When Pairwise Counterfactual Bounds Are Exact: Conflict Certificates and Temporal Compression

Pairwise counterfactual bounds can combine incompatible response mechanisms. We characterize when a supplied query attains this relaxation and construct a compact attained representation of the complete sharp function for a structured temporal class. For rational mechanism polytopes, occupied Bellman residuals and a backward no-leakage recursion give an exactness criterion and recover a rational witness, with polynomial bit cost at fixed coordinate dimension. For a structured three-state, binary-successor, same-action model with independent untied time laws and nonnegative rational one-defect terminal costs, original-domain repair, control of nonmetric histories, and first-entry counting bound the entire representation by $(392n+81)490^r$, where $r$ counts exceptional layers. The guarantee transfers to standard exact full-cone pruning. Same-support complete-atom transport gives certified enclosures, with error scaled by terminal range and entrance mass. Coverage and chronological attainment are checked separately. Fixed three-state instances require $\Omega (r)$ attained values. A public token-circulation protocol supplies native $r=0$ dynamics for a shared-response completion query. Synthetic comparisons measure a block/generic cold-cost ratio of $0.6746$ and changes in optimal route designs; supplied-query batches favor diagnosis or direct search. A growing-state construction establishes NP-hard evaluation and an exponential attained-family boundary.