hingeGramDet_t22

This module is the type-(2,2) increment of the Regge 4D full periodic-lattice star deficit kernel campaign. The hinge is the triangle with masks ${0,3,15}$ (difference masks $(3,12)$), and the deliverable is the full Freudenthal star of that hinge: four incident Kuhn simplices, flat cosines, and the flatness gate that the star angle sum is exactly $2\pi$.

The hinge Gram determinant is the scalar $4\langle a,a\rangle\langle b,b\rangle-(2\langle a,b\rangle)^2$ built from the two hinge edge vectors out of vertex 0; in squared-length coordinates it is $4 a_0 a_1-(a_0+a_1-a_4)^2$, i.e. four times the squared hinge-area factor. The flat length vector used here is the hinge-ordered assignment $(2,4,1,3,2,1,1,3,1,2)$ shared by every (2,2) star simplex.

Downstream cosine calculus needs a strictly positive Gram determinant so that the projected apex inner-product formula is well-defined and can be reduced by rewriting.

One-line wrapper: unfold the Gram-determinant definition and the flat squared-edge assignment, then discharge the resulting rational arithmetic by norm_num. Explicitly, $4\cdot 2\cdot 4-(2+4-2)^2=32-16=16$.

The identity is the first numerical gate inside the flat dihedral cosine for the (2,2) star. The parent theorem rewrites the cosine into numerator form under the hypothesis that this Gram determinant is nonzero (here 16), then evaluates the three apex-projection numerators to conclude that every incident flat cosine is exactly 0. That multiset of zeros is what feeds the flatness gate $4\cdot\arccos 0=2\pi$ and the full-star deficit-class kernel on the 15 stencil classes.

In the broader QG campaign this is the type-(2,2) counterpart of the earlier type-(1,1) seed-orbit kernel: same Gram-projection cosine calculus, new hinge orbit and full periodic star. It does not yet assemble the flat Hessian over all hinges, nor does it touch $S_{\mathrm{RS}}\to\mathrm{EH}_{4d}$ or gap-action recovery.