Generalized Murray Cost Functions and the Non-Universality of Branching Exponents

Murray's cubic branching law (α=3) rests on a two-term cost function that neglects the metabolic cost of vessel-wall tissue. Incorporating the empirical wall-thickness law h(r) = c₀ rᵖ (p ≈ 0.77) introduces a third cost term ∝ r¹⁺ᵖ and yields seven rigorous results. First, the optimal vessel radius exists and is unique for any positive parameter values. Second, a universal branching exponent exists if and only if the optimal radius is a power law in flow rate. Third, for any single maintenance term ~ r^γ, the branching exponent is exactly α = (4+γ)/2, recovering Murray (γ=2), the Da Vinci rule (γ=1), and the wall-cost limit (γ=1+p) as special cases---and providing a physical grounding for the abstract pricing exponent of Bennett's EPIC framework. Fourth, with both volume and wall costs present, no universal exponent exists; instead, a scale-dependent local exponent α^*(Q) satisfies (5+p)/2 < α^*(Q) < 3. Fifth, these bounds are independent of the flow-splitting asymmetry, proving non-universality is an inherent property of the cost function rather than junction geometry. Sixth, the wall cost rigorously breaks the degeneracy of Murray's law regarding the branching number, analytically selecting $N=2$ (bifurcation) as the unique energy-minimizing topology. Seventh, the optimal symmetric bifurcation angle is tightly bounded strictly between 74.9^∘ and 80.0^∘ for physiological parameters, generalizing Zamir's force-balance derivation. Parameter-free numerical evaluation yields α^* = 2.90 for porcine coronary arteries, within 1σ of the morphometric measurement 2.7 ± 0.2, resolving a long-standing discrepancy without invoking pulsatile wave dynamics.