Acting Is Easier Than Adapting: The Geometry and Design of Adaptation-Sufficient Representations

A representation can be sufficient for acting in the training environment yet insufficient for adapting after the environment changes. The missing object is not the nominal action but its prospective response: how the optimal action moves with the environment. For smooth continuous actions, the curvature-weighted projection of this response onto a frozen code yields a local conservation law, $\Omega ={\Gamma }_{\varphi }+{\Lambda }_{\varphi }$, \(\Omega=\Gamma_\phi+\Lambda_\phi\), decomposing prospective sensitivity into irrecoverable representation debt and executable motion. That motion is useful only to the extent that deployment evidence can identify which response to execute. An exact three-policy Gaussian frontier prices this evidence and shows that representation refinement is discounted by the resolvable fraction of newly executable motion. Representation rankings are therefore evidence-dependent and can reverse even between equal-capacity codes with identical nominal control. AP-GFAR operationalizes this principle by learning executable prospective responses and selecting representation–evidence programs from calibrated branch values, with freezing as an explicit option. Controlled reward systems verify the refinement law and ranking reversal, and equal-capacity learned encoders reproduce it. In native HalfCheetah, pre-evaluation complete-program pair-value estimates predict heterogeneous rankings under fixed interfaces; a prospectively locked, cross-fitted interface design then yields the predicted reversal across all ten declared systems. Fresh learned-POMDP and fixed-checkpoint VariBAD studies extend the evidence beyond the analytic model, showing gains from complete-program valuation over frozen/no-probe and native online deployment, respectively. Representation value is therefore a property of the representation–evidence pair, not of the representation alone.