cof32_d5
plain-language theorem explainer
The (5,5) Cayley–Menger cofactor of the complex (3,2) two-value edge tuple equals 6z−2 for every complex z. Gravity and QG workers cite it as the closed form C_qq (and C_pp on the spacelike hinge) that feeds every mixed-pair and spacelike branch/boundary certificate on the threeTwo simplex. The proof unfolds the cofactor, identifies the 5×5 minor with the PPC template, and applies the known determinant 6z−2 with positive sign.
Claim. For every $z \in \mathbb{C}$, the complex Cayley–Menger cofactor $C_{5,5}$ of the bordered $6\times 6$ matrix built from the (3,2) two-value edge tuple (timelike edges equal to $z$, spacelike edges equal to $1$) equals $6z-2$.
background
Lane B2 of the QG Seven-Gaps campaign treats the all-hinge complex-first Wick continuation of the (3,2) causal 4-simplex at the physical point $a=1$, $\alpha=1$. Edges are the two-value tuple hingeEdges32C z: each of the six cross (timelike) edges carries the complex value $z$, and each spacelike edge carries $1$. The bordered $6\times 6$ Cayley–Menger matrix of that tuple is written cmMatrixC.
A complex CM cofactor $C_{r,c}$ is the signed $5\times 5$ minor obtained by deleting row $r$ and column $c$. The sign is $(-1)^{r+c}$ as a complex number. Index $5$ is the last CM vertex slot (upper-slice vertex $4$). Module classification records that every upper-member diagonal cofactor on mixed hinges, and both diagonal cofactors on the spacelike hinge $(0,1,2)$, share the closed form $C_{pp}=C_{qq}=6z-2$.
Upstream, det_minorPPC already evaluates the symbolic PPC $5\times 5$ template to $6z-2$ (recovering the regular unit-tetrahedron value $4$ at $z=1$). The identification cmMatrixC_hingeEdges32 and the concrete submatrix lemma submatrix32_55 pin that template to the $(5,5)$ minor of the threeTwo matrix.
proof idea
Term-mode proof by direct reduction. Unfold the cofactor into sign times minor. Rewrite the ambient CM matrix via cmMatrixC_hingeEdges32 and the deleted submatrix via submatrix32_55, so the minor becomes the PPC template. Apply det_minorPPC to obtain determinant $6z-2$. The cofactor sign at $(5,5)$ is $+1$ because $5+5$ is even (decide). Finish with ring.
why it matters
This is one of the ten kernel-checked closed cofactor forms that match the executed Wick-arc trace table for threeTwo. Downstream, every mixed-pair branch certificate that touches an upper vertex (branchRegular32_pair04/14/24 and the spacelike branchRegular_threeTwo_spacelike) and the corresponding boundary continuations (boundary32_pair04/14/24) take cof32_d5 as the $C_{qq}$ input. The interior confinement theorem pentHingeCosPath_eq_moebius likewise collapses the split cosine of the spacelike hinge to the Möbius ratio $(5-6z)/(6z-2)$, whose denominator is exactly this cofactor.
In the broader Recognition gravity stack this supplies the algebraic backbone for Wick-continuing dihedral data off the Lorentzian section without leaving the complex-first CM calculus. The module notes an honest endpoint contact: at $t=0$ the spacelike-hinge cosine sits on the arccos cut at $-11/8$, so branch regularity is stated on the open interior only; the rational closed form itself remains exact at the endpoint.
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