1033 matching entries · Page 2 of 21
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.
For each δ∈(0,1), sufficiently large maximum degree Δ and finite simple n-vertex graphs of girth at least 5, Theorem 1 gives irreducible proper-q-coloring Glauber dynamics with spectral gap Ωδ(1/n) and mixing time Oδ(n² log q+n log(1/ε)) when q≥(1+δ)Δ. Theorem 3 extends the method to antiferromagnetic Potts models under its stated β and q conditions; Theorem 5 gives a local spectral-contraction criterion for general multi-spin systems. The result is rapid mixing, not the conjectured exact q≥Δ+2 threshold or uniformly optimal O(n log n) bound.
For 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.
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.
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.
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.
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.
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.
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.
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.
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.
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 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 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.
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.
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.
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.
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.
Chernikov constructs a complete simple theory with a definable group and a global type whose left-translation orbit has size at most 2^ℵ₀, although the group has no left-invariant Keisler measure. This refutes the bounded-orbit implication to definable amenability outside NIP.
For 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 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.
The electronic Coulomb ground-energy promise problem is QMA-hard for distinct rational positions of unit-charge nuclei, unary electron count and rational thresholds separated by at least an inverse polynomial, with minimization over the full three-dimensional spinful fermionic continuum.
For every 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.
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.
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 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.
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.
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 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.
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.
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.
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.
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.
Nikolenko constructs a 17-terminal common-point instance proving the universal cost-preserving unsplittable-flow constant is at least 1.282494797984843521. A 3/2 congestion ceiling applies only to common-point cells of complement mass two.
Nikolenko gives a six-terminal planar instance requiring congestion factor at least 58676765987259/50000000000000, or 1.17353531974518, under cost nonincrease. He also identifies exact local envelopes E(2)=1, E(3)=9/8 and E(4)=(299−41√41)/32.
For 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.
A counterexample on Z/3Z disproves the intended complex-valued equality of two quadratically structured function spaces in Ben Green’s Problem 57.
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.
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 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 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.
For every Boolean function on the uniform discrete cube, including biased outputs, and every product-noise correlation ρ∈[−1,1], the manuscript claims H(Ef)−E H(Tρf)≤1−H(ρ), where H(t)=sqrt(1−t²). Signed coordinates attain equality.
For every 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.
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.
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.
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arXiv · algebra-geometry — last successful metadata check: 21/10/2026, 07:20:46 UTC
arXiv · algorithms-formal — last successful metadata check: 21/10/2026, 07:19:48 UTC
arXiv · analysis — last successful metadata check: 21/10/2026, 07:21:51 UTC
arXiv · combinatorics — last successful metadata check: 21/10/2026, 07:18:49 UTC
arXiv · geometry-topology — last successful metadata check: 21/10/2026, 07:22:23 UTC
arXiv · logic-foundations — last successful metadata check: 21/10/2026, 07:23:04 UTC
arXiv · optimization-information — last successful metadata check: 21/10/2026, 07:23:10 UTC
arXiv · probability-dynamics-physics — last successful metadata check: 21/10/2026, 07:22:36 UTC
arXiv · theoretical-physics — last successful metadata check: 21/10/2026, 07:22:50 UTC
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