Safety Certificates for Selective Distillation: Routing Entropy and Verifiable Adaptation

Selective distillation leaves most histories without a teacher label. A safety certificate must therefore explain both what limits behavior there and what permits useful generalization. We study this problem using finite sequential models and architectures with enforced adaptation regions. Our main coverage result handles a base far from its teacher: under biased routing, the exact optimal supervision allocation reduces to weighted marginal gains, even when every router is eligible for supervision. At fixed fractional risk reduction, the budget exponent is routing entropy rather than the logarithm of the number of histories. We then construct gated logit envelopes that preserve a base outside a declared region while allowing large teacher-directed changes inside it, and derive parameter-only risk certificates. Experiments distinguish these structural guarantees from ordinary regularization. Four-action, two-layer transformers face rare teacher exceptions at horizons 24 and 32; query targeting, coarse sharing, and bounded residuals are varied separately. Fixed-multiplier controls on held-out tasks show that signed utility-aware local selection can match exact planning at matched completion and supervision budget. The practical recommendation is to establish the adaptation region, count rare-label access, include utility in selection, and check any behavior outside the certified envelope.