Minimal ring extensions of the integers exhibiting Kochen-Specker contextuality

This paper is a contribution to the algebraic study of contextuality in quantum theory. As an algebraic analogue of Kochen and Specker’s no-hidden-variables result, we investigate rational subrings over which the partial ring of $d\times d$ symmetric matrices ($d\ge 3$) admits no morphism to a commutative ring, which we view as an “algebraic hidden state.” For $d=3$, the minimal such ring is shown to be $\mathbb{Z}[1\mathrm{/}6]$, while for $d\ge 6$ the minimal subring is $\mathbb{Z}$ itself. The proofs rely on the construction of new sets of integer vectors in dimensions 3 and 6 that have no Kochen-Specker coloring.