscript states that amenability of M over A viewed as a K-algebra is equivalent to amenability over L. This also applies to amenable and fully amenable algebras.
Any positive n-divisor arithmetic progression of height H with log H=o(√n) has length at least (√(8/27)−o(1))n^(3/4)/√log n. This excludes the proposed arithmetic-progression bounds at (α,β)=(1/3,1/3), and whenever α<1/2 and β<3/8.
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.
Three explicit Hermitian self-dual MDS codes with parameters [12,6,7] over F25 and [18,9,10] over F121 and F169 establish AME(12,5), AME(18,11), and AME(18,13); one-party projection establishes AME(17,11) and AME(17,13). The two length-18 matrices come from nine-entry group-circulant kernels; the theorem asserts existence, not classification.
For every k, there is a finite group with a minimal generating set of involutions whose Cayley graph has chromatic number at least k.
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.
A nonzero Schwartz function has twelve distinct time-frequency shifts, eleven in one irrational translate of a half-integer lattice plus the origin, that are linearly dependent. The authors also prove a finite counterexample without a fixed twelve-shift count.
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.
Chen, Nie and Xu’s cone-inclusion Lemmas 3.1 and 3.2 show that adjoining a linear factor preserves the required right-Noetherian structure and inclusion, while f−εf′ has a strengthened right-Noetherian property and inclusion. These lemmas feed a projective uniform numerical criterion for complex Hessian-type equations.
Liu and Wang construct a 19-dimensional smooth complex projective X with B=0, nef Cartier ξ, K_X linearly equivalent to −16ξ and nef part M_X descended to 17ξ. Its generalized log-canonical ring R(X,ξ), and consequently R(X,−K_X), is not finitely generated.
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.
Das, Ferenczi and Rosendal prove that no hyperplane of the real Kalton–Peck space Z₂ admits a complex structure; hence Z₂ is not isomorphic to its hyperplanes. A symplectic operator rank-parity theorem supplies the obstruction.
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.
Zhou constructs two nonisomorphic countable ICC property-(T) groups whose group von Neumann algebras are isomorphic. A quadratic fiber shear identifies the dual actions, while the groups have distinct module structures.
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.
The retained AI-origin construction gives a C¹ circle diffeomorphism with irrational rotation number, a wandering interval and derivative modulus ω(t)=t log(1/t), the barrier case in Prasad’s Corollary 1.1.
There are full-dimensional 0/1-polytopes of dimension 4n+1 whose graph edge expansion decreases exponentially with dimension.
Liu claims that a full-dimensional compact convex body in Rⁿ with barycenter zero, zero as its only interior lattice point and volume (n+1)ⁿ/n! is an integral unimodular image of the centered simplex (n+1)Δₙ−(1,…,1).
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.
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.
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.
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).
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.
The Schubitope in R¹⁴ from columns {8,…,14} and {12,13,14}, equivalently Δ(7,14)+Δ(3,14), has Ehrhart quadratic coefficient −687323/79200. It is therefore not Ehrhart positive.
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).
For each fixed prime power q, the greatest possible least degree D_q(n) omitted by a non-covering family of n congruence classes in F_q[x] satisfies D_q(n)=n/(q−1)+O_q(1). ChatGPT supplied a truncated CRT/Brun-sieve approach used in the upper bound.
For every α>0 and three positive point charges in R^n with n≥2, the potential has at most six nondegenerate equilibria. The new step proves a four-contact statement using separation at the unique saddle of a first integral.
Bhatt and Ma construct a five-dimensional finitely generated flat normal domain over Z or any field with a codimension-one prime J for which H²_J has infinitely many minimal primes. The characteristic-two case uses a separate construction; a cone gives a six-dimensional normal local example.
A diagonal transfer-layer family gives an upper bound for the cubic-lattice dimer constant. Its (m,n)=(5,5), p=2 exact finite computation yields ℓ₃≤0.452130.
For finite integer sets A with at least two elements, Lin and Li construct families with log(|A+A|/|A|)/log(|A−A|/|A|) approaching 2. Together with Ruzsa’s upper bound, this proves that the exponent 1/2 in √σ(A)≤δ(A) is sharp.
