Post-Edit Regret for Lifted Quotients: Exact Accounting and Feasible Certificates

A state-action abstraction is deployed through a particular coarse model, optimizer, and policy lift; its realized regret reflects this full deployment, beyond the aggregation map. For finite-horizon Kullback-Leibler (KL)-regularized control, we derive an exact signed ledger for this solve-lift-deploy process, separating prior, transition, optimization, and evaluation-cost mismatch. We identify the local price of fixing the environment dynamics as an information projection, decompose it into successor tilting and action reweighting, and allocate the global price through counterfactual occupancies. A residual-control identity then combines this information cost with the original-to-surrogate cost difference. It yields a coupled, dynamics-feasible regret envelope contained in the separate-error envelope. Its raw width vanishes exactly when the combined residual is policy-invariant, equivalently a Bellman potential difference; cancellation can therefore give exact certificates even when separate error budgets remain positive. Finite-model benchmark results evaluate the separate envelope: among 370 strictly positive-width candidate pools, 54 uniquely certify the true optimum, 102 partially prune, and 214 abstain, with no false certifications. Taxi action refinement reduces exact regret by 37.1%; uniform model-error padding covers all 556 nonzero-error pools and remains informative in 494. Independent path enumeration and randomized algebraic audits support the underlying identities. Together, these results provide exact post-deployment accounting and an interpretable hierarchy of certificates for explaining abstraction effects and certifying decisions.