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: Pattern x Process x Rhythm = Form. Four properties make it a meta-law rather than a model of any one system, and they are its central claims. It is substrate-independent: the same three roles and the same threshold govern coherent form whether the substrate is physical, quantum, biological, or engineered. It is irreducible: the three roles are minimally necessary, shown by explicit construction that no two suffice and none is redundant. It is scale-invariant and nested: the functional is defined at every scale, and the same law binds each scale to the next through a cross-scale recursion. It is subsuming: in each domain's coherent limit it reduces to the standard description of that domain, so existing physics sits inside it as a limiting case rather than beside it as a rival. The three roles are jointly necessary. Pattern (P) is the structure that fixes what a system is. Process (Q) is the engine that drives it from state to state. Rhythm (R) is the timing that orders change through time. Their product, the triadic coherence functional F = PQR, is the coherent form itself, defined at each scale and multiplied across nested scales. Because it is a product, coherence collapses if any single role vanishes: the three are necessary together, not on average. The law is developed through four results. First, a Unified Threshold Law. Bounded coherent form exists precisely when F clears a critical threshold, the Spiral Coherence Number SC >= 1, reached independently by the marginal-stability fixed point of the Spiral Operator recursion and by the phase-stiffness stability boundary of a variational action, the two derivations agreeing. Second, an exact calculus. 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. 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 is bounded below by dissolution and above by structural rigidity, with three collapse modes (monotone coherence decay, contagious disalignment cascade, and structural rigidity), a reciprocal Dissolution Coherence Number, measurable early-warning signatures, and a single falsifiable prohibition: a dimensionless coherence-cost inequality that rules out a conceivable but non-existent coherent state.