The largest noncontextual subset of the 600-cell's seventy-five bases has size sixty-five

Brent W. Jonah

Published Jul 7, 2026• 1 citation

Abstract

The 60 rays and 75 orthonormal bases derived from the vertices of the 600-cell form the well-known Kochen-Specker system of Waegell and Aravind. This note determines exactly the largest number of the 75 bases that a single {0,1} ray assignment can satisfy (exactly one ray valued 1 per satisfied basis): the answer is 65, so at least ten bases must fail. This improves the lower bound of nine failures obtained by Waegell, Aravind, Megill and Pavicic (2011) from parity-proof coverings, answers the question they left open, and settles the corresponding open problem restated by Aravind, Burton, Nunez Ponasso and Richter (2026) for the 600-cell. The lower bound is an explicit 17-ray assignment verified in exact Q(sqrt5) arithmetic; the upper bound is certified both by two independent integer-programming solvers and by an explicit family of 150 parity proofs (ten orbits under the triacontagonal C15 symmetry) admitting no nine-element hitting set. The tight noncontextuality inequality is 65 versus the quantum value 75 (ratio 15/11 in the +/-1 normalization). A transfer proposition brackets the analogous 120-cell maximum in [600, 625]. Verification: run python3 verify.py with proofs150.txt in the same directory. It rebuilds the system from the two stated generators and re-verifies the construction anchors, the certificate count of 65, the ten failing bases, and the validity of all 150 parity proofs in seconds, without any solver.

DOI: 10.5281/zenodo.21249804

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