Ontic Dynamical Locality Reduces to Bell Locality

Referee: Whether ODL is genuinely weaker than static Bell locality or merely equivalent to it by restatement; the full derivation must demonstrate that the space of models satisfying ODL is strictly larger than those satisfying static Bell locality.

Authors: This is the most important comment, and we agree it is load-bearing. The key distinction is as follows. In the static Bell-local framework, the response function P(a|x,lambda) is assumed to be the full local response. In our dynamical model, the local response is decomposed into two stages: a transition kernel T(lambda -> lambda' | x, lambda) that maps the pre-measurement ontic state to a post-measurement ontic state, and then a response P(a|x,lambda'). The static Bell-local model is the special case where T is trivial (lambda' = lambda). ODL permits a strictly larger class: any local stochastic dynamics within a wing—setting-dependent transitions, memory, disturbance—are allowed, so long as they depend only on the local setting and local ontic state. The absorption step marginalizes over lambda' to produce P_eff(a|x,lambda) = sum_{lambda'} T(lambda'|x,lambda) P(a|x,lambda'), which is a proper local response function. The class of models satisfying ODL is strictly larger than the class satisfying static Bell locality because ODL permits non-trivial transition kernels that static Bell locality excludes. However, the *observable* predictions after absorption coincide with the static Bell-local form. We will make this strict-inclusion argument explicit in the revised manuscript, including a concrete example of a model that satisfies ODL but is not a static Bell-local model in the usual sense (because it has non-trivial internal dynamics). We acknowledge that if one defines 'Bell locality' broadly enough to include any marginalization over hidden internal variables, then ODL is a dynamical refinement rather than a strictly different assumption—but this is precisely the content of our result: that the dynamical refinement does not expand the observable Bell-local class. revision: partial

Referee: The absorption of transition kernels must not smuggle in distant-setting dependence through the back door; measurement independence must be shown sufficient, not merely asserted.

Authors: The referee is correct that this must be demonstrated, not merely asserted. In the full manuscript, the proof proceeds as follows. The effective response is P_eff(a|x,lambda) = sum_{lambda'} T_A(lambda'|x,lambda) P(a|x,lambda'), where lambda is the pre-measurement ontic state shared between wings. The critical step is verifying that lambda does not become correlated with the distant setting y through the absorption. Under measurement independence, P(lambda|x,y) = P(lambda), so the pre-measurement ontic state is independent of both settings. The transition kernel T_A depends on x and lambda but not on y (by ODL), and the response P(a|x,lambda') depends on x and lambda' but not on y (by ODL). Therefore P_eff(a|x,lambda) depends on x and lambda but not on y. The joint distribution factorizes as P(a,b|x,y) = integral P(lambda) P_eff(a|x,lambda) P_eff(b|y,lambda) d lambda, which is exactly the Bell-local form. The measurement independence assumption is used precisely at the step where we factor P(lambda|x,y) = P(lambda), and ODL is used to ensure that neither T nor P introduces y-dependence in wing A (or x-dependence in wing B). We will make this logical structure—measurement independence for the factoring step, ODL for the locality of each wing's effective response—more explicit in the revision, including a formal lemma isolating the sufficiency argument. revision: partial

Referee: The claim that the result holds for 'a general transition-kernel model' needs precise scoping: does it include retrocausal or temporally non-local ontic dynamics? Does it permit the ontic state space to depend on the measurement setting? The paper must state exactly what ontic temporal ontology is assumed.

Authors: This is a fair and important point. The full manuscript does specify the temporal structure, but the abstract does not, and we will revise the abstract to include the scoping. The assumed ontology is: (1) there exists a pre-measurement ontic state lambda drawn from a fixed distribution rho(lambda) at the moment of setting selection; (2) each wing has a local transition kernel T that maps lambda to a post-measurement local ontic state lambda' depending on the local setting and the local component of lambda; (3) the local response function maps the post-measurement state to an outcome. The result does NOT cover retrocausal models (where lambda depends on future settings or outcomes) or temporally non-local dynamics (where the transition in one wing depends on the future or past of the distant wing). These are excluded by ODL's requirement that transitions depend only on the local setting and local pre-measurement state. The ontic state space is fixed and does not depend on the measurement setting; setting-dependence enters only through the transition kernel. We agree that 'general transition-kernel model' in the abstract is imprecise and will replace it with 'general local transition-kernel model with a fixed pre-measurement ontic state space,' and we will add an explicit remark in the full text noting that retrocausal and temporally non-local ontologies fall outside the scope of the result and may constitute additional escape routes not covered by the three identified in the abstract. revision: yes