branchRegular32_pair03
plain-language theorem explainer
For the (3,2) causal 4-simplex at unit edge parameters, the split-form dihedral continuation at opposite vertices 0 and 3 is branch-regular on the open Wick arc interior (0,1). This is one of the six mixed-hinge certificates (hinge triangle {1,2,4}). Anyone assembling the all-hinge Wick continuation cites it. The proof is a one-line specialization of the parametric mixed-pair branch certificate using three closed-form cofactor identities.
Claim. The complex edge continuation of the $(3,2)$ causal $4$-simplex along the Wick arc at $a=1$, $\alpha=1$ is branch-regular at opposite vertices $0$ and $3$ on the open interval $(0,1)$: both diagonal Cayley-Menger cofactors stay in the complex slit plane, and the split dihedral cosine stays off the arccos cuts.
background
In the QG Seven-Gaps campaign (lane B2), one continues the Regge action of a causal 4-simplex through a complex Wick arc. The module treats the threeTwo type: three vertices on the lower slice ${0,1,2}$, two on the upper ${3,4}$; the six cross edges are timelike and follow the arc $z$, while spacelike edges stay fixed at $a^2$.
Branch regularity is the S3 model predicate: both diagonal cofactors of the Cayley-Menger matrix must lie in the complex slit plane (continuity region of principal square root), and the split cosine must stay off the arccos cuts. Mixed opposite pairs (one lower vertex, one upper) have asymmetric closed forms $C_{pp}=8z-4$ (lower), $C_{qq}=6z-2$ (upper), $C_{pq}=-1$, with area squared $z/4-1/16$.
The pair $(0,3)$ is mixed; its hinge triangle is ${1,2,4}$. The parametric class-B certificate already proves slit-plane and off-cut status on $(0,1)$ once those three cofactor identities are supplied.
proof idea
One-line term proof. Specialize the parametric mixed-pair branch certificate at vertices $p=0$, $q=3$, discharging its three hypotheses with the closed-form cofactor lemmas for this pair: the lower diagonal identity ($C_{pp}=8z-4$), the upper diagonal identity ($C_{qq}=6z-2$), and the off-diagonal identity $C_{pq}=-1$. The parametric theorem then yields both slit-plane memberships and off-arccos-cut status of the split cosine on the open arc interior.
why it matters
Feeds the B2 headline theorem, which asserts that for every hinge of the threeTwo causal 4-simplex (1 spacelike + 6 mixed + 3 upper-pair), at $a=1$, $\alpha=1$: (i) the split-form continuation is branch-regular on the full open arc interior, and (ii) the split cosine path is continuous on the closed interval $[0,1]$ and ends at the Euclidean regular-4-simplex value $-1/4$.
This declaration is the mixed-pair certificate for opposite vertices $(0,3)$ among the six asymmetric hinges. Closing all ten hinges is the second deliverable of lane B in the finishing charter. Within the Recognition gravity stack it supplies controlled complex-first Wick data for CDT building blocks; it is not itself a T0-T8 forcing step, but a concrete analytic gate on the (3,2) simplex action.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.