A Unified Variational Principle for Branching Transport Networks: Wave Impedance, Viscous Flow, and Tissue Metabolism

The branching geometry of biological transport networks is canonically characterized by a diameter scaling exponent. Traditionally, this exponent interpolates between two structural attractors: impedance matching (2) for pulsatile wave propagation and viscous-metabolic minimization (=3) for steady flow. We demonstrate that neither mechanism in isolation can predict the empirically observed ₄ₗ₏ = 2. 70 0. 20 in mammalian arterial trees. Incorporating the empirical sub-linear vessel-wall scaling h (r) rᵖ (p0. 77) into a three-term metabolic cost function rigorously breaks the universality of Murray's cubic law---a consequence of cost-function non-homogeneity established via Cauchy's functional equation---and bounds the static transport optimum to ₜ 2. 90, 2. 94. To account for the dynamic pulsatile environment, we formulate a unified network-level Lagrangian balancing wave-reflection penalties against steady transport-metabolic costs. Because the operational duty cycle between pulsatile and steady states is inherently uncertain over developmental timescales, we cast the morphological optimization as a zero-sum game between network architecture and environmental state. By the minimax theorem---proved here requiring only continuity and strict monotonicity, without global convexity assumptions---the unique saddle point (^*, ^*) satisfies the exact equal-cost condition Cₖ₀ₕ₄ (^*) = Cₓₑ₀₍ₒ₏₎ₑₓ (^*), eliminating as a free parameter. For porcine coronary arteries, this deterministic ground state yields ^* = 2. 72, while Bayesian marginalization via Monte Carlo over physiological parameter uncertainty predicts a population expectation of ^*₌₂ = 2. 86 0. 06. Without requiring fitted parameters, the framework rigorously derives the observed cardiovascular scaling and reduces exactly to Murray's law when dynamic wave modes are absent.