The Algebraic Landscape of Kochen-Specker Sets in Dimension Three

Quantum Physics [Submitted on 17 Mar 2026] The Algebraic Landscape of Kochen-Specker Sets in Dimension Three Michael Kernaghan 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 |\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 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. Comments: 36 pages, 15 tables, no figures Subjects: Quantum Physics (quant-ph) MSC classes: 81P13, 05C65, 11R04 Cite as: arXiv:2603.16988 [quant-ph]   (or arXiv:2603.16988v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2603.16988 Focus to learn more Submission history From: Michael Kernaghan Ph.D. [view email] [v1] Tue, 17 Mar 2026 17:31:59 UTC (40 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-03 References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)