threeTwoCosPath_eq_spacelike
plain-language theorem explainer
Along the Wick arc, the split cosine of the spacelike hinge of the (3,2) causal 4-simplex (opposite pair 3,4) equals the cut-free rational (5−6z)/(6z−2). Boundary-continuation and interior branch-regularity certificates for that hinge cite this identity. The proof unfolds the split cosine, inserts the physical cofactors, and cancels the complex square root.
Claim. For every real $t$, the split cosine of the spacelike hinge with opposite pair $(3,4)$ of the $(3,2)$ causal 4-simplex, evaluated on the canonical upper-half-plane Wick arc $z(t)$, equals $\frac{5-6z(t)}{6z(t)-2}$.
background
Lane B2 of the QG Seven-Gaps campaign treats all ten triangular hinges of the threeTwo causal 4-simplex by complex-first Wick continuation at the physical point $a=1$, $\alpha=1$, on the canonical upper-half-plane arc $z(t)$. Hinges are classified by opposite vertex pairs. The unique spacelike hinge is the lower triple $(0,1,2)$ opposite $(3,4)$: closed cofactors $C_{pp}=C_{qq}=6z-2$, $C_{pq}=5-6z$, with squared area $3/16$.
The split cosine is the ratio of the off-diagonal cofactor to a product of square-root diagonal cofactors (dihedral split form). The class-A claim is that this ratio collapses, everywhere on the arc, to the single rational function $(5-6z)/(6z-2)$, free of branch cuts in the expression itself. Endpoint honesty: at the Lorentzian end $t=0$ the value is $-11/8$, which lies on the arccos cut; that contact is allowed and handled by separate boundary and interior statements.
proof idea
Term-mode proof. Unfold the split-cosine and split-denominator definitions. Rewrite the edge data via the physical threeTwo continuation table. The complex Cayley–Menger index map sends vertices 3 and 4 to matrix indices 4 and 5; substitute those equalities by reflexivity. Apply the three closed cofactor identities for the $(4,4)$, $(5,5)$, and $(4,5)$ minors, then cancel the product of complex square roots against the squared denominator using the non-vanishing of the diagonal cofactor along the arc.
why it matters
This is the class-A collapse for the unique spacelike hinge of the (3,2) simplex: the algebraic identity that lets every subsequent analytic certificate work with a rational function instead of a branched split form. Downstream, the closed-interval boundary theorem uses it to prove continuity on $[0,1]$ with Lorentzian value $-11/8$ and Euclidean value $-1/4$. The interior branch-regularity theorem uses it to show the imaginary part equals $-18,\mathrm{Im},z/|6z-2|^2\neq 0$ on $(0,1)$, so the arccos cut is avoided in the open arc. Together these close the spacelike hinge in the all-hinge Wick package for the threeTwo 4-simplex under the finishing charter.
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