The Incommensurability Principle in Biological Transport

Why does the mammalian vascular tree maintain a conserved branching exponent ^* 2. 72 across a 10⁷-fold range in body mass, despite a fundamental shift from viscous to wave-dominated transport? We prove this universality cannot emerge from local optimization: any junction-level coupling of incommensurable costs requires scale-dependent fine-tuning varying by O (10²--10³) across the hierarchy. Real networks resolve this through structural heterogeneity, and vascular geometry emerges as a scale-free attractor of a network-level minimax principle. Grounding the fitness penalty in ATP stoichiometry, we prove a Topological Rigidity theorem: the optimal branching exponent depends only on dimensionless structural parameters (G, N, p, w), independent of all metabolic quantities. A self-consistency condition on the viscous--inertial energy partition yields a dual-threshold framework with Woc^fluid = 3 and Woc^wave = 3/2. The symmetric model yields ^*₌₎₃₄₋ 2. 626, in agreement with mammals near the allometric transition; morphometric heterogeneities shift large-mammal values toward 2. 72. The framework explains developmental stability of cardiovascular networks as a consequence of architecture being decoupled from biochemistry.