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Hi Lev, Thanks for this interesting contribution to the forum. I didn’t follow the controversy in your Refs [1-5] so perhaps I’m just ignorant of the context and hence missing the point. But my initial impression is that the whole question here is based on a kind of equivocation on the word “particle”. You define “counterfactual communication” as “communication without particles present in the transmission channel”. OK, so then is it “counterfactual communication” if I send morse code to my friend (on the other side of the lake) by means of ripples that propagate across the lake? Or is communication by means of information encoded in electromagnetic waves (taking here the perspective that classical Maxwellian electrodynamics is true, i.e., ignore photons) “counterfactual”? If the answer to those questions is “yes” then I suppose you are right that this simple setup involves counterfactual communication according to Bohm’s theory. But then I’d wonder why in the world I should care about this strangely and arbitrarily defined notion, “counterfactual communication”.
On the other hand, presumably, to whatever extent the idea of counterfactual communication is actually interesting, we should understand the word “particle” in your definition more broadly — basically as meaning “anything physically real”. In which case, obviously, sending messages using waves (water or electromagnetic) rather than particles, would fail to count as counterfactual communication. And in that case it seems like this setup, from the point of view of Bohm’s theory, would also fail to count — because although the Bohmian particle may not go through the arm of the interferometer that takes it past your mirror 1/2, *something* physically real (namely the “empty half” of the wave packet) does take that route. So there is a perfectly clear physical mechanism by which Bob’s choice about whether to place the mirror at 1, or 2, influences the subsequent motion of the particle.
That is, my reaction to your final question “Is it counterfactual according to the Bohmian perspective?” is “It depends on exactly what you mean — but surely there’s nothing puzzling here?!” But like I said, maybe I’m just missing what makes this puzzling…