branchRegular32_pair24
plain-language theorem explainer
Branch regularity of the Wick-continued split-form dihedral cosine holds for opposite vertices 2 and 4 of the (3,2) causal 4-simplex on the open arc interior (0,1) at the physical point a=1, alpha=1. Anyone assembling the all-hinge B2 certificate for threeTwo simplices cites this mixed-pair instance. The proof is a one-line specialization of the parametric mixed-pair branch certificate, feeding in the closed-form cofactors C_22=8z-4, C_44=6z-2, and C_24=-1.
Claim. For the complex Wick continuation of squared edge lengths of a $(3,2)$ causal $4$-simplex at $a=1$, $\alpha=1$, the split-form hinge continuation at opposite vertices $2$ and $4$ is branch-regular on the open interval $(0,1)$: both diagonal Cayley-Menger cofactors lie in the complex slit plane, and the split cosine stays off the arccos cuts.
background
Lane B2 of the QG Seven-Gaps campaign certifies split-form branch regularity and boundary continuation for all ten triangular hinges of the threeTwo causal 4-simplex (three vertices on slice $t$, two on $t+1$), at the physical point $a=1$, $\alpha=1$, along the canonical upper-half-plane Wick arc.
Opposite vertex pairs fall into three classes. Mixed pairs (one lower, one upper; six of them) determine hinges with two timelike triangle edges and asymmetric closed-form cofactors $C_{pp}=8z-4$ (lower member), $C_{qq}=6z-2$ (upper member), $C_{pq}=-1$, and area squared $z/4-1/16$. Pair $(2,4)$ is one such mixed pair; its hinge triangle is ${0,1,3}$.
Branch regularity on a parameter set means both diagonal cofactors stay in the slit plane (continuity region of principal square root) and the split cosine stays off the arccos cuts. The complex edge continuation sends timelike edges along the Wick arc and holds spacelike edges at $a^2$.
proof idea
One-line term proof: apply the parametric mixed-pair branch certificate at vertices $p=2$, $q=4$, supplying the three closed-form cofactor identities already proved by explicit $5\times 5$ minors on the hinge edge matrix. Those identities are $C_{22}=8z-4$, $C_{44}=6z-2$, and $C_{24}=-1$. The parametric theorem then places both cofactors in the open upper half-plane on the arc interior, so each principal square root lies in the open first quadrant, their product has strictly positive imaginary part, and the cosine $-1/(s_1 s_2)$ stays off the arccos cuts.
why it matters
This is one of the six mixed-pair instances required by the B2 headline theorem, which asserts branch regularity on the full open arc interior and continuous split-cosine paths on the closed interval $[0,1]$ ending at the Euclidean regular-4-simplex value $-1/4$, for every hinge of the threeTwo simplex (1 spacelike + 6 mixed + 3 upper-pair). Without each mixed-pair certificate the universal quantification over opposite vertex pairs fails. The result sits inside the complex-first Wick continuation program for causal dynamical triangulations in 4d, closing the branch-cut analysis for the asymmetric cofactor class before the product-form kill certificates and the spacelike-hinge endpoint disclosure.
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