cofactor_pq
plain-language theorem explainer
Closed form for the Cayley–Menger cofactor at CM indices (3,4) on the complex hinge edge data: it equals $1-2z$. Gravity workers cite it when collapsing the split dihedral cosine to the cut-free Möbius form $(1-2z)/(6z-2)$, and when certifying the four-one pair-(2,3) boundary and branch-regularity statements. The proof unfolds the cofactor, rewrites the hinge CM matrix and its (3,4) minor, then finishes by ring.
Claim. For every $z \in \mathbb{C}$, the Cayley–Menger cofactor of the complex hinge squared-edge data at parameter $z$, taken at matrix indices $(3,4)$, equals $1-2z$.
background
Module C11 formalizes a complex-first 4D Wick continuation of Regge hinge data (complex Cayley–Menger areas-squared and cofactor dihedral cosines) for causal 4-simplex classes. The continuation path on the timelike squared edge is the upper-half-plane arc $z(t)=\alpha a^2\exp(i\pi(1-t))$, running from the Lorentzian to the Euclidean endpoint. Scope is deliberately hinge-level only; full action-level continuation remains open (C12).
The Cayley–Menger matrix encodes squared edge lengths of a simplex; its cofactors enter the classical formulae for squared face areas and dihedral cosines. Here hingeEdgesC z is the complex squared-edge assignment for the traced hinge (spacelike edge fixed at 1, the two timelike edges equal to $z$). Indices 3 and 4 are the CM rows/cols corresponding to the vertex pair that labels this hinge in the four-one class (hinge $(0,1,4)$).
Upstream scaffolding supplies the complex CM matrix, its minors, and the signed cofactor wrapper used in the split-sqrt dihedral cosine.
proof idea
Term-mode algebraic identity. Unfold the complex cofactor into sign times minor determinant. Rewrite the CM matrix on hinge edges, replace the $(3,4)$ submatrix by its closed form, and invoke the precomputed determinant of that minor. The cofactor sign is fixed by if_neg on the parity of $3+4$ (odd, so the sign is $-1$ in the usual convention, already absorbed into the stated $1-2z$). Finish with ring.
why it matters
This cofactor is the numerator of the cut-free Möbius form of the hinge cosine. Downstream, hingeCosPath_eq_moebius quotes it to prove that on the full arc the split-sqrt cosine collapses to $(1-2z)/(6z-2)$, with the companion cofactor $6z-2$ non-vanishing on the open arc. The same triple of cofactor identities (cofactor_pp, cofactor_qq, cofactor_pq) is fed to branchRegular_pair23 and boundary_pair23 in the four-one all-hinges module, which certify branch regularity on $(0,1)$ and continuous boundary values for pair $(2,3)$ (hinge $(0,1,4)$, the landed traced hinge).
Within the Seven-Gaps campaign this is a C11 panel brick: it makes the dihedral cosine a rational function of $z$, so path-selected Wick continuation needs no branch tracking beyond the mandatory split-sqrt denominator. It does not close the ledger gap wick_action_continuation_4d; that remains C12.
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