Source
Gogioso and Pinzani (2023), LIT-788, “The Geometry of Causality”, arXiv 2303.09017v2, skimmed in NOTE-624. Sections 1, 2.3, 2.4 and the examples of 2.6.6 and 2.7 were read; the proofs (2.10) were not. The skim is the whole basis of this document.
What was actually shown
Causaltopes (Theorems 2.18–2.19, 2.25; Definition 2.12). A causal constraint is a marginal equation: two contexts sharing a lowerset of input histories must agree on it. The pseudo-empirical models (one conditional distribution per context, no cross-context constraint) form a product of simplices. Slicing it by the causality equations gives the causaltope, and empirical models on a cover correspond convex-linearly to its points. On the standard cover, equations indexed by histories suffice.
Fractions (Definitions 2.14–2.16). The supported fraction, the causal fraction over a family of spaces and the causally separable fraction are maximal-weight decompositions, so each is a linear program.
Relative separability (Proposition 2.26, §2.4). Refining a space gives inclusion of standard causaltopes. The no-signalling polytope lies inside every one, and the indiscrete one is the full pseudo-empirical polytope. So whether a model is causally separable depends on the ambient space assumed.
Lower bound (Proposition 2.29). The causally separable fraction is at least the separable local fraction. Conjecture 2.30, that local and causally separable implies local in a causally separable way, is open.
Examples (§§2.6.6, 2.7), computed. The BFW model is 0% supported by every causally complete 3-event space and 100% by each of three “one party first, two indefinite” orders. A switch with Bell-entangled control has a causally separable fraction of about 63.2% for the model shown; with GHZ control, about 41.2%. Two contextually controlled classical switches give 75%, equal to the local fraction of the controlling Bell model. Every example is fully separable once the no-signalling constraints to the measuring parties are dropped.
What could have come out otherwise. The causally compatible models could have formed a non-convex set, as they do for causal Bayesian networks with factorisable sources, which the paper sets itself against. The theorem says they do not here.
What this does not say
- That the proofs were checked. They were not read (NOTE-624).
- That quantum switches show indefinite causal order outright. The witnesses are conditional on assuming part of the causal structure; with the indiscrete ambient space everything is separable.
- What quantum theory allows. The causaltopes are theory-independent and over-approximate the quantum-realisable models.
- That the separable fraction always equals the local fraction. It is a lower bound, not always tight. In the contextually controlled switch they coincide “by construction”.
- Exact values. The fractions in §2.7 come from grids of angles and are reported “at the angular resolution” used.
Connections
- It proves, in general, Proposition 16 of the withdrawn LIT-813 (NOTE-629) as Theorem 2.25, and computes its deferred BFW claim. LIT-813’s construction of the locale of inputs fails as stated, and this paper does not cite it.
- Its examples are on the standard cover, where the solipsistic contextuality of THEORY-171 does not show, so they neither support nor test it.
- Its fractions are linear-programming values over polytopes, the setting of THEORY-165 on the contextual fraction, which this reading does not test.
Note of 2026-10-20: the proofs, checked
The proofs in §2.10 were read on promotion, with the definitions they cite from “The Topology of Causality” (arXiv 2303.07148; its numbering differs between versions).
- Theorem 2.19. A causal function on a lowerset λ assigns outputs at tip events, constant on each ∼ω class (Topology, Eq. 9). So causal functions correspond one-to-one with free assignments per class (Topology, Obs. 2.12), and pushing distributions forward gives the convex-linear bijection d ↦ ⌈d⌉.
- Theorem 2.25. Injectivity is componentwise. For surjectivity, take u in the causaltope and set e_λ := ⌊u_λ⌋_λ. The causality equations range over every lowerset μ inside two contexts, λ ∩ λ′ among them. Restriction commutes with ⌈·⌉ because each tip class in μ lies in one in λ (Lemma 2.20), so ⌈e_λ|_μ⌉ = ⌈e_λ′|_μ⌉. Theorem 2.19 on μ then gives e_λ|_μ = e_λ′|_μ, a step the printed proof leaves implicit. So e is a compatible family.
- Fractions as linear programs. A component is v with 0 ≤ v ≤ u in the cone of a causaltope, which is cut out by linear equations and v ≥ 0; its maximal mass is an LP, and so is a decomposition over several sub-spaces.
- Proposition 2.29. A separably local model is a mixture of deterministic models of separable causal functions, each arising on a causally complete Θ′ ≤ Θ, and the causal completions suffice (Topology, Def. 2.9, Obs. 2.6). So it decomposes over the completions, and scaling by p bounds the causally separable fraction below by p. The fraction is defined (Def. 2.15) when the sub-spaces share events, inputs and maximal extended histories with Θ, as under free choice.