View source for Strong forms of the Arnold conjecture

'''Strong forms of the Arnold conjecture''' predict that a Hamiltonian diffeomorphism of a closed symplectic manifold has at least as many fixed points as a smooth function, or a Morse function, on the manifold has critical points, going beyond the homological lower bounds established by Floer theory. Five manuscripts in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), dated 23 September and 5 October 2026, claim counterexamples: a Hamiltonian diffeomorphism of the complex quadric threefold with three fixed points, fewer than the four critical points of every smooth function, and nondegenerate examples below the Morse number and below the stable Morse number, with unbounded deficit in real dimension 22. The collection includes a Lean formalization of the quadric counterexample. 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:
** Arnold's fixed-point questions ask how much of the critical-point theory of a small autonomous Hamiltonian persists for a general Hamiltonian diffeomorphism. Conley and Zehnder settled the torus, Floer turned the problem into a comparison of a Hamiltonian chain complex with the topology of the manifold, Hofer and Salamon and Ono treated the weakly monotone case, and work of Fukaya and Ono, Liu and Tian, and Ruan gave the general rational homological bound.
** The stable Morse formulation is recorded by Golovko; Damian exhibited manifolds whose ordinary and stable Morse numbers differ. Dimitroglou Rizell and Golovko, and independently Pöder Balkeståhl, proved the stable Morse bound for generic Hamiltonians when the symplectic area class and the first Chern class vanish on the second homotopy group. Abouzaid and Blumberg proved homological bounds in positive characteristic, and Bai and Xu obtained an integral bound that incorporates torsion.

== The original results ==
* '''Claim.''' Theorem 1.1 of ''Three fixed points on the symplectic quadric threefold'' (listed as ''A degenerate counterexample to the critical-number Arnold bound'') states that the quadric threefold carries a smooth Hamiltonian diffeomorphism with exactly three fixed points, while every smooth function has at least four critical points, the rational cup-length also being four; at least one fixed point is degenerate. Theorem 1.1 of ''A nondegenerate counterexample to the Morse-number Arnold bound'' gives a closed symplectic twelve-manifold, with nontrivial fundamental group, and a Hamiltonian with only nondegenerate fixed points, fewer than the Morse number. Theorem 1.2 of ''Hamiltonian Fixed Points Below the Stable Morse Number'' gives a simply connected closed Kähler manifold, of real dimension 1412 in the stated example, with a nondegenerate Hamiltonian having 16 fixed points fewer than the stable Morse number, and Theorem 1.1 of ''Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two'' gives simply connected closed Kähler manifolds of real dimension 22 on which the deficit is unbounded. Theorem 1.1 of ''Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number'' gives an example whose fixed-point count equals the cyclically graded integral Floer bound and lies 32 below the stable Morse number.
* ''Method, as described:'' in the simply connected examples, Smale's theorem identifies the Morse numbers with integral homology; the separation comes from distributing torsion at two primes across different integer degrees and realizing the smaller Morse counts on the fixed components of an involution.
* '''Sources.''' The manuscripts [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026 Three fixed points on the symplectic quadric threefold], [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026 A nondegenerate counterexample to the Morse-number Arnold bound], [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-October-5-2026 Hamiltonian Fixed Points Below the Stable Morse Number], [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two-October-5-2026 Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two] and [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Sharpness-of-the-Cyclic-Integral-Floer-Bound-Below-the-Stable-Morse-Number-October-5-2026 Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number] at commit <code>1581d4c</code>.
* '''Formalization, as described by the repository.''' The [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/347.md scope note] states that the formalized counterexample is a Hamiltonian diffeomorphism of the complex quadric threefold with exactly three fixed points, at least one of them degenerate, while every smooth function on that manifold has at least four critical points; the nondegenerate Morse-number construction is not included. The Comparator statement is <code>ArnoldCounterexample.lean</code>.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 20 October 2026)
|-
| 2026-10-08 || [https://github.com/mathvet/mathvet MathVet] (GitHub; review of openai/math, tables at repository commit <code>c76f1e4</code>, 14 October 2026) || Review (fidelity of the formal statements) || Rates this family "partial": the Lean statement gives the three-fixed-point Hamiltonian diffeomorphism of the quadric threefold, with every smooth function having at least four critical points, while the nondegenerate stable-Morse counterexamples have no statement. It notes that the quadric, smooth functions and Hamiltonian isotopies are defined by hand. It records a sandboxed Comparator pass for <code>ArnoldCounterexample</code>. || MathVet states that it does not claim any theorem false and reports what was machine-checked as stated.
|-
| 2026-10-10 || [https://github.com/carlostoledo1891/cert-machine/blob/a8f92f8e74893ff9afd167fd55310ad5afbc4958/corpus/openai-math/kernel-runs/38017370880/ArnoldCounterexample.json cert-machine], openai/math lane K, run 38017370880 || Verification (two kernels) || Records that Lean's default kernel and the nanoda kernel accept the <code>ArnoldCounterexample</code> solution (1,355 s). The project reports two-kernel certification of 398 of 416 challenges of the collection. || Repository.
|-
| 2026-10-10 to 2026-10-14 || [https://github.com/leanprover/con-leche/blob/90577d9e49ee3aed45da3d421bdf5dc780192304/DESIGN.md con-leche development journal] (task #348) || Verification (second kernel) || In its run over all Comparator challenges of the collection at commit <code>fd4aeeb</code>, both con-leche and Lean's official kernel accepted <code>ArnoldCounterexample</code>; the journal lists it among the accepted challenges with a large memory gap between the two checkers. || Development journal.
|-
| 2026-10-16
| J. Jardine, Lattice24, referee reviews of [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_473.json the Morse-number manuscript], [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_469.json the stable-Morse manuscript], [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_460.json the dimension-22 manuscript] and [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_506.json the cyclic Floer 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)
| All four reviews report no gap, with no minor notes. The reviews state that every section was read, with the handle-cancellation appendices of the stable-Morse manuscripts skimmed or partly read; the review of the stable-Morse manuscript reports a script rebuilding the integral homology of the example.
| 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.
* Independently, Jiao ([https://arxiv.org/abs/2609.33626 arXiv:2609.33626], 27 September 2026) constructed a Hamiltonian diffeomorphism of the standard ''S''<sup>2</sup> × ''S''<sup>2</sup> with two fixed points, a counterexample to the degenerate version of Arnold's conjecture on that manifold; the arXiv record states that the work was done with ChatGPT.
* 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]]