Normal stability of slow manifolds in nearly-periodic Hamiltonian systems

The paper’s Free-action stability principle (Theorem 3) is proved by combining (i) a geometric inequality relating the reduced adiabatic invariant’s quadratic growth to the normal distance with an explicit ε^{d−ν} scaling from the degeneracy index, and (ii) a near-constancy estimate for a truncated reduced adiabatic invariant over polynomial times, yielding the distance bound ε^{(N+1−d+ν)/2} and the run-off-the-edge dichotomy . The candidate solution omits the crucial ε^{d−ν} factor in the quadratic control (treating the Hessian as ε-independent) and replaces the paper’s near-constancy lemma for the reduced invariant μ^* with an unproven time-derivative bound for μ that incorrectly injects the degeneracy index into the time variation. This mismatch in ε exponents leads to an overstated barrier and a variation estimate that does not generally guarantee no-crossing when d>ν. Hence the paper’s argument is correct and complete, while the candidate proof is flawed in its scaling and justification.