Tagged openai-math
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Paper Prime-factor statistics of p-1
Proves that the normalized ordered logarithms of the prime factors of p-1, counted with multiplicity, converge jointly to the Poisson–Dirichlet law \mathrm{PD}(1) as p ranges uniformly over primes up to x and…
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Paper Ostmann’s inverse Goldbach conjecture
Proves that no finite modification of the primes can be written as A+B with A,B\subseteq\mathbb Z{\ge0} each containing at least two elements. This resolves Ostmann's inverse Goldbach conjecture on additive…
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Paper The ordinary-double-point volume gap
Proves the ordinary-double-point volume-gap conjecture: every singular complex algebraic klt germ of dimension n ≥ 2, with zero boundary, has normalized volume at most 2(n-1)^n. Equality holds precisely for an…
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Paper Irrational cubic fourfolds with Hodge-theoretic and categorical K3 associations
For every sufficiently large admissible Hassett discriminant, a very general smooth complex cubic fourfold is irrational despite having both an untwisted geometric K3 category and an integral Hodge-theoretic K3…
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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 Fourier restriction for positively curved surfaces
Proves the diagonal Fourier extension conjecture for positively curved surfaces in three dimensions. For every compact smooth positively curved surface \Sigma\subset\mathbb R^3, including surfaces with boundary, the…
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Paper The exact Sobolev endpoint for Schrödinger convergence
Proves almost-everywhere convergence e^{it\Delta}f\to f as t\downarrow0 for every f\in H^{n/(2(n+1))}(\mathbb R^n) and every dimension n ≥ 2. This attains the sharp Sobolev equality case of Carleson's…
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Paper Hardness of coloring three-colorable graphs
It is NP-hard to color a three-colorable graph using any fixed number c ≥ 3 of colors. More strongly, for every fixed 0\lt \delta\lt 1/3, a deterministic polynomial-time reduction from 3SAT produces simple…
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Paper Optimal-order randomized k-server on arbitrary metrics
Establishes a randomized competitive ratio O(\log^2(k+1)) for k-server on every metric space, matching the worst-case lower-bound order. One policy serves every finite oblivious request sequence, including on…
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Paper Uniform sparsest cut: hardness and semidefinite gaps
Proves that approximating Uniform Sparsest Cut within any fixed constant factor is NP-hard, even with nonnegative rational capacities and unit demands. The Goemans–Linial semidefinite relaxation also has integrality…
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Paper The Courtade–Kumar and Hellinger conjectures
Proves the Courtade–Kumar conjecture: among Boolean functions of independent uniform bits, a single coordinate retains the most mutual information after independent bit-flip noise. A stronger theorem treats randomized…
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Paper A factor-two approximation for shortest common superstring
Gives a deterministic polynomial-time algorithm constructing a common superstring of length at most twice the optimum for every finite family of explicitly represented strings. The running time is polynomial in the full…
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Paper A cubic permanent–determinant lower bound
Proves an \Omega(n^3) lower bound for the border determinantal complexity of the n\times n permanent over ℂ. Even coefficientwise limits of determinants of affine-linear matrices require matrix size at least…
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Paper Combinatorial invariance of Kazhdan–Lusztig polynomials
Resolves the full combinatorial invariance conjecture: isomorphic Bruhat intervals in arbitrary Coxeter systems have identical equal-parameter Kazhdan–Lusztig polynomials. Thus the abstract order of the interval…
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Paper Bounded-degree coboundary expanders
Constructs arbitrarily large finite d-dimensional simplicial complexes, for every d ≥ 3, with uniformly bounded vertex degrees and uniform 𝔽2 coboundary expansion in every degree below d. Together with the known graph…
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Paper Barnette’s Hamiltonian-cycle conjecture
Proves that every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle, resolving Barnette's conjecture. Equivalently, every three-edge path in a finite simple cubic 3-vertex-connected…
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Paper Polynomial-time scheduling on three identical machines
Resolves the three-processor unit-job scheduling problem of Garey and Johnson: a deterministic polynomial-time algorithm minimizes makespan for nonpreemptive unit-length jobs with arbitrary precedence constraints on…
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Paper Planar first-passage geometry and the absence of bigeodesics
Proves that planar first-passage percolation has no doubly infinite geodesic for iid nonnegative nonatomic edge weights when the minimum of four weights has finite second moment. For exponential weights, the limit shape…
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Paper The low-temperature Sherrington–Kirkpatrick fluctuation law
For every fixed inverse temperature β > 1, determines the fluctuation scale and limiting law of the zero-field Gaussian Sherrington–Kirkpatrick log partition function. Its variance is asymptotic to $`c\beta…
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Paper Shelah's eventual categoricity and the prescribed-threshold obstruction
Proves Shelah's eventual categoricity conjecture in ZFC: for each bound on the Löwenheim–Skolem number, a uniform threshold makes categoricity of an abstract elementary class in one cardinal above that threshold imply…
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Paper Separating choiceless counting from polynomial time and witnessed choice
Confirms the Blass–Gurevich–Shelah noncapture conjecture: consistency of a linear system over 𝔽3 defines a polynomial-time query on unordered finite structures that choiceless polynomial time with counting cannot…
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Paper The Partition Principle does not imply Choice
Assuming ZF is consistent, constructs a model in which every surjective image of a set injects into that set, yet the axiom of choice fails. Choice for ordinal-indexed families still holds. From any countable transitive…
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Paper The near-boundary Birkhoff conjecture
Resolves the near-boundary Birkhoff conjecture for smooth strictly convex planar billiards of positive curvature. Such a billiard is an ellipse whenever a full grazing annulus is continuously foliated by individually…
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Paper The ionization and generalized ionization conjectures
For the full nonrelativistic Coulomb model with two electron spin states, proves that a molecule with M fixed nuclei of charges at least one and total charge Z strictly binds at most Z+CM electrons. Neutral-atom…
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Paper Exactly three mutually unbiased bases in dimension six
Proves N(6)=3, resolving Zauner's dimension-six mutually unbiased bases conjecture: three such bases exist in ℂ6, but four cannot. The exclusion is a complete certified computation under the stated binary64…
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Paper Four-dimensional disk embedding and Wall's conjecture
The unrestricted four-dimensional disk-embedding conjecture fails: framed algebraic dual spheres do not suffice to obtain disjoint locally flat spanning disks. In particular, the free group F2 is not good in the sense…
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Paper The Kervaire invariant problem at the prime three
Resolves the odd-primary Kervaire invariant problem at the prime three: exactly the standard classes with indices 0, 2, and 3 survive in the mod-three Adams spectral sequence, in stems 10, 106, and 322. Each surviving…
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Paper A smooth surface metric with no local isometric immersion in \mathbb{R}^{3}
Constructs a smooth positive-definite metric on (-1,1)^2 for which no neighborhood of the origin admits a smooth isometric immersion into ℝ3. This answers the unrestricted smooth local isometric realization problem…
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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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