First of all, the numerical stability does not depend on the round-off error. You can see the original paper of von Neumann, the discrete equation shows the stability property even if the exact precision is used, that is if you compute by hand.The round off error is a supplementary source to this framework and, depending on the spectra of the error distribution, can affect a result.

For this reason at least the double precision is mandatory in scientific commutation.

In my field when I was still researching on academics. Solitons in periodic, localized, PT, non PT... The first thing to note are the simmetries of the system, their reflect on the structure of the resulting stability matrix itself. I tip I can give is to preserve in discrete calculations, I can't give more details because I am self employed and there are publications using solitons as a way to compute and model llms in physical systems. But without offense, I have already done some similar, but using the mathematical methods, resulting equations solutions, propagation so forth without the need to create a messy composition of layered plastic materials. I am transitioning from theoretical physics, to Machine learning and AI not large models. there is then the basic example of FFT and it's inverse. Sampling, etc. Under right conditions, there are no errors in the sampling process. I am also looking for jobs, collaboration and opportunities, because well, I got bored and people are going crazy on making all these arbitrary models, using this approach let's try this and see what happens.

Back to the answer, no asymmetry on the eigenvalues are bogus, assume they are in the band. but symmetry also makes some overlapping and behaviors like under sampling. So Usually try to supper-impose different number of samples, the size of the box and play with both. Unstable eigenvalues will stay put, false positives will not.