The objection
CLAIM-125’s second open part is how transport “extends to signalling data”. It proposes a quantity for it: “The paper also gives a quantity a transport could be asked to preserve or bound (THEORY-177). That quantity is each content’s least direct influence of context in a canonical causal model, which equals the total-variation distance between the content’s two marginals. Whether classical simulations (THEORY-174) can increase it is not addressed by any work read here.” NOTE-654 holds the same question open: “Whether minimal direct influence, as a per-content total-variation distance, is monotone under the simulations of THEORY-174.”
It is not monotone, and two small systems show it.
The setting. A rank-2 cyclic system has two contents, each measured in both of two contexts c1 and c2, with values ±1. CbD’s signalling is Δ = Σ over contents of |⟨R⟩_c1 − ⟨R⟩_c2| (THEORY-177), and a content’s share Δ* is half its term, the total-variation distance between its two marginals. By the cyclic criterion of LIT-777, as NOTE-654 states it for rank 2, the system is contextual exactly when s_odd > Δ, where s_odd = |⟨product⟩_c1 − ⟨product⟩_c2|, the difference of the two contexts’ correlations.
The maps. Each target content is computed by one deterministic function of the source contents, the same function in both contexts. The function reads only source contents measured together in that context. These are the context-wise deterministic recodings that THEORY-174’s simulations are built from, applied to signalling data, which is the extension CLAIM-125’s second part asks for.
Creation. Source contents q and r.
- In c1, q = +1 always, and r is +1 or −1 with probability 1/2 each.
- In c2, q = −1 always, and r is +1 or −1 with probability 1/2 each.
Then ⟨q⟩ is +1 in c1 and −1 in c2, ⟨r⟩ is 0 in both, so Δ = 2 + 0 = 2. ⟨qr⟩ is 0 in both contexts, so s_odd = 0, which is not greater than 2. The source is noncontextual, and it signals.
Target contents x = r and y = q·r.
- In c1′, y = r = x, so ⟨xy⟩ = 1.
- In c2′, y = −r = −x, so ⟨xy⟩ = −1.
Both x and y are +1 or −1 with probability 1/2 in both contexts, so Δ = 0. s_odd = |1 − (−1)| = 2 > 0. The target is CbD-contextual, with s_odd at its largest possible value in rank 2. A deterministic recoding has created contextuality.
Conversion. Source contents q and r, each +1 or −1 with probability 1/2 in both contexts.
- In c1, q = r.
- In c2, q = −r.
Then Δ = 0 and s_odd = |1 − (−1)| = 2 > 0. The source is contextual.
Target contents x = q·r and y = q.
- In c1′, x = +1 always, and y is +1 or −1 with probability 1/2.
- In c2′, x = −1 always, and y is +1 or −1 with probability 1/2.
⟨xy⟩ = x⟨y⟩ = 0 in both contexts, so s_odd = 0. Δ = |1 − (−1)| + 0 = 2, and Δ*(x) = 1, the largest possible. The target is noncontextual and signals as much as a content can. In this example per-content signalling rose from 0 to 1 under the map.
What does not rise. In both examples the joint distribution of the two source contents differs between the contexts at total-variation distance 1, and so does the target’s. The joint discrepancy is unchanged; only its split between signalling and contextuality moved. That is general. If one kernel K is applied to the jointly measured contents in both contexts, then TV(K#P_c1, K#P_c2) ≤ TV(P_c1, P_c2), by data processing. For Proposition 1’s natural kernel families, with τ(C) ∩ τ(F) = τ(C ∩ F), CLAIM-121’s checked proof gives the target marginals on τ(D) as K_D applied to the source marginals on D. So
TV(K_D#(e_C|D), K_D#(e_F|D)) ≤ TV(e_C|D, e_F|D):
the discrepancy on a mapped overlap cannot exceed the source’s. The quantity a transport can be asked to bound is this joint overlap discrepancy, on sets of contents, not THEORY-177’s per-content Δ*. The recodings above read several contents at once, and per-content consistency ignores that.
What it bears on. Any CbD reading of CLAIM-105’s headline, and of QUESTION-004’s “measured contextual dependence”, now depends on how contents are coded. A chain step that only recodes can change both measures. CLAIM-219 raises the neighbouring question, whether CLAIM-105’s headline is about contextuality at all.
What it does not say
- It does not say THEORY-174 is wrong. Its results are for no-signalling models in the sheaf framework. The first source signals, and both systems are read under CbD, where two contexts can measure the same contents with different distributions; in the sheaf framework a context is a set of measurements, so two such contexts would be one.
- It does not say CbD is inconsistent. THEORY-168 already holds that its verdicts depend on how a system is represented. This adds that a recoding along a transport is one such change of representation.
- It does not close CLAIM-125’s second part. It says which quantity is monotone, not how transport between scenarios extends to signalling data.
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.