Sorry, but no. First, who cares that the fractional quantum Hall effect breaks Lorentz invariance? Tons of things do that—turn on a strong electric field in your lab and WHAM! Lorentz invariance broken. Great. That doesn’t mean you can’t make general statements about Lorentz-invariant systems, which our fundamental theories are. (If you actually, y’know, read literally any textbook on the spin-statistics theorem (e.g. Schwartz), you’ll see that there is a dimension restriction in the theorem statement for this very reason.) This is just a category error coming from a total lack of understanding of the theorem.
Second, I don’t know where you’ve been getting your information from, but that’s simply not true. Locality is just a rephrasing of special relativity. Causality is another, somewhat stronger version of this: it just says that things in the future can’t affect those in the past, i.e. you can only affect the future lightcone. You can also state this in terms of (anti-)commutation relations, which is what you’re referring to. These relations are usually referred to as “micro-causality”.
By the way, the reason locality/causality “shows up everywhere in physics” is because the world obeys the laws of special relativity and because special relativity = locality. It’s not mysterious, and physicists aren’t just magically so stupid that they’re blind to one of their most important theorems being a tautology. Theoretical physicists do “actually know what they’re talking about”—you obviously don’t.
Sorry, but no. First, who cares that the fractional quantum Hall effect breaks Lorentz invariance? Tons of things do that—turn on a strong electric field in your lab and WHAM! Lorentz invariance broken. Great. That doesn’t mean you can’t make general statements about Lorentz-invariant systems, which our fundamental theories are. (If you actually, y’know, read literally any textbook on the spin-statistics theorem (e.g. Schwartz), you’ll see that there is a dimension restriction in the theorem statement for this very reason.) This is just a category error coming from a total lack of understanding of the theorem.
Second, I don’t know where you’ve been getting your information from, but that’s simply not true. Locality is just a rephrasing of special relativity. Causality is another, somewhat stronger version of this: it just says that things in the future can’t affect those in the past, i.e. you can only affect the future lightcone. You can also state this in terms of (anti-)commutation relations, which is what you’re referring to. These relations are usually referred to as “micro-causality”.
By the way, the reason locality/causality “shows up everywhere in physics” is because the world obeys the laws of special relativity and because special relativity = locality. It’s not mysterious, and physicists aren’t just magically so stupid that they’re blind to one of their most important theorems being a tautology. Theoretical physicists do “actually know what they’re talking about”—you obviously don’t.