cm3_euclidean_scale
plain-language theorem explainer
The Cayley–Menger determinant of a Euclideanized causal tetrahedron factors as $(a^2)^3$ times the unit-edge value at the same $\alpha$. Gravity and CDT workers use it to reduce non-degeneracy and volume checks to the $a=1$ slice. The proof is a two-line term: rewrite by the edge-scaling identity, then apply the existing cubic scaling law for `cm3`.
Claim. For either 3d CDT causal tetrahedron type $\mathrm{ty}\in\{(3,1),(2,2)\}$ and real parameters $a,\alpha$, if $E(\mathrm{ty},a,\alpha)$ denotes the Euclideanized squared-edge 6-tuple (spacelike edges $a^2$, Wick-flipped timelike edges $\alpha a^2$), then the Cayley–Menger determinant satisfies $\mathrm{CM}_3\bigl(E(\mathrm{ty},a,\alpha)\bigr)=(a^2)^3\,\mathrm{CM}_3\bigl(E(\mathrm{ty},1,\alpha)\bigr)$.
background
This module is the Lorentzian-sector lane of the QG Seven-Gaps campaign. Prior discrete-gravity results in the repo are Euclidean; here one builds CDT-style causal tetrahedra in $D=3$, an explicit Wick map on squared edge lengths, and non-degeneracy for the Euclideanized images.
Conventions follow Ambjørn–Jurkiewicz–Loll: spatial slices are equilateral with squared length $a^2$; between slices one fills with type $(3,1)$ (3 spacelike + 3 timelike) or $(2,2)$ (2 spacelike + 4 timelike). In the Lorentzian regime timelike squared lengths are $-\alpha a^2$ with $\alpha>0$; Wick rotation is the algebraic continuation $\alpha\mapsto -\alpha$, producing the Euclideanized tuple used here.
The Cayley–Menger polynomial $\mathrm{CM}_3$ (from Geometry.CayleyMengerPolynomial) is the standard 5×5 determinant on six squared edge lengths that encodes $288 V^2$ for a tetrahedron. The upstream scaling law states that multiplying all squared edges by a common factor $\lambda$ multiplies $\mathrm{CM}_3$ by $\lambda^3$. The present statement specialises that law to the causal Euclidean edge assignment.
proof idea
Term-mode, two steps. First rewrite the left-hand side with the in-module identity that Euclidean causal edges at $(a,\alpha)$ are exactly the unit-edge tuple at $(1,\alpha)$ with every squared length multiplied by $a^2$. Then apply the existing general lemma $\mathrm{CM}_3(\lambda\cdot e)=\lambda^3,\mathrm{CM}_3(e)$ at $\lambda=a^2$ and $e=E(\mathrm{ty},1,\alpha)$. No case split on causal type is needed; both types share the same homogeneous scaling.
why it matters
Non-degeneracy and deficit-angle reality for Euclideanized causal tets are stated on an exact $\alpha$-range ($\alpha_{\min}(3,1)=1/3$, $\alpha_{\min}(2,2)=1/2$) at the physical point $\alpha=1$. Factoring out $a$ shows those thresholds are scale-invariant: volume sign and vanishing depend only on the dimensionless ratio $\alpha$. Downstream the Wick-image theorem identifies every Lorentzian-class edge tuple’s Wick image with some $E(\mathrm{ty},a,\alpha)$, so this scaling is the bridge from Lorentzian CDT data to the Euclidean $\mathrm{CM}_3$ analysis. In the broader Recognition chain this sits in the gravity lane after $D=3$ (T8) and the eight-tick discrete time structure, supplying the first certified Lorentzian simplex kinematics rather than a new dynamical law.
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