The Algebraic Landscape of Kochen-Specker Sets in Dimension Three

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 $\mathrm{\mid }x{\mathrm{\mid }}^2=2$, as in $\mathrm{\mid }\sqrt{2}{\mathrm{\mid }}^2=2$, $\mathrm{\mid }\sqrt{-2}{\mathrm{\mid }}^2=2$, or $\mathrm{\mid }\alpha {\mathrm{\mid }}^2=2$; 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 $\mathrm{\mid }x{\mathrm{\mid }}^2\ge 3$ 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 $\mathbb{Z}[(1+\sqrt{-7})\mathrm{/}2]$ (43 vectors) and the golden ratio field $\mathbb{Q}(\phi )$ (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.