I do not know if the borrow checker and Rust's type system have been formalized. There are stacked borrows and tree borrows, and other languages experimenting with features similar to borrow checking, but without formal algorithms like Algorithm W or J for Hindley-Milner, or formalizations like of Hindley-Milner and problems like typability for them, I am not sure how one can prove the complexity class of the borrow checking problem nor a specific algorithm, like https://link.springer.com/chapter/10.1007/3-540-52590-4_50 does for ML.

I could imagine you being correct about the borrow checking typability problem being NP-complete. Or an even worse complexity class. Typability in ML is EXPTIME-complete, a larger set than NP-complete https://en.wikipedia.org/wiki/EXPTIME https://dl.acm.org/doi/10.1145/96709.96748 .

I also am not sure how to figure out if the complexity class of some kind of borrow checking has something to do with the exponential compile times of some practical Rust projects after they upgraded compiler version, for instance in https://github.com/rust-lang/rust/issues/75992 .

It would be good if there was a formal description of at least one borrow checking algorithm as well as the borrow checking "problem", and maybe also analysis of the complexity class of the problem.