The claim
A81 §3.4: “Show how ambiguous communicative meaning can correspond to multiple compatible assignments, while incompatible constraints produce a different phenomenon.” A82 §8.7, for chains: “Distinguish increased ambiguity from an increase in formal contextuality.” A78 had said the theory differs from “merely saying that meanings are fuzzy, ambiguous, or context-dependent” (CLAIM-038); this says how.
Where it went
C6 has no occurrence of “ambigu”. The manuscript separates ordinary context dependence from contextuality (CLAIM-037), which is a different contrast. It treats an object as “generally a region, not a point” (§2, from A75), which is ambiguity, and gives the parity case (CASE-035), which is incompatibility. It never says that the two are opposite failures of a unique global reading. That contrast is what would let a chain be described as becoming more ambiguous without becoming more contextual (CLAIM-105).
Sharpened by Abramsky and Sadrzadeh
Abramsky and Sadrzadeh’s Semantic Unification (LIT-841, NOTE-644) treats ambiguity as a choice among several admissible covers. Each cover glues at most once. A failure to glue is inconsistency, a third thing beside ambiguity and contextuality.
So in that framework, “several compatible global assignments” means several covers, not several sections over one cover. The paper models ambiguity and context dependence only, not contextuality in the formal sense.
A qualification from the language data
Wang, Sadrzadeh, Abramsky and Cervantes (LIT-842, NOTE-645) find that ambiguity alone never yields contextuality in their phrase data. But their one contextual model, tap/box, is of Hardy type, and it still has two global assignments. So “a contextual one admits none” holds only for strong contextuality. For weaker contextuality the line between the two phenomena runs through the probabilities, not through whether any global assignment exists.
Note of 2026-10-10: what “admits none” means
“Admits none” means none consistent with the empirical supports or distributions (TERM-042). Assignments of the event sheaf always exist (CLAIM-131). An ambiguous utterance’s supports admit several global assignments. A contextual one admits none in the support (strong contextuality), or its compatible distributions have no global joint (CASE-037).
Note of 2026-10-21: a third way not to have one global account
A10 of the exchange of 2026-10-20 (§7) tabulated three ways local descriptions can fail to yield one global account. The first is “Noncompositional closure: a joint representation exists but loses a property possessed by its components” (THEORY-820). The second is “Observational nonidentifiability: multiple underlying structures share the same observable profile”. The third is a “causal-contextuality obstruction”, in which local causal assignments do not glue (THEORY-171). The second and third are this claim’s ambiguity and contextuality: several compatible global assignments, or none. CLAIM-144 adds the case of coordinates that differ without any failure to glue. The first is a third kind: the global description exists, and a property fails to survive the join (CLAIM-219, note of 2026-10-21; CLAIM-409). It recurs in A12 (Batch 5) and A14 (PR 6). Recorded here, not filed.
Note of 2026-10-30: what holds it back
Its body reads “admits none” two ways, and the same case fails the title under each. Read over supports, Wang et al.’s tap/box model (LIT-842, NOTE-645, THEORY-623) is contextual of Hardy type and still has two global assignments, so a contextual utterance can admit some. Read over distributions, as the note of 2026-10-10 does, that ambiguous phrase admits no compatible global distribution, so an ambiguous utterance need not admit several. It waits on a restatement, which is the owner’s: a merely ambiguous model is a mixture of several global assignments, a contextual one a mixture of none, and one utterance can be both.
By Fine’s theorem (Abramsky–Brandenburger), a model with no global extension is never a mixture of global assignments over the same observables, so the defeat condition cannot be met; CLAIM-238 already lists “Every pattern” as an unmeetable universal negative.