cm3_euclidean_twoTwo
plain-language theorem explainer
For a type-(2,2) causal tetrahedron after Wick rotation to Euclidean signature, the Cayley-Menger polynomial equals exactly $4(2\alpha-1)a^6$. CDT and discrete-gravity workers cite this closed form when fixing the non-degeneracy threshold $\alpha>1/2$. The proof substitutes the six squared edge lengths into the explicit CM polynomial and finishes by ring reduction.
Claim. For all real $a$ and $\alpha$, if the six squared edge lengths of a type-$(2,2)$ causal tetrahedron are Euclideanized so that the two opposite spacelike edges equal $a^2$ and the four cross (formerly timelike) edges equal $\alpha a^2$, then the Cayley-Menger polynomial in those lengths equals $4(2\alpha-1)a^6$.
background
In 3d causal dynamical triangulations (Ambjørn–Jurkiewicz–Loll), spacetime between adjacent spatial slices is filled by two tetrahedron classes. Type (2,2) places two vertices on each slice, giving two spacelike edges of squared length $a^2$ and four timelike edges. Lorentzian assignment sets those timelike edges to $-\alpha a^2$ with $\alpha>0$; the kinematical Wick rotation continues $\alpha\mapsto -\alpha$, so Euclideanized edges are $a^2$ on the spacelike pair and $\alpha a^2$ on the four cross edges.
The Cayley-Menger polynomial is the explicit cubic form in the six squared edge lengths, proportional to squared tetrahedron volume. Edge indexing follows the standard layout: edges $0=(0,1)$ and $5=(2,3)$ are the opposite spacelike pair for type (2,2); edges $1$–$4$ are the cross edges.
This module is the first certified Lorentzian layer of the QG Seven-Gaps campaign: causal-class combinatorics, Wick involution, and exact non-degeneracy ranges for Euclideanized simplices.
proof idea
Term proof by direct substitution. Six reflexivity facts pin the Euclideanized squared lengths: edges 0 and 5 equal $a^2$, edges 1–4 equal $\alpha a^2$. Unfold the Cayley-Menger polynomial, rewrite those six entries, and discharge the resulting multivariate identity in $a$ and $\alpha$ by ring.
Agrees with the hand derivation in the doc-comment: the first balanced term contributes $(4\alpha-2)a^6$; the other two balanced terms each give $2\alpha^2 a^6$ and cancel exactly against the four monomial terms, leaving $4(2\alpha-1)a^6$.
why it matters
Feeds the core non-degeneracy characterization: for $a>0$, the Euclideanized causal tetrahedron is non-degenerate (positive CM, positive squared volume) if and only if $\alpha$ exceeds the type-dependent minimum, with the threshold exact both ways. Also discharges the exactness check that CM vanishes at the type-(2,2) minimum $\alpha=1/2$.
Sits in the Lorentzian-sector lane of the Seven-Gaps gravity campaign. Prior discrete-gravity results were Euclidean; this closed form, with the companion type-(3,1) evaluation, makes the Wick-rotated non-degeneracy range fully algebraic. At the physical point $\alpha=1$ one obtains a strictly positive CM value, supporting the deficit-angle reality corollary for Euclideanized (2,2) tets. Ambient dimension is the forced $D=3$ of the CDT setting (T8), not a new derivation of dimension.
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