Exact projector-rank thresholds for state-independent contextuality

For every integer r ≥ 1 we construct a finite exclusivity graph Fr with the following two properties. First, Fr has a projective representation by pairwise distinct projectors of rank r that exhibits state-independent contextuality. Second, no projective representation of Fr by projectors of any common rank s < r, in any finite dimension, exhibits state-independent contextuality; in every such representation the maximally mixed state is noncontextual. This establishes, for every r, the “rank efficient” scenarios sought by Hohn, Xu and Kleinmann, for which only numerical evidence at r = 2, 3 was available. The graph is the disjunctive product of an odd cycle C2r+1 with the orthogonality structure of a rank-one contextuality set whose fractional chromatic number exceeds its dimension by an arbitrarily small amount. Such sets are obtained from the infinite family of Kochen–Specker parity proofs in R^4 of Elford and Lisonek, together with a coning construction that raises the dimension by one.