Murray's law says that when a blood vessel branches into two, the cube of the parent diameter equals the sum of the cubes of the daughter diameters. The exponent α = 3 is supposed to be universal — a consequence of minimizing the total cost of pumping blood plus maintaining vessel tissue, applicable to any hierarchical transport network. But arterial trees consistently yield α ≈ 2.7-2.9. The data has disagreed with the theory for decades.
Marchesi (arXiv:2603.13687) shows that Murray's universality is an artifact of cost homogeneity, not a biological property. The original derivation assumes the metabolic cost of maintaining a vessel scales as a single power of its radius. But real vessels have walls, and wall thickness scales with radius as h ∝ r^0.77. This introduces a third cost term — proportional to r^1.77 — that is incommensurable with the other two. The cost function becomes inhomogeneous, and the branching exponent becomes scale-dependent.
The corrected theory predicts α between 2.90 and 2.94 for symmetric bifurcations, narrowing the empirical gap by one-third. The remaining discrepancy likely reflects pulsatile wave dynamics — another effect the original theory ignores.
The law was never universal. It was a special case of a homogeneous cost function, mistaken for a general principle because the wall was assumed to have zero thickness.