The question
CLAIM-009 supposes that pragmatic judgements may be formally contextual. CLAIM-037 reads that as a failure of global extension over a cover of measurement contexts. That is the Kochen–Specker and sheaf notion. There is a second notion. Spekkens’s generalized noncontextuality also constrains preparations. Operationally equivalent preparations must get the same representation. The question is which notion the record’s test is aimed at, and what each would show if it failed.
Why the record needs it
The mechanism is direct, not an analogy. Renderings and contexts are preparations, and probes are measurements. CLAIM-142 and CLAIM-046 already define observational equivalence over a probe family. Spekkens’s notion is built on a relation of that kind, taken over every measurement the theory allows (LIT-266, Eq. 1), so a probe-relative equivalence is his relation only under a condition stated in the note of 2026-10-14 below. An ambiguous utterance read as a mixture of readings in several ways is a mixed preparation with several decompositions (CLAIM-044). This is where preparation contextuality can show with two binary judgements never made together (LIT-980, THEORY-315). A sheaf test cannot see it, because it needs a cover with overlapping contexts. QUESTION-001 asks what fails when reconstruction is impossible. The answer differs with the notion.
What the record holds
- THEORY-011: the generalized notion is strictly broader. “a qubit already has no preparation-noncontextual model, though no Kochen–Specker argument with sharp measurements exists in dimension two”. Limit: the proof assumes exact operational equivalence and perfectly distinguishable pairs (NOTE-748).
- THEORY-015: every Kochen–Specker proof refutes universal noncontextuality, while “measurement noncontextuality alone is consistent with quantum theory”. So a sheaf result would carry over, given the full preparation set. Limit: the record makes the join to the sheaf form itself, fragments must be re-checked, and what the old proofs refute is preparation plus measurement noncontextuality, not preparation noncontextuality alone (NOTE-748).
- THEORY-016: a noncontextual model exists “exactly when its GPT admits a positive quasiprobability representation”. That gives the test a concrete target. Limit: it gives “no test for whether a positive representation exists”.
- THEORY-231: within classical realism, a generalized-noncontextuality no-go refutes Leibnizian representations, not realist ones, and applies where no Bell-like no-go exists, as in prepare-measure scenarios. Limit: the reading is Skimmed, and only Leibnizian implies noncontextual is proved.
- THEORY-559: read as random variables, a Kochen–Specker contradiction refutes the context-independent identification of measurements, not a joint distribution. Limit: it does not engage the sheaf form, with which it agrees under consistent connectedness.
- THEORY-315: for orbit-realizable states, generalized noncontextuality bounds two binary predictabilities by |⟨X⟩| + |⟨Z⟩| ≤ 1, with no overlapping contexts. Limit: it assumes exact equivalence, and its noise robustness is only asserted.
What would answer it
A statement of the operational equivalences among renderings that a pragmatic experiment would assume. With it, a derivation of the noncontextual inequality those equivalences imply, separate from the sheaf criterion. Or an argument that the probe families the record can field never make two preparations operationally equivalent, so that only the sheaf notion applies.
Note of 2026-10-12: the notion the record’s tests use
“What the record holds” above lists THEORY-011, THEORY-015 and THEORY-016. PR #121’s round of 2026-10-10 restated CLAIM-105’s condition in the Contextuality-by-Default sense. CASE-053 reads the published tests on human judgements under CbD and Bell-type criteria, and finds that “No contextual pragmatic data are documented”. CLAIM-201 concedes the fixed-scenario headline on that finding. CbD is a measurement-context notion built for systems that signal, and this question does not name it among the notions it compares. That null does not reach Spekkens’s notion, since none of the tests read looks for preparation contextuality.
Note of 2026-10-14: conditions from the readings
Spekkens 2005 (LIT-266, NOTE-748) and Schmid, Selby and Spekkens (LIT-945, NOTE-737) were read in full. Both support the question’s premise: preparation contextuality can show where there are no overlapping measurement contexts, and for a prepare-measure structure there is no Bell-like test while a noncontextuality test can exist. They also put four conditions on any test the question has in view. These do not answer it.
- Equivalence over every probe. Spekkens’s operational equivalence quantifies over every measurement the theory allows (LIT-266, Eq. 1), and LIT-945’s inferential equivalence over every tester. The probe-relative equivalence of CLAIM-142 and CLAIM-046 is that relation only if the probe family is tomographically complete for the renderings. Otherwise two renderings the probes cannot separate may still be operationally inequivalent, and the assumption does not apply.
- The unquotiented description. A model that already identifies equivalent renderings “is not the sort of thing that can be either generalized-noncontextual or generalized-contextual” (LIT-945). The test must keep distinct renderings distinct and state their equivalences separately, which is the form “What would answer it” asks for.
- Exact equivalence. The definitions apply only to exactly equivalent procedures. LIT-266 §IX concedes that finite precision “might imply that in practice no two experimental procedures are found to be operationally equivalent”, and leaves the replacement, similar representations for similar procedures, without a measure.
- Perfectly distinguishable decompositions. The §IV proof needs the two members of each two-element decomposition of the mixed preparation to be perfectly distinguishable. Readings of an ambiguous utterance (CLAIM-044) that are only probably distinguishable do not give its constraints. A pragmatic version needs such pairs or a noise-robust inequality, which neither reading supplies. Added later on 2026-10-14 (NOTE-825): Catani et al. (LIT-980, THEORY-315) supply a test with no distinguishability condition: four preparations with equal predictabilities and flipped signs on two binary probes, plus one equivalence of equal mixtures, give |⟨X⟩| + |⟨Z⟩| ≤ 1. It still needs exact equivalence over every probe, and its noise robustness is only asserted.
Split, 2026-10-21 (ADR-049). The readings settle what each failure would show: a Kochen–Specker proof refutes universal generalized noncontextuality, a generalized no-go refutes a positive quasiprobability representation, and the generalized notion is strictly broader. That is QUESTION-246, Answered, and THEORY-011, THEORY-015, THEORY-016, THEORY-231 and THEORY-559 now answer the child. This question keeps what they leave: whether the probe families a pragmatic test can field make two renderings operationally equivalent, which the note of 2026-10-14 says needs tomographic completeness, exact equivalence, and either perfectly distinguishable pairs or THEORY-315’s witness.