Liu and Wang prove that a general pencil of plane cubics over C has exactly twelve common flex lines. They identify all twelve units of the degree-nine flex-line curve’s genus defect as ordinary nodes, each representing a line that is a flex of two distinct smooth members.
Five positive point charges in R³ can have at least 24 nondegenerate electrostatic equilibria. A small perturbation makes all critical points nondegenerate and finite in number, exceeding Maxwell’s proposed (n−1)² bound of 16 when n=5.
Hou and Liu prove pseudo-effectivity of the first Chern class for every coherent quotient of every positive tensor power of the logarithmic cotangent sheaf under their smooth proper integral complex Deligne–Mumford stack, normal-crossings divisor, and big-determinant subsheaf hypotheses.
For critical angles 0 and π/2 of multiplicity three, Yan gives αₙ=c(n+1)^(-3/20)(1+(-i)ⁿ), 0<c<1/2. It satisfies Lukic’s localized decomposition conditions but has weighted entropy −∞, refuting that direction of the unrestricted multipoint conjecture.
For a thermal attenuator with environment photon number ν=1, a rank-two input on six Fock levels has certified coherent information at least 4.7×10⁻⁴ qubits per channel use at transmissivity η=0.8, while all single-mode Gaussian inputs have at most zero. A larger witness is certified positive at η=0.7841.
For any planar convex body K and 0<W<width(K), one plank of width W covers at least as much area of K as any finite family of planks whose total width is at most W. The assessed AI-associated result is this planar reduction.
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.
Tran, Xin and Yu construct a smooth periodic three-dimensional shear flow V=(0,0,f(x₁,x₂)) and direction (2,2,1) whose homogenized G-equation flame speed has c(0)=0 and positive right derivative as curvature diffusivity d tends to zero. For small d the cutoff is inactive, so the physical cutoff cell speed increases.
For a plurisubharmonic weight on a product of polydiscs, Xia proves that the ambient multiplier ideal restricts to the fiber multiplier ideal outside a pluripolar parameter set. The paper also extends the result to holomorphic fibrations.
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.
Rethlas found a representable rank-seven binary projective-deletion matroid whose Kazhdan–Lusztig polynomial is not real-rooted, with a convolution and degree approach. Cheng and Liu later used that direction to construct representable matroids over every finite field with nonunimodal Kazhdan–Lusztig polynomials.
Dai and coauthors construct a locally finite 1-differential poset P for which the two-multichain quotient M_(P,2)(q)/F_P(q)² is nonrational over every characteristic-zero field. This refutes Stanley’s proposed rational-factor property already at k=2.
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 every isotropic log-concave random vector X and symmetric matrix M, Var⟨MX,X⟩ ≤ 2 E|∇⟨MX,X⟩|². Letwin also derives a general KLS bound of order (log n)^(1/4).
For finite-dimensional quantum states at trace distance at most δ and d=dim A, the optimal dimension-only conditional-entropy continuity modulus is h₂(δ)+δ log(d²−1) up to δ=1−d⁻², then 2 log d. Equality is attainable when dim B≥d.
For independent nonnegative random variables X₁,…,Xₙ with E[Xᵢ]≤1 and S=ΣXᵢ, Fu, Han, Wang, Yan, Zhang and Zhou prove P(S<E[S]+δ)≥δ(n/(n+δ))ⁿ for 0<δ<1, and ≥(1−1/(n+δ))ⁿ for δ≥1. The second finite-n bound is sharp and implies Feige’s unit-slack e⁻¹ conjecture.
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.
For G=SmallGroup(128,859) over the algebraic closure of F₂, depth H*(G;k)=2 while every associated-prime quotient has dimension at least 3. The group-specific certificate enumerates 75 rank-two elementary abelian subgroups and checks their centralizer types.
For each integer r≥3, Dai and coauthors construct an infinite r-differential poset that agrees with Yʳ through rank three but has rank-four size |(Yʳ)₄|−⌊r/3⌋. At r=3 the first sizes are 1, 3, 9, 22, 50, compared with 51 for Y³.
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.
The authors adapt a ChatGPT-supplied q-lift example against reciprocal flat-count convexity into matroids whose numbers of flats by rank are not unimodal. The final construction combines Larson’s graphic matroids with Whittle’s q-lift.
