I have seen it said that if we want to avoid violations of causality (I can see why we would want to avoid that!), we need the information-transfer "speed limit" of c.
I presume(?) this assumes some other stuff in the model though? Consider a 2D universe of discs moving uniformly and bouncing off each other elastically. It's not obvious to me that those violations of causality are possible in this universe, and I have not defined squared interval or anything like that.
Is it only when you bring in GR or electromagnetism that you need extra stuff to maintain causality? Is there any generality to this use of the word "causality"?
Optional philosophical part:
I am a fan of Quantum Computing researcher Scott Aaronson, who has a convention-defying spiel in his book and course. I'm going to paraphrase him / say how I see his thinking:
quantum mechanics is totally natural and inevitable and one of the best things you could possibly base a universe on. Ok, so we had it forced upon us by experiment, so it's not like we have a choice in whether or not to view the world as being based on QM. However, even if it wasn't, QM would still be extremely interesting and sensible, because if you mathematically describe certain things you might want the world to have (algebraic closure for example), you'll find QM drops right out of them. And in particular, all the folks telling you it comes out of nowhere and is just unpleasant and convoluted crap coming from experiments that you just have to deal with - they are wrong.
More generally than this thing about causality, what Scott Aaronson wants/says he has for QM, I would like for SR/speed of light/relativistic product.
One could well disagree with Aaronson about the above! It would be polite to read his book first. But even in all of that, he admits he is still not completely sure about some parts (eg https://www.scottaaronson.com/blog/?p=4021). But maybe, for SR, we DO have this "drops out of reasonable mathematically-stated requirements for a universe design"?