Emergent Closed Surfaces via Resolution-Dependent Energy Equilibrium

A foundational geometric theory proposing that the topological sphere ($S^{d-1}$) emerges not as an axiom, but as the stable equilibrium of a resolution-dependent free energy functional. This project provides the microscopic derivation of the "Marginal Spectral Universality" ($\rho \sim E^{-1}$) reported in the PhaseShift framework. This project preregisters the foundational geometric theory for the PhaseShift framework. Unlike the companion experimental study (PRL submission), this project focuses on the microscopic origin of marginal universality. We hypothesize that the topological sphere ($S^{d-1}$) is not an axiom but an emergent solution to a resolution-dependent free energy minimization problem. We further posit that the "0D defect" signature ($\rho \sim E^{-1}$) arises analytically from the critical limit of a scale-invariant inverse-square Hamiltonian ($H \sim -1/r^2$). This registration covers the theoretical derivations, the isoperimetric stability proofs, and the numerical validation of the Renormalization Group (RG) flow of the spectral dimension.