For k≥1 and δ,ε>0, Theorem 1.14 extracts an ε-tight uniform (k,δ)-ladder from a (t,2δ+ε)-tree at some depth t≤(1/ε)^(O(k/ε²)), avoiding a multicolored Ramsey step.
For G_m=⟨a,t | [t,a[a,t]^(−m)]⟩, a Baumslag–Solitar group BS(m,m+1) is a retract. In particular, G_2 is non-Hopfian and non-residually finite, while G_1 is non-automatic. Literal commutator relators therefore do not force any of these properties.
Matherne and Morales give a 12-vertex, 19-edge bipartite graph G3 whose chromatic symmetric function is reported Schur positive while its three-variable specialization omits lattice-point weight (7,4,1) between present weights (6,6,0) and (8,2,2).
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.
Matherne and Morales give two 12-vertex claw-free line graphs whose chromatic symmetric functions have [s₃₃₃₃] coefficients −64 and −40. Either graph refutes Stanley’s claim that every claw-free graph is Schur-positive.
Long gives explicit rational-coefficient polynomials in four variables defining a Poisson endomorphism of the canonical algebra P₂ in characteristic zero that is not an automorphism. An exact three-point fiber proves noninjectivity; extending the map by identity gives examples for every Pₙ with n≥2.
For every even k≥4 and prime p>k, a coloring of Z/(p^(k²/4))Z uses (k−1)^(k²/4−1)p colors and avoids nontrivial symmetrically colored k-term arithmetic progressions. At k=4 this gives 27p colors and an O(N^(1/4)) interval bound.
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.
Pohoata constructs n incident point-line pairs in [0,1]² whose off-diagonal distances are at least n^(−2/3−ε) for every ε>0 and all sufficiently large n. With the previous upper bound, the maximum separation is n^(−2/3+o(1)).
Conant proves that for every amphicheiral knot K, its Conway polynomial is congruent modulo 4 to f(z)f(−z) for some integer polynomial f. A stronger integral identity underlies this mod-4 conclusion.
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.
Kafidov claims that, for every nonnegative 16-by-16 matrix with total entry sum 16, the Dittert functional (product of row sums plus product of column sums minus permanent) is at most 2-16!/16^16, with equality only at the constant matrix. The new joint-deficit scaling lemma makes a near maximizer doubly superstochastic after a sharper dilation and excludes a boundary maximizer at dimension 16.
Theorem 1 proves that the one-sided Airy factor g has v(x) = -(log g)''(x) > 0 and v(x) ≥ 4x. Since Chernoff's density f is proportional to g(x)g(-x), its curvature κ(x) exceeds 4|x| and has a positive global infimum. This proves strong log-concavity of the density of argmax_t{W(t)-t²}.
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.
A 3-group extension G=A⋊B with elementary abelian A and B and trivial C_B(a) can have rank(Z(H)∩H′)>rank(B), where H=⟨a,B⟩.
In a non-abelian right-orderable group, the right-relatively convex subgroups need not form a sublattice of its subgroup lattice.
Every power graph of a finite group that is a cograph is chordal; equivalently, an induced four-cycle forces an induced four-vertex path.
For k ≥ 4, horizontal class transpositions of moduli at most k generate a group isomorphic to the symmetric group on lcm(2,…,k) letters.
A non-abelian group admits a weight-one Rota–Baxter operator that is surjective but not injective, answering Problem 20.125.
A simple group and a non-simple group both have order 6048 and totient sum 23984, disproving the proposed simplicity test in Problem 19.25.
For every positive n and 1 ≤ k ≤ n!, a group has n distinct elements whose n! ordered products take exactly k distinct values.
A group with exactly two maximal locally soluble normal subgroups, resolving Kourovka Problem 3.46.
Cohen and Shusterman prove the Fₖ and FPₖ virtual-surjection conjectures for discrete groups. Their homological fibre-product theorem assumes the common quotient Q is finitely presented; they also give counterexamples without that condition.
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.
Christopher D. Long reports explicit complex polynomials P and Q in three independent real Gaussian variables with E(P^m)=0 and E(QP^m)=m! for every positive integer m. A five-term degree-four example refutes the conjecture in every dimension at least three; a six-term cubic example works in four variables.
