Contribution
Before this paper, operational treatments of causal structure in quantum foundations used a causal order, definite (a partial order) or indefinite (a preorder), possibly mixed. This paper replaces the order with a space of input histories: the set of partial input assignments on which the output at some event may depend. That makes causal constraints conditional on inputs a first-class object. It works out the combinatorics of these spaces: their order, their lattice operations, composition, causal completeness and tightness. It then shows by exhaustive search that even 3 events with binary inputs admit 2644 causally complete spaces, where the literature had used 25.
Key insight
Causality is “no signalling from outside the past”, so what matters is not which events precede which but which input data an output is allowed to see. Once that is the primitive, the order of events can itself depend on inputs. Whether event B sees event C’s input can depend on what was chosen at A. A causal order is then a special, rigid case of a much larger space.
Assumptions
- Finitely many events, each with finite non-empty input and output sets. Inputs are freely chosen, so every joint input must be reachable (the free-choice condition, Definition 3.9).
- Black-box devices act locally at events. Any dependence on other events goes through the causal structure.
- The object of study is a conditional distribution on joint outputs given joint inputs. How it is obtained (experiment or theory) is outside the paper.
- A space of input histories is a finite set of partial functions that is join-prime: no history is the compatible join of others in the set. This is the normal form that makes spaces correspond to constraints.
- Causal completeness, the main object of study, is built on top of free choice by definition (Remark 3.2).
Key results
- Proposition 2.2. Ω ≤ Ω′ ⇔ Λ(Ω) ⊇ Λ(Ω′). Corollary 2.3. Λ(Ω) ∪ Λ(Ω′) ⊆ Λ(Ω ∧ Ω′), strictly in general: the union need not be closed under intersection (a four-event counterexample is given). Proposition 2.4. Λ(Ω) ∩ Λ(Ω′) = Λ(Ω ∨ Ω′): the join’s constraints are exactly those common to the orders.
- Propositions 3.2, 3.6, 3.8. Order-induced spaces Hist(Ω, I) are join-prime and satisfy free choice, and Ω ≤ Ξ ⇔ ExtHist(Ω, I) ⊇ ExtHist(Ξ, I). The spaces themselves are not ordered by inclusion (a fork and a total order on 3 events give incomparable spaces), which is why the order on spaces goes through extended histories (Definition 3.10).
- Proposition 3.10. All spaces form an infinite lattice, with Θ ∨ Θ′ = Prime(Ext Θ ∩ Ext Θ′) and Θ ∧ Θ′ = Prime(Ext Θ ∪ Ext Θ′). Spaces with fixed inputs form a finite lattice, and those satisfying free choice form a lattice between the discrete and indiscrete spaces.
- Propositions 3.11–3.12. Ω ↦ Hist(Ω, I) commutes with joins, and order-induced spaces are not closed under meets.
- Propositions 3.13–3.18. Parallel, sequential and conditional sequential compositions are spaces and match the compositions of orders. Conditional sequential composition satisfies free choice iff each part does and all branches share their input sets.
- Proposition 3.20, Definition 3.14, Proposition 3.21. Every history has at least one tip, and a non-history extended history has none. Causally complete means exactly one tip each. An order-induced space is complete iff the order is definite, since the tips of a history over ω↓ are ω’s equivalence class.
- Theorem 3.26–Corollary 3.28. The causal completions of parallel and (conditional) sequential compositions are the compositions of the completions. Example: total(A, {B, C}) has four completions, two with a fixed order of B and C and two with A’s input choosing it.
- Proposition 3.29. Causally complete spaces with fixed inputs are closed under meet, not join.
- Proposition 3.32, Theorem 3.33. Order-induced spaces are tight. The meet of two order-induced spaces on the same events is tight iff for each event one causal past contains the other.
- Theorems 3.34, 3.36, Corollary 3.35. A space equal to its own extension, with single tips, is a conditional sequential composition {ω₁} ⇝ (Θ′i), recursively. Causal switch spaces are exactly the complete spaces with Θ = Ext(Θ), and they are exactly the maximal complete spaces. With n events and k inputs each, S{n,k} = Π_{j=1..n} j^{k
- Theorem 3.37. Θ is causally complete iff every extended history with at least two events stays an extended history after deleting some one event. This drives the enumeration.
