A dynamics built from many local record-export events faces a question the moment two of them interact: in what order does the world apply its own updates? Paper 41 of this series composed the record-history kernel along one laboratory time, by declaration. Paper 45 built the covariant per-event core and proved composition on the causal partial order — at leading order only — registering the remainder as its open conjunct (o2): the multi-event composition law beyond leading order, where geometry back-reaction couples updates within a light cone, the classical-quantum analogue of the Tomonaga–Schwinger integrability condition. Paper 54 covariantized the accounting boundary and handed this panel the closure discipline. The presumed authority under audit here is the Global Integrator — the master time-orderer whose privileged foliation would be needed to compose the events into one history. This paper — the fifty-fifth, dynamics panel (second of three) of the Covariant Completion Program (Papers 54-56) — audits exactly the seated conjunct. Delivered: a Tomonaga–Schwinger lift — a surface-indexed state rhoSigma of the hybrid kernel over a declared, diamond-adapted class of Cauchy-surface sequences, with per-event Fewster–Verch updates applied slab by slab, typed at model grade against the Torre–Varadarajan obstruction (no unitary-implementability claim anywhere) ; a three-part integrability condition IC — generator commutation, placement/compensator invariance with screened increments, retarded response — with a leading-order theorem: same-endpoint foliation-path independence on the certified dephasing subclass, proved by reducing surface sequences to linear extensions of the realized event poset (a displayed interpolation-and-refinement construction; inseparable diamonds merged into nodes, the quotient proved acyclic) and connecting extensions by spacelike transpositions that IC makes harmless; an explicit out-of-class witness — two full dephasings of one nonlocal register in bases 45 degrees apart, whose two admissible orderings end trace distance 1/4 apart — showing IC earns its keep; a second-order relocation theorem, proved by expanding the composed map at the displayed orders (lambda², G*lambda, G*lambda², tree-level response) and computing every transposition-difference term — instrument commutators vanish under the minimal dressing assumption D0, stochastic cross-variations vanish under the coupling, retarded response terms cannot connect spacelike nodes — so under D0 no same-endpoint ordering term arises at those orders; whether dressed update operators, which fail exact spacelike commutation, reintroduce one is precisely the class-boundary question, relocated to the dressing frontier and left open there in both directions; and an exact rest-frame recovery of Paper 41's composed kernel with the Model M band unchanged. The realized ceiling is scoped, not certified, and the abstract says so: register conjuncts (t1) - (t4) close at the displayed orders on the declared classes; (t5) — composition beyond the displayed orders and beyond D0 (dressed update operators; iterated geometry feedback and all-orders composition; nonperturbative classical-quantum constraint consistency) — remains the named frontier, scored open here and built elsewhere. Beyond the displayed orders, "the residue is relocated" is an audit finding, not a theorem, and the paper says which is which. Verdict, from the charter-frozen domain: CovStatus (dynamics) = scoped — at model grade, on the declared surface-sequence, event, and population classes, conditional on the adopted kernel, Paper 45's certified core, D0, and H-path. No new physics is adopted; no calibration is spent; no datum is scored; no band moves; and no Global Integrator is employed anywhere on the way. Version 1. 1 applies one pre-submission external review in full (quotient acyclicity as a displayed lemma; the interpolation lemma rescoped to model grade with the Bernal–Sánchez smoothing theory imported; refinement invariance split into its definitional and dynamical halves), re-verified item by item.
Tomoyuki Uchida (Sun,) studied this question.