Antonio and Pablo Acuaviva construct a unital complex Banach algebra not isomorphic to B(X)/K(X) for any Banach space X, using large simple algebras and separable-range operator obstructions.
If T:X→Y is weakly compact and Y is separable with the bounded approximation property, then T factors through a reflexive Banach space with a Schauder basis.
Every infinite-dimensional complex normed space has unit vectors separated by at least 1+ε for some ε>0 even after either vector is multiplied by any scalar of modulus one.
For every 1<p<∞, Lp(L1) has uniform primary factorization and is therefore primary. The proof includes a scalar-compression argument on its doubly cancellative product-Haar part.
For every bounded operator T:X→Y with separable Y, T is strictly cosingular if and only if its adjoint T* is strictly singular.
Appendix A proves that choosing one element from each of n sets of size k yields at least binomial(k+n−1,n) distinct multisets, using a multiset-shadow coupling. This establishes the edgeless-graph case of the unlabeled list-color conjecture.
Ignatev uses a Claude-origin max-term asymptotic lemma twice to prove Λ(T)=[0,1] for backward shifts on ℓ^p with strictly decreasing, positive, p′-summable weights.
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.
A measurable real-line set with finite average symmetric-difference measure under unit-interval translations is commensurate with exactly one of the empty set, the whole line, a right half-line or a left half-line. For free ergodic measure-preserving flows, this implies that abstractly isomorphic L¹ full groups determine conjugacy up to a nonzero scalar time change.
An explicit width-10 shifted template gives S(k+2)≥10S(k)+2. Using existing bounds for S(6) and S(11), the authors obtain S(8)≥5362 and S(13)≥2038282.
For every fixed sequence of T=n vectors in [−1,1]^n chosen before the algorithm runs, the model-generated proof gives O(√n) maximum online prefix discrepancy. The authors also prove longer-horizon and sparse-column extensions, including O(√n) for T up to exp(cn/log^(2+η)(en)) with the stated high probability.
For the exponent-five Heisenberg group of order 125, the small unordered Davenport constant is twelve: x⁴y⁴v⁴ is product-one-free, and every thirteen-term sequence has a product-one subsequence in some ordering.
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.
Akhil Mathew’s contribution gives a noncommutative finite free group scheme of order four over a nonreduced base that is not killed by four.
Niedbala Giraudin gives an explicit irreducible degree-five Carlitz–Wieferich prime over F_(19³), refuting the proposed odd-characteristic rule that its degree must be divisible by the characteristic.
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.
For type-D ASEP with step initial data and fixed q, the standardized current of each species converges to the GUE Tracy–Widom F₂ law. Theorem 8.1 uses exact autonomy of each projected current and the known single-species step-ASEP limit.
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.
Gargava gives an exact rotated E8 lattice packing of B₂⁴ × B₂⁴ with density π⁴(1+1/√5)⁴/1024, about 9.67% above the product D4 packing. A separate determinant-preserving perturbation of the nine-dimensional densest lattice Λ9 has lower computed flat-torus height.
Shmalo's paper verifies Ding–Sun Condition 1.2, including curvature and endpoint clauses, and Huang Condition 1.3 for the zero-margin Ising perceptron. Combining these with prior conditional and universality theorems yields M_N/N→α★≈0.8330786 in probability for Gaussian and fixed standardized subgaussian disorder, including Bernoulli.
In a specified three-block Gaussian factor model with 96N true nulls and 4N nonnulls, ordinary Benjamini–Hochberg testing at level 0.01 has false discovery rate greater than 0.0104 for all sufficiently large N when applied to correlated two-sided Gaussian p-values.
Every finite loopless bridgeless undirected multigraph has a multiset of cycles covering each edge exactly twice.
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 connected cubic triangle-free graph on 14 vertices has domination number γ=4 and zero-forcing number Z=7. Since it is diamond-free, Z exceeds Davila’s proposed γ+2 bound.
GP(9,2) has a non-real-rooted independence polynomial although k is even. GP(7,3) and GP(3,1) are real-rooted although k is odd. These examples refute both directions of Pandey’s claim for generalized Petersen graphs.
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).
Five simple 3-polytopes with 10 faces violate the printed IRIS Conjecture 6.1. One has four triangular, three pentagonal and three heptagonal faces, but no hexagonal faces, below the conjectured lower bound of 1/5.
