Counterexamples to stable-Morse and strong Arnold fixed-point bounds
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Disproves stable-Morse lower bounds for nondegenerate Hamiltonian fixed points: on simply connected closed Kähler manifolds of real dimension 22, the deficit below the stable Morse number is unbounded. A separate Hamiltonian diffeomorphism of the complex quadric threefold has exactly three fixed points, fewer than the four critical points required of every smooth function. (OpenAI’s Lean note)
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Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
Five other possible matches are listed below the card, among them Theorem 1.1 in Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number and Theorem 1.1 in A nondegenerate counterexample to the Morse-number Arnold bound.
For every integer m\geq1, there exist a simply connected closed Kähler manifold (M_m,\omega_m) of real dimension 22 and a smooth one-periodic Hamiltonian H_m such that all fixed points of \phi_{H_m}^1 are nondegenerate and
We disprove the stable Morse lower bound for nondegenerate Hamiltonian fixed points with examples in fixed real dimension twenty-two. For every integer m ≥ 1, we construct a simply connected closed Kähler manifold with stable Morse number \(80+1968m\) and a Hamiltonian diffeomorphism with exactly \(80+1952m\) fixed points, all nondegenerate and with contractible orbit loops. The deficit is therefore unbounded, and the fixed-point count is at most 127/128 of the stable Morse number.
We construct a simply connected closed Kähler manifold of real dimension 3332 and minimal Chern number one whose Hamiltonian fixed-point count attains the cyclic integral Floer bound while falling below the stable Morse number. The Hamiltonian diffeomorphism has exactly \(1\,872\,232\) fixed points, all nondegenerate and with contractible orbit loops, whereas the stable Morse number is \(1\,872\,264\). Thus the stronger stable Morse bound fails by exactly 32, even when the cyclic integral bound is sharp.
We disprove the stable Morse lower bound for nondegenerate Hamiltonian fixed points by constructing a simply connected closed Kähler manifold with sixteen fewer fixed points than its stable Morse number. All the corresponding periodic orbits are contractible. The construction combines a Hamiltonian involution with integral homology torsion at two different primes; its proof uses finite-dimensional Morse theory and complex blowups.
We construct a smooth one-periodic Hamiltonian on a closed symplectic twelve-manifold whose time-one map has fewer fixed points than the manifold’s ordinary Morse number. Every fixed point is nondegenerate and has a contractible Hamiltonian trajectory. This disproves the Morse-number form of the Arnold conjecture. The deficit can be arbitrarily large among twelve-dimensional examples.
We construct a smooth Hamiltonian diffeomorphism of the complex quadric threefold with exactly three fixed points, at least one of which is degenerate. The critical number and the unit-inclusive rational cup length of this manifold are both four. Thus the example disproves the unrestricted critical-number and rational cup-length forms of the Arnold conjecture.
One Lean statement in this family
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three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Other mathematical statements in OpenAI’s Lean library
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Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).
Their note:main gives only the second claim: a Hamiltonian diffeomorphism φ of the quadric Q³ ⊂ ℂP⁴ with exactly 3 fixed points (one provably degenerate) while every smooth function has ≥ 4 critical points (4 ≤ criticalNumber). The headline nondegenerate stable-Morse counterexample on 22-dimensional Kähler manifolds with unbounded deficit has no statement (docs: ‘the separate nondegenerate Morse-number construction is not included’).
Definitions the group asks readers to compare with the paper:Q³ is a hand-built subtype of Projectivization ℂ (Fin 5 → ℂ), not a Mathlib manifold; smooth functions are those with local C^∞ lifts to the unit isotropic cone (IsSmoothFunction), criticality via horizontal lifts, criticalNumber := ⨅ h, (criticalSet h).encard, and Hamiltonian-ness via an explicit lifted isotopy HamiltonianPath with omega v w := 2·Im Σ conj(v_j) w_j tested only on horizontal vectors; all must be trusted to match the standard symplectic notions
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
The gap they name:The stable-Morse clause (unbounded deficit for nondegenerate Hamiltonian fixed points on simply connected closed Kähler 22-manifolds) is not stated.
“NARROWER — DECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note says so.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
The critical-number form of Arnold’s fixed-point conjecture predicts at least as many fixed points as the minimum number of critical points of a smooth function. The formalized counterexample is a Hamiltonian diffeomorphism of the complex quadric threefold with exactly three fixed points, while every smooth function on that manifold has at least four critical points. At least one fixed point is degenerate. The separate nondegenerate Morse-number construction is not included.
AI-drafted comparisons, with no recorded human review
Three-fixed-point Arnold counterexampleArnoldCounterexample
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a Hamiltonian diffeomorphism of the complex quadric threefold with exactly three fixed points, fewer than the four critical points every smooth function has. The nondegenerate construction of a companion paper is not included.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
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Three-fixed-point Arnold counterexampleArnoldCounterexample
Three reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
“CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: f(x,y) = e.x + (x-p).y on S^2 x S^2 has exactly three critical points, q_0 = (p,-e) degenerate (Hessian rank 2); NOT the passage to Hamiltonian fixed points on Q^3 nor Crit(Q^3) = 4”
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Papers elsewhere in the release that this family’s papers cite
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Papers elsewhere in the release that cite this family’s papers
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Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.
VibeMathed, VibeMathed: math problems solved with AI, 11 October 2026. Source
Contested: the release states that Ma’s Theorem 1.5 on Arnold fixed-point counts is contradicted by its own Theorem 1.1 (three entries flagged).
Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
The Lean statement proves only the critical-point lower bound, not the paper’s fixed-point count of three against four, though the review says the disproof itself is unaffected.
Questions awaiting review
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Family 347The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Address a question raised in an independent assessment
Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state less than the headline, and OpenAI’s Lean note says so.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
Re-run through Comparator with Lean’s kernel: accepted (the check took 751 s; peak memory 10.0 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory.
Exact calculations for finite claims, cert-machine
Their verdict
“CERTIFIED”
Claim tested or proved
Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: f(x,y) = e.x + (x-p).y on S^2 x S^2 has exactly three critical points, q_0 = (p,-e) degenerate (Hessian rank 2); NOT the passage to Hamiltonian fixed points on Q^3 nor Crit(Q^3) = 4”
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.