Manuscript XI-G@1 · XI-G.tex · section · f029bbcd

Decision, exclusions, and handoff

Decision, exclusions, and handoff · lines 1364–1419

Gate	Constructed only if	Obstructed if	Unresolved if
G1	$B_{Σ}$ explicit in Cauchy data	—	pairing only named
G2	bulk term and remainder separated	—	one-slot rewrite used as proof
G3	$u \in F_{mc, pol}$ and agrees on the shell	every reproducing distribution has a forbidden covector	pairing nonzero, extension open
G4	exclusions recorded	—	a product or $s_{int}$ is implied

Gate	Ending	Exact scope
G1	Constructed	$B_{Σ} = \int_{Σ} β ∣_{Σ} \cdot Q (j_{1}, j_{2})$ with $Q = q$ on jets, $\partial_{t}^{2} \mapsto Δ$ (Lemma lem:cauchyred); pure-trace sector printed (Corollary cor:Bsigtrace); counter-propagating symbol $q^{00} = - 2$ , $q^{0 i} = 0$ , $q^{ij} = n^{i} n^{j} - 3 δ^{ij}$ with Bianchi check (Lemma lem:qvalues); scalar slot a spectator.
G2	Constructed	$ker (w \mapsto W_{w}) \cap E^{'} = I_{sol}$ (Lemma lem:kernel); $v \in [u]_{sol} ⟺ W_{v} = W_{u}$ ; class meets $F_{mc, pol}$ iff (eq:criterion) (Theorem thm:criterion); plane-wave pairings are class invariants with leading coefficient $\frac{i}{2 λ} B_{Σ}$ (Proposition prop:planewave).
G3	Proved obstructed	F1 data: pairing $- iλ Λ_{f} + O (1)$ versus $O (λ^{- \infty})$ for every microcausal kernel; $[u]_{sol} \cap F_{mc, pol} = \emptyset$ (Theorem thm:obstruction); inherited covectors $\pm (k_{1♭}, k_{2♭}, 0)$ , $k_{1♭} + k_{2♭} = 2 d t$ (Corollary cor:covector); general detection by $b (n, p) \neq \equiv 0$ (Proposition prop:general); Einstein subshell not decided.
G4	Recorded	No bimodule no-go, no product, no differential, no loop, no limit, no jet repair.

The worst ending among G1–G3 governs: Proved obstructed, at Gate 3. The frozen-kernel theorem of XI-F is not reopened; it is inherited by the class in split form. The classical ledger is not erased.