Stefano Gogioso and Nicola Pinzani (2022; current version 2023), arXiv 2206.08911, Part 1 of a trilogy whose Parts 2 and 3 are LIT-808 and LIT-788
Key takeaways
- Causal orders and their lowersets. A causal order is a finite preorder; definite ones are partial orders. What matters operationally is the lattice of lowersets Λ(Ω). Ω ≤ Ω′ iff Λ(Ω) ⊇ Λ(Ω′) (Proposition 2.2), and Λ(Ω) ∩ Λ(Ω′) = Λ(Ω ∨ Ω′) (Proposition 2.4). But Λ(Ω) ∪ Λ(Ω′) need not be a lattice, so satisfying two orders’ constraints is not satisfying their meet’s.
- Spaces of input histories. The input histories of an order are the input assignments on the causal past of each event (Definition 3.5). In general, a space is any finite join-prime set Θ of partial functions (Definitions 3.7–3.8), and its extended histories Ext(Θ) are the compatible joins. Spaces are ordered by Θ′ ≤ Θ iff Ext(Θ′) ⊇ Ext(Θ) (refinement means more constraints) and form lattices (Proposition 3.10). Order-induced spaces are closed under join, not under meet (Propositions 3.11–3.12).
- Free choice, tips and completeness. The free-choice condition says every joint input arises by stitching histories together. A tip event of a history is an event not covered by any history below it. A space is causally complete if every history has exactly one tip. An order-induced space is complete iff the order is definite (Proposition 3.21), and completeness has a one-step deletion test (Theorem 3.37).
- Composition and switches. Parallel, sequential and conditional sequential composition are defined. The last lets one event’s input choose what comes next, as in the 3-party causal switch. All three preserve completeness and free choice under stated conditions (Propositions 3.13–3.24). The spaces with Θ = Ext(Θ) and single tips are exactly the recursively built causal switch spaces (Theorem 3.34, Corollary 3.35), and these are exactly the maxima of the complete spaces (Theorem 3.36). Their number grows more than doubly exponentially.
- Tightness. A space is tight if under every extended history each event is the tip of a unique history (Definition 3.16). Order-induced spaces are tight (Proposition 3.32). A meet of two order-induced spaces is tight iff every event’s two causal pasts are nested (Theorem 3.33). 58 of the 102 symmetry classes on 3 events are non-tight.
- The census. On 3 events with binary inputs there are 2644 causally complete spaces in 102 classes under event and input permutation. 13 classes admit no fixed definite order. After 106 days of search on 4 events, 869,529,223 spaces in 2,312,000 classes had been found, and a fitted curve projects about a billion. For comparison, earlier work used 25 of the 3-event spaces: 19 order-induced and 6 more for indefinite causality.
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. The manuscript
considered Gogioso and Pinzani 2021 (LIT-813), which its authors
withdrew in April 2024. The withdrawal names this paper (as
arXiv:2206.08911v4), The Topology of Causality and The Geometry of
Causality as superseding it. All three are filed, and LIT-813 is
Superseded by them, this one first. See the curation entry of that day.
Read on 2026-10-09 (NOTE-635). This paper is the combinatorial groundwork. It defines the spaces that replace LIT-813’s Definition 3, the “locale of inputs”, which the authors’ withdrawal says “is not fit for purpose”. The sheaf of causal functions over these spaces is in LIT-808, and the polytopes of empirical models in LIT-788.
The vocabulary has no word for causal structure or causal order. The
paper is about the combinatorics of causal constraints between events,
motivated by quantum indefinite causality. It is tagged quantum-foundations
first, for that motivation, and mathematics for its content. Neither says
what it is about. A topic for causal structure would be added by decision,
and the trilogy and LIT-813 would take it.