A Unified Spectral Theory of Machine Learning: Neural Scaling Laws, Generalization, and Architecture Design via the Latent Framework

We present a unified spectral theory of machine learning built on the Latent framework (\(\Lambda\), \(\rho\)). The Latent Number \(\rho > 1\) — the spectral decay rate of the data distribution or model operator — serves as a single organizing parameter that governs: (1) neural scaling exponents via \(\alpha = \log\rho / (2\beta)\); (2) non-vacuous generalization bounds via effective dimension \(d_{\text{eff}} = \log(1/\varepsilon)/\log\rho \ll p\); (3) the double descent peak at the \(\rho = 1\) phase boundary; (4) the feature-learning vs kernel regime transition; (5) diffusion model denoising step counts; (6) optimal transformer head counts; (7) minimax sample complexity; (8) lottery ticket spectral structure; (9) certified adversarial robustness radii; and (10) few-shot sample complexity via shared Latent structure. The deductive backbone — 143 theorems establishing the algebraic consequences of spectral decay assumptions — is machine-verified (0 errors) in the Lean 4. The modeling assumptions (geometric spectral decay of real data distributions) are stated as explicit hypotheses. The theory makes falsifiable predictions connecting \(\rho\) to every quantity of interest; we provide initial empirical evidence on image data.