Endpoint Sufficiency Behavioral Quotients
A provenance-decorated generative system may contain distinct occurrences with the same visible endpoint. The central abstraction question is therefore exact: when may provenance be forgotten without changing the lawful future? We study this question through the labeled transition system induced by lawful generation and an endpoint projection U from occurrences to visible objects. Three observation levels are separated. Enabled sufficiency preserves immediately available rule labels; trace sufficiency preserves all finite lawful rule-label traces; quotient sufficiency preserves the branching transition structure modulo endpoint equivalence. The resulting hierarchy is strict. At the linear-time level, endpoint trace sufficiency is exactly inclusion of endpoint equivalence in finite-trace equivalence. At the branching level, the canonical endpoint quotient is representative-independent exactly when endpoint equivalence is a strong bisimulation equivalence. Two canonical repairs follow. Intersecting endpoint equivalence with trace equivalence gives the greatest endpoint-respecting relation preserving finite traces. The greatest endpoint-respecting bisimulation gives the maximally coarse exact branching quotient and is universal among endpoint-respecting exact quotients. For finite systems it is computed by a terminating partition-refinement procedure initialized by endpoint classes. A self-contained application to provenance-decorated nested recursive-recombinant generation exhibits two occurrences with the same visible graph I->A->B but different enabled futures; the refinement procedure separates them in its first round. The result replaces an all-or-nothing demand to retain provenance with an exact criterion for retaining only the distinctions that remain behaviorally operative.