AreaPushforwardMatchOpen_holds
plain-language theorem explainer
On every (1,1) hinge slot, the covering-transported orbit area covector equals the classical slot area covector. Gravity analysts cite this to discharge the formerly open (1,1) area-pushforward identity in the transported all-orbit 4D Bloch fold. The proof is a one-line reindex: unfold the uniform covering definition and apply the already-proved (1,1) recovery lemma.
Claim. For every slot index $s\in\{0,\ldots,23\}$ and hinge type $t\in\{0,\ldots,9\}$ with $(s,t)$ of orbit type $(1,1)$, the area covector obtained by transporting the $(1,1)$ seed along the orbit covering permutation equals the classical slot area covector at $(s,t)$.
background
This module builds a continuum-facing multi-orbit 4D Bloch fold: each of the 24 slots transports its orbit's seed area covector and star deficit kernel by the first $S_4$ covering map orbitCoveringPerm from orbit representatives to difference-mask pairs. The lesson from earlier work is that the factorized transport used for pure $(1,1)$ must not be reused on other orbits; measured all-orbit $m^2$ along the symbol direction on the TT-plus axis is $-5/2$ raw, not the factorized $0$.
The predicate being proved packages the $(1,1)$ area pushforward identity: transported orbit area at type $(1,1)$ under the covering permutation recovers the classical slot area covector. The uniform covering pushforward of the assembly area covector (all orbits) is the sibling definition that simply applies transported orbit area to the covering permutation. Upstream, the theorem that on $(1,1)$ slots this uniform pushforward equals the classical slot area covector is already available; it reduces via the covering-permutation identity for $(1,1)$ and the div-by-4 pushforward lemmas for the area covector.
proof idea
One-line wrapper. Introduce the slot $s$, type $t$, and the hypothesis that $(s,t)$ is $(1,1)$. Rewrite the goal by unfolding the uniform covering definition of the slot-orbit area covector, then apply the upstream recovery theorem that already equates that covector to the classical slot area covector on $(1,1)$ slots. simpa closes the equality.
why it matters
Closes the formerly open Prop that named the $(1,1)$ area pushforward identity inside the transported all-orbit 4D Bloch fold. Downstream, the algebraic closer theorem for transported Regge 4D area match is exactly this result (one-line alias), and the compatibility wrapper used by older callers is the same identity under a different name. Together these feed the claim that the $(1,1)$ orbit fold recovers the classical single-orbit Bloch fold.
In the module status list this is the proved piece of covering-based transport and $(1,1)$ recovery of the classical area covector via uniform covering pushforward. It does not touch the still-open all-orbit $m^2$ Tendsto or the continuum Einstein-Hilbert isotropy residual; nor does it flip gap-action recovery. Within Recognition gravity analysis it is bookkeeping that keeps the multi-orbit fold honest on the classical slice before continuum limits are attempted.
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