boundary32_pair24
plain-language theorem explainer
For the mixed opposite pair (2,4) on the (3,2) causal 4-simplex, the split-form dihedral cosine path is continuous on the closed Wick arc [0,1], equals √6/24 at the Lorentzian end, and equals −1/4 at the Euclidean end. Cited by the all-hinge B2 continuation theorem. One-line wrapper of the mixed-pair boundary lemma with the three closed-form cofactor identities for this pair.
Claim. The split-form cosine path of opposite vertices $2$ and $4$ along the physical $(3,2)$ Wick arc is continuous on the closed interval $[0,1]$, takes the Lorentzian value $\sqrt{6}/24$ at $t=0$, and the Euclidean value $-1/4$ at $t=1$.
background
Lane B2 of the QG Seven-Gaps campaign treats complex-first Wick continuation of every triangular hinge of the threeTwo causal 4-simplex at the physical point $a=1$, $\alpha=1$, along the canonical upper-half-plane arc. Hinges are classified by opposite vertex pairs: one spacelike, six mixed (one lower-slice and one upper-slice vertex), and three upper-pair hinges.
Pair $(2,4)$ is mixed: lower slice ${0,1,2}$, upper slice ${3,4}$. Mixed hinges carry asymmetric Cayley–Menger cofactors $C_{pp}=8z-4$ (lower), $C_{qq}=6z-2$ (upper), $C_{pq}=-1$, with area-squared $z/4-1/16$. The path threeTwoCosPath p q is the split-form dihedral cosine of opposite pair $(p,q)$ along the physical threeTwo continuation edges.
The parametric mixed-pair boundary theorem already states continuity on the closed interval with Lorentzian value $\sqrt{6}/24$ (real; trace $+0.102\ldots$) and Euclidean value $-1/4$, once the three cofactor closed forms are supplied.
proof idea
One-line wrapper. Instantiate the parametric mixed-pair boundary theorem at vertices $p=2$, $q=4$, feeding the three kernel-checked cofactor identities: diagonal cofactor at the lower index equals $8z-4$, diagonal at the upper index equals $6z-2$, and the off-diagonal cofactor equals $-1$. Those three lemmas are pure 5×5 minor evaluations on the hinge edge matrix. No further analytic work.
why it matters
Feeds the B2 headline theorem wick_continuation_threeTwo_hinges, which asserts branch-regularity on the open arc and closed-interval continuity with Euclidean endpoint $-1/4$ for every hinge of the threeTwo simplex (1 spacelike + 6 mixed + 3 upper-pair). This declaration discharges one of the six mixed pairs.
In the finishing charter, closed-form boundary values at both Wick endpoints are the certificate that the split cosine continues across the Lorentzian cut contact inside the closed upper half-plane and lands on the regular Euclidean 4-simplex cosine. The Euclidean value $-1/4$ is the universal regular-simplex target shared by all ten hinges; the Lorentzian $\sqrt{6}/24$ is the mixed-class signature.
No open scaffolding: claim status is fully proved. Completes one concrete cell of the per-hinge table matching the executed Wick-arc trace.
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