ON THE DEGENERATE ARNOLD CONJECTURE ON T2m × CPn

The paper proves that if H0 on T^{2m}×CP^n satisfies max{2||H0||∞, ||∇_{CP^n}H0||∞} < 1/2, then the time-one map φ has at least 2m+n+1 contractible fixed points, by constructing a modified Hamiltonian action functional AH, verifying Condition (A) (via Lemma 3), and applying continuation-invariant Conley index and relative cup-length computations to the zero Hamiltonian; this yields the claimed bound (Theorem 1 and its proof sketch ). The candidate solution reaches the same numerical bound using Floer-theoretic spectral invariants on the monotone manifold T^{2m}×CP^n under the shorter assumption osc(H) < ℏ (ensured by 2||H0||∞ < 1/2 after normalization), leveraging the LS-type strict inequalities in the quantum-limit window and spectrality to produce CL(M)+1 distinct contractible periodic orbits. Thus the conclusions agree; the approaches are substantially different (Conley index vs. Floer spectral invariants), and the model’s method does not use the extra CP^n-gradient smallness that the paper needs to establish Condition (A) in its framework. We therefore judge both correct (different proofs).