cof32_15
plain-language theorem explainer
The complex Cayley–Menger cofactor C_{1,5} of the (3,2) two-value edge tuple equals −1 for every complex z. This is the constant off-diagonal cofactor C_{pq} for the mixed hinge opposite vertices 0 and 4. Cited by the branch-regularity and boundary certificates for that mixed pair. Proof unfolds the cofactor, substitutes the explicit hinge matrix and 5×5 minor, and reduces by ring.
Claim. For every $z \in \mathbb{C}$, the complex Cayley–Menger cofactor $C_{1,5}$ of the bordered $6\times 6$ matrix built from the (3,2) two-value edge tuple (spacelike edges $1$, timelike edges $z$) equals $-1$.
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: timelike cross-edges carry $z$, spacelike edges carry $1$.
Cayley–Menger geometry is complexified: the bordered $6\times 6$ matrix yields cofactors $C_{r,c}=(-1)^{r+c}\det M_{\widehat{r}\widehat{c}}$. Vertex indices $0..4$ map to CM rows/columns $1..5$, so $(r,c)=(1,5)$ is the opposite pair $(0,4)$.
For mixed pairs (one lower-slice, one upper-slice vertex) the module records closed forms $C_{pp}=8z-4$, $C_{qq}=6z-2$, and constant $C_{pq}=-1$. The upstream lemma det_minor32_15C already evaluates the corresponding $5\times 5$ minor determinant to $-1$; the matrix identification cmMatrixC_hingeEdges32 supplies the explicit hinge matrix.
proof idea
Term-mode proof. Unfold cofactor as sign times minor. Rewrite the CM matrix via cmMatrixC_hingeEdges32 to the explicit hinge matrix, then the deleted submatrix via submatrix32_15. Apply det_minor32_15C to get determinant $-1$. The cofactor sign is $+1$ because $1+5$ is even (if_pos by decide). The product $(+1)\cdot(-1)$ reduces by ring to $-1$.
why it matters
Fills the constant off-diagonal entry $C_{pq}=-1$ in the mixed-hinge closed-form table for opposite pair $(0,4)$. Downstream, branchRegular32_pair04 and boundary32_pair04 feed this cofactor (with the two diagonal cofactors) into the generic mixed-pair branch-regularity and boundary-continuation lemmas, certifying the split cosine path on $(0,1)$ and the endpoint values at $t=0,1$.
This is one of the six mixed hinges of the (3,2) simplex in the Wick arc campaign: without the constant $C_{pq}$, the cosine ratio and area-squared formulas cannot be discharged. It is pure algebraic geometry of the complex CM matrix, not a forcing-chain (T0–T8) step, but it closes a kernel-checked entry in the per-hinge table that the executed Wick-arc trace demands.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.