Paper 57 built the corpus's dressed region package on a fixed ambient net and scored its own ceiling in one sentence — "Anchor-carried is not background-free" — registering the deep half as (r5): full background independence and beyond weak field, "scored here, open; never attempted; closes only by a successor's exhibited construction. " Papers 69 and 70 then built the two candidate inboxes for that construction (the background-free support-functor candidates and the order-sector candidates), deposited as the artifacts U1 and U2. This paper opens the Twentieth Grouping — the Strong-Field Completion Program (Papers 71–74) — on an executed owner decision standing on those two deposits, and executes its keystone. The charter sentence is binding: "U1 and U2 are artifacts, not selection, construction, or adoption. " The corpus moves from typing to construction, and says so in a chartered program rather than by drift. Part I is the charter, complete in itself and consumed unmodified by Papers 72–74: it fixes the four panels, the verdict trichotomy CompletionStatus ∈ certified, scoped, killed, the mandatory K0, the owner-locked P-D hard lock, and the rule that successors consume definitions and normal forms only. Part I defines four elements the successors execute against — P-A (this paper's keystone), P-B (the order-sector audition over the P-A support functor), P-C1 (the quasi-local horizon predicate, classification grade), and P-D (recoverability adjudication under the hard lock) — plus the P-C2 adoption-routed branch (barred without its own 63-type mechanism), the U4 pre-classification table, and the foil-lineage table whose fourth row is this paper's foil, the Global Scaffold: the declared fixed background — ambient net, metric, certified cell assignment — persisting beneath every anchor-carried object, personified as the authority that supplies it wherever a construction cannot derive it. Part II executes P-A, the corpus's first (r5) construction attempt, run as an obstruction test. On the two rows Paper 69 typed as conditionally admissible — F1 (crossed-product/modular) and F3 (dressed-observable) — the paper declares each row's source as a genuine category with no background token, and runs the background-free support functor field-by-field against a noncanonical-embedding obstruction defined before any construction (a declared reduction to 57's fixed-net package, required functorial, natural, exact on 46's anchor classes and on 57's axioms at their honest strengths, and strict enough to recover the frozen weak-field corner without choosing an extra embedding datum), so each row terminates as "constructs" or "obstructed at declared-class grade" by the row-outcome rule, never "it was hard. " The realized landing is an obstruction, proved: F3 does not construct an exact declared-class weak-field instance of 57's package. The imported Donnelly–Giddings nonlocal algebra is a first-order deformation in Newton's constant; every star-homomorphic recovery into 57's ordinary fixed-net package annihilates its first-order (dressing) layer (Lemma λ), and — independently, at the relation level — 57's spacelike-compatibility axiom forces the images of a margin-spacelike pair to commute exactly while the imported first-order commutator of that pair is a declared nonzero word, so no first-order-exact reduction exists (Theorems T2/T2′) ; the source-side support map is relational and background-free, but the faithfulness and compactness the recovery needs are codomain-side, and they are what force the obstruction; and the two quantities the projected construction would have to equate — 57's finite geometric margin and the DG dressing tail — are not even of the same type (Theorem M−, an operator-kernel support width versus a c-number geometric margin). F1 lands a conditional noncanonical-embedding obstruction — a two-patch witness plus a type-rigidity lemma (no unital normal isomorphism between a type II₁ and a type II_∞ factor) shows no candidate functor is well-defined on the observer–record objects without adjoining a patch, an extra embedding datum — with the unconditional kill left pending a comparison-relation token semantics and the normal-form quantifier domain, both named open in-text. The ceiling is strict and the null is the result: certified is structurally unreachable — the (r5) general/deep half stays open — and killed is a first-class landing (null is a result): no framework is selected, nothing is adopted, no physics is claimed on any branch. CompletionStatus (region construction) = killed, via K3 in its DG-facing instance K3-DG (a precisification of the pre-committed K3, not a new verdict value) — Paper 57's pre-registered DG-facing kill row, discharged here with a theorem. The construction's honest yield is recorded as positive residues: a λ⁰-shadow recovery (the scaffold's shadow returns; the dressing does not), an SCS rigidity theorem (a canonical background-free support map exists if and only if the instance has a separating coincidence structure — structurally 57's own uniform/nonuniform split), the first-order decay strength a successor must target, and the unique first-order typing of the dressing algebra. The Global Scaffold wins honestly, and the paper says exactly why and exactly what would have to change. "To construct on a declared class is not to select a framework. " / "A scaffold-free reduction must recover the scaffold without consuming it. "
Tomoyuki Uchida (Tue,) studied this question.