branchRegular32_pair14
plain-language theorem explainer
Opposite vertex pair (1,4) of the (3,2) causal 4-simplex is branch-regular on the open Wick arc: both diagonal Cayley-Menger cofactors stay off the square-root cut and the split cosine stays off the arccos cuts. QG workers assembling the all-hinge threeTwo Wick certificate cite this instance. One-line specialization of the mixed-pair class-B certificate via the closed cofactor identities for this hinge.
Claim. At physical parameters $a=1$, $\alpha=1$, the complex Wick continuation of squared edge lengths for the $(3,2)$ causal $4$-simplex is branch-regular on the open arc interior $(0,1)$ at opposite vertex pair $(1,4)$ (the mixed hinge on $\{0,2,3\}$): both diagonal cofactors lie in the slit plane, and the split dihedral cosine stays off the arccos cuts.
background
Lane B2 of the QG Seven-Gaps campaign certifies split-form Wick continuation for every triangular hinge of the threeTwo causal 4-simplex (three vertices on slice $t$, two on $t+1$). Timelike edges follow the upper-half-plane arc; spacelike edges stay fixed at $a^2$. Hinges are classified by opposite vertex pair: one spacelike, six mixed (one lower and one upper vertex), three upper-pair.
Mixed pairs have asymmetric closed cofactors $C_{pp}=8z-4$ (lower member), $C_{qq}=6z-2$ (upper member), $C_{pq}=-1$, and area-squared $z/4-1/16$. Branch regularity (the S3 model predicate) demands that both diagonal cofactors stay in the complex slit plane (continuity region of principal square root) and that the split cosine ratio stays off the arccos cuts, for every parameter in a given set.
The parametric mixed-pair theorem already proves the class-B certificate once those three closed forms are supplied. Pair $(1,4)$ is one of the six mixed hinges; its cofactor identities are the kernel-checked minors cof32_d2, cof32_d5, and cof32_25.
proof idea
One-line term wrapper. Instantiate the parametric mixed-pair branch certificate at opposite indices $p=1$, $q=4$, feeding the three closed cofactor identities: diagonal lower $C_{11}=8z-4$, diagonal upper $C_{44}=6z-2$, and off-diagonal $C_{pq}=-1$. That theorem already shows both cofactors lie in the open upper half-plane on the arc interior, so each principal square root sits 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
Feeds the B2 headline theorem wick_continuation_threeTwo_hinges, which asserts branch regularity on $(0,1)$ and continuous split-cosine paths on the closed interval $[0,1]$ ending at the Euclidean regular-4-simplex value $-1/4$, for every unordered opposite pair of the threeTwo simplex (both orientations). Without each mixed-pair instance, the universal quantifier over hinges fails.
This is the second deliverable of lane B in the finishing charter: all ten triangular hinges of the (3,2) causal 4-simplex at the physical point, matching the executed per-hinge trace table. It sits inside the broader Recognition gravity stack that builds continuum limits from causal dynamical triangulations; the Wick arc and branch certificates are the analytic gate before Lorentzian signature is recovered. No open scaffold remains here: the claim is fully proved.
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