I have to say Jack, though your horological expertise is not in question, you still seem to be missing my point about resonance.
Firstly: when I referred to the frequency of the unpowered oscillator, I was of course referring to its natural resonant frequency.
Secondly: if resonant effects only occurred when the frequency of the powered oscillator matched exactly , to an infinite number of decimal places, the resonant frequency of the unpowered oscillator, then resonance would never occur, because such a precise match can only occur at the quantum scale - and then it's not resonance, it's the photoelectric effect. (Actually, even at that scale, you could argue that a discrepancies smaller than the Planck length are irrelevant or perhaps undefined.) Hence it must be possible for resonance to occur when there is a slight mismatch between frequencies. Presumably the relationship between frequency difference and strength of resonance is a normal curve. (Physicists?)
We're discussing watches here, right? Are we not interested in the application of this physical phenomenon of resonance to the achievement of better timekeeping? Then let's talk about whether resonance could be used this way.
My argument, which I'll make again using slightly different wording, is as follows:
1. Resonance can occur when the frequency of the powered oscillator is close to, but not necessarily an infinitely precise match with, the natural resonating frequency of the unpowered oscillator.
2. The powered oscillator would tend to influence the unpowered oscillator to beat at the former's rate.
3. Every action has an equal and opposite reaction. Hence the unpowered oscillator would also influence the rate of the powered oscillator.
4. Two powered oscillators with similar natural resonant frequencies, once set in motion, would influence each other. The effect would be an averaging of the individual rates.
5. In a watch, where external effects such as temperature, position, or shocks might differentially disrupt the rates of the two escapements, the above averaging effects would mitigate towards increased stability and better timekeeping.
This is fine in theory. Whether such benefits redound in practice is a matter of conjecture until someone does some tests. And since FPJ is selling bucketloads of highly priced watches despite the absence of said tests, why bother?
Oh, and if point 5 is right, then I submit that a better arrangement would be to have the two escapements mounted at right angles. Better still, have three escapements all mounted at right angles to each other. Then we definitely wouldn't be able to see the hands to tell the bleedin' time!