NOTE-737: Schmid, Selby & Spekkens — Unscrambling the omelette of causation and inference

Contribution

The paper builds causal-inferential (CI) theories. Each is a causal process theory (laboratory procedures, or functions between classical variables) joined to a classical inferential theory (Bayesian probability and Boolean logic), with rules for how the two interact and a prediction map. In this frame it restates Bell-like and noncontextuality no-go theorems as statements about which realist representations exist. It introduces Leibnizianity, the preservation of inferential equivalence, and proves that it implies the authors’ rehabilitated generalized noncontextuality. It then sets out what a nonclassical realist theory would have to satisfy to escape both kinds of theorem.

Key insight

Two processes are inferentially equivalent when they license the same predictions in every scenario. Quotienting by that equivalence, which is what GPTs and ontological models do, erases causal information. The paper’s example is a coin-flip mixture of the two constant bit functions and a coin-flip mixture of identity and flip. One has no causal link from input to output, the other has one, and both give the completely randomising channel. Noncontextuality is then a question about the map from unquotiented operational descriptions to realist ones. Does it send inferentially equivalent processes to inferentially equivalent ones? Bell’s theorem asks only whether such a map exists at all.

Assumptions

• Classical inference. The inferential theory is Bayesian probability plus Boolean logic, called “the classical theory of inference”. Alternatives are discussed but not built.
• Diagram preservation. A classical realist representation ξ is a diagram-preserving map from the operational CI theory to the classical realist one, preserving predictions and ignorability (Definition VI.1). The operational diagram is taken to encode one’s hypothesis about the fundamental causal-inferential structure, not the calculational circuit (Appendix B). Appendix B argues this costs no generality: a holist simply draws one system where the circuit has two.
• Inferential equivalence quantifies over every tester in the theory. “Two elements of P-S are inferentially equivalent if and only if they lead to exactly the same predictions, no matter which causally closed diagram they are embedded in” (Definition V.1).
• The realist causal theory is functional dynamics (structural-equation style), with states of knowledge as distributions over functions.
• Finiteness assumptions are mentioned as made, in §X.C, but were not located in the parts read.

Key results

• Inferential equivalence is a congruence (Lemma V.7), so a theory can be quotiented by it (Definitions V.8, V.11). For causally closed processes it reduces to equality of the stochastic maps they predict (Lemma V.3). For classical realist processes it has an intrinsic form, equality of the associated substochastic map (Lemma V.10, proof in Appendix F, not read).
• What quotiented theories are (§V.A.2, §V.B.2). GPTs are argued to be subtheories of quotiented operational CI theories. The proof is deferred to a forthcoming paper. Traditional ontological models are the codomain of a quotiented classical realist CI theory.
• Quotienting scrambles causation and inference (§V.C). σ_c = ½[f_0] + ½[f_1] and σ_d = ½[f_id] + ½[f_flip] “refer to completely distinct causal relations” yet are inferentially equivalent.
• Theorem VI.2. Every classical realist representation is inferentially equivalent to one that maps states of knowledge about procedures to states of knowledge about functions by a stochastic map Ξ (proof in Appendix G, not read). Whether every representation has that form is left open, with an appeal to “autonomy”.
• Leibniz’s methodological principle (§VI.C). “If an ontological theory implies the existence of two scenarios that are empirically indistinguishable in principle but ontologically distinct … then the ontological theory should be rejected and replaced with one relative to which the two scenarios are ontologically identical.” The authors say that “it is necessary to consider an epistemological generalization of Leibniz’s principle” to motivate the rehabilitated notion.
• Definition VI.5 (Leibnizianity). ξ is Leibnizian if D ∼_p E implies ξ(D) ∼_p∗ ξ(E). Proposition VI.6: this holds iff there is a representation ξ̃ of the quotiented operational theory with ξ̃ ∘ ∼_p = ∼_p∗ ∘ ξ. So ∃ Leibnizian ξ ⇒ ∃ ξ̃ (Eq. 200). The converse is Conjecture 1.
• Definition VIII.2 (generalized noncontextuality, rehabilitated). It constrains a different map, ζ, from the unquotiented operational theory to the quotiented realist theory. ζ is generalized-noncontextual if it preserves inferential equivalence: “inferentially equivalent states of knowledge about experimental procedures must be represented by the same stochastic map”. ζ is “the closest counterpart in our framework to the pre-existing notion of an ontological model”. Proposition VIII.3 and Eq. 214: ∃ generalized-noncontextual ζ ⇔ ∃ ξ̃. The notion covers every kind of process, combs and circuit fragments included, “a universal notion from the get-go”. Old no-go proofs carry over, “since all that has changed is how one conceptualizes the mathematics”.
• Eq. 215. ∃ Leibnizian ξ ⇒ ∃ generalized-noncontextual ζ ⇔ ∃ ξ̃. “Every no-go theorem for generalized-noncontextual classical realist representations yields a no-go theorem for Leibnizian classical realist representations.” The converse is Conjecture 1′. If it is false, there may be new no-go theorems from Leibnizianity alone.
• Context, and why Leibnizianity is not “noncontextuality” (§VIII.B). A process’s context is “information about a process which is irrelevant for making predictions”. Under a Leibnizian ξ, an operational process’s context may fix the context of its image, but not the image’s equivalence class. Only for ζ, whose codomain has no contexts, is the demand a context-independence.
• A quotiented theory is neither contextual nor noncontextual. In the authors’ words, “it is a category mistake to ask whether ξ̃ is generalized-noncontextual or not.” An experiment that supports a noncontextuality no-go but no Bell-like one is always consistent with some ξ, given context-dependence. Its quotiented model may admit no representation.
• Prepare-measure (§VIII.C). For a preparation with setting X and outcome A, and a measurement with setting Y and outcome B, “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 there is no Bell-like no-go for this structure, but noncontextuality no-goes exist. Existence of a generalized-noncontextual representation coincides with simplex-embeddability and with a positive quasiprobability representation of the GPT. Both are cited, not proved here.
• The comparison (§VIII.C). “Bell-like no-go theorems are based on a weaker assumption”, namely that some ξ exists. Noncontextuality no-go theorems “can be proven for a broader set of causal-inferential structures—even those involving just a single causal system”.
• Ways out (§VII.B, §VIII.D, §IX). The conventional exits from Bell’s theorem are superluminal causation and superdeterminism. Both are criticised as fine-tuned. The conventional exit from noncontextuality no-goes “requires a renouncement of Leibnizianity”. The proposed exit from both is a nonclassical realist CI theory XF-XS, which must meet five criteria (§IX.A):
  • actual rather than potential causal influence;
  • propositions attach to every system;
  • knowledge propagates through every causal mechanism;
  • a unique prediction map;
  • intrinsic and external inferential equivalence coincide. Leibnizianity is defined for such representations as before (Definition IX.2). No such theory is constructed.

