Experimental confirmation of the textbook explanation for local entropy reductions Clarification: In my original wording of this question, regrettably, I did not make it clear that I am interested in the the Boltzmann/Gibbs/statistical (BGS) interpretation of it, as opposed to the Carnot/Clausius/"classical" (CCC) interpretation.  I know that these two interpretations are supposed to be equivalent, in principle.  This equivalence is something I can only take on faith.  Be that as it may, for the sake of this question, I would like to exclude the CCC interpretation altogether, and restrict the discussion to the BGS interpretation exclusively.  In particular, when I write entropy, I mean Boltzmann's
$$S = k_B \ln W .$$

Textbook discussions of the Second Law of Thermodynamics (SLT) often stress that this law applies only to "closed systems".  Or, differently stated: if the system is not closed, its entropy can go down.
The argument is further refined as follows.  Suppose that $B$ is a closed system, and that $A$ and $A^{'}$ are "disjoint" subsystems of $B$ such that $B = A + A^{'}$.  Then, by SLT,
$$\Delta S_B = \Delta S_A + \Delta S_{A^{'}} > 0\label{a}\tag{1}$$
...but this does not rule out cases where
$$\Delta S_A < 0\label{b}\tag{2}.$$
In such case $\Delta S_{A^{'}}$ would be positive and such that $|\Delta S_{A^{'}}| > |\Delta S_A|$ thereby preserving the inequality in $(\ref{a})$.

My question is: How is this particular aspect of the theory verified experimentally?
IOW, is it possible to set up an actual (as opposed to "Gedanken-") experiment where all three quantities in $(\ref{a})$ are measured, and such that, with the measured values, both $(\ref{a})$ and $(\ref{b})$ hold?
Or is the argument given above simply a purely theoretical (i.e. experimentally unverifiable) construct to explain why apparent reductions in entropy1 are not necessarily violations of SLT?

1 ...such as in the formation of ordered crystals from a disordered collection of molecules in solution.
 A: More of an extended comment, but here are two thoughts:
1) I'm not sure I completely agree with the statement 

Textbook discussions of the Second Law of Thermodynamics (SLT) often
  stress that this law applies only to "closed systems". Or, differently
  stated: if the system is not closed, its entropy can go down.

Okay, this is correct, of course, but it is misleading because it implies that there is no comparable rule for open systems.
In fact, when we apply the second law of thermodynamics to a subsystem that can exchange energy with a larger system we get a generalization: the minimization of free energy, $F=U-TS$, or in the most general case the grand potential. So one can claim instead that in a system which can approach thermodynamic equilibrium, the free energy decreases. This rule is, as far as I know, universally valid, with the exception of rare and bounded fluctuations away from equilibrium that are negligible for a sufficiently large system. And the experimental evidence is every system described by thermodynamics.
2) That said, I completely agree that it is reasonable to want to experimentally observe systems equilibrating and verify that their behavior satisfies statistical mechanics more directly. Along these lines, you may be interested in a recent experiment studying thermalization in an isolated quantum system (1). The object of study in this system is a slightly different form of entropy than your $S=k_B \ln W$, both because a generalization of this is necessary for quantum systems and due to experimental limitations on what is directly measurable. But, nonetheless, for this related entropy (known as the second-order Renyi entropy), the authors can microscopically and directly observe all the subsystems as they increase in entropy and thermalize.
A: From the links I provided in the comments below your question it should become clear that entropy "meters" do not exist, you calculate it from other measured variables. If this does not satisfy you requisites for an experimental measurement, then your conclusion that the claims are only theoretical is justified. 
However, having said that, with your criteria of experimental measurement a large portion of physics that is now considered experimental should  be considered only theoretical, because in plenty of cases variables and other physical quantities are measured in an indirect form, you measure a few variables for which you have  "direct" measurement devices and used these values to compute that of your variable of interest. In common physical practice that is also considered a measurement. 