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>0, the Fubini number a(n), which counts ordered set partitions, equals the sum of A284005(k) for k from 0 through 2^(n−1)−1. The proposed proof counts partitions with each prescribed set of block minima.
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.
Fang, Lan and Ma identify affine binary-dihedral orbifold Gromov–Witten theory with involution-equivariant sign-sector recursion on a type-D logarithmic Toda curve, for stable descendants with n>0 and 2g−2+n>0 and free energies with g≥2 on the stated smooth chamber.
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.
The smooth Fermat quintic Calabi–Yau threefold has a faithful action of (μ₅)⁵/Δ, an elementary abelian 5-group of rank four. Since 4>15/4, this refutes Moraga’s proposed odd-dimensional extension of a rank bound.
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.
Larson gives a rank-nine binary matroid on 18 elements whose complementary basis pairs occupy disconnected symmetric-exchange classes. It refutes Farber’s two-base conjecture and White’s toric-ideal symmetric-exchange generation conjecture.
In Korsky’s upper-bound argument for north-east lattice paths with no k collinear visited vertices, the mediant of Farey neighbors raises the uniform density-increment product from 1/[8(k−1)²] to 1/[4(k−1)²]. It gives log A(k)≤(2/e+o(1))(k−1)².
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.
Fourteen binary vertices form a simplicial seven-dimensional 0/1-polytope with the maximum 2d vertices but without central symmetry, countering the proposed cross-polytope characterization.
Erdős problem #36 lower bound c_E > 0.38055470 for the asymptotic minimum-overlap constant.
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).
For a reduced isolated complex hypersurface germ of dimension at least two, or an irreducible curve germ, weighted homogeneity after coordinate change is equivalent to a logarithmic holomorphic vector field transverse in the real-Euclidean sense to every sufficiently small link in those same coordinates. An invertible ambient linear part is a third equivalent condition.
Finite planar lattice sets can have an arbitrarily large ratio of ordered rectangles to ordered isosceles triangles, including degenerate configurations. Bennett, Kovač, Nakamura and Oliveira quantify the construction in a box [-N,N]² by a ratio at least (log N)^(c/log log log N), with an analogous cardinality bound.
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.
An explicit reduced planar convex body of thickness 1, with support-function parameters m=15/16 and ε=13/20, has area 0.7862156027…>π/4. It refutes Lassak’s proposed bound area(R)≤(π/4)Δ(R)² for all reduced planar bodies.
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.
Appendix A gives a Yoshinaga half-space-criterion proof of the two local arrangement obstructions used in genus-one differential-strata non-asphericity.
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)].
Krishtal and Pfander construct a bounded invertible normal operator T on ℓ²(N) and a vector g whose normalized orbit {Tᵏg/||Tᵏg||: k≥0} is a frame.
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)).
Ajwani, Gajjala, Raman and Ray give 64 axis-parallel rectangles with packing number 16 and piercing number at least 32, disproving Wegner’s proposed bound 2ν−1. Their package construction yields piercing and clique-LP ratios approaching 5/2.
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.
For f(x,y)=(x−y)²+Mx with a fixed suitable M, there are arbitrarily large finite real sets A with |f(A,A)|≤|A|^(2−c) for some c>0, although f has neither additive nor multiplicative special form.
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.
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)).
Pozsgay gives an open-boundary spin-chain Hamiltonian built from two compatible Ising-type algebras with a hidden free-fermion solution and, for generic couplings, no exponentially growing homogeneous spectral degeneracy.
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.
Parisi and Zamponi give an analytic proof of a+b=1 from the scaling full replica-symmetry-breaking equations for infinite-dimensional hard-sphere jamming. Together with the earlier relation b=(1+c)/2, it yields the scaling identities linking force, gap and overlap exponents.
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 every n≥3, the even-degree Poincaré polynomial of the moduli space of stable n-pointed genus-zero curves is real-rooted. For n≥4, its n−3 roots are simple and negative and strictly interlace those for consecutive n.
AlphaEvolve found explicit witness words that Riestenberg, Taha, Trettel and Wienhard use to prove that the degree-six symplectic hypergeometric monodromy groups C-47 and C-55 have finite index in their ambient integral symplectic groups and are arithmetic.
