The Theory of Thermodynamic Branching: Unifying Signal Fidelity and Metabolic Transport - Under Review for Additional Data implementation

Abstract The geometry of branching neurons has historically been bracketed by two asymptotic physical limits: Rall's Law (maximizing electrical power transfer) and Murray's Law (minimizing fluid transport dissipation). Current models interpolate between these limits using heuristic weights lacking physical definition. This paper presents a Theory of Thermodynamic Branching, a first-principles derivation where the branching exponent ($\alpha$) is a dependent variable determined by minimizing a unified thermodynamic potential. The model dictates that morphology is a continuous phase transition governed by the system's dimensionless energy budget: the ratio of signaling power to metabolic maintenance power ($\eta$). We validate this ab initio approach by showing that the known cortical energy budget ($\eta \approx 0.8$) predicts the observed pyramidal neuron morphology ($\alpha \approx 2.42$) without free parameter tuning. This framework resolves the tension between electrical and metabolic constraints, offering a unified physical explanation for neural diversity.