You either have causality/time component [...]

I don't understand what you mean.

1. causality and time component
2. causality or time component
3. causality and time component are two words for the same thing
4. causality and time component are deeply connected, maybe depend on each other or imply each other
5. something else

And what does time component mean, I would guess component in the sense of a vector component, i.e. a time dimension, right? Assuming that is true, having a time dimension does probably not imply that there is causality, i.e. all events could be independent. Does having causality imply having a time dimension as causality depends on time. I would guess so, at least in the common understanding of the term.

If you get distracted by a message on your phone and hit a tree with your car, the person sending you the message caused you to be distracted which caused you to hit the tree. But did not also your driving teacher cause this because he did not get you to understand to not use your phone while driving well enough? And your parents because without them you would not exist and be driving a car? And the phone designers and manufactures because without them there would be no phone to distract you. And no car without the car manufacturer. And no tree without the person planting it exactly where you hit it?

Causality is at the very least a really tricky concept, we usually just pick something pretty arbitrarily and declare it the cause. And what about the relationships between time, space, change, events, correlation, cause, and effect - what comes first, what depends on what? Does time enable change or does change define time? Does causality cause change or does change define causality? Does causality even exist fundamentally, after all cause and effect are no where to be found in the fundamental laws of physics? Is it just a human abstraction to talk about the world? What does the theory of relativity tell us as the ordering of events is observer dependent?

[...] and can derive some basic axioms that get you existence or not existence (boolean) or you don't.

Derive axioms from what, causality and time? And does deriving axioms even make sense, are they not the things that you just postulate as true without further justification? And who said that existence - or anything for that matter - has to be binary? From the law of excluded middle which is an axiom and not even universally accepted in our universe? And what does existence even mean? Is there only one kind of existence or are there several? Does wetness exist in the same way as the keyboard I am just typing on?

If you have the former, we have turing completeness, and there we go.

What is the former and how on earth did we get from whatever the former is to Turing completeness and why does it matter?

If you don't, well, sure, whose to say there isn't a universe where you can't have both presence, absence, and a time component but it's not all that interesting.

I don't think it really matters whether it interesting or not, but this sounds a bit like you are just dismissing my main point in a single sentence - if there could be universe where our logic does not apply or is even wrong, how can we call our logic and the math on top of it true and assign some kind of Platonic existence or whatnot to it, if it is at best true relative to our universe?

Maybe I am overthinking it but I would counter that you are underthinking it.