Claims

id	claim	strength	support
C1	Inferential equivalence is a process-theory congruence, so CI theories can be quotiented by it	strong (proof read)	Lemmas V.6–V.7
C2	Quotienting erases causal distinctions	strong	the σ_c / σ_d example, §V.C
C3	A Leibnizian classical realist representation implies a generalized-noncontextual one	strong (follows from definitions as stated)	Proposition VI.6, Proposition VIII.3, Eq. 215
C4	The converse holds	conjecture	Conjecture 1, Conjecture 1′
C5	Every classical realist representation is inferentially equivalent to one built from a stochastic map on states of knowledge	strong (proof, not read)	Theorem VI.2, Appendix G
C6	Prepare-measure structures admit no Bell-like no-go but do admit noncontextuality no-goes	moderate (stated, derivation not shown in text)	§VIII.C
C7	GPTs are subtheories of quotiented operational CI theories	weak here (proof deferred to a forthcoming paper)	§V.A.2
C8	A satisfactory realist account of quantum theory may be nonclassical in its causation and inference	programmatic	§IX, by analogy with nonEuclidean geometry

Concepts

• Causal-inferential theory. A triple of process theories (causal, inferential, and their combination) with maps between them and a prediction map p (Eq. 65).
• Operational CI theory (P-S). Laboratory procedures (PROC) with classical inference (SUBSTOCH). Its processes describe only potential causal influence.
• Classical realist CI theory (F-S). Functions between classical variables (FUNC) with classical inference. Its processes describe actual causal influence, and it has a unique prediction map (Theorem IV.4).
• Inferential equivalence (∼_p). The same predictions in every causally closed embedding. It replaces the older operational equivalence of procedures, because it relates states of knowledge about procedures.
• Context. Information about a process that is irrelevant for prediction, that is, which member of its inferential equivalence class it is.
• Leibnizianity. A realist representation preserves inferential equivalence.
• Generalized noncontextuality (rehabilitated). A representation ζ into the quotiented realist theory preserves inferential equivalence.
• Quotiented theory. Processes taken up to inferential equivalence; GPTs and ontological models are of this kind.

