Auditing Structured Forecasting Losses: Error Correction and Exact-Fit Optimality

Structured forecasting losses combine error components and auxiliary statistics to guide learning. We audit whether these combinations pre- serve error correction after partial fit and favor exact agreement with fixed targets. In DBLoss, the trend-gradient coefficient vanishes at exact seasonal fit, leaving nonzero level biases stationary. On a fixed batch, a positive integrating factor yields a direction-equivalent scalar potential whose global minimizers still include these biased forecasts. Controlled recovery experiments on fifteen saved models test quadratic field attenu- ation near this boundary and show optimizer-dependent recovery. For the published RI-Loss helper, explicitly activated in our runner, we iden- tify how a finite-sample statistic favors residual-channel dispersion. We derive a sharp weight threshold for the full objective averaged over surro- gate noise. Zero residual is the unique global minimizer for nonnegative weights up to and including the threshold, and a saddle in residual space above it. Source checks, stochastic trajectories, and neural-parameter perturbations at constructed agreement test the local predictions. Match- ing dropout randomness largely removes the observed short-run effect of statistic replacement. Extensions to Time-o1 and PS Loss use controlled interventions to identify how equivalent encodings and batch-statistic scope alter learning signals. Five-source forecasting comparisons show heterogeneous predictive effects under the tested budgets. The resulting coefficient and weight conditions make error correction and exact-fit optimality directly testable.