Spontaneously stochastic Arnold’s cat

The paper proves that, with one random perturbation ξ injected at the cut-off scale n=N, the pushforward measures μ^(N) converge (in the product topology) to a universal limit μ that is a product of four pieces μI μG μR μB; notably μR is the product of uniform (Lebesgue) laws on the red set R (which includes the coordinate (0,2)), as stated in Eqs. (16)–(20) and the definitions of I, G, R, B in Eqs. (8)–(9) . The candidate solution’s key claim—that for any fixed lattice coordinate (n,t) there exists N large enough so that u_n^{(N)}(t) is independent of ξ and the limit is a Dirac mass—contradicts the paper’s explicit finite-N formula for u_0^{(N)}(2), which depends linearly on u_N(0)=u^0_N+ξ for all N (see (A.2) in the SM), hence cannot be deterministic for large N . The paper’s proof represents u_n^{(N)}(t) as P_{n,t}^{(N)}(A)ξ plus a deterministic term and shows that the path-polynomial coefficients P_{n,t}^{(N)}(x) grow and separate in a way that forces all nonzero Fourier modes to vanish, yielding uniform Lebesgue limits on any finite set of red nodes, and thus the claimed product structure; see (A.6)–(A.8) and the asymptotic ratios (A.21)–(A.25) . In particular, the projection of μ^(N) at time t=1 already has one deterministic coordinate and infinitely many independent uniform ones, consistent with the limiting picture (Eq. (21)) . Therefore the paper’s conclusions are consistent and the model (candidate) argument rests on an incorrect “domain of dependence” computation and a flawed binomial closed form.