Aftermath bot: New page: the near-boundary Birkhoff conjecture, family 147, with public follow-ups

2026-10-19T22:21:06Z

New page: the near-boundary Birkhoff conjecture, family 147, with public follow-ups

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'''The near-boundary Birkhoff conjecture''' asks whether a strictly convex planar billiard that is integrable near its boundary must be an ellipse. Two manuscripts in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), both dated 24 September 2026, claim this for tables with smooth boundary of positive curvature: when a neighbourhood of the boundary is continuously foliated by smooth convex caustics, and when a grazing annulus of the phase space is continuously foliated by invariant curves. The family has no Lean formalization. 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.

== Background ==
* As recounted in the introductions of the manuscripts:
** The conjecture is traditionally attributed to Birkhoff, and an early printed formulation appears in Poritsky's work. Lazutkin constructed a positive-measure family of caustics near the boundary for sufficiently smooth positively curved tables, and Marvizi and Melrose constructed a grazing first integral accurate up to a flat error; neither gives a full foliation. Mather's obstruction, for boundaries whose curvature vanishes somewhere, shows that positive curvature is substantive.
** Bialy proved that a foliation of the entire phase cylinder by invariant curves forces a circle. Innami, and Arnold and Bialy, treated caustics with rotation numbers approaching 1/2, and Bialy and Mironov proved rigidity for centrally symmetric tables whose foliated region reaches the four-periodic curve. Perturbative rigidity near ellipses is due to Avila, De Simoi and Kaloshin, to Kaloshin and Sorrentino, to Kaloshin, Koudjinan and Zhang, and to Koval.

== The original results ==
* '''Claim.''' Theorem 1.2 of ''Rigidity of Smooth Billiards with a Continuous Caustic Collar'' states that a bounded strictly convex planar domain whose boundary is a smooth closed curve of positive curvature is an ellipse if a neighbourhood of the boundary is foliated, through a homeomorphism, by smooth closed convex caustics; no differentiability of the foliation across its leaves is assumed. Theorem 1.1 of ''Continuous Phase Foliations Create Analytic Caustic Collars'' states the same conclusion when an invariant open annulus of the billiard map, containing a full strip at the grazing end, is foliated through a homeomorphism by essential simple closed curves, each invariant under the map.
* ''Method, as described:'' in the rigidity manuscript, rational caustics give curves of periodic orbits whose actions satisfy a high-frequency system, and an analytic Newton construction makes the boundary analytic; finite reflected chains then control the monodromy of a complex continuation. On the complexified boundary, zeros or poles of the meromorphic normal force an elliptic form, and the remaining case is excluded through growth estimates, moment bounds on the total angular change and an argument with Brownian paths. The phase-foliation manuscript proves that the invariant curves are graphs, builds an analytic collar of convex caustics from the periodic graphs at small rational rotations, and then applies the rigidity theorem of its companion.
* '''Sources.''' The manuscripts [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Rigidity-of-Smooth-Billiards-with-a-Continuous-Caustic-Collar-September-24-2026 Rigidity of Smooth Billiards with a Continuous Caustic Collar] and [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026 Continuous Phase Foliations Create Analytic Caustic Collars] at commit <code>1581d4c</code>.
* '''Formalization.''' The repository has no Lean scope note for this family.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 20 October 2026)
|-
| 2026-10-13
| Knowledge Sense, ''Exact counterexamples to the Barker–Larman problem in every dimension at least four'', manuscript dated 12 October 2026, in [https://github.com/KnowledgeSense/math/tree/b95a52a54597e8b01c1a28486f2a9b29a3a54c58/preprints/Exact-Counterexamples-to-the-Barker-Larman-Problem-in-Every-Dimension-at-Least-Four-October-12-2026 KnowledgeSense/math]
| Uses (method)
| Claims, in every dimension ''d'' ≥ 4, origin-symmetric strictly convex bodies of revolution with real-analytic boundary that are not balls but cut every hyperplane tangent to the unit ball in sections of the same volume, a negative answer to the Barker–Larman problem. The manuscript states that its reduction of a variational equation to two successive differences and a scalar compatibility condition, its quasi-Newton terms and its finite-order initializers have direct antecedents in the two OpenAI manuscripts, whose caustic-collar hypotheses it does not use.
| Manuscript in a public GitHub collection; the manuscript states that AI-assisted tools were used in its research and preparation.
|-
| 2026-10-16
| J. Jardine, Lattice24, referee reviews of [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_145.json the rigidity manuscript] and [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_423.json the phase-foliation manuscript] in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995])
| Review (AI-assisted audit)
| Both reviews report no gap. The review of the rigidity manuscript describes itself as a partial audit, in which sections 4 and 6 to 10 were not verified. The review of the phase-foliation manuscript states that all five sections were read; its one minor note is that a displayed formula in the proof of a compatibility lemma gives only a right inverse, the actual inverse involving an additional projection, which does not affect the stated norm bound.
| Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0.
|}

== Notes ==
* The earlier results are due to the authors named in the Background section, as recounted by the manuscripts.
* 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]]

The near-boundary Birkhoff conjecture - Revision history

Aftermath bot: New page: the near-boundary Birkhoff conjecture, family 147, with public follow-ups