- The census (Section 3.6–3.7). 7 complete spaces on 2 binary events; 2644 on 3, in 102 symmetry classes. 5 classes are order-induced (19 spaces), 13 classes admit no fixed definite order, and 58 classes are non-tight. On 4 events, 869,529,223 spaces had been found after 106 days, with about 10⁹ in about 3 × 10⁶ classes projected from a fitted power law.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | The constraints common to several causal orders are exactly those of their join, but jointly satisfying several orders is not the same as satisfying their meet | strong (proof and counterexample) | Propositions 2.2, 2.4, Corollary 2.3 |
| C2 | An order-induced space is causally complete iff the order is definite | strong (proof) | Proposition 3.21 |
| C3 | The maximal causally complete spaces are exactly the causal switch spaces, built by recursively letting one event’s input choose the rest | strong (proof) | Theorems 3.34, 3.36, Corollary 3.35 (3.36’s proof not read) |
| C4 | Causal completeness has a one-event deletion characterisation | strong (proof) | Theorem 3.37 |
| C5 | There are exactly 2644 causally complete spaces on 3 binary-input events, in 102 symmetry classes | moderate (exhaustive computation, not reproduced) | Section 3.6; the classification is a supplementary work |
| C6 | There are about a billion on 4 events | weak (extrapolation) | power-law fit to a search in progress |
Method
Order theory on partial functions: restriction order, compatible joins, join-prime subsets as normal forms and join-closure as their extension. Spaces are compared through their extensions, and composition is defined by joining histories. The census uses an enumeration algorithm based on Theorem 3.37, with a symmetry-reduced variant for 4 events. Both are in the supplementary classification, which was not read.
Concepts
- causal order: a finite preorder. Definite if it is antisymmetric, and indefinite if two distinct events are each ≤ the other.
- lowerset: a set of events closed under the causal past. It is the operational unit, since outputs in a lowerset cannot depend on inputs outside it.
- input history: a partial function assigning inputs to some events, on which the output at its tip event(s) may depend.
- space of input histories (Θ): a finite join-prime set of input histories. Ext(Θ): all compatible joins of its histories.
- refinement (Θ′ ≤ Θ): Ext(Θ′) ⊇ Ext(Θ); more causal constraints.
- free-choice condition: every total input assignment is an extended history.
- tip event: an event in a history’s domain covered by no history below it.
- causally complete: free choice plus exactly one tip per history. Causal completion: a maximal causally complete refinement.
- tight: under every extended history, each event is the tip of only one history below it.
- causal switch space: one event first, then, for each of its inputs, a causal switch space on the rest.
Connections
The motivation runs from Robb’s 1914 order-theoretic relativity and Malament’s theorem (causal order fixes conformal structure) to quantum indefinite causality: Hardy’s causaloid, the quantum switch, and the process matrices of Oreshkov, Costa and Brukner. The canopy of switch spaces is consistent with Oreshkov and Giarmatzi’s causal separability. The paper replaces the input structure of the authors’ withdrawn sheaf-theoretic paper (LIT-813), although it does not cite that paper. Its own reference list names only the two sequels and the classification. The sheaf of causal functions and empirical models built on these spaces are in The Topology of Causality (LIT-808), and the causal polytopes in The Geometry of Causality (LIT-788).
Bearing on the record
- LIT-813. Its NOTE found the “locale of inputs” (Definition 3, Proposition 5) defective: the given meet can leave the poset, and the poset is not distributive. Here the poset is replaced. Inputs are no longer families of input sets on lowersets but partial functions, and the topology is not asserted on them. In LIT-808 the topology is the lowerset topology of the space, which is a genuine topology and so a locale. The repair is a change of object, not a corrected proof of Proposition 5.
- The order-theoretic point in C1 is a clean, general fact: the constraints of several causal orders combine by join, but joint satisfiability is not the meet. It is worth having, but it is too narrow to be a THEORY on its own.
- The record holds no THEORY on causal structure. The one this trilogy warrants is drawn from LIT-808 (THEORY-171), and this paper supplies its definitions.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- Finite events and finite inputs only, and all examples have binary inputs.
- The census results rest on a supplementary computation that was not read or reproduced. The 4-event figure is an extrapolation from a search still running when the paper was written.
- Causal completeness is defined together with free choice, so spaces without free choice get no causal reading.
- Causal incompleteness is treated as missing information to be completed, not as a physical delocalisation of events. That is an interpretive choice, which the paper argues for and does not prove.
- No probabilities and no empirical models appear here. Those are in the sequels.
Open questions
- The exact number of causally complete spaces on 4 events, and an enumeration method that reaches 5. The authors say theirs cannot.
- A structural description of the non-tight classes, which are the majority on 3 events.
- Whether the meet of causally complete spaces, which can be non-tight, has a natural operational meaning beyond “satisfies both”.