Dear Rob,
I need to review IIR and FIR as I've not studied signal analysis for decades (except a minor review of Fourier Optics occasioned by your comments last year). My argument is, essentially, that there are no signals to propagate. As you know, outside of the radius of the spherical mass, all the mass may be considered to reside at a point at the 'origin', independent of size or density. The scale-invariant solution of my equation says that the original density distribution of the field energy, hence mass, is scale invariant hence time independent. This implies that it remains the same during inflation, and I believe, post-inflation. My working assumption is that when the field 'condenses' to mass, this does not change the distribution (except locally). In other words there is no signal, step or otherwise, to propagate that would have any dark energy effect. With no signal, IIR and FIR should be essentially equivalent as far as their effects:
Scale invariance = static distribution = no signal to propagate
If one could, say, lop off half the universe, this would change the mass distribution significantly and your analysis would be more relevant. This is similar to the problem that would occur if the sun suddenly disappeared. The step function analysis of this problem seems correct to me.
Your remarks on difference versus differential are interesting and I'll give more thought to those. I certainly agree that reality has a finite bandwidth, finite speed, finite extent -- is finite period. Thus Fourier integration over infinite ranges are clearly approximations. This is why free particles have mathematically infinite distribution, which is clearly ridiculous (although quantum field theorists I know seem not to realize this). On the other hand bound particles such as a hydrogen atom are highly localized and more susceptible to differential treatment. The mistakes, in my opinion, occur when infinite solution techniques are applied to physical reality. I believe I recall you commenting on Dickau's blog about Kauffmann's treatment of self gravitation as limiting upper bound on local energy. Whereas quantum field theory has had these infinite energies for over half a century, Kauffmann shows them to be physically unrealistic (which should have been obvious anyway).
Edwin Eugene Klingman