Why Learning Rediscovers the Closed-Form Diagonal Regularizer

Recovering modal amplitudes from a few sensors requires regularization, and the shape of the penalty decides how strongly each mode is shrunk. The best shape depends on the truncation noise that discarded modes add to every measurement, which is unobserved and could be room-specific. We identify a diagonal saturation principle: for a single snapshot, isotropic truncation noise makes the Bayes-optimal Tikhonov shape a closed-form power law Gamma_k proportional to lambda_k^|s|, set by the prior alone and independent of the domain. The off-diagonal anisotropy equals the Herfindahl index H of the discarded-mode power exactly, Weyl's eigenvalue counting law fixes the number of discarded modes and hence H, and Berry's random-wave conjecture enters only through a diagonal term D, which we measure directly. Together they predict an approximately flat loss landscape over the penalty exponent, which leaves a learned diagonal regularizer little to gain from tuning the exponent per room. On simulated acoustic rooms the closed form stays within 5.83% of a per-room power-law oracle across observation windows, and three learned diagonal architectures track its error, the closest within a fraction of a point and none more than 3pp above it, while learning qualitatively different spectra. In the truncation regime no tested method consistently improves on the closed form, including Learned Iterative Ridge, which couples modes across layers. The framework extends to heat diffusion through a known Green's-function correction with no new free parameters.