Murray's cubic branching law (=3) predicts a universal diameter scaling exponent for all hierarchical transport networks, yet arterial trees consistently yield 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) = c₀ rᵖ (p0. 77 across mammalian species) introduces a third metabolic cost term 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 < ^* (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 ^* 2. 90, 2. 94 for porcine coronary arteries---within 1--1. 2 of the morphometric value 2. 70 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.
Riccardo Marchesi (Mon,) studied this question.