Can we think about this in terms of lattice theory?
What is the graph of possible foliations? [A multiway graph] [This would be a graph of all possible time orderings for another multiway graph] [Each “time ordering” is a foliation, i.e. a reference frame]
Every node of the “orderings multiway graph” is an antichain A foliation is a “spacelike coarse graining”
Original multiway graph has “state nodes” [each node is a global state] Orderings graph has nodes which are antichains (i.e. sets of state nodes)
Given a multiway graph (or a causal graph), what are the possible foliations that one can generate, and which ones are “civilized”?
Each path in the antichains graph is a foliation; some are more civilized than others
The civilized foliations are ones whose description is compressible, perhaps lossily
[ Correspondence between foliations and spanning timelike graph ]
Space requires a notion of simultaneity
If the system is ultimately ergodic, with this definition there can be no space [e.g. ergodic wrt the atoms of space] [ or wrt tokens]
To make a persistent notion of space, we need equivalence classes in time [i.e. we need to associate each atom of space with its “successor atom”, which is not trivial because there are interactions between the atoms]
In the hypergraph system, “tokens” are relations between atoms of space In the list-numerical multiway systems, “tokens” are the lists of numbers
Complexity of finding foliations [anti-timelike surfaces]
Finding foliations is stricter than finding antichains ;; i.e. in a foliation we are chaining the antichains
How do we actually find foliations? Assuming there’s permanence to space, we can do it with arrays of light signals [relativistic analog of structured light..]
Foliation is about sequentialization (which we can interpret as time)
We normally have multiway graphs....
Our underlying model has microsequentialization already it, because every event has a past and future...
If events (AKA functions) are defined not by edges but by directed hyperedges, basically everything works the same....
Rulial multiway system
- A multiway system with all possible rules
Fibrations are like families of timelike edges; foliations are like spacelike/branchlike/rulelike space