View source for The Mahler conjectures

'''The Mahler conjectures''' are the subject of three manuscripts dated 22 September 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]). This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== The original results ==
* '''Claim.''' The symmetric and the nonsymmetric geometric Mahler conjectures hold in every dimension, with Hanner polytopes and simplices as the respective minimizers of the volume product and all equality cases classified; the corresponding sharp functional Mahler inequalities also hold. For ''n'' ≥ 2, every symmetric polar product ''K'' × ''K''° in dimension 2''n'' has Gromov width 4 (summary in the collection's contents list).
* '''Sources''' (openai/math repository, commit <code>adc7f12</code>):
** [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026 ''The symmetric Mahler conjecture and its equality cases''];
** [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026 ''The Mahler Conjecture for General Convex Bodies''];
** [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Symplectic-Balls-in-Symmetric-Polar-Products-September-22-2026 ''Symplectic Balls in Symmetric Polar Products''].
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/087.md lean/docs/087.md] states that the Lean formalization proves |''K''||''K''°| ≥ 4<sup>''n''</sup>/''n''! for every origin-symmetric convex body in every dimension, with equality exactly for invertible linear images of Hanner bodies; the general lower bound inf<sub>''z''</sub> |''K''||(''K'' − ''z'')°| ≥ (''n'' + 1)<sup>''n''+1</sup>/(''n''!)<sup>2</sup> with equality exactly for simplices; and Gromov width 4 for int ''K'' × int ''K''° when ''n'' ≥ 2 (an embedding at capacity exactly 4 is not asserted). The functional inequalities are outside the formalized statements. The comparator statements are listed in that note.
* '''Catalogue status.''' The AIM open-problem collection ([https://github.com/MColbrook/AIM MColbrook/AIM], not affiliated with the American Institute of Mathematics) records its problem [https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/086-mahler-symmetric-four-dimensions.md 086] (the symmetric Mahler inequality in four dimensions) as solved by the symmetric manuscript, after an AI statement-level comparison on 13 October 2026 (checked 14 October 2026).
* '''Earlier published results.''' H. Iriyeh and M. Shibata proved the symmetric conjecture in dimension 3, with its equality condition ([https://arxiv.org/abs/1706.01749 arXiv:1706.01749]). For background on the volume product, see the survey of M. Fradelizi, M. Meyer and A. Zvavitch ([https://arxiv.org/abs/2301.06131 arXiv:2301.06131]).

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 13 October 2026)
|-
| 2026-10-07 || K. Kitamura, [https://github.com/KitaKen1/funk-volume-kalai-flags funk-volume-kalai-flags] (GitHub) || Use || A Lean 4 proof of Kalai's full flag conjecture: every centrally symmetric ''n''-dimensional convex polytope has at least 2<sup>''n''</sup>''n''! complete flags. Also a Lean proof of the numerical symmetric Funk-volume lower bound. The README cites OpenAI's Lean module <code>OAI.Analysis.Mahler</code> (the flux, Stokes and holomorphic mass proofs) as used in the argument. || Lean repository with statements in the style of Google DeepMind's formal-conjectures project and a Lean4Web link. The README states that the work was produced with assistance from ChatGPT and OpenAI Codex (GPT-6 Astra and GPT-6.1 Sol).
|-
| 2026-10-07 (manuscript date) || R. Gounder, [https://github.com/Kohze/openai-mathematical-frontiers-review ''Mathematical Frontiers in the OpenAI Research Release''] (GitHub) || Comparative review || Includes a case study of the Mahler manuscripts. A footnote observes that in the scope note a sentence excluding the nonsymmetric Mahler conjecture conflicts with the linked general-Mahler comparator target, which states the general inequality with simplex equality. || A 25-page review. The README states that its checks "do not independently establish the underlying release theorems".
|-
| 2026-10-13 || R. Assouline, [https://rotemassouline.github.io/symmetric-mahler/ ''Notes on OpenAI's proof of the symmetric Mahler conjecture''] ([https://github.com/RotemAssouline/symmetric-mahler source]) || Exposition || Notes rewriting the details of the symmetric proof (complexification, submanifolds of the polydisc, a lower bound on the volume of a holomorphic image, the equality case), with figures and interactive applets. || Work in progress, based on conversations with Claude. The author states: "I do not claim to have verified the proof completely."
|}

== Notes ==
* The follow-ups state AI assistance as recorded in the table.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]