cofactor_pp
plain-language theorem explainer
The diagonal Cayley–Menger cofactor C_pp of the complex fourOne hinge edge data equals the linear polynomial 6z−2 for every complex timelike squared length z. Gravity and QG workers cite it whenever a dihedral cosine, branch cut, or Möbius reduction needs an explicit cofactor. The proof unfolds the cofactor, substitutes the explicit hinge matrix and its (3,3) minor, applies the closed 5×5 determinant, and finishes by ring.
Claim. For every $z \in \mathbb{C}$, the complex Cayley–Menger cofactor $C_{3,3}$ 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 a 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 is $C_{r,c}=(-1)^{r+c}\det M^{[r|c]}$, where $M$ is the bordered $6\times 6$ CM matrix of ten complex squared edge lengths. Here hingeEdgesC z is the two-value fourOne tuple: timelike edges carry $z$, spacelike edges carry $1$. Index pair $(3,3)$ is the diagonal entry tied to vertex $2$ in the CM numbering (row/col $3$).
Upstream, cmMatrixC_hingeEdges identifies that matrix with an explicit symbolic form; submatrix_pp and det_minorPPC give the $5\times 5$ minor after deleting row/col $3$, with closed determinant $6z-2$ (matching RESULTS.txt and the unit-tetrahedron value $4$ at $z=1$).
proof idea
Term-mode reduction. Unfold the cofactor into sign times minor determinant. Rewrite the CM matrix via cmMatrixC_hingeEdges, replace the deleted submatrix by submatrix_pp, and insert the closed determinant det_minorPPC ($\det=6z-2$). The cofactor sign at $(3,3)$ is $+1$ because $3+3$ is even (if_pos by decide). A final ring clears the remaining arithmetic.
why it matters
This closed form is the algebraic spine of the fourOne hinge continuation. Downstream, branchRegular_fourOne_hinge uses $\mathrm{Im}(6z-2)=6,\mathrm{Im},z>0$ on the open arc to keep both diagonal cofactors off the sqrt cut. hingeCosPath_eq_moebius collapses the split-sqrt cosine to the cut-free rational $(1-2z)/(6z-2)$ once the product of split roots returns this cofactor. endpoint_cofactor_on_sqrt_cut evaluates it at the Lorentzian end ($z=-1\Rightarrow C_{pp}=-8$ on the cut), justifying open-interior certificates. product_form_crossing multiplies the two diagonal cofactors to hit $-32$ on the cut, killing the naive single-sqrt 3D transcription and mandating the split denominator.
In WickFourOneAllHinges, boundary continuity theorems for several timelike pairs (boundary_pair02, 12, 23, 24) feed this cofactor into the path-selected cosine. Within the Seven-Gaps C11 lane it is a proved symbolic certificate, not scaffolding; the ledger gap on full action-level continuation stays open and is untouched.
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