The KA Operational Universality Framework established a scalar phi⁴ effective field theory for the coupling-matrix determinant phi=det (A). The anomalous dimension eta (k) for k>=3 flows was estimated heuristically in Paper 4, and the microscopic field content generating it was not derived. This paper resolves these gaps. We construct the microscopic multi-field action from the Onsager-Machlup path integral of the k-flow SDE and integrate out the stable modes via Hubbard-Stratonovich transformation to obtain an effective action for the k (k-1) /2 coupling fields psiᵢj (x). The free propagator is Gᵢj, kl (q) = (deltaᵢk*deltaⱼl + deltaᵢl*deltaⱼk) / (q² + mᵢj²), with mᵢj² proportional to (alphaᵢ + alphaⱼ) ², and the four-point vertex is Gamma = u * T where T is the O (k) -invariant tensor. The one-loop self-energy from the explicit tadpole diagram gives the genuine anomalous dimension: eta (k) = ( (k-2) /k) * Cₑta * epsilon + O (epsilon²), Cₑta = 1/12, with eta (k=2) =0 (Ising) and eta (k->inf) =Cₑta*epsilon (multi-flow saturation). The dynamic universality class is Model A (non-conserved order parameter) with z = 2 + eta (k) /2 + O (epsilon²). Simulation confirms etaₕat (k=3) = 0. 028 +/- 0. 006 vs theory 0. 033, and etaₕat (k=5) = 0. 047 +/- 0. 008 vs theory 0. 053 at epsilon=1.
Karimov et al. (Fri,) studied this question.