Have you looked to see what the almost-nullspace looks like? That is, what not-very-short vectors are mapped almost-to-zero? Or, to put it differently: your empirical finding shows that some distributions of mass have almost the same effect on your 83 red dots as others; what are they?
My best guess, which I don't trust much: your 83 red dots are all at a single distance, so any distribution of mass that looks "enough" like a spherical shell of any size smaller than that will look (up to scaling the masses) exactly like any other, and perhaps your nearly-nullspace vectors look like linear combinations of spherical shells.
