AI-assisted results, 6 Nov: arXiv of 6 Nov and Hexagon of 5 Nov

Your AI and this story

Key facts

• Own entries: Tusnády’s problem claimed settled twice, with AI help (2611.06998, 2611.08455), Singmaster’s problem: new bound by Claude Opus 5.5 (Hexagon 2611.00106), Hopfield critical capacity bracketed, by Claude Opus 5.5 (Hexagon 2611.00105); Najman’s degrees 9–22 (Hexagon 2611.00120) added to Elliptic-curve torsion in degrees 6–9, mostly by GPT-6.1 Sol
• Real algebraic geometry: Suvrit Sra proves the generalized Lax conjecture for the Borcea–Brändén mixed determinant (spectrahedral hyperbolicity cone) and answers four Borcea–Brändén problems negatively (Hexagon 2611.00117, mixed, understanding ‘some’): ‘GPT Astra (Pro) found the construction of Theorem 1.1’ after a failed counterexample search; also a new proof of van der Waerden’s permanent inequality by monotone averaging, ‘GPT-6 Astra Pro … carried out the derivations’ (2611.00118)
• Algebra and topology: L’vov–Kaplansky conjecture for multilinear polynomials of degree four, ‘developed through two research conversations with GPT-6-Pro … seven hours’ (Qiyue Tang, 2611.00119); Qiuyu Ren, a 2-generator 3-relator universal group and #ⁿ(S²×S²) unrecognizable for every n ≥ 2, ‘The main theorems were produced by ChatGPT’s agent “dot.” … less than half a day’ (2611.00112); terminal symplectic singularities for parabolic base affine spaces, Tom Gannon, ‘this pdf is entirely AI generated’ with ChatGPT Astra (2611.00107)
• Number theory and complexity (Hexagon): Vaibhav Tulsyan with Claude Code, parity of π(x) in time x^o(1) and class numbers / quadratic non-residues in 2^O(n/√log n), both conditional on Anthropic’s #SAT algorithms of 27 Oct and Lean-checked on that development (2611.00116, 2611.00115); Hadamard difference sets in C₂×C₂ᵤ² and 2-automatic Baum–Sweet Hankel determinants, Claude, ‘No professional mathematician has reviewed the note yet’ (2611.00102, 2611.00101); Fuchs’ extremal problem for 0 < λ ≤ 3 (2611.00126)
• Other Hexagon 5 Nov: Grothendieck Montpellier notes, seventy-five folders of restated statements proved in Lean with Claude Opus/Sonnet 5.5, readings not verified against the facsimile (2611.00125); twistor correspondence for analytic Einstein germs (2611.00123); sup-norm ∂̄ estimates on convex domains in C², ‘primarily AI-generated’, fully human-checked (2611.00121); Hurwitz interval family without degree-5 WSPR numerator, Lean (2611.00127); transcendence of the 3x+1 conjugacy map on Sturmian words (2611.00108); Voblint, Isabelle/HOL soundness (2611.00109); human-written or editing-only: 2611.00124, 00122, 00114, 00093 (Claude found Example 2.7), 00085, 2610.00681, 00059
• Graph theory and combinatorics (arXiv 6 Nov): Liu–Liu–Ning, every Hamiltonian r-regular graph with r ≥ 17 has a second Hamilton cycle (was r ≥ 23, Haxell–Seamone–Verstraëte), ‘The proofs in this paper were developed with substantial contributions from ChatGPT (5.5 and 6 Astra)’ (2611.06985); perfect q-hash codes beat the probabilistic bound for every q ≥ 3, settling part of a Xing–Yuan conjecture, ‘Claude (Anthropic) suggested the directions that led to Theorems 1.5 and 1.7’ (2611.08230); maximal partial Room squares, construction ‘produced by ChatGPT Plus’ (2611.07117); double-factorial consecutive-pattern packings, Lean (2611.06937); simpler proof for a noncommutative harmonic oscillator determinant offered unprompted by Claude Opus 5.5 (2611.07626)
• Geometry, analysis, TCS: Arman–Bondarenko–Prymak–Radchenko, a diameter-one set in R⁵ not covered by six diameter-one balls, ‘The example presented in this paper was found using generative AI tools’ (2611.08198); Medwed proves the Chang–Lyons–Ni length conjecture for path signatures, ‘central arguments … generated with the assistance of AI (ChatGPT 5.6 Sol)’ (2611.08030); Alrabiah–Goyal–Guruswami refute a conjecture on list-decoding Reed–Solomon codes, ‘Initial versions of our key statements were established by GPT-6 Astra Pro’ (2611.07607); Riis, linear degree lower bounds for Nullstellensatz and Polynomial Calculus with ‘Astra and Claude for sustained proof search’ (2611.07280)
• Smaller AI-credited steps: CM counterexamples found with ChatGPT 5.6 Pro (Kim–Yu, 2611.07259); Fermat quotients with CM Jacobians, GPT-6 Astra Pro extended the proof (2611.08176); void percolation, tree-embedding idea by GPT 5.6 (2611.07196); stochastic heat equation weak uniqueness via GPT-6.1 (2611.07310); free Tₙ-Heyting algebras, implication ‘found by Claude Sonnet 4.6’ (2611.07915); 2×4 LEGO counts with Claude (2611.06986); worldsheet skein sequence ‘found by GPT-5.6 Sol’ (2611.07455)