UNBOUNDED PROJECTOR-RANK THRESHOLDS FOR STATE-INDEPENDENT CONTEXTUALITY

For every positive integer R, this preprint constructs a finite graph that exhibits state-independent quantum contextuality using distinct projectors of one common rank. The construction gives a contextuality violation ratio of at least 81/80 for every quantum state. Yet every projective representation of the same graph with ranks between 1 and R has a noncontextual maximally mixed state, regardless of the finite dimension. The proof uses the arc graphs from Jensen Kohlmeyer and Liam Kruer’s solution to Erdős Problem 108. Their construction supplies graphs with arbitrarily large ordinary chromatic number and bounded fractional chromatic number. This combination is what makes it possible to rule out every representation below the chosen rank threshold. The result answers the arbitrary rank threshold existence question posed by Höhn, Xu, and Kleinmann in Section 8 of their paper. arXiv:2207.11183 This record includes the preprint, a standalone Lean 4 proof of the complete theorem, and supporting source files, build instructions, verification logs, and license notices.