Defining Classical Coherence: C_cl from First Principles A Mathematical Foundation for the Quantum Homeostasis Programme

Abstract Classical coherence C_cl is introduced as the normalised mutual information C_cl(A,B) ≜ I(A;B)/H(A,B) for finite discrete biological observables. The measure is bounded in [0,1], satisfies five elementary information-theoretic properties, and admits a conditional geometric interpretation as the correlational component of the Fisher–Rao metric under equal-frequency discretisation (local second-order equivalence near the independence manifold). A complete measurement protocol (equi-marginal discretisation, KPSS + Pettitt stationarity checks, change-point handling) is supplied together with sensitivity analysis across copulas, sample sizes, and noise. Thresholds are derived exclusively from system-specific calibration against an independent gold-standard (e.g., carbon flux). This yields the falsifiable Prediction P1 for homeostatic stability in mycorrhizal–host systems. Network superadditivity is left as an open problem. Version 11 deepens the operator-theoretic foundations shared with Fisher information geometry, integrates C_cl into the Architecture of Regulatory Coherence (ARC) and Coherence–Decoherence–Recoherence (CDR) cycle, and explicitly links the framework to the General Theory of Regulated Stability (GTRS) across quantum and cosmological scales. The framework supplies a rigorous classical benchmark for the Quantum Homeostasis Programme.