cof32_d1
plain-language theorem explainer
The (1,1) Cayley–Menger cofactor of the complex (3,2) two-value edge tuple equals the linear form $8z-4$. Gravity and QG auditors cite it as the closed diagonal cofactor for every hinge that includes the lower vertex corresponding to CM index 1. The proof unfolds the complex cofactor, substitutes the explicit hinge matrix and its (1,1) minor, applies the already-proved $5\times5$ determinant, and finishes by ring.
Claim. For every complex $z$, the Cayley–Menger cofactor $C_{1,1}$ of the bordered $6\times6$ matrix built from the (3,2) hinge 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 causal (3,2) 4-simplex at the physical point $a=1$, $\alpha=1$. Edges are encoded as a ten-component complex tuple: six cross (timelike) edges carry the continuation variable $z$, and the four spacelike edges stay at $1$. The bordered Cayley–Menger matrix of that tuple is the explicit $6\times6$ complex matrix used throughout the module.
A complex CM cofactor $C_{r,c}$ is the signed $5\times5$ minor obtained by deleting row $r$ and column $c$, with sign $(-1)^{r+c}$. Index $1$ is the first vertex slot in the bordered matrix (vertices run $1..5$). The module classifies hinges by opposite pairs: lower-member diagonals on mixed and upper-pair hinges are predicted to be the linear form $8z-4$, matching the executed trace table.
Upstream, the general CM matrix of the two-value tuple is identified with the explicit hinge matrix, the $(1,1)$ submatrix is the lower diagonal minor, and that minor’s determinant is already proved equal to $8z-4$.
proof idea
Term-mode proof by definition chase. Unfold the complex cofactor into sign times minor determinant. Rewrite the ambient CM matrix to the explicit hinge matrix, replace the deleted-$(1,1)$ submatrix by the named lower minor, and invoke the closed $5\times5$ determinant $8z-4$. The cofactor sign at $(1,1)$ is $+1$ because $1+1$ is even (decide), so no minus appears. A final ring normalizes the scalar expression.
why it matters
This is one of the kernel-checked cofactor receipts in §3 that lock the (3,2) hinge table to the Wick-arc trace. Downstream, every boundary and branch-regularity certificate that needs the diagonal cofactor at vertex $0$ (CM index $1$) plugs this identity in directly: upper-pair boundaries and branch certificates for pairs $(0,1)$ and $(0,2)$, and mixed-pair boundaries for $(0,3)$ and $(0,4)$, each pass cof32_d1 into the generic upper-pair or mixed-pair lemmas.
Those certificates are the split-form branch and boundary continuation deliverables for all ten triangular hinges of the (3,2) simplex on the canonical upper-half-plane arc. Without the linear closed form, the cosine paths and interior branch-regularity statements cannot be discharged. In the broader Recognition gravity stack this is pure geometric algebra supporting the complex-first Wick lane, not a forcing-chain (T0–T8) step.
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