boundary32_pair23
plain-language theorem explainer
The split-form cosine path for the mixed hinge opposite vertices 2 and 3 of the (3,2) causal 4-simplex is continuous on the closed Wick arc [0,1], equals √6/24 at the Lorentzian endpoint, and equals −1/4 at the Euclidean endpoint. The all-hinge threeTwo Wick continuation theorem cites this instance. Proof is a one-line specialization of the parametric mixed-pair boundary lemma, feeding the three closed-form cofactor identities for this pair.
Claim. Along the physical threeTwo Wick arc, the split-form cosine of the hinge opposite vertices $2$ and $3$ is continuous on $[0,1]$, equals $\sqrt{6}/24$ (real) 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 treats complex-first Wick continuation of every triangular hinge of the (3,2) causal 4-simplex at the physical point $a=1$, $\alpha=1$, on the canonical upper-half-plane arc. Hinges are classified by opposite vertex pair: one spacelike, six mixed (one lower-slice and one upper-slice vertex), and three upper-pair.
Mixed hinges have asymmetric Cayley–Menger cofactors $C_{pp}=8z-4$ (lower member), $C_{qq}=6z-2$ (upper member), $C_{pq}=-1$, and squared area $z/4-1/16$. The split-form cosine path is the dihedral cosine built from those cofactors along the continued edge data of the threeTwo type.
The parametric mixed-pair boundary theorem already states continuity on the closed interval together with the endpoint values $\sqrt{6}/24$ (Lorentzian; real) and $-1/4$ (Euclidean), once the three cofactor identities are supplied. Explicit minor computations give those identities for the matrix indices attached to vertices 2 and 3.
proof idea
One-line wrapper: apply the parametric mixed-pair boundary theorem at opposite pair $(p,q)=(2,3)$, discharging its three cofactor hypotheses by the closed-form identities $C_{33}=8z-4$, $C_{44}=6z-2$, and $C_{34}=-1$ (each proved by unfolding the cofactor, rewriting the hinge Cayley–Menger matrix to the matching 5×5 minor, and simplifying). No further analytic work is done at this site.
why it matters
This is one of the six mixed-pair boundary certificates required by the B2 headline theorem: for every hinge of the threeTwo causal 4-simplex, the split-form cosine path is continuous on the closed arc $[0,1]$ and ends at the Euclidean regular-4-simplex value $-1/4$, while branch regularity holds on the open interior. The headline packages the spacelike, mixed, and upper-pair classes into a single universal statement over all unordered opposite pairs.
In the finishing charter this closes the boundary half of the all-hinge continuation for the (3,2) simplex (lane B, second deliverable), matching the executed per-hinge trace table. It does not itself address the product-form kill certificates or the spacelike cut-contact disclosure; those sit in sibling lemmas of the same module.
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