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arXiv · 2603.16988

The Algebraic Landscape of Kochen-Specker Sets in Dimension Three

Michael Kernaghan
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Abstract

We present a computational survey of Kochen-Specker (KS) uncolorability in three-dimensional Hilbert space across coordinate alphabets drawn from quadratic, cyclotomic, golden-ratio, and cubic Pisot number fields, principally the two-symbol family A = {0, +-1, +-x}. In every tested alphabet, KS sets arise only when x supports a cancellation identity: an algebraic relation making dot products vanish exactly. Three mechanisms occur: modulus-2 cancellation (|x|^2 = 2, with the integer case 1+1=2 as the degenerate additive instance), phase cancellation (a vanishing sum of unit-modulus terms, as in 1 + omega + omega^2 = 0), and minimal-polynomial cancellation, in which the defining relation of a cubic algebraic unit supplies the vanishing sum directly, as in the supergolden ratio psi^3 = psi^2 + 1. Generators with |x|^2 >= 3 that are not roots of unity produce orthogonal triples but not KS-uncolorability in our survey. This pattern explains why constructions cluster into at least eight discrete algebraic islands among the tested fields, two of them cubic. Two yield potentially new KS graph types: the Heegner-7 ring Z[(1+sqrt(-7))/2] (43 vectors) and the golden ratio field Q(phi) (52 vectors, revealed only by cross-product completion); Z[sqrt(-2)] gives a new realization of a known Peres-type graph. Using SAT-based bipartite KS-uncolorability we verify and extend the input counts of Trandafir and Cabello across all six non-cubic islands. The two-mechanism pattern reported in versions 1-8 of this paper is refuted by the supergolden island, which motivates the third mechanism; whether the three-mechanism classification is complete remains open.

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Michael Kernaghan. 2026-03-17. The Algebraic Landscape of Kochen-Specker Sets in Dimension Three. https://arxiv.org/abs/2603.16988

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