Theory of Thermodynamic Branching

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) Phase Transition: The "Motor Neuron Anomaly" (α ≈ 1.5) is explained as a saturation-induced phase transition (V → V_max).

(3) Thermodynamic Plasticity: Neural development is derived as the relaxation of the system towards the entropic minimum (τ·dα/dt = −∇ℒ).

(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.