Route-Conditioned Prediction Extrapolation: Dependent-Data Directions, Persistence, and Certification

Route-specific fine-tuning defines a direction in prediction space, and scalar extrapolation along that direction can improve a pooled predictor. We study ${\hat{f}}_{\alpha }=\mathrm{clip}[{\hat{f}}_0+\alpha ({\hat{f}}_{\mathrm{eq}}-{\hat{f}}_0)]$, where ${\hat{f}}_0$ is a pooled checkpoint and ${\hat{f}}_{\mathrm{eq}}$ is a target-mass aggregate of route experts initialized from that checkpoint. A persistent two-state model isolates a composition-correction special case: the leading Brier risk is quadratic in $\alpha$, the optimum is $F/H$, and imperfect ergodic route errors imply ${\alpha }^{\ast }>1$. At q = 0.10 and ${\rho }_R=0.98$, the exact coefficients are 1.5368, 1.5208, 1.4205, and 1.1328 as route-error persistence increases over $\lambda =0,0.5,0.9,0.99$, and paired simulations track the corresponding correction gains. On HAR-Transitions, WISDM, PAMAP2, and PTB-XL, split-wise validation selects mean coefficients 2.725, 1.542, 3.767, and 2.008. Endpoint decomposition shows that route-specific fine-tuning dominates target-mass composition on all four real datasets; matched random-subject, temporal-block, and no-split continued-training directions are weaker than the learned route direction, while hard route gating closely matches the selected extrapolation. Direct batch-means estimates of $F/H$ are 1.58 ± 0.16, 1.61 ± 0.14, 1.46 ± 0.22, and 1.52 ± 0.08, indicating extrapolation but not its real-data magnitude. At $\delta =0.05$, the range-corrected empirical-Bernstein radius at the reported PTB-XL certification size is 0.0104 against a mean paired gain of 0.0125; the small sensor cohorts remain information-limited. This separates predictive gains from unit-aware deployment evidence without treating windows as independent samples.