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 (N⁶) to O (N³ log N) per scale step; a truncated Neumann series computes the functional inverse at O (N⁴), replacing the O (N⁹) direct inversion. The flow converges to the K41 fixed point with stability eigenvalues (8 ± sqrt (73) ) /3 and critical exponents theta₁ = 4/3 (relevant), theta₂ = -2/3, theta₃ = -2 (irrelevant). From this fixed point, without phenomenological input, the framework recovers the k^ (-5/3) energy spectrum with CK = 1. 62 ± 0. 05, She–Lévêque intermittency exponents within 1%, and multifractal dimensions D₀ = 2. 8 ± 0. 1, D₂ = 2. 7 ± 0. 1. The Reynolds stress requires no free parameters and holds to within 9% at Reₗambda = 10⁴, a consequence of fixed-point universality. The framework predicts the approach rate to K41 scaling as Reₗambda^ (-0. 09), testable in high-resolution DNS.