boundary32_pair04
plain-language theorem explainer
The split-form cosine path for mixed opposite pair (0,4) of the (3,2) causal 4-simplex is continuous on the closed unit interval, equals √6/24 at the Lorentzian endpoint, and equals −1/4 at the Euclidean endpoint. QG workers assembling the B2 all-hinge Wick certificate cite this instance among the six mixed pairs. The proof is a one-line specialization of the parametric mixed-pair boundary theorem, supplying the three closed-form cofactor identities for the relevant Cayley–Menger indices.
Claim. The split-form cosine path of opposite vertices $0$ and $4$ along the physical $(3,2)$ Wick arc is continuous on $[0,1]$, equals $\sqrt{6}/24$ (as a real embedded in $\mathbb{C}$) at the Lorentzian endpoint $t=0$, and equals $-1/4$ at the Euclidean endpoint $t=1$.
background
Lane B2 of the QG Seven-Gaps campaign continues every triangular hinge of the threeTwo causal 4-simplex along the canonical upper-half-plane Wick arc at physical parameters $a=\alpha=1$. Vertices split as lower triple ${0,1,2}$ and upper pair ${3,4}$; timelike edges are exactly the six cross edges. Opposite pairs fall into three classes. Mixed pairs (one lower, one upper) have asymmetric Cayley–Menger cofactors $C_{pp}=8z-4$ (lower member), $C_{qq}=6z-2$ (upper member), $C_{pq}=-1$, with squared area $z/4-1/16$.
The cosine path of an opposite pair is the split-form dihedral cosine of that hinge evaluated on the physical continuation edges. The parametric mixed-pair boundary theorem already gives closed-interval continuity and the two endpoint values once those three cofactor identities are supplied as hypotheses. Pair $(0,4)$ is one such mixed pair; the needed cofactors are the diagonal entries at CM indices $1$ and $5$ and the off-diagonal $(1,5)$ entry, each proved by explicit $5\times5$ minor evaluation.
proof idea
One-line wrapper applying the parametric mixed-pair boundary theorem at $(p,q)=(0,4)$. The three hypothesis slots are filled by the closed-form cofactor lemmas: diagonal cofactor at index $1$ equals $8z-4$, diagonal cofactor at index $5$ equals $6z-2$, and off-diagonal cofactor $(1,5)$ equals $-1$. Those lemmas themselves unfold the cofactor definition, rewrite through the hinge-edge Cayley–Menger matrix and the matching submatrix determinant, then finish by ring. No further analytic work is done at this site.
why it matters
This is one of the six mixed-pair instances consumed by the B2 headline theorem, which asserts that for every unordered opposite vertex pair of the threeTwo simplex (1 spacelike + 6 mixed + 3 upper-pair), the split-form continuation is branch-regular on the open arc interior and the cosine path is continuous on the closed interval $[0,1]$ ending at the Euclidean regular-4-simplex value $-1/4$. The headline packages both orientations of every pair; concrete per-pair boundary facts such as this one discharge the continuity-and-endpoints conjunct. The work sits in the gravity finishing charter for complex-first Wick continuation of causal 4-simplices, not in the T0–T8 forcing chain. Endpoint values match the executed arc-trace table (Lorentzian $\approx +0.102$, Euclidean $-1/4$).
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