2026/03/31 by Michael Kernaghan
#quant-ph
We present a computational survey of Kochen-Specker (KS) uncolorability in three-dimensional Hilbert space across two-symbol coordinate alphabets A = \0, ± 1, ± 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 |√(2)|2=2, |√(-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 ≥ 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 ℤ[(1+√(-7))/2] (43 vectors) and the golden ratio field ℚ(φ) (52 vectors, revealed only by cross-product completion); ℤ[√(-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.