pentMatrix41
plain-language theorem explainer
Bordered 6x6 Cayley-Menger matrix for a CDT (4,1) 4-simplex, with spacelike squared length p and timelike squared length q. Anyone evaluating the (4,1) volume form, Euclidean/Lorentzian edge assignment, or Wick continuation cites this two-parameter family. The body is an explicit index-by-index case match encoding the (4,1) edge-type pattern.
Claim. For real parameters $p$ (spacelike squared length) and $q$ (timelike squared length), define the bordered Cayley-Menger matrix $M(p,q)\in\mathrm{Mat}_{6\times 6}(\mathbb{R})$ of a $(4,1)$ causal 4-simplex by: $M_{00}=0$; $M_{0i}=M_{i0}=1$ for $i\ge 1$; $M_{ii}=0$ for $i=1,\ldots,5$; $M_{i5}=M_{5i}=q$ for $i=1,\ldots,4$ (apex-to-base edges); and $M_{ij}=p$ for every other off-diagonal pair among indices $1,\ldots,5$ (base spacelike edges). Rows/cols $1..5$ are vertices $0..4$, with apex vertex $4$ at index $5$.
background
This module is the 4D Lorentzian lift in the QG Seven-Gaps campaign (Phase 3a). Spatial slices are equilateral tetrahedra of squared edge length $a^2$. Between consecutive slices, spacetime is filled by two CDT 4-simplex types: $(4,1)$ (four vertices on slice $t$, one on $t+1$: six spacelike and four timelike edges) and $(3,2)$. Spacelike edges carry squared length $a^2$; timelike edges carry $-\alpha a^2$ in the Lorentzian regime ($\alpha>0$). Wick rotation is the algebraic continuation $\alpha\mapsto -\alpha$.
The classical volume of a 4-simplex is read from the bordered Cayley-Menger determinant (the $6\times 6$ matrix with a $0/1$ border row and column, zeros on the distance-block diagonal, and squared edge lengths off-diagonal). This definition freezes that matrix for the $(4,1)$ edge-type pattern into a two-parameter family $M(p,q)$, so Euclidean and Lorentzian specializations differ only by the sign of $q$.
proof idea
Pure definition by cases on matrix indices $(i,j)\in\mathrm{Fin},6\times\mathrm{Fin},6$. The border row/column is the standard Cayley-Menger pattern ($0$ at $(0,0)$, $1$ elsewhere). The $5\times 5$ distance block has zero diagonal; the four apex-base slots (indices $1..4$ paired with $5$) receive $q$; every remaining off-diagonal entry receives $p$. No lemmas are invoked.
why it matters
This is the concrete matrix object on which the $(4,1)$ volume algebra runs. Downstream, det_pentMatrix41 expands $\det M(p,q)=p^3(3p-8q)$ by a kernel-honest $6\times 6$ determinant computation; the Euclidean and Lorentzian identification theorems state that the bordered CM matrix of the Euclideanized (resp. Lorentzian) $(4,1)$ edge tuple equals $M(a^2,\alpha a^2)$ (resp. $M(a^2,-\alpha a^2)$). Those equalities feed the exact Euclidean non-degeneracy thresholds in $\alpha$ and the strict CM negativity on the Lorentzian side that the module advertises. The Wick-action lane also reuses the same two-value edge pattern when continuing physical edges along the complex arc. In the broader RS gravity stack this is the 4D CDT specialization of the dimension-parametric Cayley-Menger determinant, supporting the kinematical Wick rotation that links Euclidean and Lorentzian causal classes.
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