Rigid ternary relations in finite-dimensional Hilbert-space Grassmannians

For positive integers $1\le r<d<n$ consider subsets $S\subseteq \mathbb{G}(r,V)$ of the $r$-plane Grassmannian of an $n$-dimensional Hilbert space $V$ saturated in the sense that the $r$-plane $\eta''$ belongs to $S$ whenever it is the orthogonal projection of $\eta'\in S$ onto a $d$-plane through $\eta\in S$.

Motivated by such closure operators' natural occurrence in projective-geometry and linear preserver problems, we classify said saturated sets as precisely the disjoint unions of Grassmannian spines, with cores standing in a relation of mutual separation that can be made precise (a spine being the set of $r$-planes containing a fixed core $k$-plane $\pi$ for some $0\le k\le r$).

This generalizes the author's results describing saturated $r$-plane sets in the tame dimensional regime $2r\le d$, where the disjoint unions in question by necessity collapse to single spines.