The Theory of Thermodynamic Branching: Unifying Signal Fidelity and Metabolic Transport

The geometry of branching neurons has historically been bracketed by two asymptotic physical limits: Rall's Law (α = 1.5) for impedance matching, and Murray's Law (α = 3.0) for fluid transport. We present a Grand Unified Theory of Thermodynamic Branching, where the branching exponent α is a dynamic variable determined by minimizing a unified potential. Our framework yields four fundamental discoveries: (1) Entropic Origin: The cortical energy budget (η ≈ 0.8) is derived ab initio from Shannon Information Theory, matching the bit-per-joule optimum. (2) Bioenergetic Pivot: Contrary to classical predictions of simplification (α → 1.5), we demonstrate that high-current Motor Neurons undergo a saturation-induced expansion to a high-entropy phase (α → 3.0) to maximize metabolic transport and heat dissipation. (3) Thermodynamic Plasticity: Neural development is derived as the relaxation of the system towards the entropic minimum (τα̇ = -). (4) Biophysical Robustness: We demonstrate that this scaling law is invariant under glial metabolic coupling, spine neck impedance, and cytoskeletal hysteresis. This unifies structure, function, and learning under a single principle of entropy minimization.