Tagged openai-math
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Paper Trace cones and Razak–Jacelon stabilization
Classifies separable nuclear complex C∗-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their full topological cones of extended lower-semicontinuous tracial weights. This…
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Paper Hilbert transforms along Lipschitz directions
Proves a uniform strong L2 bound for the planar Hilbert transform along any Lipschitz unit vector field, at integration lengths bounded by an absolute multiple of its reciprocal Lipschitz constant. The estimate is…
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Paper Unitary vertex operator algebras and conformal nets
Proves the strongly rational case of the strong-locality conjecture: every simple unitary strongly rational complex vertex operator algebra generates a completely rational conformal net. Its simple modules are…
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Paper The free uniform spanning forest is a factor of IID
On every infinite connected locally finite simple unweighted graph, the free uniform spanning forest is a factor of independent vertex labels, by one isomorphism-equivariant rule using no root. Translation-invariant…
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Paper Failure of Kohn–Sham ensemble representation
Constructs a three-electron Coulomb molecule with two equal positive-integer-charge nuclei whose absolute ground-state density has no noninteracting ground-state ensemble representation by a single real spin-independent…
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Paper Potential integral density on curve character varieties
Resolves the determinant-one curve case of Litt's integral-density question. For every smooth connected complex algebraic curve and every rank, integral points become Zariski dense in every component of its SLr…
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Paper Short Egyptian fractions
Every rational a/b with 1\le a\lt b is a sum of O(\log\log b) distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian…
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Paper An asymptotic formula for the number of totients
Gives an asymptotic equivalent for the number V(x) of distinct totient values up to x, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular,…
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Paper Patterson's first moment for cubic Gauss sums
Proves unconditionally the all-primary-prime form of Patterson’s first-moment asymptotic: normalized cubic Gauss sums over primary Eisenstein primes of norm at most X, including both conjugates, have an explicit…
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Paper The weak inhomogeneous Duffin–Schaeffer conjecture
Proves that for every real shift γ and finite-valued $`\psi:\mathbb N\to September 25, 2026 · · cite as `` We prove the weak inhomogeneous…
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Paper Exact quantum factoring over a fixed finite gate set
Gives a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer with probability one. Both gate count and qubit count are polynomial in the input length, and one…
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Paper Thomason model structures in all strict higher dimensions
Resolves the Ara–Maltsiniotis conjecture: for every n ≥ 1 and n = ω, small strict globular n-categories admit proper combinatorial Thomason model structures Quillen equivalent to simplicial sets. Thus strict higher…
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Paper A quadratic bound for Jacobsthal’s function
Answers Jacobsthal's quadratic-bound question: every interval of Ck^2 consecutive integers contains an integer coprime to any prescribed positive integer with at most k distinct prime divisors, for an absolute…
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Paper Foulkes' conjecture for sixth powers and quadratic stabilization
Proves the sixth case of Foulkes’ conjecture: \mathop{\mathrm{Sym}}\nolimits ^6(\mathop{\mathrm{Sym}}\nolimits ^bV) embeds equivariantly in \mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^6V)…
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Paper Donovan's conjecture over fields and complete mixed-characteristic DVRs
Proves Donovan's conjecture: over each fixed algebraically closed field of characteristic p, blocks of finite groups with bounded defect-group order have only finitely many Morita-equivalence classes, for every prime p.…
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Paper Cycle–clique Ramsey numbers
Proves the Erdős–Faudree–Rousseau–Schelp conjecture: R(Cm,Kn)=(m-1)(n-1)+1 for every m\ge n\ge3, except R(C3,K3)=6. This is the exact threshold forcing a red m-cycle or a blue n-clique in every red–blue…
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Paper Compact counterexamples to bi-Lipschitz dimension reduction
Every infinite-dimensional real Banach space contains a compact doubling set that admits no bi-Lipschitz embedding into any finite-dimensional normed space. The doubling constant is universal. This answers the…
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Paper All-temperature pressure of orthogonally invariant Ising spin glasses
Gives an exact variational formula for the limiting pressure of orthogonally invariant Ising spin glasses at every fixed temperature, both almost surely and in expectation. The coupling matrix is a Haar-random rotation…
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Paper The sharp terminal leave in random triangle removal
Starting from the complete graph on n vertices, repeatedly delete a uniformly chosen remaining triangle. The terminal edge count is asymptotic to n^{3/2}/(2\sqrt2), with mean-square convergence after normalization…
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Paper Polynomial removal fails for ordered binary matrices
Disproves polynomial ordered binary matrix removal with one fixed 66\times66 zero–one pattern. Matrices can require many binary-entry changes to become pattern-free while their copy density is smaller than every…
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Paper Positive lower density of large prime gaps
For every fixed C > 0, a positive proportion of consecutive prime gaps exceed C\log pn, throughout every sufficiently large initial segment of the primes. The proportion may depend on C. Consequently, the indices…
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Paper A power improvement in the Heilbronn triangle lower bound
For every sufficiently large n, constructs n points in the unit square such that every triangle has area at least n^{-2+c} for one absolute c > 0. This disproves the conjectured almost-n−2 upper bound in…
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Paper The Campana–Peternell conjecture in dimension six
Proves the Campana–Peternell conjecture in complex dimension six: every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous. Manuscript September 25, 2026 · · cite as…
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Paper Weak normalization implies strong normalization in pure type systems
Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a β-normal form, every β-reduction sequence terminates. This resolves the…
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Paper Nonattainment of the three-marginal Coulomb Monge problem
An optimal three-particle Coulomb configuration need not be a deterministic function of the first particle, even for smooth identical spatial densities. The Monge and Kantorovich infima agree, but the Monge infimum is…
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Paper Bounded klt complements for Fano contractions
Proves the finite-rational-coefficient form of Shokurov’s bounded-klt-complement conjecture for ϵ-lc complex Fano-type pairs with nef anti-log-canonical divisor. For ϵ-lc Fano contractions over any algebraically closed…
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Paper Snaky in 21 Maker moves
Settles the Snaky achievement problem: Maker can force the six-cell Snaky shape within 21 of its own moves on the initially empty infinite square board. Maker moves first, each player claims one free cell per turn, and…
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Paper Semialgebraic universal covers and bounded domains
Proves the Kollár–Pardon conjecture: the semialgebraic universal covers of connected normal projective complex varieties are exactly products D\times\mathbb C^m\times F, with D bounded symmetric and F simply…
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Paper Topological triviality of \mu-constant surface singularities
Proves topological right-triviality for every holomorphic one-parameter family of isolated hypersurface singularities in ℂ3 with constant Milnor number, resolving the surface case of the μ-constant problem. After…
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Paper Counterexamples to infinite matroid intersection and packing/covering
Disproves the unrestricted infinite matroid intersection and packing/covering conjectures in ZFC, using two self-dual partitional matroids on a countably infinite ground set. The same examples answer Joó’s…
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