A Nonequilibrium Internal-Time Model of Aging: Entropy-Normalized Biological Proper Time and Repair Bifurcations

Abstract

Chronological age is an incomplete coordinate for aging. Individuals and species sharing the same calendar time can differ substantially in physiological reserve, molecular damage, mortality hazard, and remaining lifespan. The Principle of Biological Time Equivalence (PBTE) offers a thermodynamic reformulation: biological aging is governed by the accumulation of \emph{internal} physiological time rather than chronological time alone. Building on prior PBTE work, this paper defines the internal-time coordinate $θ(t)=\int_0^t f(s)\dd s$, where $t$ is chronological time and $f(s)$ is an instantaneous physiological frequency (for example heart rate or respiratory rate), so that $θ$ is the accumulated count of physiological cycles. Its entropy-normalized extension is $\Tsig(t)=\int_0^t[\sigz(s)/\sref]f(s)\dd s$, where $\sigz(s)=\ddΣ/\ddθ$ is the entropy produced per physiological cycle (the entropy cost per biological tick), $Σ$ is cumulative entropy production, and $\sref$ is a fixed reference entropy cost per cycle used as a normalizing unit. The normalized PBTE age $\APBTE(t)=\Tsig(t)/\Nref$ measures the fraction of a reference entropy--cycle budget consumed, where $\Nref$ is the reference number of entropy-weighted cycles available over a lifetime. The manuscript is explicitly theoretical: no empirical cohort is analyzed, and the numerical demonstrations are synthetic stress tests rather than validation.