Machine-Verified Spectral Intelligence: From Data Geometry to Holographic Bounds via Lean 4

We present the first complete, machine-verified proof chain connecting four fundamental phenomena in deep learning theory: neural scaling laws, attention as spectral filtering, grokking as a spectral phase transition, and holographic entropy bounds on learned representations. All proofs are formalized in Lean 4 with Mathlib and compile without axioms beyond the standard foundation. The chain establishes that a single parameter — the spectral decay exponent s determined by data regularity class — controls scaling law exponents, attention depth requirements, the timing and sharpness of grokking transitions, and the information-theoretic capacity of learned representations. The holographic capstone proves that spectral representation entropy satisfies a Bekenstein-type bound, placing a physical limit on what any finite-precision learner can encode. We validate the theory empirically: training transformers on modular arithmetic across five vocabulary sizes, we find that total compute to grokking scales as \(p^{1.22}\) (\(R^2 = 0.963\), 15 runs), with the extracted spectral exponent matching the range predicted by scaling law theory. To our knowledge, this is the first formally verified end-to-end theory linking data geometry to learning dynamics and information capacity, accompanied by quantitative empirical confirmation.