hingeMatrixC
plain-language theorem explainer
Explicit bordered 6×6 complex Cayley–Menger matrix for the four-one causal edge tuple with spacelike squared lengths 1 and one complex timelike entry z. Anyone computing complex hinge areas or dihedral cosines along the Wick arc cites this matrix. The body is pure index case analysis mirroring the real pentMatrix41 at (p,q)=(1,z).
Claim. For $z\in\mathbb{C}$, define the bordered $6\times 6$ complex matrix $H(z)$ of the four-one edge tuple (spacelike squared lengths $1$, one timelike squared length $z$): the $(0,0)$ entry is $0$, the first row and column (except the corner) are $1$, the diagonal of the $5\times 5$ vertex block is $0$, the four apex–base off-diagonals equal $z$ symmetrically, and all remaining entries equal $1$. Rows/cols $1..5$ index vertices $0..4$, with apex vertex $4$ at index $5$.
background
Module C11 formalizes complex-first 4D Wick continuation of Regge hinge data (Cayley–Menger areas-squared and cofactor 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 $z(0)=-\alpha a^2$ to Euclidean $z(1)=+\alpha a^2$. Scope is hinge-data only; full action-level continuation remains open (C12).
A Cayley–Menger matrix encodes squared edge lengths of a simplex in bordered form so that principal minors give volumes and cofactors give dihedral cosines. Here the four-one tuple has nine spacelike edges of squared length $1$ and one complex timelike edge $z$, matching CausalSimplex4D.pentMatrix41 at $(p,q)=(1,z)$. Indices: row/col $0$ is the CM border; $1..5$ are the five vertices with apex at $5$.
proof idea
Definition by exhaustive match on $(i,j)\in\mathrm{Fin},6\times\mathrm{Fin},6$. Border corner $(0,0)\mapsto 0$; border strip $\mapsto 1$; five vertex-diagonal entries $\mapsto 0$; the four symmetric apex–base pairs $(1,5),(2,5),(3,5),(4,5)$ and swaps $\mapsto z$; every other entry $\mapsto 1$. No lemmas; pure pattern table.
why it matters
Anchor matrix for the entire complex CM pipeline in this module. The theorem cmMatrixC_hingeEdges identifies the abstract CM construction on hingeEdgesC z with this explicit form. Downstream minors minorPPC and the submatrix_pp / submatrix_pq / submatrix_qq equalities extract the $5\times 5$ blocks used for areas-squared and split-sqrt dihedral cosines. WickFourOneAllHinges reuses it for branch-regularity and opposite-pair symmetry on the full open arc. It is the complex mirror of the real four-one pent matrix and the concrete carrier of the C11 hinge-data continuation; it does not close the open ledger gap wick_action_continuation_4d.
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