A Multi-Scale Kolmogorov Turbulence Scaling Law for Dark Matter Phenomenology Across Three Orders of Magnitude

We report the identification of a multi-scale scaling relation between the critical transitionvelocity of dark matter phenomenology and the characteristic scale of the gravitating system,spanning galaxies, groups, and clusters. Using 175 galaxies from the SPARC database and50 spectroscopically confirmed clusters from the Hectospec Cluster Survey (HeCS), ourunbinned power-law fit to all 225 individual data points (175 SPARC galaxies + 50 HeCSclusters, 223 degrees of freedom) yields vcrit ∝ Rβ with β = 0.348 ± 0.05, a combinedsignificance of p < 10−20, and R2 = 0.997 across three orders of magnitude in spatial scale(10–1000kpc). The measured exponent is consistent with the Kolmogorov (1941) turbulenceprediction βK = 1/3 to within 0.6σ, suggesting that dark matter phenomenology may begoverned by an inertial-range energy cascade in a higher-dimensional medium. Independentmorphological analysis of the Bullet Cluster (1E0657−558) weak lensing convergence mapreveals a statistically significant elongation (axis ratio 1.24) of the dark matter distributionalong the collision axis at 10–30kpc smoothing scales, consistent with a hypersonic wakegeometry at Mach ≈ 6. We frame these results within the Kinetic Transit Theory (KTT)hypothesis, in which apparent dark matter arises from velocity-dependent hydrodynamic dragin a four-dimensional viscous bulk fluid, and outline falsifiable predictions for multi-clustermorphological tests.