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

What OpenAI claims

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)

Papers (5)

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

   5 October 2026 · Preprint, not peer reviewed · PDF

   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.

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

   5 October 2026 · Preprint, not peer reviewed · PDF

   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.

3. Hamiltonian Fixed Points Below the Stable Morse Number

   5 October 2026 · Preprint, not peer reviewed · PDF

   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.

4. A nondegenerate counterexample to the Morse-number Arnold bound

   23 September 2026 · Preprint, not peer reviewed · PDF

   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.

5. Three fixed points on the symplectic quadric threefold

   23 September 2026 · Preprint, not peer reviewed · PDF

   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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.