alphaMin_twoTwo

This module is the Lorentzian-sector lane of the QG Seven-Gaps campaign. It equips 3d CDT tetrahedra with causal edge typing and a kinematical Wick map on squared lengths. Spatial slices are equilateral triangulations with spacelike squared length $a^2$; between slices one fills with two types: (3,1) (three vertices on $t$, one on $t+1$) and (2,2) (two vertices on each slice). Timelike squared lengths are $-\alpha a^2$ with $\alpha>0$; Wick rotation is the continuation $\alpha\mapsto -\alpha$.

The function $\alpha_{\min}$ assigns to each causal type the exact Euclidean non-degeneracy threshold obtained by hand Cayley–Menger computation: $\alpha_{\min}(3,1)=1/3$ and $\alpha_{\min}(2,2)=1/2$. Non-degeneracy means the Cayley–Menger polynomial $\mathrm{cm}_3$ of the Euclideanized edge multiset is strictly positive (positive squared volume).

One-line definitional reflexivity. The piecewise definition of $\alpha_{\min}$ already sets the (2,2) branch to $1/2$, so rfl closes the equality.

Pins the larger of the two 3d thresholds, which forces the joint Euclidean regime. Downstream, threshold exactness (cm3_euclidean_degenerate_at_min) rewrites at $\alpha=\alpha_{\min}$ and obtains $\mathrm{cm}_3=0$ for (2,2) via this identity plus the closed-form $\mathrm{cm}_3$ for that type. The core iff theorem (cm3_euclidean_pos_iff) uses it to prove $\mathrm{cm}_3>0$ exactly when $\alpha>1/2$ for (2,2). The joint statement (cm3_euclidean_pos_joint) then packages both types as simultaneously non-degenerate on $\alpha>1/2$, the standard 3d CDT Euclidean-regime bound. In the Recognition gravity stack this is the certified kinematical floor before deficit-angle reality at the physical point $\alpha=1$.