A Single--Index Theory of Optimal Branching: Murray Laws, Gilbert Networks, and Young--Herring Junctions
Murray-type flux-radius laws, Gilbert-type concave transport costs, and Young-Herring triple-junction angle balances are usually treated as separate theories. This work shows that, within a natural class of quadratic, scale-free ledgers for branched networks, all three are different faces of a single structure controlled by one dimensionless index chi. Each edge carries a flux Q, an effective radius r, and a per-length ledger P(Q,r) encoding transport dissipation and structural burden. Under locality, evenness in Q, linear-response (quadratic) dependence, and an exact homogeneity ansatz in (Q,r), any admissible ledger reduces in the scale-free regime to the two-term form P(Q,r) = a Q^2 r^{-p} + b r^m. Local optimality then implies simultaneously: (i) a flux-radius power law with generalized Murray closures at degree-3 nodes; (ii) a Young-Herring-type vector balance with radius weights r^m and a fixed symmetric Y-junction angle; and (iii) an effective flux-only cost of Gilbert/branched-transport type with exponent beta. The exponents alpha and beta, the symmetric angle, and the split between transport and structural cost are all set by chi = m/(m+p) = beta/2. A rigidity theorem shows conversely that any quadratic ledger that yields power-law optimal radii and power-law flux-only cost on an open scaling cone must belong to this two-term family and obey the same Murray-Gilbert-Young dictionary. Examples for Poiseuille, diffusive, and geophysical trees illustrate how chi can be inferred from geometry and used as a falsifiable order parameter for scale-free branching architectures.