The record-history kernel Pₖer adopted earlier in this series reads its source object — the record-export current's localization, rhoJ = sumₖ nuₖ wₖ (x) — off declared weak-field coordinate cells. That convenience was registered as a debt: Paper 35 set up and left open "the genuine dressed/diffeomorphism-invariant region-algebra construction, " Paper 42 barred discharge claims on "dressed sources (35), " and Paper 45 — which certified the kernel's causal core under a minimal dressing assumption D0 (apparatus-anchored regions on a fixed background) — assigned this panel the dressed replacement as its open conjunct (o3). This paper is the permitted attempt on that debt, for the kernel's source object specifically: not the general region-algebra problem of quantum gravity, which stays open on every branch and is said to stay open. The question audited: does what the kernel's sources are require a unique, background-independent, strictly local algebra per source — the Global Local Algebra, barred in any case by the imported Donnelly-Giddings obstruction — or does a declared family of admissible dressings, plus demonstrated invariance of exactly the functionals that enter experiments, do all the work the physics needs? The load-bearing objects are fixed first. A dressed source package assigns each admissible dressing a support and normalized profile densities on the weak-field background, with dressing acting on localization only — the intensities, operators, and eligibility thresholds are apparatus-side and dressing-invariant by typing. Three dressing classes are constructed — anchor-record (on a non-circular anchor base fixed by Paper 42's a-priori eligibility certificate), relational matter-field (partial-observable level sets), and world-tube/causal-diamond (the relational reading whose objects Paper 45's covariant candidates consume) — each with domain, map, regularity, a covariance argument, a named reduction map to Paper 41's cells, and a failure branch. Within the family, the audit is run on the transport-bounded convention subclass C^ (v): dressings whose profiles satisfy the declared Wasserstein-1 bound W₁ (wₖᵃlpha, wₖ) <= ellc relative to the certified cell profile. That bound is a declared subclass membership condition, not a theorem about all admissible dressings — the paper says so plainly, displays per class what would verify membership, and treats the derivation of membership from each construction's first principles as open work. The flagship is the invariance audit. Four functional rows exhaust what the kernel's empirical content reads off rhoJ. The order-face pair — Paper 43's excess-law coefficient and the certified-export count — is proved identically invariant across the entire admissible family, subclass irrelevant, by a one-line typing argument: those observables contain no localization data at all. The drift moment is invariant on C^ (v) with the bridge from intensity profiles to gravimeter response displayed (Kantorovich-Rubinstein plus the Newtonian kernel give a dressing residual of about 2e-18 at Paper 42's co-location benchmark — nine to sixteen orders below every declared gravimetry tolerance). The geometry-noise source is audited at Paper 41's actual bilocal covariance form on a declared positive test class (non-negative, unit-sup, grain-scale, lower-overlap — so the relative bound is licensed), with Paper 43's response scalar re-checked at sixteen orders of margin; no audited cell consumes Paper 44, which remains context only. The verdict is scoped, not discharged. What is certified: on the named convention class C^ (v), at weak-field order and the declared tolerances and operating points, every entering functional is invariant — identically for the order face (family-wide), with displayed margins for the exposed rows — so the kernel may legitimately run with its source object typed as relational on a named convention class, and Paper 45's (o3) receives exactly that: a convention class, not a completed dressing. Why not discharged: the transport bound that delimits C^ (v) is the subclass's definition rather than a derived property of the constructed dressings, so paying the debt in full would require the membership derivations (open, per class), an audit of admissible dressings beyond the subclass, and the general region-algebra construction (Paper 35's deep half, untouched). Supports may remain admissibly open while every audited value holds — non-uniqueness of where and invariance of what enters experiments are compatible, and that combination, on its named class, is what replaces the Global Local Algebra. Nothing is adopted, calibrated, or scored; no band moves; the substantive canonical-dressing face is declined, not refuted.
Tomoyuki Uchida (Fri,) studied this question.