cofactor_qq
plain-language theorem explainer
The Cayley–Menger cofactor C_{4,4} of the complex fourOne hinge edge data equals 6z−2 for every complex timelike squared length z. Gravity and Regge-calculus workers cite it when evaluating dihedral cosines along the Wick arc. The proof unfolds the cofactor, identifies the 5×5 minor with the precomputed diagonal block, and applies the closed-form determinant 6z−2 with positive sign.
Claim. For every $z \in \mathbb{C}$, the complex Cayley–Menger cofactor $C_{4,4}$ of the bordered $6\times 6$ matrix built from the fourOne hinge edge tuple (timelike squared lengths equal to $z$, spacelike equal to $1$) equals $6z-2$.
background
Module C11 formalizes complex-first 4D Wick continuation of Regge hinge data (areas-squared and dihedral cosines) for causal 4-simplex classes. The path is the upper-half-plane arc $z(t)=\alpha a^2\exp(i\pi(1-t))$ from Lorentzian to Euclidean endpoints. Scope is hinge-data only; full action-level continuation remains open (C12).
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 $\det$ of the submatrix deleting row $r$ and column $c$. The two-value edge tuple hingeEdgesC z sets every timelike edge of the fourOne causal type to $z$ and every spacelike edge to $1$.
Upstream, cmMatrixC_hingeEdges identifies that matrix with an explicit symbolic form; det_minorPPC proves the diagonal minor deleting the appropriate row/column has determinant $6z-2$ (matching RESULTS.txt; at $z=1$ this recovers the regular unit tetrahedron value $4$).
proof idea
Term-mode rewrite chain. Unfold the cofactor into sign times minor determinant. Rewrite the CM matrix via cmMatrixC_hingeEdges, identify the deleted submatrix with the precomputed diagonal minor via submatrix_qq, and replace its determinant by det_minorPPC ($6z-2$). The cofactor sign at indices $(4,4)$ is $+1$ because $4+4$ is even (if_pos by decide). Finish with ring.
why it matters
Supplies the closed form $C_{qq}=6z-2$ needed for every split-form dihedral cosine that touches vertex 3 on the fourOne hinge. Downstream, hingeCosPath_eq_moebius collapses the split cosine to the cut-free Möbius ratio $(1-2z)/(6z-2)$; branchRegular_fourOne_hinge uses $\mathrm{Im}(6z-2)=6,\mathrm{Im},z>0$ on the open arc to keep both cofactors off the square-root cut; product_form_crossing multiplies $C_{pp}C_{qq}$ to exhibit the gate FAIL at $-32$ on the slit, forcing the split denominator. In WickFourOneAllHinges it feeds boundary continuity for pairs $(0,3)$, $(1,3)$, $(2,3)$, $(3,4)$ and their branch-regular certificates. Landmark: C11 panel-locked Wick hinge continuation; does not close the ledger gap wick_action_continuation_4d.
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