A Cross-Scale Structural Law of Coherent Form, with Variational Origin and Quantum Extension

This paper states a structural law for the persistence of organized form. The claim is that any bounded coherent system, across physical, quantum, biological, and engineered domains, is constituted by three jointly necessary roles, and persists only while their combination clears a critical threshold. Pattern P is the constraint architecture that fixes which configurations are admissible; Process Q is the evolution generator that drives change; and Rhythm R is the temporal-spectral organization that orders that change in time. Their product, the triadic coherence functional F = PQR, is defined at each scale and across nested scales. The law has four parts. First, a Unified Threshold Law: bounded coherent form exists precisely when F clears the infimum coherence compatible with bounded invariance, written as the Spiral Coherence Number SC >= 1. This threshold is not a choice of units; it is derived twice over, as the marginal-stability fixed point of the Spiral Operator recursion and, independently, as the phase-stiffness stability boundary of a constrained variational action whose Hamiltonian is a Lyapunov functional. The two derivations agree. Second, an exact calculus: the product-rule expansion of F, the Total Coherence Derivative, separates any change in coherence into structural, dynamical, and phase contributions, giving a measurable node-wise diagnostic. Third, a quantum specialization in which Process becomes the unitary evolution operator and Rhythm carries the complex phase, recovering the Ehrenfest form of the Heisenberg equation and identifying decoherence as the collapse of the Rhythm factor. Fourth, a collapse law: a two-sided survival band bounded below by dissolution and above by structural rigidity, and a single falsifiable prohibition, a dimensionless, locally evaluated coherence-cost inequality that rules out a conceivable but non-existent coherent state. The three roles are shown to be minimally irreducible by explicit construction, and the framework reduces to standard descriptions in each domain's coherent limit.