The Algebraic Landscape of Kochen-Specker Sets in Dimension Three
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Quantum Physics arXiv:2603.16988 (quant-ph) [Submitted on 17 Mar 2026] Title:The Algebraic Landscape of Kochen-Specker Sets in Dimension Three Authors:Michael Kernaghan View a PDF of the paper titled The Algebraic Landscape of Kochen-Specker Sets in Dimension Three, by Michael Kernaghan View PDF HTML (experimental) 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 $|\alpha|^2=2$; 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: 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 arXiv-issued DOI via DataCite (pending registration) Submission history From: Michael Kernaghan Ph.D. [view email] [v1] Tue, 17 Mar 2026 17:31:59 UTC (40 KB) Full-text links: Access Paper: View a PDF of the paper titled The Algebraic Landscape of Kochen-Specker Sets in Dimension Three, by Michael KernaghanView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-03 References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: 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 Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) Links to Code Toggle Papers with Code (What is Papers with Code?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)