Quantitative Gaussian limits of periodic O(N) models at critical and low temperatures

We prove a quantitative Gaussian approximation for local spin coordinates in the zero-field nearest-neighbour O(N) model on periodic cubic lattices in dimension d ≥ 3. For a fixed spatial window and the first p ≤ N coordinates, the unnormalized Euclidean Wasserstein distance satisfies W1 ≤ C√p(N−1/2 + sM), uniformly on compact parameter ranges with positive directional couplings and inverse temperatures at or above the large-N critical value βc,J; sM is the finite-volume saddle mass. The Gaussian target has independent coordinate columns, each β−1/2 times an anisotropic Gaussian free field plus an independent Gaussian constant whose variance vanishes at criticality. On fixed windows containing a nearest-neighbour star, finite-spin lower bounds give the exact joint convergence criterion p = o(N) and √p sM → 0 as M,N → ∞. The bound also gives convergence for every choice of periodic infinite-volume accumulation states precisely when p = o(N). At a fixed positive gap from the limiting critical inverse temperature, we also obtain sharper rates with explicit dependence on the number of observed coordinates. In dimensions d ≥ 5 this bound is C(p/N + √p M−d). For fixed p, its order N−1 + M−d is sharp on star-containing windows, and the accumulation-state error has order N−1 on every nonempty window. This first preprint version includes the main article, complete Supplementary Material (Appendices A–G), and compilable LaTeX sources.