Geometric Substrate Theory: From Shell Spectrum to Thermodynamic Geometry

We develop a derivation-first geometric substrate theory in which all macroscopic quantities are constructed from the spectral data of a shell operator K together with its geometric deformation structure. The framework is derivation-driven, struc-turally constrained, and avoids heuristic instanton gases or fitted constants. Starting from the eigenpairs {kn, fn} of K, we construct: • A variational Gradient Relaxation Path (GRP) branch Uvar(q) that provides a controlled upper bound to the substrate energy as a function of the projection q (Milestone 1).• A substrate renormalization group (SRG) flow α(τ ) with UV and IR fixed points,defined from the variational energy landscape (Milestone 2).• An explicit transfer matrix constructed from the shell spectrum, followed by a quadratic response theorem for general finite L, showing that α−1(ϵ) is strictly increasing, strictly convex, and crosses any fixed value exactly once (Milestone 3).• An emergent SU(2) spin structure arising from holonomy (Milestone 4).• A thermodynamic geometry sector in which response functions and coarse-grained free-energy structure arise from the substrate, with a working hypothesis relating transfer-matrix and loop-gas descriptions (Milestone 5). All constructions are derived from the shell spectrum and transfer-matrix deformation, with explicit hypotheses stated where needed (positivity, spectral gap, coarse-grained identification).