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 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 $|α|²=2$; the integer case $1+1=2$ is the degenerate additive instance or phase cancellation a vanishing sum of unit-modulus terms, as in $1+ω+ω²=0$. 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.