Admissibility under Declared Transformations: Invariant Verdicts, Compatible Quotients, and Resolution Limits

Abstract What must a declared representation preserve for judgments to remain invariant under redescription and transformation? This paper gives exact conditions for compatible representations and identifies the obstruction when those conditions cannot be jointly satisfied. For a declared equivalence R and an observation kernel K, the eligible representations are exactly the equivalences E satisfying R ⊆ E ⊆ K. They exist precisely when R ⊆ K and then form a complete lattice in their induced order. Requiring compatibility with a transformation catalogue narrows this family to an interval of congruences between the least catalogue congruence containing R and contextual indistinguishability Θ. Existence is equivalent to R ⊆ Θ; uniqueness holds exactly when the endpoints coincide. Nonexistence has a finite-context witness: a declared-interchangeable pair whose observations differ after a finite sequence of transformations. Effective verification requires separately stated presentation assumptions. The paper also distinguishes named-action descent, may/must abstraction and path lifting; separates preservation from authorization; and shows why sound admissibility rules need not be deviation thresholds. For finite conflict orders, order-representable resolutions correspond to linear extensions, equivalently maximal chains of the ideal lattice, rather than to its individual elements. These results characterize compatibility and its limits under explicit declarations. They do not establish unrestricted decidability, computational affordability, physical correspondence or a uniquely forced architecture. This paper supersedes A Forcing Theorem for Admissibility Under Transformation (2026). Appendix A records the predecessor’s claims and their dispositions. This version replaces the forced-architecture interpretation with explicitly scoped compatibility, invariance and obstruction results. The accompanying verification supplement contains executable checks, recorded outputs, the editable source and a payload manifest.