What is Time?

To begin: When we use the word "time", we are referring to multiple different things, which we treat as if they are the same thing.  There are two broad categories, however, which I think it is particularly useful to differentiate between: Time as timing, and time as history.

Here's the thing: There is absolutely no reason why these two things should be the same thing, and indeed, given what we know about the universe, maybe it should surprise us if they are, in fact, the same thing.  From an informational perspective, entropy is already, basically, kind of, a compression algorithm on the history of the universe; it increases because there's an ever-increasing amount of history to store.  Given that entropy is storing the history of the universe, albeit in an unrecoverable kind of way, why should we expect history to also be stored in some kind of multi-dimensional pattern covering the entirety of the universe, past and present?

Indeed, if you think about general relativity seriously, if history (which includes past, present, and future) is stored in some kind of dimensional pattern, the future (and past) should exert influence on the present.  There are some causal issues here.  And it'd be kind of weird for history to be represented twice, both in an entropy-compressed form, and in some kind of linear dimension form.

So maybe time-as-timing is entirely separate from time-as-history, which leads to the question, if time, in the sense of the fourth dimension required by general relativity, is purely a matter of timing, and not entirely a matter of a spacial dimension along which the past and future are stored, what shape would we expect time-as-timing to take?

A loop is the simplest answer there, but a spiral has certain qualities to recommend it, assuming certain properties can be managed; in particular, if you consider both chiralities of a spiral, there are points of intersection, but mostly divergence, which I think has some interesting ramifications.  But more generally, with loop-like structures, there is a characteristic behavior where going backwards in time is the same as going forwards in time, and you don't have coordinate mismatch problems.

What are coordinate mismatch problems?

Well, here's the thing: If time stores past and future, how can the time of two objects ever come out of alignment?  To meet, two objects have to be at the same points, in space and time. That is, time-as-storage-mechanism-of-history has some disagreement with relativity, in that you shouldn't be able to have two objects meet having different internal clocks; they should meet at the same time.