Persistence and Splitting of Bounded Plücker-Ratio Rays under Grassmannian Label Embeddings

We study the behaviour of bounded Plücker-ratio rays on the positive Grassmannian Gr>0(3,n) under increasing label embeddings. Starting from the nine-label classification, we prove that each of its 54 nonbasic primitive rays remains an extreme ray after any sequence of label insertions. An elementary five-label ray instead acquires a complete exposed simplicial face whose dimension is the product of the lengths of two subdivided arcs. The face is saturated in the original incidence lattice and admits unique integral ray coordinates. Consequently, for n ≥ 9, the primitive extreme rays and Hilbert-basis elements with at most nine essential labels coincide, and their number is n(n−3)(n−4)/2 + 54·C(n,9), where C(n,9) is the binomial coefficient. The proofs combine a one-gap exact sequence, an all-coordinate positive-chart transport identity, a dimension-independent face-composition argument and finite original-coordinate certificates. No assertion is made about the complete Hilbert basis at higher essential support.