A harmonic mean theorem for optimal branching exponents in transport networks

Optimal branching in transport networks has been analysed mechanism by mechanism since Murray's law (branching exponent x=3), but no combining rule exists when several dissipation mechanisms coexist. We prove that the optimal exponent is the weighted harmonic mean of the single-mechanism exponents, weighted by each mechanism's share of the dissipation budget. The result holds for any smooth dissipation function and any smooth convex structural cost. A bounding consequence establishes a strict floor x>5/2 for any Poiseuille-plus-pulsatility combination at every finite Womersley number, providing a parameter-free test that distinguishes dissipation-based from impedance-based optimization of biological networks. The predictions are robust to wall compliance and physiological flow variability; the latter preferentially amplifies entrance losses, yielding a testable prediction that vascular beds with high flow variability should exhibit systematically lower branching exponents. Read inversely, the theorem predicts that roughly half of coronary pumping power is dissipated by entrance effects. An order-by-order analysis of the human pulmonary arterial tree predicts the observed transition from x≈2.1 to x≈3, in quantitative agreement with cross-species morphometric data. The same entrance-driven mechanism reproduces the two-regime architecture of allometric scaling theory as asymptotic limits of a single cost function.