@IosifPinelis - I don't have a fully worked-out example, but if I were to go about constructing one, I'd start with something that's constant in $s$, except for a switch in behavior at a certain point, say, $L=\theta (-(t+s)/2) A + \theta((t+s)/2) B$, where the operators $A,B$ are time-independent. If we don't insist on the types of operators specified in the OP, we could use Pauli matrices, say, $A=\sigma_{x} $, $B=\sigma_{y} $ to get something that can be worked out explicitly. Insisting on the OP's form of operator, it gets messy, because we need the $\nabla^{2} $ to have it self-adjoint.

At its core, this question seems to be a variation of this companion question. I.e., in general, (1) does not imply (2). If $[L_t ,L_{t'}]=0$, then (1) does imply (2).

My impression is that neither definition is usually used, since most people never worry about this, i.e., never bother to define it. Would you worry about this if the metric were $(+,-,-,-)$? Maybe this is an example of an issue where the choice $(+,-,-,-)$ is preferable because it reduces confusion.

Hm - not sure where to point you in the mathematical literature. Objects of this type are standard in physics textbooks, where they are related to time evolution - that's where I'm familiar with them from. But maybe you don't want to start rummaging around there. The wikipedia entry on "ordered exponential" seems ok and the references there might provide a lead.

The square bracket is the commutator, $[B,C]=BC-CB$.

Why do you surmise the representation in your question to be true? It seems to me to hold only for $A(t)$ constant in $t$.

Ah, you're talking specifically about the fixed point(s) already, that explains it. Might be worth clarifying that, in case other readers get stuck on this point ...

Yes, and that is what confuses me - in a renormalizable theory, the coupling runs. So, what is the meaning of "the coupling terms remain unchanged"? The coupling of a renormalizable theory can run to 0 (asymptotic freedom), or to $\infty $ (at least perturbatively, i.e., a Landau pole) ...

I'm not sure what is meant to be unchanged/decreasing/increasing here. In a renormalizable theory, the values of the couplings change with energy. The point of renormalizability is that there are only a fixed number of such couplings that need to be adjusted. Non-renormalizability means that the number of such couplings proliferates. Super-renormalizability means that the couplings don't run (just a few infinite constants might have to be subtracted).

Yes, I think we've converged. Sorry about the intermediate mixups.

Ok, $x=2\sqrt{f(1-f)}\ / \pi - (1/\pi ) \arctan(\sqrt{(1-f)/f}\ (2f-1)/(2f-2))+1/2$.