  We introduce and explore the notion of "spaces of input histories", a broad
family of combinatorial objects which can be used to model input-dependent,
dynamical causal order. We motivate our definition with reference to
traditional partial order- and preorder-based notions of causal order, adopted
by the majority of previous literature on the subject, and we proceed to
explore the novel landscape of combinatorial complexity made available by our
generalisation of those notions.
  In the process, we discover that the fine-grained structure of causality is
significantly more complex than we might have previously believed: in the
simplest case of binary inputs, the number of available "causally complete"
spaces grows from 7 on 2 events, to 2644 on 3 events, to an unknown number on 4
events (likely around a billion). For perspective, previous literature on
non-locality and contextuality used a single one of the 2644 available spaces
on 3 events, work on definite causality used 19 spaces, derived from partial
orders, and work on indefinite causality used only 6 more, for a grand total of
25.
  This paper is the first instalment in a trilogy: the sheaf-theoretic
treatment of causal distributions is detailed in Part 2, "The Topology of
Causality" [arXiv:2303.07148], while the polytopes formed by the associated
empirical models are studied in Part 3, "The Geometry of Causality"
[arXiv:2303.09017]. An exhaustive classification of the 2644 causally complete
spaces on 3 events with binary inputs is provided in the supplementary work
"Classification of causally complete spaces on 3 events with binary inputs",
together with the algorithm used for the classification and partial results
from the ongoing search on 4 events.
