Metabolic Scaling as a Jamming Transition: Deriving the 1.25 Biological Safety Factor from Thermodynamic Geometry

Biological systems operate under strict energetic constraints, yet the physical origin of the universal ”safety factors” governing metabolic robustness remains an open question. In this work, we derive the biological Safety Factor (ΦSF ≈ 1.25) as a necessary consequence of the jamming transition in the cytoplasm. By modeling the cell as a granular thermodynamic system near the glass transition, we derive the metabolic scaling as a representation of an optimal transport trajectory that minimizes thermodynamic action within a Nonequilibrium Steady State (NESS). We analytically show that the operational density of the cytoplasm is bounded by the Mode-Coupling Theory (MCT) crossover, necessitating a 25% safety margin to prevent the divergence of structural relaxation times (critical slowing down). Furthermore, we extend this geometric framework to evolutionary timescales, recovering the Sharov-Gordon complexity horizon as a thermodynamic speed limit on information fixation. These results suggest that biological robustness is not merely an evolutionary adaptation but a requirement of thermodynamic transport in crowded media.