Stefano Gogioso and Nicola Pinzani (2021), in M. Backens and C. Heunen (eds), Quantum Physics and Logic (QPL 2021), EPTCS 343, 301–324 — ARXIV-2103.13771 (withdrawn, v3)
Key takeaways
- Scenarios. A definite causal scenario is a finite poset Ω of events, with finite input and output sets at each event (Definition 1). The discrete order is exactly the Abramsky–Brandenburger non-locality scenario (Definition 2, Proposition 9).
- Causal sections. Sections over a family of input sets indexed by a lower set are causal functions: the output at ω depends only on inputs at events ≤ ω (Definitions 6–7). Explicitly, a product over ω of functions from the inputs in ω’s down-set to O_ω (Eq. 21; worked for a four-event diamond in Section 5).
- Empirical models are causal conditional distributions (Proposition 15). For every lower set λ, the marginal on λ does not depend on inputs outside λ. The probabilistic ones form a polytope: a product of simplices cut by the linear causality equations (Proposition 16).
- Locality (Definition 18, Proposition 21). A model is local if and only if it is a mixture, with weights in the semiring R, of deterministic causal functions: a classical hidden-variable model in which outputs depend only on past inputs. The quantum diamond example (Section 8) is presented as a model, not analysed for locality.
- Indefinite causality, sketched (Section 9). Pre-orders replace partial orders. For the Baumeler–Feix–Wolf model, an “Ω-causal fraction” is 0 for every partial order and 1 for four pre-orders. Results are stated, with proofs deferred to a later paper.
- Withdrawn. The authors withdrew it on arXiv in April 2024. Its Definition 3 should be replaced by the “spaces of input histories” of arXiv 2206.08911 (LIT-800).
Standing in the record
Filed on 2026-10-09 from the manuscript bibliography of 2026-10-09 (work
what-survives-translation) at the owner’s request: a work the manuscript
considered and dropped from its final reference list. See the curation entry
of that day.
Read on 2026-10-09 (NOTE-629) and set to Rejected. The reason was not
the reading’s judgement of its idea. The authors withdrew the paper and
named its successors, and the reading found the defect they name.
Later the same day the three successors were filed and read, and this entry
became Superseded by them:
- The Combinatorics of Causality (LIT-800, arXiv 2206.08911) replaces the locale of inputs with spaces of input histories.
- The Topology of Causality (LIT-808, 2303.07148) redoes this paper’s sheaf of causal functions over the lowerset topology of those spaces. That topology is a genuine locale. It shows that the sheaf property holds for every space induced by a causal order, so the definite case of this paper survives, but fails for some dynamical structures (THEORY-171).
- The Geometry of Causality (LIT-788, 2303.09017) proves the polytope of Proposition 16 in general and computes the BFW causal fractions this paper’s Section 9 deferred.
A reader who wants the sheaf-theoretic treatment of causal order should go to LIT-808 first. None of the three cites this paper.
It extends LIT-016’s framework: on the discrete order it reduces to it
exactly (Proposition 9). The vocabulary has no topic for causal structure
or causal order. It is tagged contextuality first because its
contribution is to the sheaf-theoretic contextuality framework.