Let me add a remark about CPT theorem and its possible applications - most physicists believe this symmetry is satisfied, it also contains time symmetry. So can we prove that causality only works past -> future? (as in Euler-Lagrange, in contrast to the least action principle).

For example laser causes excitation of target later - so shouldn't CPT analogue of laser cause deexcitation of target earlier?

While building "CPT analogue of laser" might seem extremely difficult, for free electron laser (FEL) it looks quite simple:

Hi Barry, Excellent question.

My feeling is that the concept of Past, Present, and Future have no reality beyond a social agreement. Wolfram's principles of Physics would suggest to me instantaneous 'relationship' exists in the present, like a snapshot of the cascade. Your memory of events and sensory assimilation are happening in the instantaneous now and are part of that snapshot.

Time for the most part since the rise of civilization has been at the center of human behavioral evolution. The concept of time is a shared agreement built into the syntax of language. Many past and present cultures do not have language for the concept of time. Time is really only just one way of socially communicating physical reality to allow greater symbiosis with physical tools and manipulation of more stable forms of informational and spatial cascades we call 'matter' and 'life'.

You have only to look at the anthropology of time to understand that it is a shared social contract in the instant now. The artifacts of our obsession with time are colossal but they only exist in the instantaneous present as stable energetic and spacial recursions. By 8 Billion people agreeing on what time is in the present we can imagine or dream the idea of the past in the present. Time is intrinsic to our language so escaping the concept of time is difficult. The fact is that many cultures do not have past, present, and future descriptive verbs. The Piraha tribe that lives in the Amazon rain forest base their language on whistling and humming. They have no concept of time. Everything exists in the present for them.

The Hopi also have a language that lacks verb tenses, and their language avoids all linear constructions in time. The Hopi appear to have no sense of linear time. Their religious beliefs include a cyclic view of time, similar to ancient Hindu and Buddhist belief in the “wheel of time”.

For the Amondawa of the Amazon, language has no word for "time", or indeed of time periods such as "month" or "year". Instead of aging they assume different names in different stages of their lives or as they achieve different status within the community. They experience no need for "mapping" between concepts of time passage and movement through space. Ideas such as an event having "passed" or being "well ahead" of another are familiar from many languages, forming the basis of what is known as the "mapping hypothesis". When the Amondawa learn Portuguese - which is happening more all the time - they have no problem acquiring and using these mappings from the language. To them, time is just an unnecessary idea. There are many anthropological examples of cultures with no concept of time.

Young children for the most part have to learn the concept of time. You have to explain to them what time is and draw them into the social contract. What psychologists have discovered is that there is no simple, undifferentiated type of time knowledge. Instead, multiple forms of time emerge at different stages as a function of the development of underlying cognitive processes. Young children’s time judgments are context-dependent and closely bound up with the situation within which time is experienced.

At the age of seven years, their time judgments improve because they acquire a symbolic representation of time. They represent time as something absolute that flows uniformly, and this enables them to measure the duration of events independently of their specific characteristics. This representation of time, which is close to Newton’s conceptualization, allows children to think about time per se, and resist their tendency to distort it. The whole social relationship with time is learned.

Why can't the universe as a whole re-appear every instant? According to Wolfram Physics it can. It may be all physical reality is one note. Our experience of it makes it into a song, the memory of that song is part of the now.

Particular hypergraph states are always reference-frame dependent. One possible ("cosmological)" reference frame is updating the entire hypergraph at once at each step, but other choices are possible as well.

And we are not generally using the word "time" to describe evolution steps, we either call them steps or generations.

For any single branch's causal graph, any space-like slice through it (including horizontal slices in between layers corresponding to steps) would produce a unique space hypergraph. But, of course, multiway branching is possible, in which case you would not get a unique graph at any given step.

Thank you for your reply. I did not understand that the evolution of the graph is not somehow relate to time as observed from inside.

How then, does the evolution of the system by running the rule relate to "time" as perceived by an observer inside the system? How is the Arrow of Time represented, and how is the portion of the graph representing the past protected from further "evolution" by future events?

An interesting and subtle question!

Of course, our model is only deterministic up to measurement. As detailed in my quantum mechanics paper (https://www.wolframphysics.org/technical-documents/) different choices of quantum observation frame, corresponding to different choices of measurement sequence, generally distinct causal graphs (in much the same way as different foliation choices in the causal graph generally yield distinct spatial hypergraphs). So I'd argue that our model implies a refinement of the block universe concept, in which it is not spacetime (i.e the causal graph), but rather branchtime (i.e. the multiway causal graph), that is the invariant structure containing all past, present and future events.