I'm asking this question to spur a discussion that might lead somewhere interesting.
If there is a time's arrow (infinite aperiodic evolution), the descriptive complexity of the Universe will increase as a logarithmic function (in base 2) of time (here is the proof of my theorem). Does something that is increasing its descriptive complexity all the time deserve to be called dead? I do not think so.
Another possibility is that time is cyclic. In this case, the evolution of the Universe will be like the finite cellular automata studied by S. Wolfram, e.g., rule 30 and rule 90. After the heat death, there will be a big bang again.
Finally, there is the possibility that the Universe will halt and everything will be frozen forever.
The heat death of the universe, if it exists in any real sense, would imply that updates no longer modify the graph in any meaningful sense, perhaps leading to a static final state where each application of the rule leads to no change. At this point what happens? If the total universe is casually invariant to the degree that all possible branches lead to this single final termination state, perhaps the rule resolves and the universe simply halts? Maybe there is a mechanism to check for too many loops with no updates to the graph and terminates the program? We would need to know more about what exactly is going on with nodes and edges, what the underlying nature of a connecting hyperedge was, and if there is some kind of pruning of halted/stuck universes. The Demon gardener of rule space snipping universes that fail to update for a few 10^100 cycles?
Perhaps the king of rule space asked his wizard to solve the halting problem and now they are sitting around drinking and computing every possible universe that derives from the idea of causality, mostly just for fun but also to impress the blob of fuzzy logic floating on the currents of primordial chaos that they both want to mate with. Every time a universe spends more than a billion years in heat death they throw it in the river.
I image a cyclic evolution that ends with a solitary or ”singular” point/vacuum. That would be a distinct case of gravity potential. Assuming a quantum action of the quantum actor, circulation would accumulate its surface tension or ”charge”. It would divide as from energy density. Time-like aspect would be linear as axis is ”compressed”. Process is unitary and terminated by division at t0.
I suspect this relates to math theorems of eternal recurrence that holds for dimensions 1, 2 but not 3. Polya’s Random Walk if I recall correctly. But it requires the vacuum to self-charge and that is disallowed in theory because it makes quantum theory ”trivial”. As for me ”trivial” implies ”probably valid”