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

arXiv:2603.13687v1 Announce Type: new
Abstract: Murray’s cubic branching law ($alpha=3$) predicts a universal diameter scaling exponent for all hierarchical transport networks, yet arterial trees yield $alpha sim 2.7-2.9$. We show that this discrepancy has a structural origin: Murray’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’s functional equation, homogeneity is necessary and sufficient for a universal branching exponent to exist; its absence implies non-universality, and Murray’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