When Does Fourier Diagonalize the Physics? Calibrated Meters for Fixed versus Learned Representations in Classification and in Operator Learning, and What the Learned Part Is Worth

Code and paper. A gated coupling of a fixed transform dictionary and a learned basis (from the companion study 'Fixed, Learned, or Both?') is read as a calibrated meter in two settings. Classification: the gate follows the information gap in bits between the learned and fixed features with a rectified saturating law (r = 0.99 on a family with a gap controlled by design, 0.98 on held-out benchmark tasks); energy-based (codec) selection discards discriminative coefficients; the task-trained basis is not symbolically readable. Physics (1-D operator learning): the off-diagonality of a fixed Fourier basis is a heterogeneity meter; a Sturm-Liouville family discovered without an oracle returns the physical coefficient (corr 1.0000) and flags its own mis-specification; the DMD residual (a data-driven Laplace transform) is a nonlinearity meter and a gated learned branch fires with the same rectified law; structure beats depth; sparse regression names the term exactly; the named term, integrated rather than applied in one explicit step, extrapolates in horizon and amplitude and never creates energy, while the black box and the one-step surrogate diverge. All figures regenerated by deterministic CPU scripts; every number in the paper is generated from the results files.