branchRegular32_pair02
plain-language theorem explainer
Branch regularity holds for opposite vertex pair (0,2) of the (3,2) causal 4-simplex along the open Wick-arc interior at unit edge lengths. The hinge triangle is (1,3,4), an upper-pair hinge inside the lower triple. Anyone assembling the full all-hinge Wick continuation certificate cites this instance. The proof is a one-line specialization of the parametric upper-pair class certificate, fed the three closed-form cofactor identities for this pair.
Claim. The split-form hinge continuation of the $(3,2)$ causal $4$-simplex along the complex Wick arc at $a=1$, $\alpha=1$ is branch-regular for opposite vertices $0$ and $2$ on the open interval $(0,1)$: both diagonal Cayley-Menger cofactors stay in the complex slit plane, and the split dihedral cosine stays off the arccos cuts.
background
In the QG Seven-Gaps campaign (lane B2), the Regge action of a causal 4-simplex is continued from Euclidean to Lorentzian signature along a complex arc. The $(3,2)$ type places three vertices on the lower time slice and two on the upper; its six cross edges are timelike and track the arc coordinate $z$, while spacelike edges stay fixed at $a^2=1$.
Branch regularity is the model S3 predicate: for every parameter $t$ in a set $s$, 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 stays off the arccos branch cuts.
Opposite vertex pairs fall into three hinge classes. Pair $(0,2)$ is upper-pair: both endpoints sit in the lower triple, so the hinge triangle is $(1,3,4)$. Closed forms are $C_{pp}=C_{qq}=8z-4$, $C_{pq}=3-4z$, with area squared $z/4-1/16$. The parametric class-C certificate already proves branch regularity once those three cofactor identities are supplied.
proof idea
One-line wrapper applying the parametric upper-pair branch certificate at vertices $p=0$, $q=2$, feeding three cofactor lemmas: the two diagonal cofactors equal $8z-4$, and the off-diagonal cofactor equals $3-4z$. Those identities are obtained by expanding the relevant $5\times5$ Cayley-Menger minors on the hinge edge matrix and simplifying. The parametric theorem then discharges slit-plane membership of $8z-4$ on the open arc and nonvanishing imaginary part of the collapsed cosine (trace margin $0.4167$).
why it matters
This is one of three upper-pair instances feeding the B2 headline theorem, which asserts that for every hinge of the $(3,2)$ simplex (1 spacelike + 6 mixed + 3 upper-pair), at $a=1$, $\alpha=1$: (i) the split-form continuation is branch-regular on the full open arc interior, and (ii) the split cosine path is continuous on the closed interval $[0,1]$ and ends at the Euclidean regular-4-simplex value $-1/4$.
The upper-pair class carries the worst interior-attained margin of the whole Wick-arc trace, so discharging it is the tightest interior check among the ten hinges. Together with the spacelike and mixed certificates it closes lane B2 of the finishing charter: all-hinge complex-first Wick continuation at the physical point.
Within Recognition Science gravity this is discrete-geometry infrastructure for the continuum limit of causal dynamical triangulations, supporting the quantum-gravity side of the Seven-Gaps campaign rather than a direct T0-T8 forcing step.
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