branchRegular32_pair12
plain-language theorem explainer
On the open Wick arc interior, the split-form dihedral continuation for opposite vertices 1 and 2 of the (3,2) causal 4-simplex stays branch-regular: both diagonal Cayley–Menger cofactors avoid the square-root cut and the split cosine avoids the arccos cuts. This is the upper-pair hinge (0,3,4). Anyone assembling the all-hinge B2 certificate cites it. The proof is a one-line specialization of the parametric upper-pair class certificate with the three closed cofactor identities.
Claim. For the complex Wick continuation of squared edge lengths of the $(3,2)$ causal 4-simplex at $a=1$, $\alpha=1$, the opposite-vertex pair $(1,2)$ (upper-pair hinge $(0,3,4)$) is branch-regular on the open parameter interval $(0,1)$: 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 treats the complex-first Wick continuation of every triangular hinge of the $(3,2)$ causal 4-simplex (three vertices on one CDT slice, two on the next). Timelike edges follow the canonical upper-half-plane arc; spacelike edges stay fixed at $a^2$.
Branch regularity (BranchRegularOn) is the model S3 predicate: for every $t$ in a parameter set, both diagonal Cayley–Menger cofactors of the continued edge tuple lie in $\mathbb{C}$'s slit plane (continuity region of complex square root), and the split-form dihedral cosine stays off the arccos branch cuts.
Opposite pairs inside the lower triple ${0,1,2}$ give the three upper-pair hinges. Pair $(1,2)$ is the hinge $(0,3,4)$, with closed cofactors $C_{pp}=C_{qq}=8z-4$ and $C_{pq}=3-4z$. The parametric class theorem already proves branch regularity once those three identities are supplied.
proof idea
One-line wrapper. Instantiate the parametric upper-pair certificate branchRegular_threeTwo_upper_pair at opposite indices $p=1$, $q=2$, feeding the three kernel-checked cofactor lemmas: diagonal cofactor at index 2 equals $8z-4$ (cof32_d2), diagonal at 3 equals $8z-4$ (cof32_d3), and the off-diagonal cofactor equals $3-4z$ (cof32_23). Those identities match the class-C hypotheses, so the class theorem discharges slit-plane membership and off-arccos-cut for the physical continuation on $(0,1)$.
why it matters
Feeds the B2 headline theorem wick_continuation_threeTwo_hinges, which asserts branch regularity on the full open arc interior and continuous closed-interval cosine paths ending at the Euclidean regular-4-simplex value $-1/4$ for every hinge of the $(3,2)$ simplex (1 spacelike + 6 mixed + 3 upper-pair).
This declaration is one of the three upper-pair instances. Together with the mixed and spacelike certificates it closes the split-form branch side of the all-hinge Wick continuation at the physical point $a=\alpha=1$. In the gravity lane this is discrete-geometry infrastructure for a controlled Lorentzian-to-Euclidean continuation of CDT building blocks, not a continuum Einstein equation derivation. The spacelike hinge's allowed arccos-cut contact at $t=0$ is handled separately; this upper-pair case has no such endpoint subtlety on the open interior.
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