My studies keep leading me back to the seemingly intractable regions of chaos. I wonder if I will ever see something in all of this that will lead me to any clear path to a solution.
Having cleared up the way of constructing the gaps for m=c2-r between the first two Remainder Series quantities, r and r+2c+1 and consecutive perfect squares of corresponding oddness/evenness, I determined that the nice, suggestive “constellations” of the modulo values led me to nothing other than the place where examining the Zone Boundary lines had led me before. And when I attempted to graph the “new constellations” (see below) based on the before-and-after gaps for any particular m, even that nice “patternicity” dissolved, leaving me with arrangements of dots that resembled nothing so much as the scattering created by bifurcation and chaos in dynamical systems. Not too encouraging.
I eagerly await the next inspiration that will hit me, even though all such up to now have been met with great excitement but led me to a point not very much further along any revelatory road.