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OpenAI releases mathematical results produced by its internal frontier model

OpenAI says an independent advisory group at the Institute for Advanced Study helped inform how it released the results.

NidalNI
Josh BickettJB
maharshiMA
3 Sources, 9h ago, first seen 9h ago

TLDR

OpenAI says it is releasing a broad range of mathematical results produced by an internal frontier model. It says advice and public recommendations from the Advisory Group on Mathematics and Artificial Intelligence at the Institute for Advanced Study informed the release. A user discussing the results claims they span 372 families and 722 papers, and says some still need checking.

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3 Sources, first seen 9h ago

77 likes7 comments48 saves7 reposts

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3 Sources

Nidal@nselmiSome top results from the OpenAI math drop: NUMBER THEORY & ARITHMETIC GEOMETRY • Quasi-Riemann [003]: Zeta and every Dirichlet L-function are zero-free for Re(s) > 7/8, with corresponding Hecke results and uniform exclusion of Landau–Siegel zeros. • Hodge and Tate [032]: Rational Hodge holds for CM abelian varieties and products of projective complex K3 surfaces, yielding Tate over finite fields and Hodge standard in every characteristic for abelian varieties. • BSD and Goldfeld [002, 006]: Full BSD holds at low Selmer corank and for density-one quadratic twists of every elliptic curve over Q; twist ranks 0 and 1 each have density ½, with mean rank tending to ½. • Hilbert’s tenth over Q [004]: No algorithm decides whether an arbitrary integer-coefficient polynomial has a rational solution. • Imaginary-quadratic modularity [030]: Every elliptic curve over every imaginary quadratic field is modular, with matching local parameters everywhere. • Catalan’s constant [005]: Catalan’s constant is irrational. • Irrationality exponent of π [017]: The exponent is exactly 2, implying convergence of the Flint–Hills series. ALGORITHMS & COMPUTATIONAL COMPLEXITY • Unique Games [102]: Khot’s conjecture holds, alongside direct proofs of optimal Max-Cut and Vertex Cover hardness and constant-factor hardness for Min-UnCut and directed feedback vertex set. • Log-space derandomization [103]: L = RL = BPL; bounded-error randomness adds no computational power in logarithmic space. • Integer multiplication [109]: Deterministic multiplication takes O(n(log n)^(1−κ)), with κ = 2⁻¹⁸², breaking the conjectured n log n bound in the multitape bit model. • Matrix multiplication [107]: ω ≤ 2.25 over C, ω < 2.258 outside finitely many positive characteristics, and ω < 2.371055 over every fixed field. • Exact matching [120]: Randomized maximum-cardinality matching takes (n+m)^(1+o(1)) word time in general simple graphs, with success probability at least ⅔. • Subset Sum [138]: A randomized classical algorithm solves worst-case polynomial-bit instances in O(2^(0.49n)) word-RAM time, with success probability at least ⅔. • Polynomial factorization [142]: Dense polynomials over prime fields factor deterministically in polynomial bit time, without GRH or auxiliary oracles. COMBINATORICS & DISCRETE GEOMETRY • Erdős and Szemerédi [159]: Divergent reciprocal sums force arbitrarily long arithmetic progressions, with quasipolynomial bounds on progression-free sets. • Hadwiger and Colin de Verdière [157]: Both fractional coloring bounds fail, while list-chromatic number remains bounded by a universal constant times the largest clique-minor order. • Plane coloring [158]: Every five-coloring contains a monochromatic unit-distance pair; the plane’s chromatic number is 6 or 7. • Borsuk [156]: A compact subset of R⁹ cannot be