The Algebraic Landscape of Kochen-Specker Sets in Dimension Three
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2026
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| author | Kernaghan, Michael |
| author_facet | Kernaghan, Michael |
| contents | 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 $|α|^2=2$; 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 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 $\mathbb{Q}(φ)$ (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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16988 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Algebraic Landscape of Kochen-Specker Sets in Dimension Three Kernaghan, Michael Quantum Physics 81P13, 05C65, 11R04 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 $|α|^2=2$; 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 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 $\mathbb{Q}(φ)$ (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. |
| title | The Algebraic Landscape of Kochen-Specker Sets in Dimension Three |
| topic | Quantum Physics 81P13, 05C65, 11R04 |
| url | https://arxiv.org/abs/2603.16988 |