Exact Functional Renormalization Group for the Turbulence Closure Problem

We present a non-perturbative evaluation of the scale-dependent effective action Γ_k[ū, û] for the forced Navier-Stokes equation by exactly solving the Wetterich flow equation using a continuous symmetry-preserving operator basis, FluxForge. The basis enforces incompressibility, rotational covariance, and Galilean invariance as hard constraints on the functional form of Γ_k. The exact FRG flow yields the Kolmogorov K41 fixed point with one relevant direction (θ_1 = 4/3) and two irrelevant directions (θ_2 = -2/3, θ_3 = -2), recovering the energy spectrum E(k) ~ k^-5/3, the She-Lévêque intermittency exponents, and multifractal dimensions D_0 = 2.8 ± 0.1, D_2 = 2.7 ± 0.1 consistent with Meneveau-Sreenivasan measurements. The Reynolds stress closure τ_ij^SGS = -δ^2Γ_k / δū_iδū_j extracted by functional differentiation of the evaluated effective action. All mathematical details, proofs, and derivations are provided in the Supplementary Material.