Extreme-ray Generation of Bounded Plücker Ratios on Gr(3,n)

For n ≥ 5, consider the semigroup of column-balanced integer exponent vectors defining bounded monomial ratios of Plücker coordinates on the positive Grassmannian Gr>0(3,n). We prove that this semigroup is generated by the lattice-primitive extreme-ray exponents of its real bounded-ratio cone if and only if 5 ≤ n ≤ 9. The positive cases follow from exact finite cone and Hilbert-basis certificates. For n = 10, two explicit bounded ratios span an exposed two-dimensional face whose lattice-primitive generators have lattice index two; their integral midpoint is indecomposable but lies on neither extreme ray. Boundedness is proved by finite positive Plücker-identity trees, and a uniform terminal extension preserves the complete face in every higher dimension. Thus the obstruction is arithmetic rather than a failure of real conic generation. The argument does not require a classification of the higher-dimensional cones or their Hilbert bases.