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 $\dot{e}_p = \sigma_0 f$, linking entropy production rate to heart rate via a mass-specific parameter $\sigma_0 \propto M^0$. Integration yields a finite dissipative budget $\Sigma = \sigma_0 N^\star$, identifying $N^\star = \Sigma/\sigma_0$ as the correct primitive conserved quantity; lifetime energy per unit mass is a derived consequence valid only under simultaneous constancy of body temperature and $\sigma_0$. 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 $\Phi_C = \Phi_{\mathrm{duty}} \cdot \Phi_{\mathrm{thermal}} \cdot \Phi_{\mathrm{mito}} \cdot \Phi_{\mathrm{haz}}$, 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 $\sigma_0$). The decisive test is calorimetric measurement of $\sigma_0 = P/(TfM)$ across three body-mass decades.