hingeGramDet_near

This module is the QG kernel increment for the type-(1,2) triangle hinge ${0,e_0,e_0+e_1+e_2}$ and its full periodic Freudenthal star (two unit cubes, four incident 4-simplices). It imports the committed Gram-projection cosine calculus and never redefines that API.

The hinge Gram determinant is the cleared factor $4\langle a,a\rangle\langle b,b\rangle-(2\langle a,b\rangle)^2$ built from the two hinge edge vectors from vertex 0; in coordinates it is $4\cdot a_0\cdot a_1-(a_0+a_1-a_4)^2$. The near-orbit flat squared-edge tuple is the fixed $10$-component assignment $(1,3,2,4,2,1,3,1,1,2)$ used for one of the two local squared-length orbits on the star.

Together these feed the numerator form of the dihedral cosine, which must be zero on every flat simplex of the star before the angle sum $4\cdot\pi/2=2\pi$ can be stated.

One-line wrapper: unfold the Gram determinant definition and the near-flat edge assignment, then close by norm_num. No intermediate lemmas are required beyond those two definitions.

Parent use is cosDihedral_near_flat, which rewrites the cosine into numerator form under the non-vanishing Gram gate supplied here, then evaluates the apex numerators to obtain cosine $0$ on the near orbit. That zero is one half of deliverable A.2 (flat cosine multiset on all four star simplices) and is required for the flatness gate (star angle sum exactly $2\pi$).

In the broader campaign this is a kernel-checked brick toward the full-star deficit class kernel on the 15 stencil classes; it does not yet close the complement type-(2,1) orbit, the flat Hessian over all hinges, or S_RS_converges_EH_4d. Within Recognition gravity it sits in the Regge discrete curvature layer that must match continuum Einstein–Hilbert stationarity once all hinge orbits are filled.