Even in the smooth case, what do you do if it turns out that $A=0$?

The parametrization in the OP is also clearly symmetric in $a$ and $b$, one just has to couple the exchange of $a$ and $b$ with $s\rightarrow 1-s$. Whereas in your result, it's coupled with exchanging two variables, $u_2 $ and $u_3 $. In that sense, the structure is quite different. Interesting that one can use Feynman's trick to go into diverging directions here.

@Upax - I doubt the dilation operator is of use for your problem. Even in the case of simple dilations (your expression is more complicated), its use is mostly in establishing general properties, not in practical calculation (taking infinitely many derivatives?). If you're going to integrate, the straightforward thing to do is substitute $y=Az/(z-1)$ and hope you can do something with the prefactor ...

@Upax - More than you'll probably want to know is in G. Dattoli, P. Ottaviani, A. Torre and L. Vasquez, Rivista del Nuovo Cimento Vol. 20, N. 2 (1997) 3. It's discussed right at the beginning. Note that there's nothing specific to the hypergeometric function here, so mine is not a very illuminating answer - you get what you ask for ...

Whether resonance happens only in some regions of the system depends entirely on whether the system supports localized normal modes. It is not determined by localizing $f$. You don't excite regions of space, you excite normal modes. It's a different basis.

About your follow-up questions - just expand the driving force $f$ into the normal modes of the system. Then the problem decomposes into independent problems for each normal mode that each behave like the single harmonic oscillator you start out with in your post, and for each of them, you can assess whether the resonance condition is satisfied. You can excite any number of modes; if you want just one of them, just make sure that $f$ is such that the resonance condition is satisfied for only one of the modes.

@semisimpleton - it seems to me you have the ingredients: To solve the wave equation, you need $\nabla^{2} g = -k^2 g$ as well as $A(t)=ae^{i\omega t} $, and to make these fit together, $\omega = |k|$. Now, $A(t)$ satisfies a standard harmonic oscillator equation with $\omega = |k|$. This is without any external force; now you can couple to an external driving force like any other oscillator, and you get resonance around $\omega = |k|$.

Reality has a nasty habit of resisting formalization into a neat theorem that fits into an MO answer. Everything we're saying here is an approximate model of reality, to which the corrections have to be considered case by case. To some extent, you already gave the mathematification (up to the ingredient that coupled harmonic oscillators can be diagonalized), and I answered the remaining question as to why this is the right one. This type of question will often not have a clean formal answer, because that's not how reality is.

@TomCopeland - hmm, but that statement sounds to me like it's conflating two different meanings of 'classical'.

The Ising model would usually be considered a quantum system, so you've studied quantum statistical mechanics already!

@IgorKhavkine - oh, the very first thing I say is that it's a question about physics, i.e., already from that point of view, I do not think it has a good chance to be answered here (I absolutely agree with you on that). I'm just adding that it doesn't have a good chance over there either, even if the topic is better aligned.

It is indeed a question about physics, but would need more foundation and focus to have a good chance to be answered in a physics forum.