Renormalisation Group Analysis of the KA Operational Universality Class: Fixed Points, Critical Exponents, and Lead-Time Scaling - From RG Analogy to Leading-Order Derivation (OP2)

The KA Operational Universality Conjecture predicts that systems satisfying axioms A1-A5 exhibit metric singularity precursor divergence dFKA/dt -> +infinity as kappa -> kappac^-. Paper 2 proposed an RG-analog fixed-point structure and motivated lead-time scaling Deltaₜ ~ lambda^-1 ln (kappac/kappa₀) from an asymptotic argument. Open Problem 2 asked for a rigorous RG derivation. We provide this derivation at leading order. Starting from the Wilsonian effective action for fluctuations phi = det (A (kappa) ) near the critical manifold Mc, we derive one-loop RG beta-functions and identify the Wilson-Fisher fixed point at r*=-epsilon/6+O (epsilon²), u*=2*epsilon/3+O (epsilon²) (dimensionless couplings). This gives the correlation length exponent nu = 1/2 + epsilon/12 + O (epsilon²) and effective anomalous dimension eta (k) = tr (dₖappa Sigma*^-1 * dₖappa Sigma*) /k + O (1/k²) for k >= 3 flows. The lead-time formula Deltaₜ = (nu/kappadot) *ln (kappac/kappa₀) is derived from RG flow, identifying the phenomenological constant of Papers 1-2 as the correlation length exponent nu. Simulation confirms nuₕat = 0. 503-0. 541 across k=2, 3, 4, 5 flow regimes (R² > 0. 98). Lead-time predictions match Paper 3 calibrated domains to within 2%. Full higher-loop calculations remain open (OP2b).