Contribution
It gives the geometric counterpart of the topological framework of LIT-808. The empirical models on any cover of any space of input histories form a polytope, the causaltope, described compactly as a product of simplices cut by causality equations rather than by its exponentially many facets. Fractions supported by sub-spaces, and the causally separable fraction, become linear programs. Causal separability becomes relative to an assumed ambient causal structure. Worked quantum examples tie the separable fraction to the contextual (local) fraction.
Key insight
A causal constraint is a marginal equation: two contexts that share a lowerset of input histories must agree on it. So instead of characterising causal correlations by inequalities (facets), slice the easy polytope of conditional distributions with equations read off the space’s topology. Then every question of the form “how much of this model can that causal structure explain” is a linear program.
Assumptions
- Finite events, inputs and outputs, and a space of input histories with a cover, as in LIT-800 and LIT-808.
- Comparison of causaltopes (Proposition 2.26) needs spaces on the same events and inputs with the same maximal extended histories, for instance both satisfying free choice. Section 2.9 extends separability beyond the standard cover.
- Theory-independent: quantum realisability is not characterised. The quantum examples are used as inputs.
Key results
- Theorems 2.18–2.19. Distributions on extended causal functions over a context correspond convex-linearly to distributions on output histories, with one output per tip-equivalence class.
- Definition 2.12, Propositions 2.21–2.22. The causality equations equate marginals of context pairs on common lowersets. A chain of adjacent pairs suffices, and on the standard cover equations indexed by histories suffice.
- Theorem 2.25. Empirical models on a cover correspond convex-linearly to points of the causaltope.
- Proposition 2.26. Refinement of spaces gives inclusion of standard causaltopes. The no-signalling polytope lies inside every one, and the indiscrete one is the full pseudo-empirical polytope (Observations 2.27–2.28).
- Definitions 2.14–2.16. Supported fraction, causal fraction over a family, causally separable fraction (over the causal completions).
- Proposition 2.29. The causally separable fraction is at least the separable local fraction. Conjecture 2.30. Local and causally separable implies local in a causally separable way.
- Examples. BFW: 0% on all causally complete 3-event spaces, 0% on the no-signalling causaltope (26-dimensional), and 100% on each of the three “one party first, two indefinite” orders and on their 38-dimensional intersection. Fixing Alice’s output leaves deterministic inseparable functions requiring two-way signalling between Bob and Charlie. Switch with Bell-entangled control: a causally separable fraction of about 63.2% for the model shown. The local fraction lower-bounds it at every angle, not always tightly. GHZ control: about 41.2%. Two switches with Bell-entangled control: about 47.7%, with the separable fraction equal to the separable local fraction. Two contextually controlled classical switches: 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.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Empirical models on any cover of any space of input histories are exactly the points of the causaltope | strong (proof, not read) | Theorem 2.25 |
| C2 | Supported and causally separable fractions are computable by linear programming | strong (by construction) | Definitions 2.14–2.16 and the computed examples |
| C3 | The causally separable fraction is bounded below by the separable local fraction | strong (proof, not read) | Proposition 2.29 |
| C4 | Local and causally separable implies local in a causally separable way | conjecture | Conjecture 2.30 |
| C5 | Entangled or contextually controlled quantum switches witness indefinite order relative to an ambient space with known no-signalling constraints, but not relative to the indiscrete space | moderate (computed examples) | Section 2.7 |
| C6 | Causal inseparability in BFW stems from a cyclic order structure | weak (the authors’ “intuition”, supported by the conditioned models) | Section 2.6.6 |
Method
Convex geometry and linear programming. Polytopes are given as solution sets of linear equations within products of simplices. Fractions are maximal-weight decompositions, computed by LP. The landscapes in 2.7 are computed numerically over grids of measurement angles, and one takes 15 hours on 30 cores.
Concepts
- pseudo-empirical model: a family of conditional distributions, one per context, with no cross-context constraint.
- causality equations: equalities of marginals on a common lowerset of two contexts.
- causaltope: the pseudo-empirical models satisfying the causality equations. Standard on the standard cover, solipsistic on the solipsistic one.
- supported fraction / causal fraction: the largest weight of a component of the model lying in a sub-space’s causaltope, or in the convex hull of several.
- causally separable fraction: the causal fraction over a space’s causal completions.
- contextual causality: the authors’ name for causal inseparability correlated with, or implied by, non-locality or contextuality.
Connections
It generalises the no-signalling polytopes of Barrett and co-authors and Popescu–Rohrlich-style theory-independent no-signalling. It sets itself against causal Bayesian networks (Fritz) and multipartite locality with factorisable sources, where convexity can fail. It refines the causal-inequality approach to indefinite order (Oreshkov, Costa and Brukner; Branciard et al.; Oreshkov and Giarmatzi) by making separability relative. It treats the OCB and BFW processes. A version of its first switch example was studied by others with facet inequalities computed in PANDA. It uses the contextual fraction of Abramsky and co-authors. It proves the polytope and causal-fraction statements that the withdrawn LIT-813 made with proofs deferred, which it does not cite.
Bearing on the record
- LIT-813. Its Proposition 16 (a polytope cut by linear causality equations) is Theorem 2.25 here, in general. Its Section 9 claim (BFW has causal fraction 0 for every definite order and 1 for four pre-orders), which its NOTE marked “not supported here (proof deferred)”, is now computed. The model has 0% support on every causally complete space and 100% on each of three indefinite orders where one party acts first. The fourth pre-order of the earlier count is presumably the indiscrete one, which supports everything trivially. The paper does not say this, and it is my reconciliation.
- THEORY-171 rests on LIT-808. This paper’s examples are all on the standard cover, where solipsistic contextuality does not show, so they neither support nor test it.
- THEORY-165 (contextual fraction Lipschitz within a scenario): the fractions here are LP values over polytopes, which fits that THEORY’s setting. The reading does not test it.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- Skimmed: proofs not read, and Sections 2.8–2.9 followed only for their statements.
- Quantum realisability is outside scope, so the causaltopes over-approximate what quantum theory allows.
- The relative notion of separability depends on assuming part of the causal structure. The witnesses of indefinite order are witnesses conditional on that assumption, as the authors say.
- The causal fractions in 2.7 are read from grids of angles at finite resolution, and the minima are reported “at the angular resolution” used.
- Remark 2.4 and the text before Proposition 2.29 cite “Theorem 4.51 p.77” and “Definition 4.38 p.70” of The Topology of Causality. Neither number exists in that paper’s current version (v2), whose results are numbered 2.x. They look like leftovers from the combined v1 of 2206.08911.
Open questions
- Conjecture 2.30.
- A characterisation of the quantum-realisable subset of a causaltope, by semidefinite hierarchies or otherwise.
- Whether the tight correlation between separable and local fractions in the contextually controlled switch is generic or by construction. The authors say it holds “by construction” in that example.
Note of 2026-10-20: the proofs read
The proofs in §2.10.11–2.10.18 were read for THEORY-580’s promotion. They are short and correct given the definitions in “The Topology of Causality” (arXiv 2303.07148): causal functions are assignments constant on tip-equivalence classes (its Eq. 9 and Obs. 2.12), and separable functions are those arising on a causally complete subspace, of which the causal completions suffice (its Def. 2.9 and Obs. 2.6). The one step left implicit, in the proof of Theorem 2.25, is that ⌈·⌉ is injective on the lowerset μ the restrictions land in; Theorem 2.19 applied to μ gives it. The citation numbers 4.x in the paper refer to an older version of that companion, as the Limitations above say.