NON-INTEGRABILITY AND CHAOS FOR NATURAL HAMILTONIAN SYSTEMS WITH A RANDOM POTENTIAL

The paper’s main theorem asserts that, with probability tending to 1 as L→∞, the random natural Hamiltonian XL on T^d simultaneously exhibits a horseshoe and a positive-volume set of d-dimensional invariant tori inside each cube of a fixed N-grid, and moreover gives volumetric expectation bounds (≳L^d horseshoes and ≳1 inner volume) on a fixed energy window; see the statement of Theorem 1.1 and surrounding discussion of (1.2)–(1.4) and the scaling intuition . The proof strategy proceeds by (i) constructing an explicit analytic ‘seed’ potential on R^d that has both a robust Smale horseshoe and positive-volume KAM tori (Proposition 3.1) , (ii) approximating this seed by a band-limited potential (Lemma 3.2) , (iii) proving a positive density of such structures almost surely for the stationary random field on R^d (Theorem 3.5 via Definition 3.3) , and (iv) transferring to the torus by a local rescaling argument and weak convergence of the rescaled measures (Equation (5.2) and the estimates in Section 5) . However, as written, the theorem and its corollary make no explicit restriction on dimension d; for d=1 an autonomous natural Hamiltonian on T^1×R cannot exhibit a Smale horseshoe, so the statement (as printed) cannot hold in that case. The seed construction (rotor plus d−1 pendula) and the KAM input also implicitly require d≥2 (cf. Proposition 3.1’s setup) . Hence the paper lacks an essential hypothesis (“assume d≥2”). The candidate model solution correctly flags and fixes this missing dimension restriction, and outlines a plausible alternative proof based on planting microscopic copies of a fixed smooth seed via a small-ball Gaussian jet argument plus a second-moment estimate. It aligns with the paper’s scaling picture and conclusions (including the expectation bounds) but leaves technical gaps. In particular, the model’s claim that C^r-smooth KAM persistence near an elliptic equilibrium holds with r≥3 is not compatible with the differentiability thresholds invoked in the paper (r>2d+1 in Proposition 3.1) , and the reduction from approximate jets at finitely many points to uniform C^2-closeness on a ball—and the decorrelation control for the Bernoulli “good ball” indicators—would need careful justification. Consequently, both the paper (missing d≥2) and the model (technical gaps) are incomplete.