The Incommensurability Principle in Biological Transport

Why does the mammalian vascular tree maintain a conserved branching exponent ${\alpha }^{\ast }\approx 2.72$ across a ${10}^7$-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 $O({10}^2$–${10}^3)$ 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 $(G,N,p,{\alpha }_w)$, independent of all metabolic quantities. A self-consistency condition on the viscous–inertial energy partition yields a dual-threshold framework with ${\mathrm{W}\mathrm{o}}_c^{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{i}\mathrm{d}}=\sqrt{3}$ and ${\mathrm{W}\mathrm{o}}_c^{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}}=3\mathrm{/}\sqrt{2}$. The symmetric model yields ${\alpha }_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}^{\ast }\approx 2.626$, 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.