branchRegular32_pair04
plain-language theorem explainer
Branch regularity of the split-form dihedral continuation holds for opposite vertices 0 and 4 of the (3,2) causal 4-simplex on the open Wick-arc interior. Anyone assembling the all-hinge threeTwo Wick certificate cites this mixed-pair instance. The proof is a one-line specialization of the parametric mixed-pair branch lemma, feeding the closed-form cofactors C_{00}=8z-4, C_{44}=6z-2, and C_{04}=-1.
Claim. For the $(3,2)$ causal 4-simplex at physical parameters $a=1$, $\alpha=1$, the complex Wick-arc edge continuation is branch-regular for opposite vertices $0$ and $4$ on the open interval $(0,1)$: both diagonal Cayley-Menger cofactors stay in the complex slit plane, and the split-form dihedral cosine stays off the arccos cuts.
background
Lane B2 of the QG Seven-Gaps campaign certifies complex-first Wick continuation for every triangular hinge of the threeTwo causal 4-simplex (three vertices on the lower slice, two on the upper). Timelike edges follow the upper-half-plane arc; spacelike edges stay fixed at $a^2$. Opposite vertex pairs fall into three classes; pair $(0,4)$ is mixed (one lower, one upper), so the hinge triangle is ${1,2,3}$ with two timelike edges.
Branch regularity on a parameter set means both diagonal cofactors of the Cayley-Menger matrix stay in the slit plane (the continuity region of principal square root) and the split cosine stays off the arccos cuts. For mixed pairs the closed forms are asymmetric: lower cofactor $8z-4$, upper cofactor $6z-2$, off-diagonal $-1$.
The parametric mixed-pair lemma already proves that whenever those three closed forms hold, both cofactors lie in the open upper half-plane on the arc interior and the cosine $-1/(s_1 s_2)$ avoids the arccos cuts. The three cofactor identities for indices $(1,1)$, $(5,5)$, and $(1,5)$ (vertex-index convention) are kernel-checked 5x5 minor evaluations.
proof idea
One-line term proof: instantiate the parametric mixed-pair branch certificate at opposite vertices $0$ and $4$, supplying the three closed-form cofactor lemmas (diagonal lower $8z-4$, diagonal upper $6z-2$, off-diagonal $-1$). No extra analytic work; the class-B argument (upper-half-plane cofactors, first-quadrant square roots, positive-imaginary product, cosine off cuts) is inherited wholesale.
why it matters
This is one of the six mixed-pair legs of the B2 headline theorem: for every unordered opposite pair of the threeTwo simplex, branch regularity on the open arc interior plus continuous split-cosine path on the closed interval ending at the Euclidean regular value $-1/4$. Without each concrete pair certificate the universal quantifier over hinges fails.
In the broader Recognition gravity stack this closes the complex-first Wick lane for the $(3,2)$ building block of 4d causal dynamical triangulations, the dual of the $(4,1)$ type. It does not itself touch the forcing chain (T0-T8) or the Recognition Composition Law; it is infrastructure for the continuum limit of the discrete gravitational path integral once the arc continuation is under analytic control.
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