Triacontagonal proofs of the Bell-Kochen-Specker theorem

TL;DR

This work shows that Coxeter’s triacontagonal projections of the $600$-cell, $120$-cell, and Gosset’s polytope $4_{21}$ yield Kochen–Specker diagrams from which parity proofs of the Bell–Kochen–Specker theorem can be efficiently extracted. By discarding antipodal redundancy and encoding orthogonality relations as orbits of fifteen-ray pentadecagons, the authors introduce a generator-based word representation that captures entire bases sets; parity proofs correspond to odd-length words, obtained via a nullspace analysis of a pentadecagon–generator incidence matrix. The approach yields fifteen-fold symmetric parity proofs across all three polytopes, including direct $30_2-15_4$ proofs for each and enormous families of proofs generated by combining generators (e.g., $2^{29}$ for the $600$-cell, $2^{30}$ for the $120$-cell, and $2^{130}$ for Gosset’s polytope). Additionally, the paper discusses algorithmic reconstruction of proofs, contrasts these results with previous methods, and establishes that the $600$-cell is not rigid, with implications for experimental realizations and future investigations into contextual sets and rigidity. The work thus bridges Coxeter geometry and quantum contextuality, offering a compact, symmetry-driven framework to generate and analyze KS proofs with potential practical applications. $30_2-15_4$ parity proofs and fifteen-fold symmetry play central roles throughout, enabling compact representations via generator words such as $abcde$ and their combinations.

Abstract

Coxeter pointed out that a number of polytopes can be projected orthogonally into two dimensions in such a way that their vertices lie on a number of concentric regular triacontagons (or 30-gons). Among them are the 600-cell and 120-cell in four dimensions and Gosset's polytope in eight dimensions. We show how these projections can be modified into Kochen-Secker diagrams from which parity proofs of the Bell-Kochen-Specker theorem are easily extracted. Our construction trivially yields parity proofs of fifteen bases for all theree polytopes and also allows many other proofs of the same type to be constructed for two of them. The defining feature of these proofs is that they have a fifteen-fold symmetry about the center of the Kochen-Specker diagram and thus involve both rays and bases that are multiples of fifteen. Any proof of this type can be written as a word made up of an odd number of distinct letters, each representing an orbit of fifteen bases. Knowing a word makes it possible to write down all the features of the associated proof without first having to recover its bases. A comparison is made with earlier approaches that have been used to obtain parity proofs in these polytopes, and two questions related to possible applications of these polytopes are raised.