Status of the local hidden variable underlying mechanism

Determine whether the local hidden variable or underlying mechanism associated with the generalized Local realism framework based on extended probability should be regarded as a classical variable or a quantum variable.

Background

The generalized Bell–CHSH inequality is satisfied using an extended probability, but the paper distinguishes this result from establishing a conventional local hidden-variable model. The authors discuss three possible positions concerning counterfactual definiteness across incompatible consistent-history frameworks: complete allowance, conditional allowance, or prohibition.

Within this discussion, the unresolved issue is whether the underlying local hidden variable or mechanism should itself be interpreted classically or quantum mechanically. This matters because generalized determinism and counterfactual definiteness depend on whether a single probability or extended-probability space can consistently describe incompatible observables.

References

Since there is no direct influence between Alice and Bob based on the history framework, whether the underlying the local hidden variable ($\lambda$), or underlying mechanism, should be a classical or quantum variable remains as an open question in this point.

— No violation on a generalisation of Leggett-Garg inequality and Bell-CHSH inequality with extended probability  (2609.24169 - Kajonsombat et al., 21 Sep 2026) in Section IV, paragraph discussing counterfactual definiteness and generalized Determinism

Nevertheless, how valid is the generalisation of this kind? Honestly, it should be asked ``How valid is the extended probability?" instead.

— No violation on a generalisation of Leggett-Garg inequality and Bell-CHSH inequality with extended probability  (2609.24169 - Kajonsombat et al., 21 Sep 2026) in Conclusion, final paragraph