# 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):

- [Three fixed points on the symplectic quadric threefold](../../preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/paper.pdf)

## 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](../ComparatorChallenges/ArnoldCounterexample.lean) |