AI-assisted results, 12 Nov: Hexagon

Your AI and this story

Key facts

• Petty’s polar projection conjecture (Sihan Chen, 2611.00401): in dimension n ≥ 3 the polar projection body is a dilate of K exactly for origin-centred ellipsoids, ‘settling Petty’s polar projection conjecture’, and also the translative constant volume conjecture of Horváth and Lángi; provenance: ‘The final key derivation and ideas were completed by AI’; the author ‘verified and re-derived’ the derivation (ChatGPT GPT-6 Astra). Companion: fixed points of the centroid body operator are exactly origin-centred ellipsoids (2611.00335, GPT-6.1 Sol)
• Ionization (Alec Shelley, Charles F. Yang, Daniel M. Tartakovsky, 2611.00325): helium binds at most two electrons and lithium at most four, both sharp; Lieb (1984) proved fewer than 2Z+1, and ‘the sharp upper bound has remained unknown for any nucleus other than hydrogen’. Paper: arguments ‘drafted with use of gpt-6-astra and fable-5.2, then audited, simplified, and assembled using Pharos’ (built on Rethlas), with a main agent and seven proof workers; computer-assisted; human understanding ‘some’
• Tejasvi Singh Tomar, 25 deposits (2611.00371–00395, each also on Zenodo; mixed text, understanding ‘all’; ‘Generative AI tools were used to find and prove the results, to write the programs and the formal proofs, and to draft the text’): kissing numbers τ₅ ≤ 43 and τ₇ ≤ 132 (were 44 and 134) from exact Lasserre-hierarchy certificates, also proved in Lean (00376); Steiner ratio ≥ 0.8564 (was 0.8559; Gilbert–Pollak conjecture √3/2 open; 00379); thrackles have at most 4(n−1)/3 edges (was 18(n−1)/13; Conway’s conjecture n; 00383); Lebesgue universal covering ≥ 0.8366 (00382); rank of 4×4 matrix multiplication over F₂ ≥ 35 (00372); Smale’s sixth problem for six bodies, generic masses, Lean (00377); planar four-body central configuration counts (00378); 24-cell conjecture in two classes (00390); no automorphism of order 2, 3 or 5 for an extremal [72,36,16] code, so its group is trivial (00380); PPT² conjecture for spin-factor maps (00386, 00387); surface-code threshold lower bounds (00371); which fine-grained lower bounds survive the refutation of 3SUM and APSP (00388); also 00373, 00374, 00375, 00381, 00385, 00389, 00391, 00392, 00393, 00394, 00395
• Langlands over function fields (Boyang Hu, 2611.00398, 00399, 00402): every semisimple elliptic parameter of a split group over a global function field is the parameter of a cusp form (a converse to V. Lafforgue), for classical groups in all characteristics and exceptional groups above stated bounds; ‘The mathematical arguments of the three papers of this series were developed with GPT-6 Astra Pro’, rewritten with Claude Fable 5.2 and reviewed ‘line by line, by GPT and Claude models’; human understanding ‘some’
• Qiyue Tang (GPT-6-Pro runs, understanding ‘none’): the Humphreys support conjecture for p > h (00364, 98 pages, ~506 minutes of research run), the quantum Kazhdan–Lusztig conjecture for quantum loop algebras (00322), Lusztig’s Fourier matrix conjecture for noncrystallographic Coxeter groups (00362, computer-assisted), Bonnafé–Geck–Iancu–Lam conjectures A, A⁺, B on type B cells (00405); Rethlas-OSV reviews ‘planned’
• Yair Lavi (human understanding ‘all’; ‘The proof of this conjecture was carried out by GPT-5.6-sol, GPT-6 Astra, and Claude Fable 5, under the guidance of the author’): the Marcus–Minc permanent conjecture (00352), Kim and Roush’s 1981 conjecture max per(I−A) = 3·2^(k−2) in odd order (00355), a counterexample to the Foregger–Sinkhorn tie-point conjecture in dimension 8 (00354), his own 2018 conjectures on stochastic matrices of bounded rank (00353)
• Other named problems: diameter n(n−1)/2 of the Cayley graph of S_n with LRX generators, ‘a conjecture open since at least 2011’, proof obtained by an agentic system on GPT-6 Astra (Uskov, Chervov, Levkovich-Maslyuk et al., 00333); Euler’s 1747 conjecture on sums of two integers with prime factors ≡ 1 mod 4, Codex (Kohlmeyer et al., 00370); Kominers, a Fibonacci analogue of the Fifteen Theorem, ‘GPT-6 Astra identified a key step’, Lean (00357); Stanikzai, α(J(13,5,3)) = 33, so χ(R¹²) ≥ 39, Claude Code (00358); optimal packing of eleven unit squares, formalized Voronoi-cell proof, ‘The initial proof was generated by Astra’ (Manasseh Ahmed, 00359); Venkat, Erdős’s Mersenne-divisor doubling ratio ≥ 3 by Opus 5.5 (00366); Kim, ≥ 0.672468 of zeta zeros isolated, understanding ‘none’ (00318); twinheterodox, at least one of β(4), …, β(32) irrational (00346, 00336, Codex)
• Primarily AI-generated, no human understanding: Jaswal’s Claude Code research system, an effective André–Oort theorem (00339) and a proof that Z is Diophantine in Q via congruent-number twists (00331); Lopatto, supersymmetric analysis of a critical random band matrix, two volumes ‘with GPT 6 Astra, GPT 6.1 Sol, and Claude Opus 5.5’ (00367, 00368); MacGregor, tight 39-point code in the octonionic projective plane, Lean (00340); Lewis, genuinely multipartite bound entanglement of three qubits (00329); Slutsky, the Borel Ruziewicz problem, ‘obtained by AI agents … Konstantin Slutsky organized the AI agents but claims no mathematical contributions’ (00309)