cmCofactorSignC
plain-language theorem explainer
Complex cofactor sign $(-1)^{r+c}$ on the bordered $6\times 6$ Cayley-Menger index set $\mathrm{Fin}\,6$. Anyone writing complex CM cofactors or closed-form hinge dihedral identities in the C11 Wick lane cites it. The body is a one-line parity branch returning $1$ or $-1$ in $\mathbb{C}$.
Claim. For row and column indices $r,c\in\{0,\ldots,5\}$, the complex cofactor sign is $+1$ when $r+c$ is even and $-1$ when $r+c$ is odd; equivalently $(-1)^{r+c}\in\mathbb{C}$.
background
The module formalizes a complex-first 4D Wick continuation of Regge hinge data (areas-squared and cofactor dihedral cosines) for causal 4-simplex classes. The continuation path is the upper-half-plane arc $z(t)=\alpha a^2\exp(i\pi(1-t))$ from Lorentzian to Euclidean squared edges. Scope is hinge-data only; full action-level continuation remains open (C12).
Classical Cayley-Menger geometry encodes simplex volumes and dihedral cosines via cofactors of a bordered Gram-type matrix of squared edge lengths. The real precursor cmCofactor3 multiplies a real minor by the sign $(-1)^{r+c}$. Here the matrix is $6\times 6$ (five vertices plus the CM border), so indices run in $\mathrm{Fin},6$, and the sign is valued in $\mathbb{C}$ to match the complex edge data SqEdges10C.
This definition isolates that pure sign factor so every complex cofactor is sign * minor.
proof idea
Pure definition, not a proof. On inputs $r,c:\mathrm{Fin},6$ it tests whether $r.\mathrm{val}+c.\mathrm{val}$ is even; if so it returns $1:\mathbb{C}$, otherwise $-1:\mathbb{C}$. No lemmas are applied. Downstream code unfolds it and discharges the parity branch by decide (e.g. even diagonal pairs, odd off-diagonal pairs).
why it matters
Building block for the complex Cayley-Menger cofactor $C_{r,c}$ (cmCofactorC), which multiplies this sign by the complex minor. That cofactor is the numerator/denominator ingredient of complex dihedral cosines along the Wick arc.
Closed-form hinge theorems depend on it directly: cofactor_pp and cofactor_qq get $C_{pp}=C_{qq}=6z-2$ via the even-parity branch; cofactor_pq gets $C_{pq}=1-2z$ via the odd branch. The four-one all-hinges lane reuses it for symmetry (cmCofactorC_symm, using commutativity of addition on the exponents) and further closed forms such as cofactor41_12.
In the QG Seven-Gaps C11 campaign this is scaffolding for PATH-SELECTED boundary continuation of hinge data. It does not close the ledger gap wick_action_continuation_4d; that stays open pending interior-hinge complexes (C12).
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.