Polynomial-Time Distribution Learning for Centered Diagonal Gaussian Mixtures

We give an improper distribution learner for arbitrary-weight mixtures of centered Gaussian distributions whose covariance matrices commute and lie between the identity and twice the identity. The common orthogonal eigenbasis is unknown. The number of observations, arithmetic operations, stored real words, and instructions in the returned sampling program is bounded by a universal polynomial in the dimension, the number of components, inverse total-variation accuracy, and logarithmic inverse failure probability. The model permits exact real arithmetic, square roots, logarithms, exponentials, comparisons, and independent continuous uniform draws; all loops have deterministic caps. No finite-bit sampling claim is made. The basis-discovery step uses the observable fourth moment to construct a trace-minimizing sum-of-squares majorant and a minimum-Frobenius positive semidefinite Gram operator. This operator approximates the latent covariance-commutator energy. Projecting an isotropic Gaussian symmetric matrix into its low-energy space, then certifying a deliberately generated eigenvalue gap, gives a stable approximate common basis. The procedure requires neither component weights nor covariance separation. An approximate-basis guarantee is transferred to distribution learning by fresh-sample coupling at constant confidence, followed by approximate-density comparison among independently trained candidates. The known-axis diagonal learner and its proof are included as a complete dependency.