We observe numerically, and explain analytically, a previously unknown phenomenon of quantum-Hall-like jumps in saturation spectral rigidity in the semiclassical spectrum of a modified Kepler problem as a function of the interval center. These jumps correspond to integer decreases of the radial winding numbers in classical periodic motion. We also observe and explain single-harmonic-dominated oscillations of the level number variance with the width of the energy interval. The level number variance becomes effectively zero for the interval widths defined by the frequency of the shortest periodic orbit. This signifies that there are virtually no variations from sample to sample in the number of levels on such intervals.