boundary32_pair12
plain-language theorem explainer
For the upper-pair hinge opposite vertices 1 and 2 of the (3,2) causal 4-simplex, the split-form cosine path is continuous on the closed unit interval, equals -7/12 at the Lorentzian end, and equals -1/4 at the Euclidean end. Cited by anyone assembling the all-hinge Wick continuation for threeTwo. Proof is a one-line specialization of the generic upper-pair boundary lemma with matching cofactor certificates.
Claim. Along the canonical Wick arc at physical parameters $a=1$, $\alpha=1$, the split-form cosine of the triangular hinge opposite the edge joining vertices $1$ and $2$ of the $(3,2)$ causal $4$-simplex is continuous on $[0,1]$, equals $-(7/12)$ at $t=0$, and equals $-(1/4)$ at $t=1$.
background
Lane B2 of the QG Seven-Gaps campaign treats complex-first Wick continuation for every triangular hinge of the (3,2) causal 4-simplex (lower slice ${0,1,2}$, upper slice ${3,4}$; six cross edges timelike). Hinges are indexed by unordered opposite vertex pairs and fall into three classes: one spacelike, six mixed, three upper-pair.
Upper-pair hinges come from opposite pairs inside the lower triple. They share the closed forms $C_{pp}=C_{qq}=8z-4$, $C_{pq}=3-4z$, and $\mathrm{areaSq}=z/4-1/16$, kernel-checked by explicit $5\times 5$ minors. The cosine path tracks the split-form cosine of that dihedral data along the canonical upper-half-plane arc at $a=1$, $\alpha=1$.
Every hinge is expected to land at the Euclidean regular-4-simplex cosine $-(1/4)$ when $t=1$. Lorentzian endpoint values differ by class; for this upper-pair the value is $-(7/12)$.
proof idea
One-line term wrapper. Instantiates the generic upper-pair boundary lemma at opposite indices $1$ and $2$, feeding the three cofactor certificates that pin the closed-form minors for this hinge class (diagonal cofactors at the two opposite vertices and the off-diagonal $2$--$3$ cofactor). That lemma already packages continuity on $[0,1]$ and the two endpoint evaluations.
why it matters
Feeds the B2 headline wick_continuation_threeTwo_hinges, which asserts, for every hinge of threeTwo (spacelike, mixed, and upper-pair, both orientations): branch-regularity of the split-form continuation on the open arc interior, plus continuity of the cosine path on the closed interval ending at the Euclidean regular value $-(1/4)$.
This declaration is one of the three upper-pair boundary bricks in that conjunction. Matching the Euclidean $-(1/4)$ at $t=1$ is the geometric sanity check that the Wick arc really reconnects Lorentzian causal geometry to the regular 4-simplex. It closes part of the executed Wick-arc trace for causal simplices in the gravity finishing charter; no open scaffold remains on this pair.
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