cof32_45
plain-language theorem explainer
The complex Cayley–Menger cofactor C_{4,5} of the (3,2) two-value edge tuple equals 5−6z. Gravity and QG workers cite it for the spacelike hinge of the threeTwo 4-simplex, where this is the off-diagonal entry C_{pq} in the closed split cosine. The proof unfolds the cofactor, substitutes the explicit 5×5 minor determinant 6z−5, multiplies by the odd-parity sign −1, and simplifies by ring.
Claim. For every complex $z$, the Cayley–Menger cofactor $C_{4,5}$ of the bordered $6\times 6$ matrix built from the threeTwo edge tuple (timelike edges equal to $z$, spacelike edges equal to $1$) equals $5-6z$.
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 $z$, and each spacelike edge carries $1$.
The complex Cayley–Menger cofactor $C_{r,c}$ is the signed $5\times 5$ minor of the bordered $6\times 6$ CM matrix: sign $(-1)^{r+c}$ times the determinant after deleting row $r$ and column $c$. Indices $4$ and $5$ are the CM rows for vertices $3$ and $4$, the opposite pair of the spacelike hinge $(0,1,2)$.
Upstream, cmMatrixC_hingeEdges32 identifies the CM matrix with the explicit hinge matrix, and det_minor32_45C evaluates the corresponding minor determinant as $6z-5$. The module class table records the closed forms for this hinge: $C_{pp}=C_{qq}=6z-2$ and $C_{pq}=5-6z$.
proof idea
Term-mode proof by unfolding and rewrite. Unfold the cofactor into sign times minor determinant. Rewrite the CM matrix via cmMatrixC_hingeEdges32, the deleted submatrix via submatrix32_45, and the minor determinant via det_minor32_45C (value $6z-5$). The parity test Even(4+5) fails, so the sign is $-1$. The identity $-1\cdot(6z-5)=5-6z$ is discharged by ring.
why it matters
This is the off-diagonal closed form for the unique spacelike hinge of the threeTwo simplex (opposite pair $(3,4)$). Downstream, threeTwoCosPath_eq_spacelike uses it to collapse the split cosine to the rational Möbius form $(5-6z)/(6z-2)$ everywhere on the arc; pentHingeCosPath_eq_moebius feeds the same numerator into the interior-hinge confinement path.
In the module’s honest endpoint disclosure, that Möbius form is exact even at the Lorentzian endpoint $t=0$, where the cosine hits the arccos cut at $-11/8$ (classical boost angle). The cofactor identity is therefore a kernel-checked building block of the branch certificate on the open interior and of the cut-free rational collapse at the boundary. It sits in the gravity/Wick lane of the Seven-Gaps finishing charter, not in the T0–T8 forcing chain itself.
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