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.
released 2026-10-05 | 2 theorems · 4 lemmas · 12 proofs · 8,992 words |
PLAY LEVEL 1 »(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.
released 2026-10-05 | 2 theorems · 3 lemmas · 10 proofs · 6,134 words |
PLAY LEVEL 2 »(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.
released 2026-10-05 | 2 theorems · 3 lemmas · 12 proofs · 9,014 words |
PLAY LEVEL 3 »(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.
released 2026-09-23 | 1 theorem · 6 lemmas · 9 proofs · 9,160 words |
PLAY LEVEL 4 »(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.
released 2026-09-23 | 1 theorem · 2 lemmas · 8 proofs · 4,483 words |
PLAY LEVEL 5 »(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.