submatrix32_11
plain-language theorem explainer
Deleting row and column index 1 from the complex bordered (3,2) hinge matrix recovers the canonical lower-diagonal 5x5 minor. Causal-simplex and Wick-continuation workers cite this when reading lower-vertex cofactors. The proof is a 5x5 entrywise case split with definitional equality.
Claim. For every $z\in\mathbb{C}$, the $5\times 5$ submatrix of the bordered $(3,2)$ hinge matrix obtained by deleting the row and column of index $1$ equals the explicit lower-diagonal minor (the common matrix for any lower-slice vertex deletion).
background
Lane B2 of the QG Seven-Gaps campaign continues all ten triangular hinges of the (3,2) causal 4-simplex on the complex upper-half arc. The hinge matrix is the explicit bordered $6\times 6$ complex matrix of that tuple: rows/cols 1..3 are lower-slice vertices 0..2, rows/cols 4..5 are upper-slice vertices 3..4, with spacelike weight 1 and timelike weight $z$ (mirror of the real pentagon matrix at $p=1$, $q=z$).
The lower diagonal minor is the $5\times 5$ matrix obtained by deleting any one lower vertex. The three such deletions give the same explicit matrix: a $2+2$ tetrahedron with two spacelike and four timelike edges. Cayley-Menger cofactors of the hinge edge data are assembled from these minors and their determinants.
proof idea
Extensionality on the remaining $5\times 5$ index set, then exhaustive finite case analysis on both indices. Each of the twenty-five entries agrees by definitional equality with the pattern-matching clauses of the lower-minor matrix. No ring identity or determinant lemma is used; the two matrix definitions share the same case table after the index-1 deletion.
why it matters
Direct input to the cofactor theorem that the $(1,1)$ Cayley-Menger cofactor of the (3,2) hinge edges equals $8z-4$. That closed form is the lower-member asymmetric cofactor $C_{pp}=8z-4$ in the module classification (mixed lower-upper pairs and upper-pair hinges). Those cofactors, with the matching area-squared formulas, underwrite the split-form branch certificates that close Wick continuation for all ten hinges at the physical point $a=1$, $\alpha=1$. The result is pure linear-algebra scaffolding inside the Seven-Gaps finishing charter, not a continuum or curvature claim.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.