How to Reuse a Representation

An invertible feature change can preserve every classifier logit yet change which inputs a newly fitted out-of-distribution (OOD) rule accepts. This raises an information question for representation reuse: what must remain available for the decisions and subsequent operations of a downstream procedure? We attach representations to the structures their consumers use and check how fitting, selection and calibration carry through to final decisions. In no-intercept quadratic fitting with summed loss, a fixed full-rank design X, fixed penalties and a retained X^⊤ y, current label agreement for all real targets y and queries determines the Hessian up to a positive scale. Requiring agreement both before and after appending one prescribed nonzero input with target zero determines the full Hessian (d≥2). In controlled same-logit OOD comparisons, mild feature rescalings change fully reselected and recalibrated ASH-B accept/reject decisions by a median 6.42% across 24 model/OOD comparisons. Compatible constructions preserve their own decision rules on valid tested domains. What a representation can safely discard depends on its declared use.