Vortex Dynamics Locks Power-Law Exponents in Transport Networks: A Fixed Point at α = √2 from Competing Conservation Laws

Power-law exponents in fluid transport networks are systematically narrower than inself-organized critical (SOC) systems. Across 22 vortex/fluid systems the exponent clusterstightly at a = 1.381 ± 0.069, while 18 SOC systems yield a = 1.515 ± 0.327 — anorder-of-magnitude larger variance (Levene F = 14.1, p = 0.0006). We derive this 'vortexlocking' from two competing conservation laws: Murray's cubic scaling (viscous energyminimization, aM = 3/2) and Kelvin's circulation theorem (topological invariance, aK = 4/3).Both bounds emerge from Hack's scaling law t = 2 - h, with hM = 1/2 and hK = 2/3respectively. When both constraints act with equal logarithmic weight — as in intermediateReynolds number flows — the cost function minimizes at their geometric mean: a* =sqrt(3/2 x 4/3) = sqrt(2) ~ 1.4142. The observed mean lies 0.48s from this prediction. Threefalsifiable tests are proposed, including a Reynolds-number dependence of a and a 2D analogwith predicted a* = sqrt(4/3) ~ 1.155.Keywords: power-law scaling, vortex dynamics, Murray's law, Kelvin's theorem, Hack'slaw, self-organized criticality, universality class, fractal networks