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LIT-788: The Geometry of Causality

3 min read

Properties5
StatusActive — skimmed 2026-10-09 (NOTE-624); worth reading as the geometric half of the trilogy and the successor of the withdrawn LIT-813's polytope and causal-fraction results: the empirical models on any cover of a space of input histories are exactly the points of a "causaltope", the product of simplices sliced by linear causality equations derived from the space's topology, so supported fractions and causally separable fractions are linear programs. Causal separability is defined relative to an ambient space rather than to the indiscrete one, which lets quantum switches with entangled or contextual control witness indefinite causal order once known no-signalling constraints are imposed (and not otherwise); the causally separable fraction is bounded below by the separable local fraction, and in the switch examples it tracks, and sometimes equals, the local fraction ("contextual causality"). The BFW model is 0% supported by every causally complete 3-event space and 100% by each of three indefinite orders. Read in part: introduction, causaltopes, standard causal separability and the examples.
Tagscausality, contextuality, quantum-foundations, mathematics
SupersedesLIT-813 — The Sheaf-Theoretic Structure of Definite Causality
Cited as “paper” byNOTE-624 — The Geometry of Causality
Cited as “source” byTHEORY-580 — On any cover of a space of input histories, the empirical models compatible with a causal structure are exactly the points of a polytope cut from a product of simplices by linear causality equations, so how much of a model a causal structure can explain is a linear program; causal separability is relative to an assumed ambient structure, and the causally separable fraction is never less than the separable local fraction

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Stefano Gogioso and Nicola Pinzani (2023), arXiv 2303.09017, Part 3 of the trilogy after LIT-800 and LIT-808

Key takeaways

  • Causaltopes. For a space of input histories Θ and a cover, take the product over contexts of simplices of distributions on output histories (“pseudo-empirical models”, Definition 2.10). Then impose causality equations: for any two contexts containing a lowerset μ, their marginals on μ agree (Definition 2.12, with reductions in Propositions 2.21–2.22). The result is the causaltope (Definition 2.13). It is in convex-linear bijection with the empirical models of LIT-808 (Theorem 2.25).
  • Hierarchy. For spaces on the same events and inputs with the same maximal histories, Θ ≤ Θ′ gives Caus_std(Θ) ⊆ Caus_std(Θ′) (Proposition 2.26). The no-signalling polytope is the bottom and the indiscrete space’s polytope is the whole product of simplices.
  • Causal fractions as linear programs. The fraction of a model supported by a sub-space, or jointly by a family of sub-spaces, is the largest weight of a component in their causaltopes (Definitions 2.14–2.15). The causally separable fraction is the fraction over the causal completions (Definition 2.16).
  • Relative separability. Separability is relative to an ambient space, not fixed to the indiscrete space as in the causal-inequality literature. This is strictly finer, and it cuts the search: for a 6-party example, 16 known completions instead of 16,511,297,126,400 for the indiscrete space.
  • Contextual causality. The causally separable fraction is at least the separable local fraction (Proposition 2.29). The converse is a conjecture (2.30). For a switch with Bell-entangled control the separable fraction is about 63.2% at the plotted angles, and the bound is not tight everywhere. For two contextually controlled classical switches the separable fraction is 75% and equals the local fraction of the Bell model controlling them. Every such example becomes causally separable if the no-signalling constraints to the measuring parties are dropped.
  • BFW. The Baumeler–Feix–Wolf model is 0% supported by every causally complete 3-event space. It is 100% supported by each of the three indefinite orders in which one party acts first, and the intersection of those three causaltopes (38-dimensional) supports all of it.

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, as a successor of Gogioso and Pinzani 2021 (LIT-813). The withdrawn paper’s Proposition 16 (empirical models form a polytope cut by linear causality equations) and its Section 9 (causal fractions for pre-orders, the BFW example, with proofs deferred to a later paper) are carried and proved here. LIT-813 is now Superseded by the trilogy. See the curation entry of that day.

Skimmed on 2026-10-09 (NOTE-624). The reading covers the definitions, the main theorem and the examples, but not the generalisation to non-standard covers in full, nor any proofs. Its BFW result sharpens the withdrawn paper’s: there, the Ω-causal fraction was 0 for every partial order; here, it is 0 for every causally complete space, which includes the dynamical ones.

The vocabulary has no topic for causal structure. The paper is tagged contextuality first because its examples and its central bound tie causal separability to the contextual fraction.

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