cof32_d3
plain-language theorem explainer
The (3,3) Cayley-Menger cofactor of the complex (3,2) two-value edge tuple equals the closed form $8z-4$. Gravity and QG workers cite it when assembling split-form cosines and branch certificates for every hinge that touches the lower-slice vertex of index 2. The proof unfolds the cofactor, identifies the 5x5 minor with the precomputed lower diagonal block, inserts the even-parity sign $+1$, and finishes by ring.
Claim. For every complex parameter $z$, the Cayley-Menger cofactor $C_{3,3}$ 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 $8z-4$.
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: each of the six cross (timelike) edges carries the complex value $z$, and each spacelike edge is fixed at $1$. The bordered Cayley-Menger matrix is the standard $6\times 6$ complex matrix whose $5\times 5$ principal minors encode squared volumes and dihedral data.
A cofactor $C_{r,c}$ is the signed $5\times 5$ minor obtained by deleting row $r$ and column $c$, with sign $(-1)^{r+c}$. Index $3$ is the CM slot for the lower-slice vertex of combinatorial index $2$. Module classification places the diagonal cofactors of lower vertices in the closed-form family $C_{pp}=8z-4$, matching the executed per-hinge trace table.
Upstream, the general complex CM matrix of this two-value tuple is identified with an explicit symbolic matrix, and the lower diagonal $5\times 5$ block is already known to have determinant $8z-4$.
proof idea
Term-mode proof. Unfold the cofactor into sign times minor. Rewrite the ambient CM matrix to the explicit hinge matrix, identify the deleted-$(3,3)$ submatrix with the precomputed lower diagonal block, and replace that block's determinant by the closed form $8z-4$. The parity test $3+3$ even forces the cofactor sign to $+1$. A final ring normalizes the scalar expression.
why it matters
This identity is one of the ten kernel-checked closed cofactors that underwrite the split-form branch certificates and boundary continuations for all triangular hinges of the (3,2) simplex. Downstream it is fed directly into the upper-pair boundary and branch-regularity theorems for opposite pairs $(0,2)$ and $(1,2)$, and into the mixed-pair boundary theorems for $(2,3)$ and $(2,4)$. Those parents assemble the continuous cosine path on $[0,1]$ and the open-interval branch-regularity predicate required by the Wick-arc finishing charter. In the broader Recognition gravity stack this is pure geometric scaffolding for the complex-first continuation, not a forcing-chain step; it closes the symbolic gap between the Cayley-Menger matrix and the per-hinge cosine table already recorded in the executed trace.
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