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

Claims counterexamples to strong forms of Arnold's fixed-point conjecture; the three-point quadric example is in Lean.

5 papers September 23, 2026 Explained by AI

Arnold's conjecture links two counts: fixed points of a mechanical motion on a space, and critical points of a function on it. Strong forms predict that any such motion has at least as many fixed points as some topological count. Five papers build spaces where these strong forms fail. The cleanest example is a motion on a six-dimensional quadric with only three fixed points, where any function needs four. Lean, a computer proof checker, has a formal statement of that example.

First step

1986

Vladimir Arnold, who posed the fixed-point questions

Significance

Major

It claims counterexamples to several standard strong forms of Arnold's conjecture, one of them with a Lean statement.

Trust

Formal proof listed

Field

Differential geometry

5 papers

The explanation

Take a round sphere and turn it a small angle about its vertical axis. Only the north and south poles stay put, so this motion has two fixed points. Now take the height function on the sphere. Its critical points, where the slope is zero, are the top and the bottom, again two. This is no accident. For a short motion made from a function this way, the fixed points are exactly the function's critical pointsSource 1. A motion of this kind, generated by an energy function as in classical mechanics, is a Hamiltonian diffeomorphism.

Arnold's conjecture asks whether every Hamiltonian diffeomorphism keeps at least the minimum number of critical points forced by the topologySource 2. There are several ways to count that minimum. The critical number counts critical points of any smooth function. The Morse number allows only well-behaved, nondegenerate functions. The stable Morse number also allows extra auxiliary variablesSource 3. Conley and Zehnder proved the torus case, and Fortune proved a bound on complex projective spaceSource 4. Floer's methods gave lower bounds from homology, a tool that counts holesSource 5.

The fifth paper gives the cleanest example. On the complex quadric threefold, a six-dimensional space cut out by one equation, it builds a motion with exactly three fixed points. Every smooth function there has at least four critical pointsSource 6. At least one fixed point is degenerate, meaning not well-behavedSource 7. The count of three is sharp, because Gong had already shown that every such motion on this space has at least three fixed pointsSource 8.

The other papers handle the nondegenerate versions, where every fixed point is well-behaved. The fourth paper builds a twelve-dimensional example with fewer fixed points than the Morse numberSource 9. The third gives a space with sixteen fewer fixed points than its stable Morse numberSource 10. The first makes the gap grow without limit in dimension 22. For each m it builds a space with stable Morse number 80 + 1968m and a motion with 80 + 1952m fixed pointsSource 11. For m = 1 that is 2048 against 2032, a gap of 16. For m = 10 it is 19760 against 19600, a gap of 160. The ratio 2032/2048 is exactly 127/128. The second paper, in dimension 3332, has 1,872,232 fixed points against a stable Morse number of 1,872,264, a gap of exactly 32Source 12. The key idea is torsion. It is a finer part of homology that appears only when holes are counted with whole numbers, not fractions. Torsion can force more critical points than the count of holes taken with fractionsSource 13. These examples show that fixed points need not feel that extra pressure.

How sure is this? Lean, a computer proof checker, has a formal statement of the quadric example: three fixed points, while every function has at least four. So the fifth paper proves that result, as far as the Lean statement matches the paper. The nondegenerate constructions are not in the Lean scope. They remain claims. No specialist review is published.

What it doesn't show. This does not disprove the homological, Floer-theoretic lower bounds for nondegenerate fixed points; the papers present their examples as compatible with those bounds. Only the quadric example has a Lean statement.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major. The quadric counterexample has a Lean statement; the nondegenerate constructions in very high dimensions are claims only. Floer-theoretic forms of Arnold's conjecture are untouched. No specialist review is published.

Words used

Hamiltonian diffeomorphism: The result of running, for one unit of time, a motion generated by a changing energy function.
Fixed point: A point that a map sends to itself.
Critical point: A point where a function's slope is zero in every direction.
Nondegenerate: Well-behaved: for a fixed point, its linear part has no eigenvalue 1.
Morse number: The least number of critical points of a function whose critical points are all nondegenerate.
Stable Morse number: The Morse number when extra variables with a fixed quadratic shape far out are allowed.
Symplectic manifold: A space with a rule for signed area in pairs of directions, the setting of classical mechanics.
Torsion: Parts of homology that vanish when counted with fractions but matter when counted with whole numbers.

The original papers (5)

Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two (PDF, opens in a new tab)
October 5, 2026
Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number (PDF, opens in a new tab)
October 5, 2026
Hamiltonian Fixed Points Below the Stable Morse Number (PDF, opens in a new tab)
October 5, 2026
A nondegenerate counterexample to the Morse-number Arnold bound (PDF, opens in a new tab)
September 23, 2026
Three fixed points on the symplectic quadric threefold (PDF, opens in a new tab)
September 23, 2026 Formal proof listed

Lean notes

All quoted lines (13)

For a sufficiently short autonomous Hamiltonian flow on a closed symplectic manifold, its fixed points are exactly the critical points of its Hamiltonian.
Three fixed points on the symplectic quadric threefold Back to text
The conjecture asks whether an arbitrary Hamiltonian diffeomorphism must retain the minimum number of critical points forced by the topology of the manifold.
Three fixed points on the symplectic quadric threefold Back to text
Its stable Morse number $\SM(W)$ is the minimum of $\#\Crit(F)$ over integers $k\geq0$ and smooth Morse functions $F:W\times\R^k\to\R$ such that, for $k>0$ , $F(x,z)=Q(z)\qquad\text{for all sufficiently large }|z|,$ where $Q$ is a nondegenerate quadratic form of arbitrary signature.
Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two Back to text
Conley and Zehnder proved that every Hamiltonian diffeomorphism of the standard symplectic torus $T^{2n}$ has at least $2n+1$ fixed points, and Fortune obtained the bound $n+1$ on standard $\CP^n$ .
Three fixed points on the symplectic quadric threefold Back to text
Floer's pseudoholomorphic-curve methods established homological Morse inequalities for nondegenerate Hamiltonian fixed points, including the monotone setting.
Three fixed points on the symplectic quadric threefold Back to text
There is a smooth Hamiltonian diffeomorphism $\phi$ of the closed connected symplectic six-manifold $Q^3$ such that $\#\Fix(\phi)=3<4=\Crit(Q^3) =\operatorname{cuplength}(Q^3;\mathbb Q).$
Three fixed points on the symplectic quadric threefold Back to text
At least one fixed point of $\phi$ is degenerate.
Three fixed points on the symplectic quadric threefold Back to text
The count of three is sharp.
Three fixed points on the symplectic quadric threefold Back to text
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.
A nondegenerate counterexample to the Morse-number Arnold bound Back to text
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.
Hamiltonian Fixed Points Below the Stable Morse Number Back to text
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 $\begin{aligned}\SM(M_m)&=80+1968m,\\ \#\Fix(\phi_{H_m}^1) &=\#\Fix_0(\phi_{H_m}^1;H_m)=80+1952m.\end{aligned}$
Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two Back to text
The Hamiltonian diffeomorphism has exactly $1\,872\,232$ fixed points, all nondegenerate and with contractible orbit loops, whereas the stable Morse number is $1\,872\,264$ .
Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number Back to text
The resulting rational homological lower bound and the geometric Morse bounds are different assertions: integral torsion can force more critical points than the total rational Betti number.
Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two Back to text

Other sources

openai/math, Lean scope note for family 347 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.
openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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