The Algebraic Landscape of Kochen-Specker Sets in Dimension Three

• URL: http://arxiv.org/abs/2603.16988v4
• Date: Wed, 25 Mar 2026 01:53:48 GMT
• Title: The Algebraic Landscape of Kochen-Specker Sets in Dimension Three
• Abstract summary: We present a computational survey of Kochen-Specker uncolorability in three-dimensional Hilbert space across two-symbol coordinate alphabets.<n>In every tested alphabet, KS sets arise only when $x$ supports one of two cancellation mechanisms.<n>Two yield potentially new KS graph types.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 alphabet, 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 $|α|^2=2$; the integer case $1+1=2$ is the degenerate additive instance) or phase cancellation (a vanishing sum of unit-modulus terms, as in $1+ω+ω^2=0$). 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 six discrete algebraic islands among the tested fields. 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 and extend the input counts of Trandafir and Cabello for bipartite perfect quantum strategies across all six islands. Whether the two-mechanism pattern extends to all number fields remains an open question.

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