Motion Closes the Rank Gap: Bayes-Optimal Mobile Sound-Field Estimation

Recovering a room's sound field from microphone recordings is basic to room acoustics and spatial audio. Most methods improve the estimator, with better priors, kernels and learned physics models. Unfortunately, a static array of m microphones resolves at most m independent directions of a K-mode field, so for K>m, absent sparsity, no estimator, noise level or recording length recovers the rest. In contrast, we treat the loss as one of acquisition. Moving the same array makes all K directions identifiable at equal observation time. Under a Bayes-optimal model its risk then falls inversely with observation time, with a constant we derive in closed form as a trace. That constant is governed by a contraction factor, which approaches one in some rooms as K grows. Therefore we prove which rooms keep it bounded away from one for every K. Facets in all four axis directions, each at least one grid spacing longer than the aperture, suffice, and the length threshold is sharp. Simulations with no fitted parameter match the predicted constant, and at the longer observation times the moving array beats the best static position across the room under independent estimators as well as our own. On a real 16-microphone array in a furnished room, under a pre-registered protocol, 16 surveyed positions resolve more independent directions than any static position can, and the held-out error falls from one position to 16 and ends below the best-predicted static position.