Result 347, Differential geometry

Counterexamples to stable-Morse and strong Arnold fixed-point bounds

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.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

How many points must return to their starting positions under motion generated by a Hamiltonian? These manuscripts report counterexamples to proposed bounds linking that count to the minimum number of critical points of functions on the underlying space.

What changes?

In real dimension 22, the manuscript reports simply connected closed Kähler manifolds: compact complex-geometric spaces without boundary or noncontractible loops. For every integer m at least one, their stable Morse number is 80 plus 1968 times m, but a Hamiltonian diffeomorphism has exactly 80 plus 1952 times m fixed points. All are nondegenerate, with contractible orbit loops. The deficit is 16 times m, hence unbounded in fixed dimension; the count is at most 127/128 of the stable Morse number.

What does that help mathematicians do?

The stable Morse number minimizes the number of nondegenerate critical points when auxiliary quadratic variables are allowed. Falling below it rules out using that function-based invariant as a universal fixed-point lower bound. Separately, the complex quadric threefold reportedly admits exactly three fixed points although every smooth function needs at least four critical points. At least one fixed point is degenerate, so this example addresses unrestricted bounds, not bounds restricted to nondegenerate fixed points.

Are there practical applications?

The immediate value is foundational for symplectic geometry and Hamiltonian dynamics. Nondegenerate fixed points have no fixed direction in the linearized return map, so the dimension-22 counterexamples cannot be dismissed as an artifact of degeneracy. They constrain which topological quantities can support universal fixed-point guarantees, rather than supplying a practical simulation method.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

5 manuscripts

Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two

October 5, 2026 19 pages

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+1968m80+1968m and a Hamiltonian diffeomorphism with exactly 80+1952m80+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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two-October-5-2026,
  author = {{OpenAI}},
  title = {{Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two-October-5-2026/hamiltonian-fixed-points-below-stable-morse-number.pdf}{OAI:Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two-October-5-2026}},
  year = {2026}
}

Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number

October 5, 2026 16 pages

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 2321\,872\,232 fixed points, all nondegenerate and with contractible orbit loops, whereas the stable Morse number is 1 872 2641\,872\,264. Thus the stronger stable Morse bound fails by exactly 32, even when the cyclic integral bound is sharp.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Sharpness-of-the-Cyclic-Integral-Floer-Bound-Below-the-Stable-Morse-Number-October-5-2026,
  author = {{OpenAI}},
  title = {{Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Sharpness-of-the-Cyclic-Integral-Floer-Bound-Below-the-Stable-Morse-Number-October-5-2026/sharp-cyclic-integral-floer-bound-below-stable-morse-number.pdf}{OAI:Sharpness-of-the-Cyclic-Integral-Floer-Bound-Below-the-Stable-Morse-Number-October-5-2026}},
  year = {2026}
}

Hamiltonian Fixed Points Below the Stable Morse Number

October 5, 2026 19 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-October-5-2026,
  author = {{OpenAI}},
  title = {{Hamiltonian Fixed Points Below the Stable Morse Number}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-October-5-2026/paper.pdf}{OAI:Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-October-5-2026}},
  year = {2026}
}

A nondegenerate counterexample to the Morse-number Arnold bound

September 23, 2026 21 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026,
  author = {{OpenAI}},
  title = {{A nondegenerate counterexample to the Morse-number Arnold bound}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/paper.pdf}{OAI:A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026}},
  year = {2026}
}

Three fixed points on the symplectic quadric threefold

September 23, 2026 12 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026,
  author = {{OpenAI}},
  title = {{Three fixed points on the symplectic quadric threefold}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/paper.pdf}{OAI:A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/347.md.

Counterexamples to stable-Morse and strong Arnold fixed-point bounds

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

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.

Comparator links

Result Comparator statement
Three-fixed-point Arnold counterexample ArnoldCounterexample.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.