branchRegular32_pair01

Lane B2 of the QG Seven-Gaps campaign continues the Regge action of causal dynamical triangulations from Euclidean to Lorentzian signature along a complex Wick arc. The $(3,2)$ causal 4-simplex has three vertices on one time slice and two on the next; its ten triangular hinges fall into three classes by opposite vertex pair.

Branch regularity (the S3 model predicate) demands that, for every parameter $t$ in a given set, both diagonal cofactors of the Cayley-Menger matrix lie in the complex slit plane (exact continuity region of the principal square root) and the split-form dihedral cosine avoids the arccos branch cuts.

Upper-pair hinges come from opposite pairs inside the lower triple ${0,1,2}$. For pair $(0,1)$ the hinge is ${2,3,4}$, with closed forms $C_{pp}=C_{qq}=8z-4$ and $C_{pq}=3-4z$. The parametric class certificate already proves these keep the continuation branch-regular on the open arc interior whenever the cofactor identities hold.

One-line term application of the parametric upper-pair branch certificate at vertices $0$ and $1$, supplying the three closed-form cofactor identities: both diagonal cofactors equal $8z-4$ and the off-diagonal equals $3-4z$. Those identities are kernel-checked evaluations of $5\times5$ Cayley-Menger minors on the hinge edge matrix. No extra case splits.

Discharges one of the three upper-pair cases required by the B2 headline theorem, which asserts branch regularity on $(0,1)$ and continuous closed-interval cosine paths ending at the Euclidean regular value $-1/4$ for every hinge of the $(3,2)$ simplex (spacelike + mixed + upper-pair, both orientations). Together with the spacelike and mixed-pair certificates it closes the all-hinge complex-first Wick continuation at the physical point $a=1$, $\alpha=1$. The module is part of the gravity Seven-Gaps finishing charter; the result is local CDT simplex geometry, not a T0-T8 forcing step, but it is a concrete prerequisite for a well-defined Lorentzian path-integral measure on causal triangulations.