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).
Liu and Pikhurko use a Ramanujan-graph spectral 3-graph gadget to build a smoothly decreasing sequence of Lagrangians. With their separate recursive and lifting constructions, they show that for every r≥3, Turán densities of possibly infinite forbidden r-graph families contain a terminal interval [1−δ_r,1].
Bérczi and Fehér report that, for every k, the elementary-basis coefficients of the kth Chern class of Sym^d(C^n) are universal rational polynomials in T_r(n,d)=binom(d+n−1,n+r−1), for 1≤r≤k.
For circle-valued maps, Theorems 1.3–1.4 give degree inequalities with constants of order p−1 as p approaches 1 and order δ as δ approaches 0. The latter covers continuous maps for 0<δ≤√3.
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.
Every pure O-sequence of codimension three and type two is log-concave: h(i−1)h(i+1)≤h(i)^2 at every interior degree.
A counterexample on Z/3Z disproves the intended complex-valued equality of two quadratically structured function spaces in Ben Green’s Problem 57.
Korsky proves that every sufficiently large prescribed length L admits full-chord sets in each compact convex body with finite piecewise C² boundary and discrepancy O((log L)^(3/2)). For the disk, every full-chord set has discrepancy Ω(log L).
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.
A flat complex family made by substituting T=sE into Singh’s determinantal family has a klt-type special fiber and nearby fibers that are not of klt type. It gives a negative answer to Han and Jiang’s Problem 4.6.
Liu and Qin prove a uniform index I for projective log-canonical foliated triples of dimension at most three with K_F+B numerically trivial and coefficients in a fixed rational DCC set, so I(K_F+B) is linearly trivial. Rethlas contributed the non-algebraically-integrable case split and Lie-theoretic route.
A surface qdlt log Calabi–Yau pair with three big boundary components has a square dual complex, rather than the boundary of a two-dimensional standard simplex. This answers part (2) of Mauri and Moraga’s Question 9.9 negatively.
A flat projective family over an open subset of the complex affine line has reduced Cohen–Macaulay fibers and smooth generic fiber, yet h¹(O) is 1 at the special fiber and 0 elsewhere.
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.
Liu reports effective divisors D_n on one fixed smooth projective surface with D_n²=4 and h⁰(mD_n)=1 for every 1≤m<n²/4+1, for arbitrarily large odd n. Thus no uniform movable multiple depending only on that surface exists.
The human-verified writeup gives a fixed ε>0 and growing planar point sets with at least n^(1+ε) unit distances.
For the switch–walk–switch lamplighter walk with Z₂ lamps over the infinite d-regular tree (d≥3), p₂ₙ(e,e)=ρ_d^{2n} exp[−(π²(log(d−1))²+o(1))n/(log n)²], where ρ_d=2√(d−1)/d.
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.
Appendix B Proposition B.1 gives a convex polyhedral Gaussian partition with prescribed cell masses and projected barycenters when a centered finite-support X is convexly dominated by ρG for 0<ρ<1. The GPT proof is an alternative martingale-coupling step, not the paper’s main Talagrand theorem.
Defant’s manuscript claims that, for every alternating sign matrix A, gradedness of [e,A], the corresponding equal-length condition and equidimensionality of all X_B in that interval are equivalent to Cohen–Macaulayness of all those ASM varieties. The new direction establishes Escobar–Klein–Weigandt Conjecture 3.21.
Defant’s manuscript displays an interval in the weak order on six-by-six alternating sign matrices whose Möbius value is 1, although it has none of the specified parabolic forms. Its open-interval order complex is therefore not contractible, contrary to the Hamaker–Reiner dichotomy.
For p>2, a common-set/private-set construction shows Carbery’s proposed exponent c=2 fails. Grok also supplied the kernel inequality used to establish the optimal necessary constraint c≤p′.
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 n=4 and p=2, the AlphaEvolve-found cosine correlation matrix Σᵢⱼ=cos(π(i−j)/n) has a strictly smaller expected squared minimum absolute Gaussian coordinate than the regular-simplex correlation matrix.
