Source
Karvonen, Categories of Empirical Models (LIT-846), Definition 3.9 and Theorems 4.1, 4.3, 4.5 and 4.8, as read in NOTE-651. Abramsky, Barbosa, Karvonen and Mansfield, A comonadic view of simulation and quantum resources (LIT-847), Proposition 7 and Theorems 17, 20–22, as read in NOTE-647. Barbosa, Karvonen and Mansfield, Closing Bell (LIT-845, arXiv v2), Theorems 29, 37, 40 and 44 and §4.5, as read in NOTE-650.
What was actually shown
Take two finite measurement scenarios S and T, with possibly unrelated covers. A classical procedure from S to T answers each measurement x of T by measuring a set π(x) of S’s measurements and post-processing the outcomes. π must send every context of T into a context of S, so the agent never attempts an incompatible set. Shared classical randomness is allowed, as a stochastic outcome map (Karvonen), as an auxiliary non-contextual model (the comonadic paper, which also allows adaptive choices), or as a mixture of deterministic procedures (Closing Bell). Every no-signalling empirical model on S is then pushed forward to one on T.
Three results hold across all three formulations. First, a model is non-contextual exactly when it can be simulated from the model on the empty scenario (LIT-846 Theorem 4.1; LIT-845 Theorem 29, which extends this to logical and strong contextuality). Second, if d simulates e then NCF(d) ≤ NCF(e): the contextual fraction cannot increase (Theorem 4.5; Theorem 21), and strong contextuality is reflected backwards (Theorem 4.3). Third, a model can be used to simulate two independent copies of itself only if it is non-contextual (Theorem 4.8; Theorem 22). The comonadic paper adds that adaptive simulation is the same preorder as convertibility by the free operations, which include translation of measurements along any simplicial map and conditional measurement (Theorem 20).
Closing Bell answers the converse question for non-adaptive procedures. A map F from S’s models to T’s models preserves convex combinations if it is a simulation (Lemma 24), and is then fixed by its values on deterministic models (Theorem 37). It is induced by a procedure exactly when F = F_e for some non-contextual model e of the hom scenario [S, T] (T’s contexts, with procedures into S as outcomes), satisfying the predicate that each context uses a context of S (Theorem 44, as corrected in v2). Deciding this is a linear program. A family of local procedures that agree on overlaps but do not glue into one global procedure is a contextual model of [S, T].
What could have come out otherwise: a cover-changing map built from locally classical pieces could have produced a global inconsistency the source lacked. It cannot when the pieces form one procedure, and Closing Bell shows that whether they do is itself a contextuality question.
What this does not say
- Nothing about signalling data. Every result assumes no-signalling; Karvonen’s pushforward is undefined without it (Remark 3.2). Contextuality-by-Default systems and corpus or language-model data (LIT-842, LIT-844) are outside its reach.
- Nothing about information preserved. The simulation preorder orders resources by contextuality. It is not shown to relate to Blackwell’s informativeness order (THEORY-156), and no result says when a transport preserves what a decision-maker could learn from the source.
- Not that contextual transports are understood. “Contextual simulations”, models of [S, T] that do not glue, are named in LIT-845 (§6.7) and not studied; this theory says only that such a family is not a classical procedure; the map it induces may still be one (note of 2026-10-29).
- Not the adaptive characterisation. Theorem 40 fails to transfer to adaptive procedures because its key lemma fails (§6.2); the adaptive form of Theorem 44 is asserted without proof. It is THEORY-1170, Proposed.
- Not a statement about language or translation. The use made of it for the manuscript’s transport problem (CLAIM-100) is the readers’, in the NOTEs, and not the authors’.
Note of 2026-10-19: the monotonicity half filed apart
The three results that “hold across all three formulations” above are now THEORY-720, Active, sourced to LIT-846 and LIT-847, whose proofs NOTE-651 and NOTE-647 followed. Its no-cloning result cites LIT-847 Theorem 22 rather than LIT-846 Theorem 4.8, which rests on a monoidal structure the paper asserts without checking. What keeps this THEORY Proposed is the characterisation: which maps between model sets are classical procedures.
Note of 2026-10-29: Theorem 44 re-derived
The corrected statement follows from the definitions with Corollary 43, in two short steps. Write a deterministic procedure f = (π, α) as a global assignment x ↦ ⟨π(x), α_x⟩ of the hom scenario [S, T]. It satisfies g_{S,T} on every context, because π sends each context of T into a context of S. For a model e on [S, T] that satisfies g, set F_e(d)_σ(t) = Σ_a e_σ(a) · d|_τ{s : α_x(s|U_x) = t_x for x ∈ σ}, with τ any context of S containing the U_x. No-signalling makes this independent of τ.
- If F = Σ_k p_k EMP(f_k), let e be the marginals of p on the contexts of T. It is non-contextual by construction, and its supports satisfy g. The summand for a context depends only on f_k’s restriction to it, so F_e = F.
- If e is non-contextual, some distribution q on global assignments of [S, T] has e’s marginals. Each global assignment in q’s support restricts on each context into e’s support, which satisfies g, so each is a deterministic procedure, and F_e = Σ_f q(f) EMP(f).
Neither step uses convexity or Theorem 37. They enter only in deciding the condition for a map given as a black box: Theorem 37 fixes a convex map by its values on deterministic models, and the condition becomes a linear program over deterministic procedures.
The map e ↦ F_e is not injective, so the theorem’s “some” matters. A contextual model of [S, T] can induce a classical map, for instance one that writes the constant 0 as two different local procedures in a strongly contextual pattern. So a contextual simulation in §6.7’s sense is a family of local procedures that does not glue, not necessarily a map that no classical procedure induces.
A script run at the review checked this on the binary triangle as both S and T: 52,472 deterministic procedures, 12,152 distinct maps. F_p and F_{e_p} agreed exactly on 20 random procedures and ten no-signalling models, the strongly contextual one included. The linear program recognised random classical maps (residual below 10⁻¹³), and the procedure it recovered reproduced the map on contextual models, which is Theorem 37 at work. Two contextual hom models whose maps send every model to the contextual triangle model were rejected (residual 1.6), and the non-injective case was accepted. As a control on the well-definedness check, a signalling source made the result depend on the choice of τ.