Connections

The definition of generalized noncontextuality it rehabilitates is Spekkens 2005 (LIT-266), cited as [6]. Eq. 214 is presented as the analogue of Proposition 1 of Schmid, Selby, Pusey and Spekkens’s structure theorem (LIT-003, cited as [20]): an operational theory has a generalized-noncontextual model iff its GPT has an ontological model. The statement of Leibniz’s principle is taken from Spekkens 2019 (arXiv:1909.04628, cited as [26], not held), which argues that Leibniz proposed it and Einstein endorsed it. The causal side follows structural equation models (Pearl). The fine-tuning arguments against the conventional exits from Bell’s theorem are Wood and Spekkens’s (cited as [21], not held). The paper does not mention Kochen–Specker or sheaf-theoretic contextuality.

Bearing on the record

• LIT-945’s filing claim is accurate, with the direction stated. The paper “characterizes generalized noncontextuality process-theoretically and derives it from a principle of Leibnizianity” in this sense: existence of a Leibnizian classical realist representation implies existence of a generalized-noncontextual one (Eq. 215). So a noncontextuality no-go refutes Leibnizian representations. Whether a Leibnizianity failure implies a noncontextuality failure is conjectured, not shown.
• QUESTION-067: what refuting Spekkens’s notion commits the record to.
  • Within classical realism, it commits the record to renouncing Leibnizianity, the conventional way out. The alternative is a realist account whose causation or inference is nonclassical (§VIII.D, §IX).
  • It does not show that no classical realist account exists. That is the stronger, Bell-like conclusion, and §VIII.C notes that noncontextuality no-goes rest on the stronger assumption.
• Support for the question’s premise. For a prepare-measure structure, the only constraint on classical realist distributions is that the measurement setting is independent of the preparation’s setting and outcome. So no causal-compatibility (Bell-like) test exists there, while a noncontextuality test can. This matches QUESTION-067’s claim that a single binary judgement can show preparation contextuality. The paper speaks of Bell-like and causal-compatibility constraints, not of sheaf covers. The step to the sheaf notion still rests on LIT-266 and the record’s own join in THEORY-015. Correction (2026-10-14, NOTE-825): the paper shows only that a prepare-measure scenario has no Bell-like test, not that a single binary judgement suffices. Catani et al.’s witness (LIT-980) needs two binary judgements never made together and four preparations.
• Two constraints on any pragmatic test.
  • Inferential equivalence quantifies over every tester in the operational theory. A probe-relative equivalence, as in CLAIM-142 and CLAIM-046, is this relation only when the probe family exhausts the testers, that is, is tomographically complete for the renderings. The paper treats tomographic completeness as a defining feature of GPTs (§V.A.2).
  • The noncontextuality question can be put only of 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”. The test must keep distinct renderings distinct and state their equivalences separately. This is the form QUESTION-067’s “What would answer it” asks for.
• THEORY-016. The paper restates, citing LIT-003 and others, that existence of a generalized-noncontextual representation coincides with simplex-embeddability and a positive quasiprobability representation. It is a restatement, not an independent proof. It agrees with THEORY-016.
• THEORY-011’s “What this does not say”. It ascribes the choice of notion to “Spekkens’s Leibnizian methodological principle”. That wording belongs to this paper (§VI.C, after its [26]), not to LIT-266. LIT-266 argues from “ontological economy”. This paper is the better citation for it. It also says the 2005 notion needs an epistemological generalization of Leibniz’s principle to be motivated in its rehabilitated form.
• No instruction for machine-learning practice; nothing for the anthology.

Limitations

• Only classical inference is built. The nonclassical realist theory that is the paper’s goal is specified by desiderata, not constructed.
• The converse of the Leibnizianity ⇒ noncontextuality implication (Conjecture 1 and 1′) and the uniqueness of the classical normal form are open. Whether all classical realist representations take the form of Eq. 190 is also open.
• The identification of GPTs with quotiented operational CI theories is deferred to a later paper.
• Diagram preservation is harmless only given the authors’ reading of operational diagrams as causal hypotheses (Appendix B). Under the usual calculational reading, it builds in assumptions such as ontic separability.
• The framework describes a single agent’s reasoning (§X.C).
• No noise-robust or experimental treatment of noncontextuality is given here; §VIII.C defers to earlier work.

Open questions

• Does a no-go for Leibnizian representations always yield a no-go for generalized noncontextuality (Conjecture 1′)? A Leibnizianity no-go with a generalized-noncontextual ζ would refute it.
• What axioms for a quantum realist CI theory (QF-QS) meet the five criteria of §IX.A and represent operational quantum theory Leibnizianly?
• For this reading: §II–IV and the proofs in Appendices E–G were not read. A Read note should check Theorem VI.2’s proof and the prepare-measure realizability claim of §VIII.C, which is stated without a derivation in the text.