Ah, ok, I think I've figured it out. Quantum mechanics is computable. But the Schrödinger equation is very versatile and can be applied to particles that don't actually exist in our universe just as easily as those that do. So physicists come up with "quantum field theories" that aim to describe only the exact universe we have or some simplification of it, and exclude the forces and particles that don't exist. Those quantum field theories are often not rigorously mathematically formalized, and we don't necessarily know how to compute them. Scott Aarronson's second post was talking about a QFT, not quantum mechanics itself.

So this will resolve YES unless I'm mistaken about what's going on here. Perhaps @ScottLawrence would be willing to confirm whether I'm off the mark?

@IsaacKing Miscellaneous comments, attempting to go roughly in order of increasing relevance to the question. I don't check manifold anymore but if you @ me in the response I'll get an email and be sure to answer any questions.

1. Quantum field theories do not attempt to exclude particles that don't exist. There are plenty of QFTs---even interesting ones---that are not believed to reflect our universe in any meaningful way. For a trivial example, Yang-Mills with four colors. For a less trivial example, the famous [N=4 SYM theory](https://en.wikipedia.org/wiki/N_%3D_4_supersymmetric_Yang%E2%80%93Mills_theory). There are also QFTs used to model various phenomena in condensed matter physics; maybe the most famous is the Ising model in three (spacetime) dimensions.

2. The goal of a "true" quantum field theory (terminology invented for this post) is to simultaneously accommodate both special relativity and quantum mechanics. We can also describe "partial" quantum field theories that only approximately accommodate special relativity. For instance, if you go to small enough distance scales, you might see a lattice. Depending on context you might refer to these objects as "effective field theories" or "regularized field theories" or "lattice field theories".

3. It is often said (and your post flirts with this framing) that nobody knows how to rigorously formalize a QFT in 4 dimensions. This is somewhere between very misleading and false. There is a procedure for defining a ("true", special-relativistic) QFT which involves taking a limit. What nobody knows how to do (see the Yang-Mills millennium prize problem) is prove that the limit exists. It is widely believed that it does, however, and the entire field of lattice QFT (and lattice QCD in particular) is based around this fact.

4. The case of field theories where you see a lattice at small distances is a special one from the point of view of computational techniques. Lattice QCD computes the properties of nucleons and (small) nuclei via a monte carlo method in the lattice-regularized version of QCD. This version isn't really QCD because it lacks Lorentz invariance (i.e. violates special relativity). In order to access QCD, we take a particular limit (the limit where the lattice spacing becomes small). In this sense QCD, at least, is clearly "computable".

5. Here's an uncomputable problem which is quantum-mechanical in nature: deciding whether a certain QM system has a "gap" (meaning that there's a finite difference in the energies between the lowest- and second-lowest-energy states). See https://arxiv.org/abs/1502.04573.

6. Aaronson's post is addressing technical challenges with putting fermions on a lattice. I have never seen anyone even suggest that these technical challenges were due to uncomputable problems hiding within the physics of fermions. It is typically the case that these challenges make simulating fermions inconvenient and expensive (perhaps occasionally exponentially expensive in system size), but never (in any case that I'm aware of) impossible in the "uncomputable" sense.

7. I won't say too much more about lattice fermions because it's a technical area I've generally avoided whenever I could! However, the stuff about "fermions at the boundary" Aaronson alludes to is almost certainly this: https://en.wikipedia.org/wiki/Domain_wall_fermion

Probably I missed a bunch and moreover I'll bet that what I did write was unclear. Feel free to ask questions (or gesture vaguely at "this area") and I'll do my best to clarify.

On the topic of resolution: I think you should try to define very precisely what it would mean to you for quantum mechanics to be (un)computable.

@ScottLawrence Thanks! I didn't know you don't use Manifold anymore, I'll avoid tagging you in the future so as not to bother you unless it's something important. I hope you won't mind one follow-up to this:

1. The paper you linked as an example of uncomputability only applies to infinitely large objects, right? It doesn't mean any physically-realizable object has uncomputable behavior? (Going by another Scott Aaronson post.)

2. My question was spawned by a very bad philosophy paper that claims the brain of any organism is Turing-incomputable. If this conclusion were true it would disprove the physical Church-Turing thesis, but since that thesis is speculative in the first place, that's not a maximally compelling condemnation of the paper. So I was curious whether I can reasonably say that this paper ostensibly disproves quantum mechanics, since that's an even more ridiculous claim. Of course I know that quantum mechanics has already been partially "disproven" due to its incompatibility with GR, but the incompatible behaviors only happen near systems like black holes that are hard for us to examine. If the brain of a nematode violates QM, that would be a much more actionable research direction for the search for a theory of everything. So my actual question is "does this paper logically imply that there are trillions of easily accessible physical systems on the Earth that violate quantum mechanics, thus clearly demonstrating how ridiculous the paper is?"

3. On that note, when it's said that QM and GR are incompatible, does that mean that both must be wrong, or only that at least one does? Is it possible that QM does hold over all exotic physical systems like singularities, and it's GM that breaks? Or do we know for sure that QM is wrong, either because we know it will return an answer that's incompatible with our observations, or because it doesn't return an answer at all for some systems?