Archive

Kourovka Problem 3.46: Exactly two maximal locally soluble normal subgroups

• Mathematics

A group with exactly two maximal locally soluble normal subgroups, resolving Kourovka Problem 3.46.

AI-led proof of Rényi and Tsallis entropy monotonicity for Bernoulli sums

• Mathematics
• Theoretical Computer Science

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.

Short Egyptian fractions

OpenAI

• Mathematics

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.

GPT-6 Astra proves the rational-exponents conjecture

• Mathematics

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.

Anthropic reports: Claude reconstructs NIST elliptic-curve seed generation

• Mathematics
• Theoretical Computer Science

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 Goldbach exceptional set: an exponent 7/12 and conditional refinement

János Pintz, Ingo Althöfer

• Mathematics

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.

The Blume-Capel Model as a φ⁶ Field Theory: A Monte Carlo and Flow Model Study

Ejaaz Merali, Ryan Larson, Ehsan Khatami, Richard T. Scalettar

• Theoretical Physics

Merali, Larson, Khatami and Scalettar report phase diagrams for a two-dimensional soft-spin φ⁶ Blume–Capel model as spin softness varies. Normalizing-flow samples with finite-size scaling locate continuous, first-order and tricritical transitions with accuracy comparable to canonical and extended-ensemble Monte Carlo on the same model.

Positivity and negativity for h-bases for n

Wouter van Doorn, Pietro Monticone, Melvyn B. Nathanson

• Mathematics

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.

Explicit positive lower density for the Collatz Conjecture

Kenny Lau

• Mathematics
• Theoretical Computer Science

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.

A counterexample to the generalized Viterbo conjecture on the spectral distance in a disc cotangent bundle

Filip Broćić, Baptiste Serraille, Ibrahim Trifa

• Mathematics
• Theoretical Physics

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.

Discovering state-of-the-art quantum circuits with large language models

Andi Gu, Alek H. Kemeny, Susanne F. Yelin, Mikhail D. Lukin

• Mathematics
• Theoretical Computer Science
• Theoretical Physics

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.

A quasi-Riemann hypothesis for Dedekind zeta functions

Dean Cureton, Jesse Thorner

• Mathematics
• Theoretical Computer Science

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.

AI-assisted sharp support criterion for variance-proxy sub-Gaussian laws

• Mathematics
• Theoretical Computer Science

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.

Non-Abelian quantum Hall states at strong Landau level mixing: finding anyons with first-principles AI

Pierre-Antoine Graham, Ahmed Abouelkomsan, Liang Fu

• Theoretical Physics

Graham, Abouelkomsan and Fu report neural-network variational Monte Carlo evidence for Ising-type non-Abelian order in fully spin-polarized two-dimensional Coulomb electrons at filling ν=3/2, including Landau-level mixing. Ground-state degeneracies, entanglement and trial-state overlaps favor an anti-Pfaffian phase in the studied liquid regime.

Scoped AI contribution: Strinz–Claude reports: Möbius invariance of spherical Ramsey sets

• Mathematics
• Theoretical Computer Science

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.

Vu-Le reports: uniqueness and stability of Gerver’s moving sofa

• Mathematics
• Theoretical Computer Science

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.

An Upper Bound of 9/4 for the Matrix Multiplication Exponent

OpenAI

• Mathematics
• Theoretical Computer Science

The manuscript reports that the complex matrix-multiplication exponent is at most 9/4.

Navot: AI-assisted extensions of the 9/4 matrix multiplication bound to fields and rings

• Mathematics
• Theoretical Computer Science

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.

Vu-Le reports: optimality and uniqueness of the ambidextrous sofa

• Mathematics

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.

Honeycomb Crystal in a Mass-Asymmetric Electron-Hole Bilayer

Shengdu Chai, Wangqian Miao, Yuqiang Li, Lei Wang, Xi Dai, X. C. Xie

• Theoretical Physics

Chai and collaborators report neural-network variational Monte Carlo evidence that mobile-electron correlations favor a honeycomb electron–hole crystal over triangular order at intermediate densities in a strongly mass-asymmetric bilayer.

A Central Limit Theorem for High-Temperature SYK Model

Yoshinori Kamijima, Shuta Nakajima, Ryoichiro Noda, Kôhei Sasaya

• Mathematics
• Theoretical Physics

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.

Lie groups are countably representable

Nicolas Monod, Andreas Thom

• Mathematics

The model-origin deformation/Puiseux construction with prescribed residues used in the countable-representability argument, separating the authors’ general Lie-group extensions.

Physical Yukawa Couplings in Heterotic String Compactifications

Giorgi Butbaia, Damián Mayorga Peña, Justin Tan, Per Berglund, Tristan Hübsch, Vishnu Jejjala, Challenger Mishra

• Theoretical Physics

Butbaia and collaborators report normalized Yukawa-coupling calculations in specified standard-embedding heterotic compactifications, comparing geometric and period-integral methods with calculations using a machine-learned approximate Ricci-flat metric and harmonic representatives.

Transformer Wave Function for two dimensional frustrated magnets: emergence of a Spin-Liquid Phase in the Shastry-Sutherland Model

Luciano Loris Viteritti, Riccardo Rende, Alberto Parola, Sebastian Goldt, Federico Becca

• Theoretical Physics

Viteritti, Rende, Parola, Goldt and Becca report numerical evidence for an intermediate gapless spin-liquid phase in the two-dimensional Shastry–Sutherland model, using transformer variational wave functions and extrapolations of order parameters and triplet excitation energies.

