cof32_25
plain-language theorem explainer
For every complex z, the Cayley-Menger cofactor C_{2,5} of the (3,2) two-value edge tuple equals -1. This is the constant off-diagonal cofactor C_pq for the mixed opposite pair (1,4). Branch-regularity and boundary certificates for that pair cite it. The proof unfolds the cofactor, substitutes the explicit 5x5 minor (det = 1) and the odd-parity sign, then closes by ring.
Claim. For every $z \in \mathbb{C}$, the complex Cayley-Menger cofactor $C_{2,5}$ of the bordered $6\times 6$ matrix built from the threeTwo two-value edge tuple (spacelike squared lengths $1$, timelike squared lengths $z$) equals $-1$.
background
Lane B2 of the QG Seven-Gaps campaign continues the (3,2) causal 4-simplex along the complex-first Wick arc. The threeTwo type has lower slice ${0,1,2}$ and upper slice ${3,4}$; the six cross edges are timelike. The two-value edge tuple hingeEdges32C z sets every timelike squared length to $z$ and every spacelike squared length to $1$.
Dihedral data are read from cofactors of the bordered $6\times 6$ Cayley-Menger matrix. The complex cofactor $C_{r,c}$ is the signed $5\times 5$ minor after deleting row $r$ and column $c$, with sign $(-1)^{r+c}$. CM indices $1..5$ label the five simplex vertices; so $(r,c)=(2,5)$ is the off-diagonal entry for opposite pair $(1,4)$.
Mixed opposite pairs (one lower vertex, one upper) have closed forms $C_{pp}=8z-4$, $C_{qq}=6z-2$, and constant $C_{pq}=-1$. The present identity is that constant off-diagonal claim for pair $(1,4)$. Upstream, the CM matrix of the two-value tuple is identified with an explicit matrix, and the corresponding $5\times 5$ minor is proved to have determinant $1$.
proof idea
Unfold the cofactor into sign times minor determinant. Rewrite the CM matrix via the explicit hinge-matrix identity, then replace the deleted submatrix by the named minor minor32_25C. The determinant of that minor is the constant $1$ (det_minor32_25C). The cofactor sign is $-1$ because $2+5=7$ is odd (if_neg on Even, discharged by decide). A final ring multiplies $(-1)\cdot 1=-1$.
why it matters
Among the ten triangular hinges of the (3,2) simplex, six are mixed (two timelike triangle edges). Their off-diagonal cofactor is the constant $-1$, independent of the Wick parameter $z$. This theorem discharges that entry for opposite pair $(1,4)$, the mixed hinge on vertices $(0,2,3)$.
It is fed directly into branchRegular32_pair14 (split-form branch certificate on the open arc interior) and boundary32_pair14 (continuous boundary values of the cosine path at the Lorentzian and Euclidean endpoints). Those are two of the per-pair certificates that close Lane B2: all-hinge complex-first Wick continuation at the physical point $a=1$, $\alpha=1$.
The constant $C_{pq}=-1$ is the algebraic reason the mixed-pair cosine stays a rational function of $z$ with no higher-degree numerator, matching the executed per-hinge table in the Wick-arc trace.
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