Associative Memory and Generative Diffusion in the Zero-noise Limit

The paper’s four main claims (i)–(iv) about zero-noise limits for gradient flows, structural/topological stability, generic bifurcations in one- and two-parameter families, and probabilistic tracking by small-noise diffusions are stated correctly and largely supported by standard results. However, parts of the exposition (notably the proof of Proposition 9 and the corollary on stochastic stability) conflate general small random perturbations with gradient diffusions and rely on Boltzmann–Gibbs structure without making this restriction explicit. The model’s solution proves the same targets with a different, more robust route (invariance plus Lyapunov monotonicity and stable-manifold arguments), and flags the needed regularity/closure conditions for pushforwards. Hence both reach the correct conclusions; the paper would benefit from clarifying assumptions, but the core statements stand.