Bumpy Metric Theorem in the Sense of Mané for Non-Convex Hamiltonians

The paper establishes, via a local normal form and a finite-dimensional control/transversality scheme, that for generic potentials the zero-energy level is regular and every zero-energy periodic orbit with a neat time has a restricted linearized return map outside any prescribed meagre conjugacy-invariant subset Υ ⊂ Sp(2d); in particular, nondegeneracy follows when Υ contains matrices with eigenvalue 1 (Theorems 1–2) . It further provides a generic avoidance-of-ΣH mechanism under Hypothesis 1 (Theorem 3), leading to Theorem 4 . By contrast, the model’s proof relies on an Abraham–Sard–Smale universal-section argument and a global surjectivity claim for the differential of u ↦ L(θ,H+u) onto sp(2d). This approach omits key ingredients emphasized by the paper: the necessity to work with admissible potentials to keep the orbit fixed when varying the return map , and the presence of an algebraic exceptional set KD in the linear-control step that must be explicitly excluded to obtain surjectivity . The model also glosses over Fréchet/Banach issues and does not justify the cokernel surjectivity needed for Sard–Smale on the constrained universal section. Consequently, the paper’s argument is sound and complete for the stated results, while the model’s solution is incomplete and makes unjustified surjectivity and transversality claims.