covered by ten sets of strictly smaller diameter. GROUPS, TOPOLOGY & OPERATOR ALGEBRAS • Kaplansky and related counterexamples [196, 197]: Torsion-free group algebras violate zero-divisor and direct-finiteness conjectures, with companion constructions producing nonsofic groups and refuting surjunctivity and determinant conjectures. • Baum–Connes and Kadison–Kaplansky [285]: Coefficient-free reduced Baum–Connes fails in rational injectivity and surjectivity; a torsion-free group’s reduced C*-algebra contains a nontrivial projection. • Thompson’s group F [248]: The group of dyadic piecewise-linear interval homeomorphisms is nonamenable. • Free group factors [287]: All interpolated free group factors are isomorphic, including L(F∞), with common fundamental group R>0. • Cannon [246]: Hyperbolic groups with boundary S² act properly and cocompactly by isometries on hyperbolic three-space; torsion-free cases are closed hyperbolic three-manifold groups. • Hilbert–Smith [304]: A locally compact second-countable Hausdorff group acting faithfully and continuously on a connected finite-dimensional manifold without boundary is a Lie group. • Grothendieck homotopy hypothesis [312]: ∞-groupoids associated with every Grothendieck coherator in the Ara–Henry convention recover the homotopy theory of spaces. GEOMETRY & HARMONIC ANALYSIS • Mahler [087]: Hanner polytopes and simplices minimize symmetric and nonsymmetric volume products in every dimension, with all equality cases classified. • Symplectic width [087]: Every symmetric polar product K × K° has Gromov width 4 in dimension 2n, for n ≥ 2. • Kakeya [074]: The maximal conjecture holds in three dimensions, and every four-dimensional Kakeya set has Hausdorff dimension 4. • Falconer [073]: Every compact subset of Rᵈ with Hausdorff dimension above d/2 determines distances of positive measure, for every d ≥ 2. • Yau uniformization [338]: Every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to Cⁿ. PDE & MATHEMATICAL PHYSICS • Relativistic Vlasov–Maxwell [362]: The 3D one-species system has unique global smooth solutions for admissible smooth data with compactly supported particle density and finite-energy fields with bounded derivatives of every order. • De Giorgi [375]: Entire solutions of Δu = u³−u valued in (−1,1) and strictly monotone in one direction in R⁸ depend on one linear coordinate. • Hot spots [369]: First nonconstant Neumann eigenfunctions on smooth bounded simply connected planar domains have no interior critical points, placing all global extrema on the boundary. • Forced Navier–Stokes computation [376]: Smoothly forced viscous flows simulate arbitrary Turing machines; a designated particle reaches a fixed region exactly when the machine halts. • Strong cosmic censorship [264]: Near rotating subextremal two-ended Kerr data, generic smooth vacuum data admit no future continuous nondegenerate extension with locally square-integrable weak connection.9h
Josh Bickett@josh_bickettOpenAI basically gave an unreleased model ~4,000 unsolved math problems and let it think for ~3 hours on each one. It came back with 372 families of results and 722 papers. Claims include progress/resolutions on the quasi-Riemann hypothesis, Hilbert’s 10th problem over Q, Catalan’s constant, Goldfeld’s conjecture, Deligne–Drinfeld, and a bunch of stuff I’m not qualified to pronounce. Some are formally verified in Lean, some still need checking. If even a good fraction of this survives verification, this is completely insane.8h
maharshi@maharshiishort 15s explainer videos made with H3 max for 300+ results from openai's math catalogue: https://gokayfem.github.io/openai-math-explained/1h
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    3 Sources

