Singular perturbation in heavy ball dynamics

The paper proves the equivalence: boundedness of all gradient-flow trajectories ⇔ uniform boundedness (for small mass ε) of all heavy-ball trajectories, under f ∈ C^{1,1}_{loc}, lower bounded, and definable. The proof uses (a) a compactness/finite-horizon regular-perturbation lemma to pass from heavy-ball to gradient flow on any [0,T] and (b) a definability-based KL/length argument plus a finite induction on critical values (via definable Morse–Sard) to go from gradient flow to heavy-ball boundedness. See Theorem 1 and its proof, together with Lemmas 1–3 . By contrast, the candidate solution relies on a “uniform-in-time Tikhonov theorem” to claim sup_{t≥δ} ||x_ε(t)−x(t)|| ≤ C(ε+e^{-c δ/ε}) without the strong stability hypotheses needed for infinite-interval singular perturbations; this is incompatible with the paper’s explicit example where heavy-ball and gradient-flow limits differ (hence no uniform convergence on [δ,∞)) . The candidate also asserts definability is unnecessary, whereas the paper’s (⇐) direction uses definability crucially (KL inequality and finiteness of critical values).