On the Hofer-Zehnder Conjecture for Non-Contractible Periodic Orbits in Hamiltonian Dynamics

The paper claims Theorem 1.1 for all closed weakly monotone symplectic manifolds and sketches an argument via Z_p-equivariant Floer theory and Tate homology, culminating in a prime-gap continuation step to force simple non-contractible periodic orbits in classes p·γ (stated explicitly in Theorem 1.1) and implemented in the monotone case with the energy-shift estimate 2(p′−p)||H||<pC and Baker–Harman–Pintz bound p′−p=o(p) . However, for the weakly monotone case the manuscript acknowledges that the requisite Z_p-equivariant Floer theory has not yet been constructed and then outlines a construction, appealing to homological perturbation and asserting that the local Z_p-equivariant pair-of-pants product isomorphism from [30] carries over; key steps are asserted but not fully proved, e.g., the extension of the local isomorphism and E_1-page isomorphism in the promoted setting . In contrast, the candidate solution confines itself to established contexts (atoroidal or toroidally (negative) monotone) and prime-gap bridging using filtered Floer homology and persistence of local Floer homology under admissible iterations, and it flags the fully general weakly monotone case as open as of 2021-02-10. Given the manuscript’s reliance on unproved extensions of equivariant Floer machinery beyond the monotone case, we judge the paper’s argument incomplete in the claimed generality, while the model’s stance is aligned with the state of rigor at that time.