branchRegular32_pair23
plain-language theorem explainer
Opposite vertices 2 and 3 of the (3,2) causal 4-simplex determine a mixed hinge (triangle 0-1-4). At unit edge data on the open Wick-arc interior, the split-form dihedral continuation is branch-regular: both diagonal Cayley-Menger cofactors stay off the square-root cut and the split cosine stays off the arccos cuts. Cited by anyone assembling the all-hinge B2 certificate for threeTwo. Proof is a one-line specialization of the parametric mixed-pair branch lemma via three closed cofactor identities.
Claim. For the $(3,2)$ causal 4-simplex at $a=1$, $\alpha=1$, the complex Wick-arc continuation of squared edge lengths is branch-regular at opposite vertices $p=2$, $q=3$ on the open interval $(0,1)$: both diagonal Cayley-Menger 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 continues the Regge action of a causal 4-simplex along a complex Wick arc from Lorentzian to Euclidean signature. The threeTwo type places three vertices on the lower time slice and two on the upper; the six cross-edges are timelike.
Branch regularity is the S3 model predicate: for every arc parameter $t$ in a set $s$, both diagonal cofactors of the complex Cayley-Menger matrix stay in the slit plane (exact continuity region of the principal square root), and the split cosine of the hinge dihedral stays off the arccos cuts. Edge data come from the complex continuation map: timelike edges follow the arc, spacelike edges stay at $a^2$.
Opposite pair $(2,3)$ is mixed (one lower, one upper member), so the hinge triangle is ${0,1,4}$. Mixed hinges have the asymmetric closed forms $C_{pp}=8z-4$ (lower), $C_{qq}=6z-2$ (upper), $C_{pq}=-1$, matching the module's per-hinge table.
proof idea
One-line term wrapper. Instantiates the parametric class-B certificate for mixed threeTwo pairs at opposite indices $p=2$, $q=3$, supplying the three closed-form cofactor lemmas (diagonal identities $8z-4$ and $6z-2$, off-diagonal identity $-1$). Those lemmas are explicit 5x5 minor evaluations on the hinge edge matrix.
The parametric theorem already shows that on the open upper half-plane both principal square roots lie in the open first quadrant, their product has strictly positive imaginary part, and the cosine $-1/(s_1 s_2)$ avoids the arccos cuts throughout the open arc interior.
why it matters
Discharges one of the six mixed-pair instances consumed by the B2 headline theorem, which asserts: for every opposite vertex pair of the threeTwo simplex (1 spacelike + 6 mixed + 3 upper-pair), at $a=1$, $\alpha=1$, (i) split-form branch regularity on the full open arc interior, and (ii) continuous split-cosine path on the closed interval $[0,1]$ ending at the Euclidean regular-4-simplex value $-1/4$.
In the Recognition gravity stack this is finishing-charter work: a well-defined complex-first Wick path for every hinge of the causal 4-simplex is required before discrete bulk actions can be treated as analytically controlled inputs to continuum or mass-ladder arguments. It does not touch the T0-T8 forcing chain; it closes an analytic gap inside the SevenGaps gravity lane.
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