Solving an Open Problem in Theoretical Physics using AI-Assisted Discovery

Michael P. Brenner, Vincent Cohen-Addad, David Woodruff

• Theoretical Physics

Brenner, Cohen-Addad and Woodruff report six analytical methods for the integral controlling radiation from a specified Garfinkle–Vachaspati cosmic-string loop family, including exact spectral representations and a large-harmonic asymptotic formula with angular dependence.

Learning the Simplicity of Scattering Amplitudes

Clifford Cheung, Aurélien Dersy, Matthew D. Schwartz

• Theoretical Physics

Cheung, Dersy and Schwartz report a transformer-based simplifier that reduces long spinor-helicity expressions and produces new compact five-point scalar and graviton amplitude expressions, alongside recovering the known five-gluon Parke–Taylor expression.

Resummation of the C-Parameter Sudakov Shoulder Using Effective Field Theory

Matthew D. Schwartz

• Theoretical Physics

Matthew Schwartz reports a soft-collinear effective-theory factorization theorem for the C-parameter distribution near C = 3/4, computes the one-loop jet and soft ingredients, and presents matched next-to-leading-logarithmic plus next-to-leading-order predictions.

Single-minus gluon tree amplitudes are nonzero

Alfredo Guevara, Alexandru Lupsasca, David Skinner, Andrew Strominger, Kevin Weil

• Theoretical Physics

Guevara, Lupsasca, Skinner, Strominger and Weil report nonzero tree-level single-minus gluon amplitudes for special half-collinear configurations in Klein space or complexified momenta, with a closed product formula in a specified decay region.

Presenting a proof of the Kannan-Lovasz-Simonovits conjecture

Pierre Bizeul, Boaz Klartag, Joseph Lehec

• Mathematics

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.

A Strongly Aperiodic Monotile in Three Dimensions

Ioannis Tsiokos

• Mathematics

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.

Finite time blowup for Navier–Stokes

OpenAI

• Mathematics

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.

Proudfoot–Perrault reports: a laser-method exponent below 2.3710449

• Mathematics
• Theoretical Computer Science

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.

Morris–Claude reports: Snaky is a first-player win on 15×15

• Mathematics

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.

More than two thirds of the zeta zeros are simple and on the critical line

Levent Alpöge, Ralph Furman

• Mathematics

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.

No percolation at criticality on quasi-transitive graphs

OpenAI

• Mathematics

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.

A proof of an identity for the critical exponents of jamming

Giorgio Parisi, Francesco Zamponi

• Mathematics
• Theoretical Physics

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.

The (3, 4, ∞) modular family of 2-tori, completed at its three special points, is a complex structure on S⁶.

• Mathematics

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.

Small Counterexamples to the Gaussian Moments Conjecture

Christopher D. Long

• Mathematics

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.

An explicit power saving for the exact discrete Fourier transform

OpenAI

• Mathematics
• Theoretical Computer Science

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.

AlphaEvolve: A coding agent for scientific and algorithmic discovery

Alexander Novikov, Ngân Vũ, Marvin Eisenberger, Emilien Dupont, Po-Sen Huang, Adam Zsolt Wagner, Sergey Shirobokov, Borislav Kozlovskii, Francisco J. R. Ruiz, Abbas Mehrabian, M. Pawan Kumar, Abigail See, Swarat Chaudhuri, George Holland, Alex Davies, Sebastian Nowozin, Pushmeet Kohli, Matej Balog

• Mathematics
• Theoretical Computer Science

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.

The Mahler Conjecture for General Convex Bodies

OpenAI

• Mathematics

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.

A counterexample to Hadwiger's conjecture

OpenAI

• Mathematics

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.

Faster sorting algorithms discovered using deep reinforcement learning

Daniel J. Mankowitz, Andrea Michi, Anton Zhernov, Marco Gelmi, Marco Selvi, Cosmin Paduraru, Edouard Leurent, Shariq Iqbal, Jean-Baptiste Lespiau, Alex Ahern, Thomas Köppe, Kevin Millikin, Stephen Gaffney, Sophie Elster, Jackson Broshear, Chris Gamble, Kieran Milan, Robert Tung, Minjae Hwang, Taylan Cemgil, Mohammadamin Barekatain, Yujia Li, Amol Mandhane, Thomas Hubert, Julian Schrittwieser, Demis Hassabis, Pushmeet Kohli, Martin Riedmiller, Oriol Vinyals, David Silver

• Mathematics
• Theoretical Computer Science

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.

AI-assisted counterexample to Grothendieck’s group-scheme question

• Mathematics

Akhil Mathew’s contribution gives a noncommutative finite free group scheme of order four over a nonreduced base that is not killed by four.

Finite time blowup for the Euler equation

OpenAI

• Mathematics

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.

Milne's rationality conjecture for abelian varieties

OpenAI

• Mathematics

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.

Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank

OpenAI

• Mathematics

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.

Hilbert’s tenth problem over the rational numbers

OpenAI

• Mathematics
• Theoretical Computer Science

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.

Goldfeld's analytic density conjecture and the 2-converse for elliptic curves

OpenAI

• Mathematics

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.

Ordinary two-point correlations of multiplicative functions

OpenAI

• Mathematics

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.