The Coherence Formula, Proven General: A Domain-General Measurement Law, Its Instantiation Protocol, and Four Worked Proofs of Applicability

This paper is a proof that a single equation — effective dimensionality equals a ceiling dimensionality times a coherence gate, written D_eff = D_max × Φ(C), where the gate Φ(C) is the normalized participation ratio of a system's modal distribution — is a well-defined, domain-general measurement law, and a demonstration that the same mathematics runs, unchanged in structure, across systems that share no physics. It separates cleanly what is proven from what is empirical. The mathematical core is elementary and exact: the participation-ratio identity, the coherence bounds (the gate lies between one over N and one), the collapse and spread limits, and the reduction to quantum purity in the diagonal case are all proved directly. Applicability to a given system is not claimed as a theorem but delivered as a five-step protocol that fixes what a state is, how the modal distribution is read from data, and — the one step that must be earned rather than assumed — how the ceiling is justified. The protocol is then run end to end in four domains chosen to span a ladder of evidential standing: quantum states, where the identification is exact and definitional; neural population activity, where an unrelated field already computes the identical statistic; the stratospheric polar vortex, where the instantiated read is validated and leads a physical transition; and financial markets, where the read is coincident and a leading-indicator claim was tested and falsified. The four are not offered as evidence that the framework is true everywhere; they are a proof that one equation is instantiable in different systems with honestly different standing, and the refutation condition is stated for every domain. The central statistic — quadratic concentration and its reciprocal, the participation ratio — is credited to prior literature; what is new is the coherence reading placed on it and the instantiation protocol.