cof32_23
plain-language theorem explainer
The (2,3) Cayley–Menger cofactor of the complex-first (3,2) hinge edge matrix equals the closed form 3−4z for every complex z. Gravity and Regge–Wick workers cite it when certifying upper-pair hinges of the threeTwo 4-simplex. The proof unfolds the cofactor, substitutes the hinge matrix and its 5×5 minor, applies the odd-parity sign, and finishes by ring.
Claim. For every $z \in \mathbb{C}$, the complex Cayley–Menger cofactor $C_{2,3}$ of the bordered $6\times 6$ matrix built from the (3,2) hinge squared-edge data at $z$ equals $3-4z$.
background
Lane B2 of the QG Seven-Gaps campaign continues the (3,2) causal 4-simplex (lower slice ${0,1,2}$, upper ${3,4}$) along the canonical upper-half-plane arc of the complex-first Wick module, at the physical point $a=\alpha=1$. Hinges are classified by opposite vertex pairs. The three pairs inside the lower triple give the upper-pair hinges $(0,3,4)$, $(1,3,4)$, $(2,3,4)$, with closed forms $C_{pp}=C_{qq}=8z-4$, $C_{pq}=3-4z$, and $\mathrm{areaSq}=z/4-1/16$.
The complex cofactor $C_{r,c}$ is the signed determinant of the complementary $5\times 5$ minor of the bordered Cayley–Menger matrix (sign $(-1)^{r+c}$). Upstream, that cofactor and its sign are defined in the WickActionComplexFirst layer; the hinge edge packing and the concrete submatrix/minor identities for indices $(2,3)$ live in this module. All closed forms are kernel-checked against explicit minors to match the executed per-hinge trace table.
proof idea
Term-mode proof by direct expansion. Unfold the cofactor into sign times minor. Rewrite via the hinge-edge matrix identity, the $(2,3)$ submatrix lemma, and the evaluated $5\times 5$ minor determinant. The parity test $\neg\mathrm{Even}(2+3)$ forces the sign to $-1$. A final ring reduces the resulting polynomial identity to $3-4z$.
why it matters
Supplies the off-diagonal closed form $C_{pq}=3-4z$ for every upper-pair hinge of the (3,2) simplex. Downstream, branchRegular32_pair12 and boundary32_pair12 feed this cofactor (with the two diagonal cofactors $8z-4$) into the generic upper-pair branch-regularity and boundary-continuation lemmas for opposite pair $(1,2)$, i.e. hinge $(0,3,4)$. The same pattern covers the other two lower-triple pairs. Together these close the split-form branch certificates and continuous boundary values on the Wick arc for all ten triangular hinges, a finishing-charter deliverable of the Seven-Gaps gravity lane. The identity is pure complex algebra; it does not itself address the spacelike-hinge arccos-cut endpoint disclosure, which is handled separately for hinge $(0,1,2)$.
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