摘要: The branching geometry of biological transport networks is canonically characterized by a diameter scaling exponent α. Traditionally, this exponent interpolates between two structural attractors: impedance matching (α∼2) for pulsatile wave propagation and viscous-metabolic minimization (α=3) for steady flow. We demonstrate that neither mechanism in isolation can predict the empirically observed αexp=2.70±0.20 in mammalian arterial trees. Incorporating the empirical sub-linear vessel-wall scaling h(r)∝r^p (p=0.77) into a three-term metabolic cost function rigorously breaks the universality of Murray’s cubic law — a consequence of cost-function inhomogeneity established via Cauchy’s functional equation — and bounds the static transport optimum to αt∈[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 η 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 von Neumann’s minimax theorem — for which we provide a direct constructive proof exploiting the strict monotonicity of the cost curves — this game admits a unique saddle point (α∗,η∗) satisfying an exact equal-cost condition, from which the empirical exponent emerges as the robust optimal compromise between competing thermodynamic demands. We further prove that N=2 (binary branching) uniquely maximizes the network stiffness ratio κ_eff(N), establishing the universal preference for bifurcations not as an anatomical assumption but as a derived property of the unified wave-transport framework. Numerical evaluation on the porcine coronary tree (G=11 generations) yields α∗=2.72, in quantitative agreement with morphometric data. Sensitivity analysis confirms that this prediction is structurally robust to metabolic parameter variation (|Δα∗|<0.01 across the physiological range of viscosity and wall metabolic rates), depending critically only on the histological scaling exponent p — the single parameter with direct anatomical grounding. Specifically, the prediction is analytically insensitive to the exact value of the wall-thickness pre-factor c0, making the framework a zero-parameter derivation from fundamental scaling principles.