Nonbinary multiplication-phase cocycles in projector-context constructions

Determine whether a nontrivial multiplication-phase cocycle analogous to the Pauli multiplication cocycle can produce an essential contextuality obstruction beyond the binary case, rather than having the obstruction already arise at the incidence level of a projector-context arrangement.

Background

The appendix compares two sources of contextuality obstructions. For Pauli-based binary linear constraint systems, the context phase vector can be interpreted as the pullback of a nontrivial multiplication 2-cocycle, and group cohomology describes additional phase data associated with symmetries.

By contrast, the projector-context constructions used in the main paper are phase-invariant: their classical obstruction is already captured by the incidence relations, without requiring a multiplication cocycle. The authors leave unresolved whether an analogous nontrivial phase-cocycle mechanism exists beyond the binary setting and can play an independent role in producing contextuality or quantum-classical gaps.

References

While we leave open whether a nontrivial multiplication-phase cocycle analogous to the Pauli case can play such a role beyond the binary case, we point out that - beyond being equivalent descriptions of the same underlying contextual obstruction (see Prop.~\ref{prop: b <-> beta} and Prop.~\ref{prop: phase function}) - Pauli-based quantum solutions of binary LCS can be integrated into the language of projector-context arrangements.

— A complete classification of the existence of finite-dimensional quantum solutions to inconsistent linear constraint systems  (2609.16490 - Frembs, 15 Sep 2026) in Appendix, Section “Quantum solutions from n-qubit Pauli operators,” subsection “Projector reduction,” final discussion

This comparison does not establish a correspondence between our stencil phase data and their cohomological obstruction classes; determining whether such a correspondence exists is left to future work.

— Complete parameterization and parameter-space topology of discrete Wigner representations on $d \times d$ phase space  (2609.39119 - Antonopoulos et al., 30 Sep 2026) in Section 6, Discussion and conclusions