Rigorous Foundations of the Spectral RG Functional: Kernel Uniqueness, Boundary Invariance, and Variational Structure

This paper provides a mathematically rigorous foundation for the spectral renormalization group (RG) framework. It proves uniqueness of the admissible gap kernel, establishes strict boundary invariance, derives the local Fisher--Rao metric structure, proves Γ-convergence of regularized spectral functionals, and verifies full consistency of the projected gradient and Hessian on the eigenvalue simplex. The framework is internally closed and requires no external parameters.