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I think you have an over simplified view of the problem.
The PJ is not a point source of sound, rather there is a fairly large surface that will emit sound. There's many components between the fan blade and fan motor (and indeed all the other blades and motors) that will together add a serious number of harmonics and distortions.
The PJ then distributes this sound energy with a fairly omnidirectional erm distribution. The sound that you hear will be a complex mix of single-path (direct to ear) and multi-path (reflected from surfaces within the room).
To cancel the sound that you hear, you'd need to consider how all path sources would mix and then (and only then) consider how you're going to *accurately* reproduce that sound (and the easy bit) at 180 degrees out of phase.
This is the reason that noise-cancelling headphones work - being placed in the ear canal (or by the ear) they only need to deal with the (largely) single path noise (i.e. the single-path and multi-path are already mixed once in the ear).
Personally, I prefer the earlier solution(s) - quieter PJ, bigger Amp.
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