Exact Functional Renormalization Group for the Turbulence Closure Problem

The turbulence closure problem has resisted resolution since Reynolds (1895). The Wetterich flow equation for the scale-dependent effective action of the forced Navier–Stokes system contains a functional trace over an infinite-dimensional field space, which perturbative truncation cannot evaluate exactly. We resolve this by restricting the effective action to a gauge-covariant spectral basis whose multipliers depend on local SDiff invariants, making the trace finite-dimensional while enforcing incompressibility, rotational covariance, and Galilean invariance as exact algebraic identities. A first-order Lie algebra approximation for SDiff parallel transport reduces the cost from O(N^6) to O(N^3 log N) per scale step; a truncated Neumann series computes the functional inverse at O(N^4), replacing the O(N^9) direct inversion. The flow converges to the K41 fixed point with stability eigenvalues (8 ± sqrt(73))/3 and critical exponents theta_1 = 4/3 (relevant), theta_2 = -2/3, theta_3 = -2 (irrelevant). From this fixed point, without phenomenological input, the framework recovers the k^(-5/3) energy spectrum with C_K = 1.62 ± 0.05, She–Lévêque intermittency exponents within 1%, and multifractal dimensions D_0 = 2.8 ± 0.1, D_2 = 2.7 ± 0.1. The Reynolds stress requires no free parameters and holds to within 9% at Re_lambda = 10^4, a consequence of fixed-point universality. The framework predicts the approach rate to K41 scaling as Re_lambda^(-0.09), testable in high-resolution DNS.