This paper formalizes a structural law governing the complete lifecycle of any bounded coherent form, its emergence, persistence, and collapse, across all scales of organization. Three irreducible, jointly necessary structural roles are identified: Pattern P, the constraint architecture defining admissible configurations; Process Q, the evolution generator governing state transitions; and Rhythm R, the temporal-spectral organization governing stability. From these, a triadic coherence functional F = PQR is defined across nested scales. A Unified Threshold Law is derived from a persistence criterion: bounded coherent form exists precisely when triadic coherence clears the infimum coherence compatible with bounded invariance, expressed as the Spiral Coherence Number SC >= 1. The total differential of F is expanded via the product rule into a Total Coherence Derivative, the paper's primary mathematical contribution, mapping structural shift, dynamical evolution, and phase modulation onto the three components. A structural argument for the role of F as a temporal ordering parameter is given, distinguishing it from a mere coherence index. Quantum mechanics is embedded as the micro-scale specialization of this framework: the Process variable becomes a unitary evolution operator, and the Rhythm variable carries the complex phase structure of the Spiral Operator. The Ehrenfest form of the Heisenberg equation of motion emerges as the quantum form of the Process differential. Quantum phase coherence, formalized via the Kuramoto order parameter, is identified with the Rhythm variable, and its loss under environmental coupling is the quantum-scale instance of the general coherence collapse law. The framework is presented as a structural constraint law governing persistent coherent form across domains. The threshold is established by two independent derivations that agree: a persistence criterion that fixes it as the infimum of coherence compatible with bounded invariance, and a constrained (Dirac-Hamiltonian) variational reconstruction whose Hamiltonian acts as a Lyapunov functional and yields the sharp phase stiffness stability condition. The survival law is two-sided, bounded below by dissolution and above by a structural rigidity boundary. A corrected, dimensionless, locally evaluated coherence-cost inequality rules out a conceivable but non-existent coherent state.
Andrew Lee Johnson (Wed,) studied this question.