The degenerate Arnold conjecture: critical-number and rational cup-length lower bounds for Hamiltonian fixed points

Theorem 1.1: on the complex quadric threefold there is a smooth Hamiltonian diffeomorphism with exactly three fixed points, at least one degenerate, while every smooth function has at least four critical points and the rational cup length is four; this disproves both unrestricted degenerate bounds, and contractibility conditions cannot rescue them. The construction composes a Hamiltonian involution fixing $Q^2\cong S^2\times S^2$ with a short invariant flow, using a three-critical-point function on $S^2\times S^2$. The count three is sharp by Gong (2025). It says nothing against the nondegenerate homological (Floer) bounds. Jiao (arXiv, 27 Sep 2026, after this paper) independently reports a two-fixed-point example on $S^2\times S^2$.

The release says: 'Ma [Theorem 1.5] asserted that the minimum number of fixed points over all Hamiltonian diffeomorphisms always equals Crit(M). Theorem 1.1 contradicts that assertion' (degenerate paper) and 'Ma's preprint asserts that the minimum number of fixed points of a nondegenerate Hamiltonian diffeomorphism always equals the ordinary Morse number ... Theorem 1.1 contradicts that unrestricted assertion' (Morse-number paper). The other side: R. Ma, Proofs on Arnold conjectures, arXiv 0808.0613v7 (2013), Theorem 1.5, claims these lower bounds hold in general. Ma's preprint is unpublished; no referee has ruled between them. Theorem 1.1: on the complex quadric threefold there is a smooth Hamiltonian diffeomorphism with exactly three fixed points, at least one degenerate, while every smooth function has at least four critical points and the rational cup length is four; this disproves both unrestricted degenerate bounds, and contractibility conditions cannot rescue them. The construction composes a Hamiltonian involution fixing $Q^2\cong S^2\times S^2$ with a short invariant flow, using a three-critical-point function on $S^2\times S^2$. The count three is sharp by Gong (2025). It says nothing against the nondegenerate homological (Floer) bounds. Jiao (arXiv, 27 Sep 2026, after this paper) independently reports a two-fixed-point example on $S^2\times S^2$.