Spectral Transition from a Quartic Functional to the Emergence of Mass

This work develops a fully self-contained quartic variational framework in which mass arises as a purely spectral phenomenon. Starting from the static equation, the dynamic flow, the Hessian operator, and the decision map, the structure forms a closed and parameter-free system with no external constants, scales, or background geometry. The lowest eigenvalue of the Hessian defines a unique internal curvature, which induces a single admissible mass scale. All higher masses emerge from the rigid spectral hierarchy enforced by the variational structure and by the Class S conditions of autosufficiency, internal closure, and spectral stability. The resulting transition from a quartic functional to a complete mass spectrum requires no modelling assumptions, couplings, or external inputs. Mass is shown to be an inevitable consequence of the internal spectral geometry of the functional.