Paper 55 of this series proved that the certified record-history kernel composes foliation-independently at the displayed orders under D0 — undressed update operators, Kraus operators in mutually commuting spacelike algebras — and relocated the dressed case in both directions: "dressed update operators (beyond D0) — the class-boundary question — not settled here in either direction: physical dressed operators fail exact spacelike commutation (Donnelly the paper's thesis, annotations, and verdict all carry that qualifier. The audited authority is the Global Integrability Oracle: the guarantor that the world's many dressed local updates compose consistently, assumed rather than earned. The paper replaces the pronouncement with a computation. Delivered: the dressed slab calculus — Paper 55's one-node factorization lifted to the package, with the dressing correction displayed per factor in three typed components (support re-anchoring, collar-bounded; the retarded dressed-response correction; the constraint-tail conditioning at the kernel's classical-quantum grade), complete positivity, adaptedness, and localization proved, with the common-domain fact typed at the channel level (apparatus channel type vs support realization — no canonical local-algebra element identity asserted across anchorings, consistent with Paper 57's sandwich) ; the residual theorem — the transposition computation extended to the full dressed pair table (sixteen cross-factor pairs, displayed as a 4x4 matrix with an order dictionary, each computed, plus the surface-tied booking schemes whose mid-composition differences cancel at the endpoint) — landing every in-class same-endpoint spacelike transposition residual in the zero cell of the pre-committed typology zero, pure-convention, order-effective, strict map identity for every admissible representative, the trichotomy exhaustive and decidable via a fully specified dressing-convention quotient; the representative-equivalence lemma owning the presentation content (the pure-convention cell is inhabited by presentation data, not endpoint residuals) ; Paper 45's (o4) operator families entering as first-class rows (their composition-facing half decided — no self-adjointness consumed; their measurement-scheme half returned open-named) ; the stability theorem (Paper 55's Theorems 1-2 persist under dressing at both equality grades) ; an out-of-class witness displaying the order-effective cell's shape (a quantum-tail toy phase, with the Weyl-convention note: the factor-2 ordering ratio is convention-independent in magnitude) ; and exact recovery of the undressed calculus with the Model M band unchanged. The verdict is scoped, not certified: conjuncts (d1) - (d4) close at the displayed orders on the declared classes; (d5) — iterated feedback, all-orders composition, the nonperturbative classical-quantum constraint sector, and the quantum-field dressing boundary — is scored open and built elsewhere. FrontierStatus (composition) = scoped at model grade. Paper 55's (t5) closes the kernel-grade half of its dressed-operator row by annotation; the Oracle is unemployed: consistency was computed, not pronounced. Nothing is adopted, calibrated, or scored; no band moves; the substantive integrability-of-nature face is declined, not refuted.
Tomoyuki Uchida (Sun,) studied this question.