For every d≥1 and nonnegative r,s,t with r+s+t=dn, the uniform distributions on the lattice-cube slices L_d(r), L_d(s), and L_d(t) are jointly mixable. Keevash and Lim use this theorem to obtain the optimal limiting sum-free density of [n+1]^d and an O(n^(d−1)) upper error.
For k≥1, a primitive set A of integers with at least k prime factors, and any set Q of odd primes, the sum f(A(Q)) is at most f(Nₖ(Q)), where f sums 1/(a log a).
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).
If A has positive upper doubly logarithmic density Δ, then A contains an infinite strictly increasing divisibility chain whose upper count through x, divided by log log x, is at least Δ.
Portier proves that an edge-minimal generically globally rigid graph in Rᵈ with at least d+3 vertices contains no K_(d+2) subgraph. Equivalently, the only minimally globally rigid graph containing that clique is K_(d+2) itself.
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 every n≥2, g₄(n)=3: any A⊆{1,…,2n} with at least n+3 elements contains all six pairwise sums of four distinct integers, and the constant cannot be lowered. The four integers need not be positive or members of A.
For semifree noncommutative differential graded algebras over a nontrivial computable unital commutative ring, stable tame isomorphism, quasi-isomorphism and derived Morita equivalence are undecidable.
For the largest measure M(R) of a measurable subset of a radius-R planar disk with no positive integer distance between distinct points, M(R)≤C√R. Together with Sárközy’s earlier lower construction, this yields M(R)=R^(1/2+o(1)).
For the discrete-time switch–walk–switch walk on Z₂ wr Z, started with all lamps off and walker positions 0 or 2, the total-variation distance between the time-t laws is comparable to t^(−1/2), with constants independent of t.
For a one-parameter complex projective degeneration with simple-normal-crossing central fiber and smooth nearby fiber X_t, QED proves that H¹(X,Z) maps onto the monodromy-invariant subgroup H¹(X_t,Z)^T.
Turturean’s write-up claims h(n)=(1/2+o(1))log* n and H(n)=(1+o(1))log* n for almost all n, so H(n)/h(n) tends to 2. A GPT-5.5 Pro subset-product successor lemma supports the composite-divisor lower bound.
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 nondecreasing F tending to infinity, an infinite-measure measurable set S of positive reals with no distinct-point integer quotient and |S∩(0,x)|≥F(x) eventually exists exactly when ∫₁∞ F(x)/x² dx is finite.
For infinitely many n=q+1 with q a prime power, Ho constructs n points in R^(n²−n) whose distances for distinct pairs differ by at least one but whose diameter is at most (1−1/π²+o(1))n².
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.
There is no universal Erdős–Turán angular-discrepancy bound of order √(ν(f) log M(f)) with polynomial degree replaced by the number ν(f) of nonzero terms. For each N, a polynomial with ν=N+2 and M<3 has a positive real zero of multiplicity N+1, giving discrepancy at least N+1/2 on the stated short arc.
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.
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.
For integers r≥3, k≥4 and n≥72, the maximum ordinary-line count Fᵣ,ₖ(n) is at least n²/12−(10/3)n. The elliptic-curve construction has no four collinear points and a bipartite ordinary-line graph.
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 regular adjacent-transposition chains on S_n with n≥3, the second-largest eigenvalue has algebraic multiplicity M equal to the number N of neutral labels, except when N=n−2 or n, when M=n−1.
There exists a weakly quasi-complete Noetherian local ring that is not quasi-complete.
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.
For every real α, the sequence of fractional parts of α times consecutive primes is not well-distributed in the uniform-in-starting-index interval sense. This does not say ordinary equidistribution fails.
An explicit basis A of order two has the property that every partition A=A₁⊔A₂ leaves at least one of A₁+A₁ or A₂+A₂ with arbitrarily long gaps. The construction uses powers-of-five blocks and special points that force missing intervals.
For a real degree-n trigonometric polynomial with 2n distinct zeros on a period, Tao’s Theorem 1.13(ii) gives an integral of its Lebesgue-type sum at least 8, with equality for a sinusoid. The bound follows by multiplying two factors in Lemma 1.14.
Patel’s revised paper claims that for every genus g≥2, the Hodge bundle on the moduli stack of smooth complex genus-g curves has no nontrivial subbundle. The proof uses traces of involutions and curves with commuting involutions to constrain a hypothetical subbundle’s rank.
