Multiscale Anti-Localization of GOE Spectra by Low-Complexity Hamiltonian Families

Abstract

Let Lambda_D be the ordered spectrum of a D x D Gaussian orthogonal ensemble matrix, scaled so that its equilibrium measure is the semicircle law on [-1,1]. We prove that every deterministic real linear subspace V_D of Herm(D) with subpolynomial dimension p_D = D^{o(1)} has a universal multiscale spectral approximation floor: with probability at least 1 - C D^{-c’}, its normalized squared spectral distance from Lambda_D is at least c log(D)/D^2. The result is uniform in the deterministic subspace. For D = 2^n, it applies in particular to the complete noncommuting k-local Pauli family for every fixed k, including the full two-local Hamiltonian class and the diagonal quadratic Ising subfamily. The proof attaches a growing vector of coupled Chebyshev responses to each spectrum. A quantitative multivariate GOE central limit theorem makes these responses approximately Gaussian in M = D^{1/20} modes. The bounded spectral part of a p_D-dimensional Hamiltonian family has response covering entropy O(p_D log D), whereas a weighted Gaussian small-ball estimate above mode p_D has cost Omega(p_D log^2 D). A local semicircle-frame estimate, including an edge-strip bound, transfers response separation back to Euclidean spectral distance. Noncommutativity is handled by the trace Frechet identity D[Tr phi(H)][K] = Tr(phi’(H)K). For the optimal equal-weight semicircle quantizer m_D, let Delta_D(V_D) denote its normalized squared distance from the model spectrum. We further prove the two-scale law E C_D(V_D) asymp Delta_D(V_D) + log(D)/D^2, which separates deterministic model-form deficiency from irreducible GOE multiscale fluctuation.