Spectral Determinant Unification: A Closed Geometric Solution to Gravity, Gauge Couplings, and Fermion Mass Generation

This work presents a mathematically closed spectral–geometric framework in which the full low–energy structure of fundamental physics emerges from a single determinant functional of an elliptic Dirac operator. Within one unified variational principle, the theory provides explicit and constructive solutions to three central problems of modern theoretical physics: • emergence of Einstein gravity from spectral curvature invariants• determination of gauge couplings and renormalization flow• derivation of fermion and Higgs mass spectra from operator eigenvalues All effective constants arise as heat–kernel coefficients and spectral invariants.No phenomenological parameters are inserted by hand. The formulation is rigorous and fully self-contained, based on: • heat–kernel asymptotics (Seeley–DeWitt expansion)• zeta–regularized determinants• functional trace methods• spectral extremization principles Closed analytic expressions for the characteristic length scale and effective couplings are derived and evaluated numerically, yielding values consistent with Planck/GUT scales. The framework can be interpreted as a spectral completion of effective field theory and provides a compact, parameter–minimal route toward geometric unification. This document is written in theorem–proof style and is intended as a reference for researchers working on: spectral geometry, quantum gravity, spectral action, effective field theory, and mathematical physics.