Spectral Transition from a Quartic Functional to the Emergence of Mass

This work develops a fully self-contained quartic variational framework in which all physical scales, spectral quantities, and dynamical structures emerge from a single parameter-free functional. The static equation, dynamic equation, Hessian operator, and decision map form a closed algebraic system with no external constants, units, or background geometry. I prove that the lowest Hessian eigenvalue defines a unique internal curvature scale, and that all emergent quantities — length, time, frequency, mass, and energy — are fixed functions of this curvature. The full Hessian spectrum satisfies autosufficiency, spectral stability, internal closure, rigidity, and generativity (Class S). No alternative branches, parameters, deformations, or interpretations are admissible. The framework therefore provides a mathematically inevitable transition from a quartic functional to a complete emergent hierarchy of masses and frequencies. This establishes a globally deterministic variational origin for spectral structure and eliminates the need for postulated constants, external theories, or model-based constructions.