Boundary Rigidity for Equi-isoclinic Planes

Consider a family of real two-dimensional subspaces with common isoclinic projection length 1/2. We prove that if one triangle of normalized cross-Gram matrices has negative determinant, then the entire family spans at most six dimensions and contains at most nine planes, independently of the initial ambient dimension. We give an exhaustive finite description relative to such a triangle and show that the nine-member families in this branch form one unordered congruence class. The proof uses the null relation of a singular anchor Gram matrix, eight extension types, and four explicit nonnegative polynomial identities. If no triangle has negative determinant, the family is the realification of complex equiangular lines. The known maximum of nine planes in R6 is credited to Et-Taoui. This is a complete proof of a restricted boundary-parameter theorem, not a resolution of the general fixed-angle conjectures in AIM-COMBINATORICS-0014. The preprint is AI-assisted, self-audited and unrefereed; absolute priority is not certified.