Construction of Kochen-Specker Sets from Mutually Unbiased Bases

We present a systematic, constructive analysis of Kochen-Specker contextuality, emphasizing the foundational importance of complete orthogonal bases (contexts). First, in three dimensions, we generate a complete inventory of 165 rays and 130 bases from mutually unbiased bases. This unified framework reveals that several known constructions are equivalent manifestations of a minimal 69-ray, 50-context Kochen–Specker nucleus and uncovers a striking 40-4-4 generative asymmetry among the mutual unbiased bases, which we explain via the algebraic exclusivity of the Fourier basis. Second; in higher dimensions (D=4, 5), we develop explicit “forcing gadgets” that use orthogonality constraints to compel a central vector into a state of maximal unbiasedness. We demonstrate that our 20-vector gadget in D=4 and the 18-vector Cabello set are informationally equivalent subsets of the Peres–Mermin eigensystem, yet differ in their contextuality due to the choice of basis completions. Our findings establish that contexts, not merely intertwining vectors, are the crucial carriers of Kochen-Specker type logic and are indispensable for a rigorous assessment of quantum contextuality.