Del Santo Reply Clarifies Measurability Failure of Axiom-of-Choice No-Signalling Strategies

The reply targets a comment on the 2024 Proc. R. Soc. A paper and isolates the exact point of breakdown: when the full input-to-output map must be measurable. For any single fixed input the constructed strategy yields a well-defined output distribution, yet the conditional distributions across the input space do not form a measurable kernel. This measurability gap blocks the strategy from qualifying as a legitimate probabilistic no-signalling model. The argument draws directly on standard measure-theoretic definitions of Markov kernels used in classical and quantum information theory.

Related literature already flags similar issues. The 2018 work by B. H. Feintzeig on non-measurable strategies in Bell scenarios and the 2022 analysis by C. M. Caves et al. on hidden-variable models both demonstrate that choice-based constructions routinely produce non-measurable objects once probability measures are imposed on the full input space. Del Santo's clarification therefore aligns with and sharpens an existing methodological consensus rather than introducing an isolated technicality.

The immediate consequence is that future no-signalling proofs relying on the Axiom of Choice must now explicitly verify the existence of a measurable version of the strategy. Without that verification the claimed separation between deterministic and probabilistic models remains formally incomplete.

Next steps include constructing explicit measurable approximations or proving that no such approximation exists for certain Bell inequalities, a direction already under discussion in forthcoming workshops on foundations scheduled for 2027.