The claim
A93 §1: “Research opportunity: Extend these structures to directed transport between observational scenarios, including changing covers and stochastic maps between local outcome spaces. That would make the theory about more than detecting contextuality within a fixed scenario. It would let us investigate how compatibility structures evolve during translation and repeated reconstruction.” A94, bridging Fritz, Blackwell and Abramsky–Brandenburger: “Under what conditions can a directed stochastic transport preserve the decision-relevant information of a context-indexed empirical model while changing its observational cover or global compatibility structure?”, “one of the central mathematical problems”.
Where it stands
The manuscript’s Propositions 1 and 2 (CLAIM-121, CLAIM-070) cover transports that commute with restriction or come from one global kernel. §6: “some postprocessing classes preserve noncontextuality, whereas cover-changing or incompatible maps demand separate analysis.” Fritz (LIT-772), the bridge’s first term, is in the references and uncited. The owner’s answer is CLAIM-001.
C7 Appendix A, omitted from the extracted manuscript, comes closest: “If mapped overlaps are larger than images of source overlaps, the stated conditions must cover those extra target variables too; this is why the geometric assumptions on τ matter”, and “measure departures from this class rather than treating arbitrary changes of measurement cover as intrinsic contextuality”. The crystallized argument’s §16 qualified the owner’s diffusion answer (CLAIM-001): “probabilistically combining conditional scores is not the same thing as proving sheaf-theoretic gluing”. It is the first of its five questions (QUESTION-022).
Partial prior art
Gogioso and Pinzani’s Topology and Geometry of causality (LIT-808, LIT-788) define empirical models on any open cover. They give a lattice of covers with restriction to finer ones, and a hierarchy of spaces whose polytopes nest. All of that is within one family of spaces, not a directed stochastic transport between scenarios, so the problem stays open.
Answered in part by the simulations literature
Simulations between empirical models already define directed, stochastic transport between scenarios whose covers change:
- Karvonen 2018 (LIT-846);
- Abramsky, Barbosa, Karvonen and Mansfield 2019 (LIT-847);
- Barbosa, Karvonen and Mansfield, “Closing Bell” (LIT-845).
Each target measurement is mapped to a jointly measurable set of source measurements. These works prove that such transport preserves noncontextuality and cannot raise the noncontextual fraction (THEORY-174). Closing Bell’s Theorem 44, as corrected in arXiv v2, characterises exactly which maps are classical transports. Locally compatible pieces that fail to glue are “contextual simulations”.
That meets the global-compatibility half of this claim’s defeat condition. It does not meet the half about decision-relevant information, and it does not cover signalling data. The claim should be narrowed to those, and these works listed as prior art ahead of Gogioso and Pinzani.
Superseded
Narrowed on 2026-10-09 by CLAIM-125, at the owner’s request. The new claim holds open only the two parts the simulations literature leaves: when a cover-changing transport preserves decision-relevant information, and how transport extends to signalling data. It lists the simulations works as prior art ahead of Gogioso and Pinzani.