Microscopic Multi-Field Formulation of the KA Coupling-Degradation Framework: Propagator Algebra, Anomalous Dimension, and Dynamic Universality

The KA Operational Universality Framework established a scalar phi^4 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 psi_ij(x). The free propagator is G_ij,kl(q) = (delta_ik*delta_jl + delta_il*delta_jk) / (q^2 + m_ij^2), with m_ij^2 proportional to (alpha_i + alpha_j)^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) * C_eta * epsilon + O(epsilon^2), C_eta = 1/12, with eta(k=2)=0 (Ising) and eta(k->inf)=C_eta*epsilon (multi-flow saturation). The dynamic universality class is Model A (non-conserved order parameter) with z = 2 + eta(k)/2 + O(epsilon^2). Simulation confirms eta_hat(k=3) = 0.028 +/- 0.006 vs theory 0.033, and eta_hat(k=5) = 0.047 +/- 0.008 vs theory 0.053 at epsilon=1.