cmMatrixN_euclidean_threeTwo
plain-language theorem explainer
The bordered Cayley-Menger matrix of a Euclideanized type-(3,2) causal 4-simplex equals the two-parameter matrix with entries fixed by squared lengths p=a^2 and q=alpha a^2. CDT and discrete-gravity workers cite it when reducing the 4-simplex volume determinant on the (3,2) class. The proof is entrywise: finite case analysis on the 6x6 indices collapses every slot to definitional equality.
Claim. For all real $a$ and $\alpha$, the bordered Cayley-Menger matrix built from the Euclideanized squared edge lengths of a type-$(3,2)$ causal 4-simplex (three vertices on one spatial slice, two on the next) equals the two-parameter matrix with parameters $p=a^2$ and $q=\alpha a^2$.
background
In 4D causal dynamical triangulations (Ambjørn-Jurkiewicz-Loll conventions), spacetime between adjacent spatial slices is filled by two 4-simplex types. Type (3,2) places three vertices on slice $t$ and two on $t+1$, giving four spacelike edges of squared length $a^2$ and six timelike edges. Euclideanization replaces Lorentzian timelike squared lengths $-\alpha a^2$ by $+\alpha a^2$ (the inverse of the Wick map $\alpha\mapsto -\alpha$ on the causal class).
The dimension-parametric Cayley-Menger matrix is the bordered $(n+2)\times(n+2)$ array of squared distances whose determinant yields simplex volume data. For a 4-simplex one obtains a $6\times 6$ matrix. This module packages the ten squared edge lengths in lexicographic order, converts them to a full distance table, and feeds that table into the bordered matrix.
The local setting is the QG Seven-Gaps Lorentzian-sector Phase 3a lift of the 3D causal-simplex Wick machinery: combinatorial edge-type counts, an involutive Wick map, and exact cm4 evaluations on both causal classes.
proof idea
Pure matrix extensionality. After equating the two $6\times 6$ matrices entrywise, nested finite case splits on the row and column indices exhaust all thirty-six pairs. Each residual goal is reflexivity: the Euclideanized (3,2) edge assignment and the two-parameter target matrix are definitionally identical on every border entry and every squared-distance slot. No algebraic lemmas are required.
why it matters
This matrix identity is the sole rewrite step feeding the Euclidean type-(3,2) Cayley-Menger determinant theorem, which evaluates $\mathrm{cm}_4=(12\alpha-7)a^8$. That closed form cross-checks the AJL volume $V(3,2)=(a^4/96)\sqrt{12\alpha-7}$ via $9216 V^2=(12\alpha-7)a^8$. In the Seven-Gaps campaign it supplies the exact Euclidean non-degeneracy threshold $\alpha>7/12$ for type (3,2), with degeneracy on the threshold and strict cm4 negativity on the Lorentzian side. It is the D=4 matrix-level counterpart of the kernel-checked 3D causal-simplex Wick lane, and it anchors later comparisons between Euclidean positivity and Lorentzian signature for both (4,1) and (3,2) classes.
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