I'm not sure this is a fully satisfying answer. For any observable I could ask, for example, whether you can design a series $n={1,2,...}$ of experiments to measure that observable with precision ${1/2^n}.$ Then there's a reasonable question as to whether the possibility to 'measure zero' is in this sense is different between classical and quantum theories. This is a totally reasonable set of physical data to think about.

Note also that the 'best fit' model of DESI is way outside the region of parameter space considered in the Linder papers you referenced, or seen in their 'realistic' models. So I think it's hard to call it physically motivated still.

Do you know somewhere it's shown that the DESI preference for their $w_0 w_a$ model is driven by the lowest redshift bins? I would think the preference for a rapidly changing equation of state must come from bins where that deviation would be felt. If indeed that preference is driven by $z\sim 0$ I'd feel better about it.

It's also difficulty to take seriously the model DESI used of the dark energy equation of state because their 'best fit' has $w<-1$ in the recent past, $z\sim 1$ of something. And it has been shown that this behavior cannot appear in a causal, Lorentz-invariant quantum field theory Hartman, Kundu, Tajdini (2016), the framework of which we know describes our world quite well. Perhaps more precise data will still show some deviation from the behavior of a cosmological constant, but it won't end up being like this.

This isn't really a physics answer, as it does not disambiguate between chemistry really being independent of the details of short-distance particle physics vs. chemistry being so complicated that we don't understand how to connect it to short-distance particle physics.

No, this is not correct. See for example this answer physics.stackexchange.com/a/353157/386743

@ohwilleke I certainly agree, and the point of understanding a spurion analysis of the SM is that the strong CP problem very generically occurs when you write a UV theory which explains the features of the SM. And as you said "So, if we are thinking about a UV theory that supposes that this should not be so, maybe we are barking up the wrong tree in that whole approach." which means that indeed we must be sure our UV theory includes a solution to the strong CP problem.

@ohwilleke Physicists have always been interested in discovering more-fundamental UV theories which explain the features of the IR theory, from QED explaining the cluttered periodic table to QCD explaining the otherwise ad-hoc patterns of nuclear physics. With the SM we do know UV theories which explain its structure, e.g. GUTs and theories of flavor. And of course we know we need further physics for neutrinos, DM, baryogenesis, the Landau pole, etc. I think imagining the SM structure could be fundamental to arbitrary UV scales is silly.

This is a mischaracterization of why particle physicists care about naturalness. The strong CP problem is sharp in a UV theory where theta bar is an output, rather than an input. For example if CP is an exact symmetry in the UV--the puzzle is generating large CP violation in e.g. CKM while keeping theta bar small, since it generically gets renormalized to O(1). We can understand the strong CP problem generally by spurion analysis of the SM which warns us of this issue. Of course if it was only the SM into the UV there's no strong CP problem, but we would like a UV theory which explains the SM.

There is a $U(1)\simeq SO(2)$ subgroup of the remaining symmetry in this case for N>2. But N=2 does constitute an example of no remaining symmetry. Indeed you can introduce scalars to just spontaneously break all the symmetries.

Well, if your theory obeys charge conjugation symmetry then $C$ applies to every process. I don't know the nuclear physics language so I may not follow everything you mean. I think you should think about an off-shell photon turning into the $C$-odd superposition of those states, as $\langle \gamma | \left( |\bar p \Delta \rangle - |p \bar \Delta\rangle \right) \neq 0$.

Perhaps you're missing Srednicki's rule 6 which says for each internal line labeled with momentum $p$ you get a factor of the propagator with momentum $p$. Those particular momenta are dictated by momentum conservation, since $k + \ell - (k+\ell)=0$. But you could split up the momenta in different ways, e.g. $\ell+k/2$ on the upper leg and $\ell - k/2$ on the lower leg, as that still satisfies $k + (\ell - k/2) - (\ell + k/2)=0$. That's fine, these are equivalent by a change of variables of $\ell$ and will produce the same result.

I can't say I follow your objection. We agree there is lots of evidence that quantum field theory describes our universe, so if we can use mathematics and theoretical physics to prove something about the physics of all quantum field theories, we learn something about our universe. And something much more general than just how physics works with weakly coupled interactions which have classical limits.