So the most physically grounded example I know of is the undecidability of some spectral gaps[0] (is there a discrete jump between the ground state and the excited states), which as far as I can tell doesn't rely on self-reference.[0]: https://arxiv.org/pdf/1502.04573.pdf
calf|2 years ago
> Rahul #97: No, because the other point is that whether a spectral gap goes to zero as the lattice size goes to infinity is the kind of thing that real condensed-matter physicists and quantum field theorists ask all the time. It’s something they accept every day as a good mathematical idealization for what they care about. So it’s worthwhile to know that their problem secretly contains the halting problem—that for large enough constant d, there can never be a clean algorithmic criterion to tell you which nearest-neighbor qudit Hamiltonians are gapped and which are gapless.
https://scottaaronson.blog/?p=2586#comment-975416
If I read this he's clearly implying the spectral gaps problem is done by reduction to the halting problem.
(Edit, in the blog entry he explains: "Cubitt et al.’s theorem now says the following: for some fixed, constant local dimension d, there is no algorithm that takes as input the local Hamiltonian h (say, as a d2×d2 matrix of algebraic numbers), and that decides whether the material is gapped or gapless. Indeed, you can reduce the halting problem to that problem, in such a way that the material will be gapped if your Turing machine halts, or gapless if it runs forever.")