Coupling Power and the Regulatory Ratio: An Exact Dissipation-Performance Relation for Driven Dissipative Systems

This paper connects two rigorous but separate literatures — state-space stability geometry (basin stability, viability theory, stability thresholds) and stochastic thermodynamics (information-entropy bounds, sensory-efficiency tradeoffs, thermodynamic uncertainty relations) — using a recently confirmed variance scaling law for driven dissipative systems. A pre-registered prospective simulation test (SIP-PREREG-02) established that the stationary variance of a first-order stochastic regulator obeys Var(ẽ) = σ_q² τ_σ² / (ρ(ρ+1)), where ρ = τ_σ/τ_R is the ratio of the noise relaxation timescale to the regulatory timescale; the parameter-free core prediction Var(ρ=1)/Var(ρ=2) = 3.0 was confirmed (95% bootstrap CI [2.941, 3.033]). From this confirmed law, two exact results are derived: (1) the coupling power — the rate of energy transfer from the noise source into the regulated response — as the closed-form, monotone function Π(ρ) = σ_q²/(1+ρ); and (2) an exact dissipation-performance relation linking the regulatory ratio to the ratio of coupling power and response variance. Within the Ornstein-Uhlenbeck model this relation is an algebraic identity; whether it can serve as an observational signature depends on the independent measurability of its terms, which is discussed explicitly. The paper then conjectures — without deriving — a two-component cost structure (a derived, decreasing coupling cost competing with a conjectured, increasing regulatory-maintenance cost) that, if it exists, would yield an optimal regulatory ratio ρ*, structurally analogous to the coherence-decoherence-recoherence (CDR) threshold of the General Theory of Regulated Stability. The maintenance term is not obtained here and the open step is flagged; the qualitative two-component structure is noted as already established for cellular sensing. The contribution is positioned as a step toward bridging thermodynamic cost to state-space stability geometry through a pre-registered, confirmed performance measure, not a completion of that bridge.