Spectral Geometric Prism Model: A Unified Field Framework via Finite-Action Constraints

We propose the Spectral Geometric Prism Model (SGPM), a unified field frameworkbuilt upon finite-action topological constraints to resolve infrared divergences arisingfrom noncompact physical spacetime. Single compactification via finite Yang–Millsaction yields a topological homeomorphism between finite-action gauge configurationspaces over R4 and smooth gauge fields on S4. Under this isomorphism, standardspectral triples can be rigorously extended to compact four-manifolds. The heat-kernelasymptotic expansion of a single universal spectral trace functional simultaneously recovers gravitational Einstein–Hilbert dynamics, full Standard Model Yang–Mills gaugesectors, and Higgs mass generation terms, realizing a single master equation unifyingall four fundamental interactions. All core algebraic trace identities, Dirac operatorsquaring reductions, and field decay bounds are formally verified within the Lean 4proof assistant to eliminate human algebraic error. We fully address low-energy renormalization decoupling, intrinsic type-I neutrino seesaw mass generation, and clearlyoutline all open mathematical and physical limitations of the framework. Supplementary materials include LaTeX source code and Lean 4 formal proof code for trace identities and algebraic verifications.