Isn't continuous Fourier Transform simply the limit of Fourier series as the fundamental frequency approaches zero though? That's what I meant when I said "all

Isn't continuous Fourier Transform simply the limit of Fourier series as the fundamental frequency approaches zero though? That's what I meant when I said "all of free space", it's the same idea with the emergence of "continuous" band structures in bulk materials -- although their sizes aren't strictly infinite, for all practical purposes it's long enough that the energy levels are sufficiently close-together to be considered a continuum. And yes, I stand corrected. My analogy with overtones bandwidth vs spatial localization instead pertains to the position-momentum tradeoff, it is indeed incorrect to overgeneralize it to what the classical limit means. The classical limit, as you correctly pointed out, is more about the tradeoff between the two being practically negligible in the length/impulse scale being dealt with in the classical regime -- effectively a "rounding error", so to speak.