Hilbert–Sobolev regularization

Hilbert–Sobolev Regularization is an advanced mathematical book that develops a unified theory of regularization as a constitutional science of structured intelligence. It shows how Hilbert geometry, Sobolev smoothness, operator theory, spectral filtering, variational design, adaptive regularization, PDE-constrained learning, neural fields, and cognition can be understood under one framework for governing admissible function worlds. The book moves from foundational functional analysis to frontier research programs, presenting regularization not as a technical patch, but as a deep law for building stable, interpretable, high-value systems in science, AI, and advanced reasoning.