The Algebraic Landscape of Kochen-Specker Sets in Dimension Three

TL;DR

This paper computationally surveys Kochen-Specker uncolorability in 3D Hilbert space across various algebraic fields, revealing patterns linked to algebraic cancellation mechanisms and identifying new potential KS graph types.

Contribution

It uncovers algebraic patterns underlying KS sets in different number fields and introduces new KS graph types, expanding understanding of algebraic structures in quantum contextuality.

Findings

KS sets arise only with specific algebraic cancellation mechanisms.

Identifies new potential KS graph types in Heegner-7 and golden ratio fields.

Verifies and bounds KS uncolorability counts using SAT-based methods.

Abstract

We present a computational survey of Kochen-Specker (KS) uncolorability in three-dimensional Hilbert space across two-symbol coordinate alphabets $\mathcal{A} = \{0, \pm 1, \pm x\}$ drawn from quadratic, cyclotomic, and golden-ratio number fields. In every tested raw alphabet (before cross-product completion), KS sets arise only when $x$ supports one of two cancellation mechanisms: modulus-2 cancellation (the generator satisfies $|x|^2 = 2$, as in $|\sqrt{2}|^2=2$, $|\sqrt{-2}|^2=2$, or $|\alpha|^2=2$; the integer case $1+1=2$ is the degenerate additive instance) or phase cancellation (a vanishing sum of unit-modulus terms, as in $1+\omega+\omega^2=0$). Alphabets whose generators have $|x|^2 \geq 3$ and are not roots of unity produce orthogonal triples but not KS-uncolorability in our survey. This empirical pattern explains why constructions cluster into at least six discrete algebraic islands among the tested fields (with a seventh, cubic island confirmed at higher cost). Two yield potentially new KS graph types: the Heegner-7 ring $\mathbb{Z}[(1+\sqrt{-7})/2]$ (43 vectors) and the golden ratio field $\mathbb{Q}(\varphi)$ (52 vectors, revealed only by cross-product completion); $\mathbb{Z}[\sqrt{-2}]$ provides a new algebraic realization of a known Peres-type graph. Using SAT-based bipartite KS-uncolorability, we verify the input counts of Trandafir and Cabello for three islands (exact) and establish upper bounds for three others. The golden ratio island is a boundary case: its raw alphabet satisfies neither mechanism, but cross-product completion introduces effective modulus-2 cancellations. Whether the two-mechanism pattern extends to all number fields remains an open question.