    Nidal@nselmiSome top results from the OpenAI math drop: NUMBER THEORY & ARITHMETIC GEOMETRY • Quasi-Riemann [003]: Zeta and every Dirichlet L-function are zero-free for Re(s) > 7/8, with corresponding Hecke results and uniform exclusion of Landau–Siegel zeros. • Hodge and Tate [032]: Rational Hodge holds for CM abelian varieties and products of projective complex K3 surfaces, yielding Tate over finite fields and Hodge standard in every characteristic for abelian varieties. • BSD and Goldfeld [002, 006]: Full BSD holds at low Selmer corank and for density-one quadratic twists of every elliptic curve over Q; twist ranks 0 and 1 each have density ½, with mean rank tending to ½. • Hilbert’s tenth over Q [004]: No algorithm decides whether an arbitrary integer-coefficient polynomial has a rational solution. • Imaginary-quadratic modularity [030]: Every elliptic curve over every imaginary quadratic field is modular, with matching local parameters everywhere. • Catalan’s constant [005]: Catalan’s constant is irrational. • Irrationality exponent of π [017]: The exponent is exactly 2, implying convergence of the Flint–Hills series. ALGORITHMS & COMPUTATIONAL COMPLEXITY • Unique Games [102]: Khot’s conjecture holds, alongside direct proofs of optimal Max-Cut and Vertex Cover hardness and constant-factor hardness for Min-UnCut and directed feedback vertex set. • Log-space derandomization [103]: L = RL = BPL; bounded-error randomness adds no computational power in logarithmic space. • Integer multiplication [109]: Deterministic multiplication takes O(n(log n)^(1−κ)), with κ = 2⁻¹⁸², breaking the conjectured n log n bound in the multitape bit model. • Matrix multiplication [107]: ω ≤ 2.25 over C, ω < 2.258 outside finitely many positive characteristics, and ω < 2.371055 over every fixed field. • Exact matching [120]: Randomized maximum-cardinality matching takes (n+m)^(1+o(1)) word time in general simple graphs, with success probability at least ⅔. • Subset Sum [138]: A randomized classical algorithm solves worst-case polynomial-bit instances in O(2^(0.49n)) word-RAM time, with success probability at least ⅔. • Polynomial factorization [142]: Dense polynomials over prime fields factor deterministically in polynomial bit time, without GRH or auxiliary oracles. COMBINATORICS & DISCRETE GEOMETRY • Erdős and Szemerédi [159]: Divergent reciprocal sums force arbitrarily long arithmetic progressions, with quasipolynomial bounds on progression-free sets. • Hadwiger and Colin de Verdière [157]: Both fractional coloring bounds fail, while list-chromatic number remains bounded by a universal constant times the largest clique-minor order. • Plane coloring [158]: Every five-coloring contains a monochromatic unit-distance pair; the plane’s chromatic number is 6 or 7. • Borsuk [156]: A compact subset of R⁹ cannot be covered by ten sets of strictly smaller diameter. GROUPS, TOPOLOGY & OPERATOR ALGEBRAS • Kaplansky and related counterexamples [196, 197]: Torsion-free group algebras violate zero-divisor and direct-finiteness conjectures, with companion constructions producing nonsofic groups and refuting surjunctivity and determinant conjectures. • Baum–Connes and Kadison–Kaplansky [285]: Coefficient-free reduced Baum–Connes fails in rational injectivity and surjectivity; a torsion-free group’s reduced C*-algebra contains a nontrivial projection. • Thompson’s group F [248]: The group of dyadic piecewise-linear interval homeomorphisms is nonamenable. • Free group factors [287]: All interpolated free group factors are isomorphic, including L(F∞), with common fundamental group R>0. • Cannon [246]: Hyperbolic groups with boundary S² act properly and cocompactly by isometries on hyperbolic three-space; torsion-free cases are closed hyperbolic three-manifold groups. • Hilbert–Smith [304]: A locally compact second-countable Hausdorff group acting faithfully and continuously on a connected finite-dimensional manifold without boundary is a Lie group. • Grothendieck homotopy hypothesis [312]: ∞-groupoids associated with every Grothendieck coherator in the Ara–Henry convention recover the homotopy theory of spaces. GEOMETRY & HARMONIC ANALYSIS • Mahler [087]: Hanner polytopes and simplices minimize symmetric and nonsymmetric volume products in every dimension, with all equality cases classified. • Symplectic width [087]: Every symmetric polar product K × K° has Gromov width 4 in dimension 2n, for n ≥ 2. • Kakeya [074]: The maximal conjecture holds in three dimensions, and every four-dimensional Kakeya set has Hausdorff dimension 4. • Falconer [073]: Every compact subset of Rᵈ with Hausdorff dimension above d/2 determines distances of positive measure, for every d ≥ 2. • Yau uniformization [338]: Every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to Cⁿ. PDE & MATHEMATICAL PHYSICS • Relativistic Vlasov–Maxwell [362]: The 3D one-species system has unique global smooth solutions for admissible smooth data with compactly supported particle density and finite-energy fields with bounded derivatives of every order. • De Giorgi [375]: Entire solutions of Δu = u³−u valued in (−1,1) and strictly monotone in one direction in R⁸ depend on one linear coordinate. • Hot spots [369]: First nonconstant Neumann eigenfunctions on smooth bounded simply connected planar domains have no interior critical points, placing all global extrema on the boundary. • Forced Navier–Stokes computation [376]: Smoothly forced viscous flows simulate arbitrary Turing machines; a designated particle reaches a fixed region exactly when the machine halts. • Strong cosmic censorship [264]: Near rotating subextremal two-ended Kerr data, generic smooth vacuum data admit no future continuous nondegenerate extension with locally square-integrable weak connection.9h
    Josh Bickett@josh_bickettOpenAI basically gave an unreleased model ~4,000 unsolved math problems and let it think for ~3 hours on each one. It came back with 372 families of results and 722 papers. Claims include progress/resolutions on the quasi-Riemann hypothesis, Hilbert’s 10th problem over Q, Catalan’s constant, Goldfeld’s conjecture, Deligne–Drinfeld, and a bunch of stuff I’m not qualified to pronounce. Some are formally verified in Lean, some still need checking. If even a good fraction of this survives verification, this is completely insane.8h
    maharshi@maharshiishort 15s explainer videos made with H3 max for 300+ results from openai's math catalogue: https://gokayfem.github.io/openai-math-explained/1h
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