The claim
A84 proposed three anchoring propositions; this is the first. C6 Proposition B1: “For finite scenarios with a compatible stochastic transport K_C, if the source empirical laws are overlap-consistent, then the transported family is overlap-consistent. Proof: restrict a transported context distribution; use kernel restriction-compatibility to exchange pushforward and restriction; then use source overlap consistency; repeat from the second context.” Manuscript §6, Proposition 1, with the same sketch.
What it does not say
C6: “This proposition concerns overlap consistency, not preservation of noncontextuality under all imaginable maps.” The manuscript: “not automatically global noncontextuality”.
What was checked
C7 Appendix A’s proof, read on 2026-10-09: “For any two contexts C,F with overlap D, the target marginal from K_C#e_C on τ(D) equals K_D#(ρ^C_D#e_C). Source consistency makes the latter K_D#e_D, which also equals the corresponding target marginal from F.” Each step holds: the first is the naturality assumption (marginalizing after K_C equals K_D after marginalizing), the second is source overlap consistency, and the third is the same argument from F. Two conditions are needed and both are stated in C7. Kernels K_D must exist for each overlap D, which “for every inclusion D⊆C” supplies if the family is closed under overlaps. The conclusion is consistency on τ(D), which is the mapped overlap only when τ(C) ∩ τ(F) = τ(C ∩ F). Appendix A says so itself: “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.” The manuscript’s “consistent on the mapped overlaps” is right under that reading. Active.
Prior art
This proposition is a special case of Karvonen’s “Categories of Empirical Models” (LIT-846). His natural transformation σ is exactly the family of local kernels commuting with restriction, and his Lemma 3.12 states the limit that local kernels need not glue. The record’s reader drew this mapping; Karvonen does not mention translation. The manuscript should cite it, and this claim now names it in its grounds, with the THEORY that states the general result (THEORY-174): classical simulations between scenarios, cover-changing ones included, never create contextuality. The open remainder is CLAIM-125.
Note of 2026-10-27: grounded in the proved half
The status review of this date (ADR-057) moved this claim’s ground from THEORY-174 to THEORY-720. The claim uses the classical simulations and their monotonicity, that they never create contextuality, which is proved in refereed venues and Active as THEORY-720. It does not use THEORY-174’s characterization, that a map is such a simulation exactly when a non-contextual model of the hom scenario induces it. That half waits on an independent proof of the corrected Theorem 44 and stays Proposed.