Thanks for the piece-by-piece treatment!

> that the universe must abide by logical necessities (and even more so, by logical necessities the way we perceive them)

This axiom can be defined better, to account for the fact that we only perceive the universe through our minds, and this opens a whole can of philosophical worms - but in practice, this axiom does hold, and if it didn't, we could only pack our bags and go back to caves, as nothing would make sense and there would be no point to thinking.

(I didn't dig too deep in that can of worms, but I vaguely suspect the justification for such axiom could be found in anthropic reasoning.)

> QM has given us many non-classical-logic compliant observations.

Well, classical logic is pretty limited and it isn't be-all, end-all of logics, and not exactly what we use in daily reasoning anyway. I admit to not having much knowledge on the more complex types of logics, but the problems I recall seeing boil down to misapplying logic as an abstraction. For instance, I just stumbled upon the following example of "classical" vs. "resource-aware" logic:

"suppose you're standing at a vending machine that dispenses candy for a dollar and also dispenses coke for a dollar. You might write that as 1$⇒candy and 1$⇒coke. But then inferring that 1$⇒candy&coke is clearly wrong. The dollar gets "used up"."[0]. The post introduces resource-aware logic as a solution, but to me the problem seems to be modelling vending machines as (classical) logical implications; it's kind of dumb if you think about it.

(I'm also aware that probabilistic models are a better way to describe the observed world in practice than formal logic.)

> Then, there's the assumption that: "it exists <=> something would be different about the world if it didn't exist". That's again an assumption. I could just as well consider that (the fact of its existence turning from false to true aside) the world could be the same whether something exists or not.

I don't see how it could be any other way. Without asserting that there is a non-zero diff between universe with something and universe without something, "existence" is an empty label without meaning.

> This presupposes when it should instead prove: that existence necessitates difference in the world.

"Existence" is a term we define, and again, a different definition seems meaningless.

> whether something can be studied by science or not shouldn't be a criterion for its existence.

I didn't mean that current science defines what exists or not. I meant that existence of something has observable impact on universe. Observable in principle - not right now by us, but theoretically observable with sufficient knowledge and technology level. And if something is observable like this, then it falls into domain of science - it could be studied, and conclusions could be drawn. My point is that the set of things that exist is equal to the set of things that are of interest to science. This is not a time or technology-dependent claim.

> Fourth, one could argue that something could still exist, have impact, and not be observable.

How? "Have impact" is synonymous to "in principle observable".

> E.g. some entities or energies at the sub-Plank scale level could hold the universe together and be essential substrate. But we just have no way to know about it, because we can't look at that level (and our theories can't even go at that level).

I think you could only say such entities exist in a meaningful sense if their non-existence would make the universe look different. This way you could indirectly infer their existence through considering the things you've observed and not observed. If the sub-structure of the universe was completely screened away, so that whatever happens there, no bit of information reaches our universe, then I see no meaning in the question of whether or not that sub-structure, or entities inhabiting it, exists.

Again, by "observable" I at not point meant "observable by us, today". I meant "observable in principle".

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[0] - https://physics.stackexchange.com/a/279493