Biological Time Equivalence in Vertebrates: Thermodynamic Framework, Comparative Tests, and Clade-Specific Deviations

Abstract: The product of resting heart rate and maximum lifespan is approximately constant across adult warm-blooded vertebrates, $N^\star = f_H L \approx 10^9$ cardiac cycles, a regularity documented since Rubner (1908) but lacking a thermodynamic derivation. We derive $N^\star$ from the non-equilibrium second law by treating the adult organism as a metabolic non-equilibrium steady state (NESS) and introducing the closure $\dot{e}p = σ_0 f$, linking entropy production rate to heart rate via a mass-specific parameter $σ_0 \propto M^0$. Integration yields a finite dissipative budget $Σ= σ_0 N^\star$, identifying $N^\star = Σ/σ_0$ as the correct primitive conserved quantity; lifetime energy per unit mass is a derived consequence valid only under simultaneous constancy of body temperature and $σ_0$. Phylogenetically independent contrasts on 112 endotherm species yield a $\log f_H$--$\log L$ slope of $-0.99 \pm 0.04$ ($p=0.84$ against $-1$); the West--Brown--Enquist null of zero inter-clade variation is rejected ($F=12.7$, $p<0.001$). A factored multiplier $Φ_C = Φ{\mathrm{duty}} \cdot Φ{\mathrm{thermal}} \cdot Φ{\mathrm{mito}} \cdot Φ_{\mathrm{haz}}$, calibrated from independently measured physiology, accounts for longevity deviations across four warm-blooded clades. The integral of physiological frequency defines a biological proper time classifying longevity mechanisms as time dilation (reduce $f$) or budget expansion (reduce $σ_0$). The decisive test is calorimetric measurement of $σ_0 = P/(TfM)$ across three body-mass decades.