recognitionMeshDualEntryCoupling4DStatus_flags
plain-language theorem explainer
Status snapshot for the Wave B residual R4 mesh dual-entry coupling: R4 is marked closed, the ledger-named recognition-ratio binding stays open (owned by R5), and the gap1 bridge flag remains false. Gravity and QG completion auditors cite it to pin honesty flags on the residual DAG. Proof is a one-line decidability check on the concrete boolean record.
Claim. The R4 mesh dual-entry coupling status record satisfies $r4Closed = \top$, the recognition-ratio ledger binding remains open, and $gap1BridgeDerived = \bot$.
background
Wave B of the QG full-completion session attacks the residual TypedResidual_DeficitSourceConstitutiveCoupling_from_enrichment. Residual R4 assembles banked pieces R1 (mesh geometric deficit), R2 (mesh hinge kappa with source-dominated regime), and R3 (dual-entry strain state) into an inhabited DeficitSourceConstitutiveCoupling on the carrier $H = \mathbb{R}$, then applies the blocker's conditional recognition-ratio derivation.
The status record is a three-flag honesty board for that module: whether R4 itself is closed, whether a ledger-named standalone recognition-ratio Prop binding is still open (that binding is R5's job, not R4's), and whether the gap1 bridge has been derived. Upstream, the status definition hard-codes those three booleans; this theorem only reifies them as a proved conjunction.
Convention in the module: deficit iff debit-leads ($0 < h$), mirroring the Regge-side geometric-deficit convention. Carrier is the reshaped real line from R1/R2, not an encoded Freudenthal triangulation.
proof idea
One-line wrapper: decide on the three boolean fields of the concrete status definition. No algebraic work; the definition already sets $r4Closed$ true, ledger-binding-open true, and $gap1BridgeDerived$ false, and Boolean equality is decidable.
why it matters
Pins the honesty contract of residual R4 in the QG Wave B gap1 residual DAG. Module doc is explicit: R4 does not flip $gap1_bridge_derived$, does not introduce a ledger-named standalone recognition-ratio Prop (that is R5), and leaves R0a/R0b validation name-bindings open. Downstream consumers of the residual stack can therefore treat R4 as closed for the dual-entry constitutive coupling while knowing the gap1 bridge and the SevenGaps recognition-ratio ledger binding remain future work. No used-by edges yet; the theorem is a status seal rather than a lemma in a longer derivation chain. Framework role is bookkeeping inside the gravity analysis layer, not a T0–T8 forcing step.
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