This is extremely important question already for classical field theories: should we solve them by Euler-Lagrange evolution (presentism), or maybe by the least action principle (eternalism/block universe) - history of the Universe optimizing action?

While naively they are equivalent: mathematically we can translate between such solutions, the ones originally found with each of them have essentially different properties - e.g. only the latter is time/CPT symmetric, superdeterministic: outcomes depending on future measurements.

For example for the general relativity, which seems close to your approach, Euler-Lagrange would mean some "spacetime unrolling" having little sense - everybody find spacetime minimizing Einstein–Hilbert action instead. Then QFT replaces single action optimizing history, with their Feynman ensemble.

From QM perspective, originally solving with Euler-Lagrange it is local, realistic model - cannot violate Bell inequalities (in contrast to nature). Solving with the least action principle instead, such solution is superdeterministic, gets Born rule allowing to violate Bell - literally one amplitude from propagator from -infinity, second from +infinity, like e.g. in S-matrix.

Here is analogous Born rule, Bell violation already in Ising model - using similarity between Boltzmann and Feynman path ensembles: https://arxiv.org/pdf/0910.2724

I would say yes. Look up "rulial multiway graph" and you will see that there is a structure that does all rules on each step. Half of the rules would be to go "back" one step while the other half go "forward" a step, so not only is the present being generated by the rules, but the past and future are constantly being (re)generated.

If I understood your reply - you are saying that the stepwise evolution of the hypergraph is deterministic, and unique for the whole universe. Is it correct to say that here is no concept of reference frame at this level of representation?

But - to any entity inside the universe - this stepwise evolution (represented by the hypergraph) is not directly visible - but is visible as a derived causal graph which represents spacetime in the reference frame of the observing entity.

If this is the case, would it not be easier to use a different terminology describe the t=0 => t=N evolution steps as something other than time?

Am I correct in assuming that any causal graph Time T will always point back to a unique Hypergraph step? They are not necessarily contiguous, or in the same order as the hypergraph?

I should clarify that the hypergraph, which is the state of the model, represents only space at a particular instance in time (in some reference frame). And rules operate only on that hypergraph.

From these rule applications, one can compute a causal graph, which represents spacetime. But rules don't operate on the causal graph directly, which is why the past is "protected".

Whether the past is "real", I think, is a philosophical question that our model does not answer, and it seems to me it fundamentally depends on what "real" means. To compute the future evolution of the hypergraph (on any of the multiway branches) past is not needed, so in that sense, it is not real. Whether the past can be reconstructed from the present would be rule-dependent. However, the causal (i.e., spacetime) graph can be computed from the evolution. So the past does exist in that sense.

If the universe is a deterministically evolving hypergraph - is the state of previous cycles preserved?

This is another way of asking (in this model) if the past is real? Is the future real? or is this model an example of Presentism rather than a Block Universe?

I was listening to the working session on April 30th, and Stephen seemed to be saying that Time is special, and the idea of a block future universe (where the future is "calculated" in advance) can't happen because of computational irreducibility.

Did I misunderstand the discussion?