Conditional-Risk Optimization and KKT Geometry of a Regime- and Horizon-Adaptive Probabilistic Ensemble for Local Weather Forecasting

A regime- and horizon-adaptive probabilistic ensemble is proposed for local multi-horizon weather forecasting. The theoretical formulation links conditional second-moment error matrices to posterior probabilities of latent atmospheric-state components, while the Soft_Final scheme combines horizon adaptation, regime-dependent weighting, stabilization, and soft posterior mixing. For the posterior-weighted quadratic model, we derive an exact excess-risk decomposition, a Karush–Kuhn–Tucker (KKT) optimality certificate, an active-boundary decomposition on the probability simplex, and deterministic spectral stability bounds. The experimental dataset comprises 5184 hourly meteorological observations and 1296 forecast records across nine lead times; confirmatory inference is based on six station–season blocks. After Holm correction, statistically significant superiority of Soft_Final over the globally fixed static ensemble is not established, whereas ablation analysis indicates that horizon adaptation contributes more consistently than additional regime conditioning. Probabilistic performance exhibits a substantive trade-off among coverage, interval width, and the continuous ranked probability score (CRPS). The main contribution is a rigorous separation of model optimality, operational approximation, and out-of-sample predictive error. Same-protocol comparisons with ensemble model output statistics (EMOS), Bayesian model averaging (BMA), quantile regression forests (QRFs), and residual conformal post-processing demonstrate competitive but non-uniform performance rather than universal dominance.