Source
Schmid, Selby and Spekkens (2020), LIT-945, arXiv:2009.03297v3, read in part in NOTE-737, whose status is Skimmed. Read closely: §V (inferential equivalence and quotienting), §VI (classical realist representations, Leibnizianity, Definition VI.5, Proposition VI.6), §VIII (generalized noncontextuality, Definition VIII.2, Proposition VIII.3, Eqs. 214–215, §VIII.C), §IX and Appendices A–B. Not read: §I.A, §§II–IV, where the theories are constructed, and the proofs in Appendices C–G. Spekkens (2005), LIT-266, read in full in NOTE-748, for the definition rehabilitated here.
What was actually shown
Two maps, one demand. A classical realist representation ξ takes the unquotiented operational theory into the unquotiented classical realist theory. It is Leibnizian if it maps inferentially equivalent processes to inferentially equivalent ones (Definition VI.5), a formalisation of an epistemological generalisation of Leibniz’s methodological principle (§VI.C). A generalized-noncontextual representation ζ makes the same preservation demand of a map into the quotiented realist theory (Definition VIII.2), the counterpart of the ontological models of LIT-266.
The chain (Eq. 215). ∃ Leibnizian ξ ⇒ ∃ generalized-noncontextual ζ ⇔ ∃ representation ξ̃ of the quotiented operational theory (Proposition VI.6, Eq. 200; Proposition VIII.3, Eq. 214). So “Every no-go theorem for generalized-noncontextual classical realist representations yields a no-go theorem for Leibnizian classical realist representations.” The converse, that a no-go for Leibnizian representations yields one for noncontextual ones, is Conjecture 1′. If it fails, Leibnizianity alone may yield new no-go theorems.
What the no-go leaves. The conventional exit from a noncontextuality no-go “requires a renouncement of Leibnizianity”, not of classical realism (§VIII.D). An experiment that supports a noncontextuality no-go and no Bell-like one is always consistent with some non-Leibnizian ξ. The authors’ own exit is not to give up Leibnizianity but to seek a realist theory with nonclassical causation and inference, specified by five criteria and not constructed (§IX.A).
Against Bell-like no-goes (§VIII.C). A Bell-like no-go denies every classical realist representation consistent with a causal structure: “Bell-like no-go theorems are based on a weaker assumption.” Noncontextuality no-goes “can be proven for a broader set of causal-inferential structures—even those involving just a single causal system”. For a prepare-measure structure, “the only restrictions on F-S-realizable distributions over X, Y, A and B are that Y must be independent of A and X”, so no Bell-like no-go exists there, while noncontextuality no-goes do.
A condition on the question asked. A quotiented theory is neither contextual nor noncontextual: “it is a category mistake to ask whether ξ̃ is generalized-noncontextual or not.” The question can be put only of an unquotiented description in which equivalent procedures stay distinct.
What this does not say
- It rests on a partial reading. The NOTE is Skimmed. The propositions of the chain were read as stated; their proofs, and the construction of the causal-inferential theories they are about (§§II–IV), were not. The prepare-measure realizability claim is stated without a derivation in the text read.
- It does not say that refuting noncontextuality refutes Leibnizianity in the other direction. Only Leibnizian ⇒ noncontextual is proved. A failure of Leibnizianity is not shown to produce a noncontextuality no-go.
- It does not say that a noncontextuality no-go refutes realism. Within classical realism it refutes Leibnizian representations. Denying every classical realist representation is the stronger, Bell-like conclusion, which needs a causal structure that a single system does not supply.
- It does not say Leibnizianity must be given up. The authors argue for a nonclassical realist theory instead, and build none. Whether one exists that represents quantum theory Leibnizianly is open.
- It does not treat the Kochen–Specker or sheaf notions. The relation of those to generalized noncontextuality is THEORY-015’s, from LIT-266, and the join to the sheaf form is the record’s own.
- It does not say that LIT-266 appealed to Leibniz. LIT-266 argues from ontological economy. The Leibnizian grounding is this paper’s, and needs, in the authors’ words, an “epistemological generalization” of the principle.
Connections
- QUESTION-067 (answers, in part). It says what refuting Spekkens’s notion would commit the record to: within classical realism, giving up Leibnizianity, not realism. It puts two conditions on any pragmatic test. Inferential equivalence quantifies over every tester, so a probe-relative equivalence (CLAIM-142, CLAIM-046) is this relation only if the probe family is tomographically complete for the renderings. And the test must use the unquotiented description, keeping distinct renderings distinct and stating their equivalences separately. That prepare-measure structures admit noncontextuality no-goes and no Bell-like ones supports the question’s premise that preparation contextuality can show where there are no overlapping measurement contexts. It does not support a single binary judgement: LIT-980’s witness needs two binary judgements never made together and four preparations (NOTE-825).
- THEORY-011. That account’s choice-of-notion caveat cites this paper for the Leibnizian principle. This account states what the principle does in the no-go: it is the assumption a noncontextuality refutation removes.
- THEORY-015. Compatible and about a different step. THEORY-015 places Kochen–Specker contextuality inside generalized noncontextuality; this account places generalized noncontextuality relative to Leibnizianity and to classical realism.
- THEORY-016. The paper restates, citing LIT-003, that a generalized-noncontextual representation exists exactly when a positive quasiprobability representation does. A restatement, not an independent proof; nothing here extends that account.
- QUESTION-001. Bears through QUESTION-067, which refines it.
Note of 2026-10-19: the chain re-derived
From the promotion audit (batch 026, ADR-047 v2). The definitions are as stated above from §§V, VI and VIII. The construction of the theories in §§II–IV was not read.
- (D1) Inferential equivalence ∼ is a congruence, preserved by composition. So quotienting by it is a composition-preserving map q.
- (D2) A classical realist representation ξ is Leibnizian when P ∼ P′ implies ξ(P) ∼ ξ(P′).
- (D3) ζ into the quotiented realist theory is generalized-noncontextual when P ∼ P′ implies ζ(P) = ζ(P′).
- (D4) A Bell-like no-go for a causal structure denies every classical realist representation consistent with it.
- Leibnizian implies noncontextual. Take ζ = q ∘ ξ. It preserves composition. If P ∼ P′, then ξ(P) ∼ ξ(P′), so q(ξ(P)) = q(ξ(P′)). Hence every noncontextuality no-go is a no-go for Leibnizian representations. The converse would need every noncontextual ζ to factor through a Leibnizian ξ. That is Conjecture 1′, and it is not shown.
- Realism survives. Suppose an experiment admits a noncontextuality no-go and no Bell-like one. Then some classical realist ξ consistent with its causal structure exists, and by step 1 it is not Leibnizian.
- No Bell-like no-go in prepare-measure. Take preparation setting X
and outcome A, a system Λ, an exogenous measurement setting Y and
outcome B.
- Y is exogenous, so p(x, a, y) = p(x, a) p(y) is necessary.
- It is also sufficient. Send Λ = (x, a), with a drawn from p(a | x), and draw b from p(b | x, a, y). The joint is then exactly p.
- No statistics violate this constraint, so no Bell-like no-go exists.
- Preparation-noncontextuality no-goes do exist there, for a single system (LIT-266, NOTE-748).
- The stronger assumption. A noncontextuality no-go rules out only Leibnizian-or-noncontextual representations, a smaller class than every classical realist representation consistent with the causal structure. So it assumes more, and reaches where a Bell-like no-go cannot.
If §§II–IV define ∼ so that it is not a congruence, step 1 fails. The THEORY’s account of §V gives no sign of that.