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QUESTION-246: In quantum theory, what does a failure of Kochen–Specker noncontextuality refute, as against a failure of Spekkens's generalized noncontextuality, and which of the two notions is broader?

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Properties5
StatusAnswered
Tagscontextuality, quantum-foundations
Linepragmatic-transport
RefinesQUESTION-067 — For a test of pragmatic judgements, under what probe families do renderings count as operationally equivalent, so that a test of Spekkens's generalized noncontextuality applies and not only one of the sheaf or Kochen–Specker kind?
Answered byTHEORY-011 — Generalized contextuality is strictly broader than Kochen–Specker contextuality: a qubit already has no preparation-noncontextual model, though no Kochen–Specker argument with sharp measurements exists in dimension two, THEORY-015 — Kochen–Specker noncontextuality is measurement noncontextuality plus outcome determinism for sharp measurements, and preparation noncontextuality implies that determinism, so every Kochen–Specker proof refutes universal generalized noncontextuality while measurement noncontextuality alone is consistent with quantum theory, THEORY-016 — An operational theory admits a generalized-noncontextual model exactly when its GPT admits a positive quasiprobability representation, and for a tomographically local theory any diagram-preserving such model is an exact frame with exactly as many ontic states as the GPT's dimension, THEORY-231 — Within classical realism, a generalized-noncontextuality no-go refutes Leibnizian representations, not realist representations as such; it rests on a stronger assumption than a Bell-like no-go and so applies where none exists, as in prepare-measure scenarios, but only Leibnizianity implying noncontextuality is proved, THEORY-559 — In Kolmogorovian probability, joint distributedness is transitive, so representing a measurement by one random variable in every context it occurs in already puts all measurements of a connected system on one probability space; within that representation a Kochen–Specker or Bell contradiction refutes the context-independent identification of measurements, not the existence of a joint distribution

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Answer

This is the part of QUESTION-067 that asks “what each would show if it failed”, split off on 2026-10-21 (ADR-049). Five Active THEORYs answer it.

  • What a Kochen–Specker failure refutes. THEORY-015, from Spekkens 2005 (LIT-266). Kochen–Specker noncontextuality is measurement noncontextuality plus outcome determinism for sharp measurements, and preparation noncontextuality implies that determinism. So every Kochen–Specker proof refutes universal generalized noncontextuality, that is, preparation and measurement noncontextuality together. Measurement noncontextuality alone is consistent with quantum theory.
  • The same failure, read as random variables. THEORY-559. If each measurement is one random variable in every context, all of them already sit on one probability space. A Kochen–Specker or Bell contradiction then refutes that context-independent identification, not the existence of a joint distribution.
  • What a generalized failure refutes. THEORY-016: a generalized noncontextual model exists exactly when the theory’s GPT has a positive quasiprobability representation, so a failure rules that out. THEORY-231: within classical realism, such a no-go refutes Leibnizian representations, not realist ones as such.
  • Which is broader. The generalized notion, strictly. THEORY-011: a qubit has no preparation-noncontextual model, though no Kochen–Specker argument with sharp measurements exists in dimension two. THEORY-231 adds that a generalized no-go applies where no Bell-like one exists, as in prepare-measure scenarios.

Limits

THEORY-015’s implication uses quantum theory’s full preparation set, and must be re-checked in a fragment. Its statement for the sheaf form is the record’s own join, not a source’s. THEORY-559 notes that the sheaf account and the random-variable one agree under consistent connectedness. THEORY-011’s qubit proof assumes exact operational equivalence and perfectly distinguishable pairs. THEORY-231 rests on a Skimmed reading, and proves only that Leibnizian implies noncontextual.

What it leaves to QUESTION-067

The answer is about the notions. It does not say which one a test of pragmatic judgements would reach. That turns on whether the record’s probe families make two renderings operationally equivalent, which needs them to be tomographically complete for the renderings. THEORY-315’s witness, which needs exact equivalence over every probe, stays with the parent.

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