Non-Normal Operators, Dynamical Multifractality, and the Functional Renormalization Group of Turbulence Scaling Laws

This paper constructs a field theory of turbulence scaling laws from a non-normal operator core. The Thermo-Optical Guidance (TOG) operator L(k)=Dk2I+AJL(k)=Dk2I+AJ with nilpotent JJ (J2=0J2=0) is coupled to Navier–Stokes advection via a marginal nonlocal vertex deformation A∣k∣αA∣k∣α with α=1α=1. The resulting two-coupling RG flow (g,A)(g,A) admits a stable interacting fixed point only after including a nonlinear saturation term −sA2−sA2, which converts a marginal manifold into a genuine IR-attractive fixed point. Key results include: Coupled RG flow: Beta functions βg=Cg2(1+c1A)βg=Cg2(1+c1A), βA=(α−1)A+c2gA+c3g−sA2βA=(α−1)A+c2gA+c3g−sA2 produce a stable fixed point with g∗≠0g∗=0, A∗≠0A∗=0 for α=1α=1 and s>0s>0. Critical exponents: Dynamic exponent z<2z<2, third-order structure exponent ζ3=1+O(A∗,g∗)≠1ζ3=1+O(A∗,g∗)=1, and nonzero intermittency correction Δint≠0Δint=0. Multifractal spectrum: At one loop, D(h)D(h) is log-normal; at two loops, a cubic correction D(h)=3−(h−h0)2/(4μ)+κ(h−h0)3D(h)=3−(h−h0)2/(4μ)+κ(h−h0)3 emerges, breaking the Gaussian closure and generating skewed multifractality. Infinite cumulant hierarchy: The RG generates all higher cumulants with an∼n!σ−nan∼n!σ−n at strong coupling, signaling a multifractal phase transition where D(h)D(h) becomes non-analytic. Functional RG equation: The running multifractal spectrum satisfies ∂ℓD=−∂h(χD)+D0∂h2D+A2∂h2(1/(1+(∂hD)2))−s(∂hD)3∂ℓD=−∂h(χD)+D0∂h2D+A2∂h2(1/(1+(∂hD)2))−s(∂hD)3, a nonlinear reaction–diffusion equation in Hölder space. Chaotic dynamics: Linear stability analysis yields a positive maximal Lyapunov exponent Λmax∼A4/D0Λmax∼A4/D0 in regions of negative slope, with positive definite entropy production. Non-equilibrium invariant measure: Detailed balance is broken by the non-gradient TOG term; the stationary state is characterized by circulating probability currents in function space. The theory defines a dynamically generated skewed multifractal universality class with RG-controlled intermittency, continuously connected to Kolmogorov scaling in the weak-coupling limit, and exhibiting second-order turbulence where the scaling laws themselves evolve chaotically.