The diagonal cubic X³+Y³+Z³+ζ₃T³=0 over Q₂(ζ₃) has trivial R-equivalence. Gemini 3 Deep Think supplied rigorous methods for the gap-closing Lemmas 3.4–3.5 in the human-led proof.
For smooth positive steady single-species Vlasov–Maxwell–Landau solutions on the three-torus with Coulomb collisions, Schwartz velocity decay and a polynomial score bound, the stated Lean theorem yields a spatially uniform Maxwellian at ion density, zero electric field and constant magnetic field.
A 1,280-word binary code of length 19 and minimum distance five inside a five-punctured extended Golay code extends Cohn and Li’s 10,668-point configuration. It gives the lower bound k(19)≥11,948.
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.
A centrally symmetric simplicial six-dimensional polytope with 48 vertices has an antipodal boundary quotient that triangulates real projective five-space on 24 vertices.
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 ε>0, there is an r such that integers 2^k+n with n having at most r prime factors have lower density at least 1−ε. The claim neither establishes a natural density nor fixes one r for all ε.
A repaired GPT-5.2 Pro argument uses prime cyclic groups to refute f(N)≤log₂N+o(log log N) for the random subset-sum covering threshold. Ma and Tang later prove the stronger lower bound f(p)≥log₂p+(1/(2 log 2)+o(1))log log p; the constant and quantitative refinement are their work.
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.
Lee and Seo prove a degreewise lower bound for the multivariate independence polynomial at nonnegative vertex weights and a two-proper-colour semiproper partition-function bound. The model-origin Lemma 2.1 and dual-set ideas are retained central steps; the equality characterization in their first theorem is outside the Lean target.
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.
Aletheia constructed two closed, infinite, countable compact subsets of the complex plane with the same zero transfinite diameter but different values of the polynomial sublevel-area quantity μ(F).
Aletheia constructed planar n-point sets with no four cocircular in which every point determines fewer than 3n/4 distinct distances. These refute the near-n-distances question under that hypothesis.
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.
For independent W(q,r) contamination and the stated Hölder-smooth clean and contaminated densities in p dimensions, revised Theorem 4.1 gives the squared-loss pointwise minimax rate max{n^(−2s/(2s+p)), ε^(2s/(s+1+p/r))}. A smoothed kernel estimator attains the upper bound.
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.
Wang and collaborators report a second unstable self-similar Córdoba–Córdoba–Fontelos (CCF) profile, stable and three unstable incompressible porous media (IPM) profiles, and stable and three unstable 2D Boussinesq profiles. They test residuals and linearized modes on dense grids. A fourth Boussinesq profile remains an unvalidated candidate.
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.
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.
AlphaEvolve’s authors report an eleven-dimensional kissing configuration with 593 outer spheres.
AlphaEvolve found a step-function construction giving the Erdős minimum-overlap constant C5 an upper bound of 0.380924.
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.
The FunSearch paper reports new large cap-set constructions improving the authors’ stated finite-dimensional and asymptotic baselines.
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.
The selected contribution is the model’s empirical identification of cusp translations as salient for knot signature, not the human-defined natural slope or the later theorem.
The selected contribution is the 48-vertex tree of §2.4, not Wagner’s subsequent infinite-family theorem.
AI contributions across the mathematics fields in the catalog, including algebra, analysis, probability, combinatorics, geometry, topology, number theory, algorithms, optimization, and formal proof.
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.
Pointwise convergence of ergodic averages for two commuting transformations
The normal order of the number of iterations of Euler’s function
Revisiting Separators in String Graphs, Region Intersection Graphs, and Induced-Minor-Free Graphs
Erdős Problem 111: bipartizing finite subgraphs of uncountably chromatic graphs
Every two-coloring of the plane misses at most one triangle, which is equilateral
AI-led proof gives a sharp global squared-law transport bound
OpenAI reports the exact three-state tree reconstruction threshold
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
Two AI-assisted constructions settle proper hat guessing for K7−e
AI-assisted construction settles proper hat guessing for K5−e
ChatGPT ideas underpin all-dimensional sum-free lattice-cube result
AI-assisted construction settles proper hat guessing for K6−e
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