TailCast: Release-Order Tail Extrapolation for Frozen Time-Series Foundation Models

Time-series foundation models forecast zero-shot, yet they expose their predictive distribution only on a finite grid of native quantiles—Chronos-Bolt and TimesFM2.5 stop at 0.90, Chronos-2 at 0.99—while tail decisions are defined above it. Extending the profile is hard because the frozen model supplies no local shape there and validating outcomes are rare by construction, and because a label may arrive only after later forecasts have gone out. Existing post-hoc layers estimate the tail from scarce exceedances, discarding the conditional shape the backbone already encodes. We introduce TailCast, which treats that conditional quantile profile as an issue-specific anchor and learns only a small residual correction to it, continuing the top native spacing in log-survival coordinates above the grid. The residual is estimated at two adaptation scales, whose thirteen candidates are combined by a learning-rate-free rule over released pinball losses, each token scored against the state that issued it; a weighted isotonic projection, provably nonincreasing for the reported score, returns the ladder. Across four frozen backbones, two benchmark groups holding 696 cases, and three scenarios, TailCast attains the lowest tail-weighted pinball loss of the thirteen methods compared, by 6.7% and 8.9% relative to the strongest of the other twelve on each group, issuing finite, noncrossing ladders under every delay and missingness condition tested. Because it reads only native quantiles, targets, and release timestamps, the layer drops onto any frozen backbone without retraining, bringing such tail levels within reach of operational alerting.