Beyond Murray's Law: Non-Universal Branching Exponents from Vessel-Wall Metabolic Costs

Abstract

Murray&#39;s cubic branching law ($α=3$) predicts a universal diameter scaling exponent for all hierarchical transport networks, yet arterial trees yield $α\sim 2.7-2.9$. We show that this discrepancy has a structural origin: Murray&#39;s universality is an artifact of cost homogeneity, not a biological property. Incorporating the empirical vessel-wall thickness law $h(r)=c_0 r^p$ ($p \approx 0.77$) introduces a third metabolic cost term $\propto r^{1+p}$ that renders the cost function inhomogeneous with incommensurate scaling exponents. By Cauchy&#39;s functional equation, homogeneity is necessary and sufficient for a universal branching exponent to exist; its absence implies non-universality, and Murray&#39;s law is identified as a singular degeneracy of the cost-function family rather than a general principle. We prove that the resulting scale-dependent exponent satisfies the strict bounds $(5+p)/2 < α^*(Q) < 3$ independently of flow asymmetry (Theorem 4, Corollary 5). The static wall-tissue mechanism bounds the symmetric bifurcation exponent to $α_t \in [2.90, 2.94]$ from measured parameters, marking a first-order symmetry breaking from Murray&#39;s law that narrows the empirical gap by one-third. The remaining discrepancy with the cardiovascular mean ($α_{exp} \approx 2.70$) is not a model failure but a mathematical necessity that signals the independent contribution of pulsatile wave dynamics. Additionally, the wall cost breaks Murray&#39;s topological degeneracy, bounding the optimal branching number to small finite integers; binary bifurcation emerges as the physiologically selected minimum under steric constraints.