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

Murray's cubic branching law ($\alpha=3$) predicts a universal diameter scaling exponent for all hierarchical transport networks, yet arterial trees consistently yield $\alpha \approx 2.7$--$2.9$. We show that this discrepancy has a structural origin: Murray's universality is an artifact of his cost function's homogeneity, not a property of biological networks. Incorporating the empirical vessel-wall thickness law $h(r) = c_0 r^p$ ($p\approx0.77$ across mammalian species) introduces a third metabolic cost term $\propto r^{1+p}$ that renders the cost function quasi-homogeneous but not homogeneous. By Cauchy's functional equation, homogeneity is both necessary and sufficient for a universal branching exponent to exist; its absence rigorously implies non-universality. We prove that the resulting scale-dependent exponent satisfies the strict bounds $(5+p)/2 < \alpha^*(Q) < 3$ independently of flow asymmetry (Theorem 4, Corollary 5), that Murray's law is the unique member of this cost-function family admitting a universal exponent (Corollary 6), and that the wall cost strictly breaks Murray's topological degeneracy, bounding the optimal branching number to small finite integers and excluding star-like topologies; binary bifurcation emerges as the physiologically selected minimum under additional steric constraints detailed in Theorem 10. The non-universality is structurally stable: it persists under generation-dependent wall scaling, active smooth-muscle tone, and non-Newtonian viscosity corrections. Parameter-free evaluation yields $\alpha^* \in [2.90, 2.94]$ for porcine coronary arteries---within $1$--$1.2\sigma$ of the morphometric value $2.70 \pm 0.20$, reducing the gap from Murray's cubic law by one third. The predicted bifurcation angle bound 74.9◦ < 2θ∗ < 80.2◦ is independently confirmed by three-dimensional coronary morphometry, with no parameters fitted to angle data. The residual gap between the static prediction and the empirical mean points to the role of pulsatile wave dynamics as a complementary architectural constraint beyond the static cost function analyzed here.