The Price of a Label: Static Identity in Heads on Frozen Time-Series Models

Adding a store or platform label to a frozen time-series foundation model raises three separate questions: whether the label reaches the model (access), whether a fitted head gains from it (fitted value), and whether the gain carries to new data (transfer). We give an exact erasure condition: a static channel that enters only through centered values, with no bypass, loses its identity exactly. On 70 public retail series, three of five released forecasters give identical forecasts under every tested relabelling, and a fourth model's response is traced to unobserved entries and removed by neutralizing them. The standard permuted-label ablation can mislead: on one public retail subset at a frozen checkpoint, true store codes beat permuted codes over ten paired seeds yet lose to a label-free input code shared by all series, which keeps the lower mean loss in five further seeds, where true versus permuted is unresolved. On 3,087 development-seen fundraising campaigns, equal-width heads using true labels improve later seven-day path loss beyond group-mean and dispersion calibration by 0.0154 and 0.0048 nats on two frozen representations, with favorable adjusted seed intervals; for transfer, the same law yields no store-code contrast with a favorable adjusted interval on a public retail panel, so fitted value is population-specific. On 3,022 public crowdfunding campaigns, ordinary history recalibration is a strong reference, and adding centered category and country indicators lowers the mean loss by a further 0.03254 nats. A favorable true-versus-permuted contrast is therefore not by itself evidence that identity helps: label ablations need shared-code and ordinary-calibration references.