Are you familiar with the Gödel metric? Time travel may well be impossible, but at least within the context of general relativity it is plenty well-defined to reason about.
Does Gödel metric say anything about prohibition of paradoxes?
I have found that assigment of 0 to a paradoxical configuration is an opinion or assumption and not a result or a theorem.
For example the electron wave function negative solutions were dismissed as unrealistic math artefacts at one point and later adopted as a valid way to reason about positrons. Would it have been correct to say that “obviously electrons have a negative charge?. In modern terms you can distinguish between claims of “leptons have negative charge” and “electrons have negative charge”. But if the only lepton you know is an electron is “positive electron” a valid thing?
While things add up to normality unusual circusmstances can exhibit unsual phenomena.
The real question here is what mechanics + GR “says” about paradoxes; there’s nothing special about the Gödel metric other than that it’s a specific example of a system containing closed time-like loops.
The answer is that mechanics + GR cannot represent a system containing a paradox, at all. We just have a bunch of particles and/or fields moving around a space with a metric. The local laws of mechanics + GR constrain their behavior. A “paradox” would, for instance, assert that there is a particle at (x, t) with velocity v, but also not a particle at (x, t) with velocity v—the underlying theory can’t even represent that.
We don’t know how to integrate QFT with GR, but conceptually a similar problem should arise: we just have some quantum fields with complex amplitudes at each point in spacetime. A paradox would assign two different amplitudes to the field at the same point. Again, our physical models can’t even represent that: the whole point of a field is that it assigns an amplitude at each point in spacetime.
We could maybe imagine some sort of multivalued state of the universe, but at that point our “time machine” isn’t actually doing time travel at all—it’s just moving around in a somewhat-larger multiverse.
As we lack the means to represent the different options we probably do not have a law that paradoxes will be avoided (partly because we do not have a technical analogoue for “paradox”)
In the extended ontology what corresponds to old time would be an open question. That is if you have a multivalued state in the past and some of the values of that are effects of (partial) values in the future it’s still pretty much “time travel”.
I also thought that qunatum mechanics is pretty chill with superposition. Could not one extend the model by having a different imaginary unit and then have a superposition of amplitudes? And I thought getting a sure eigenvalue is a special case. Isn’t the non-eigenvalue case already covering a simultanoues attribution of multiple real values? I case there are two cases 1) we do not represent that currently in our models or 2) Our representations used in our models can not represent that.