Directed radial inversion energy and uniform offset-gap estimates

We study radial gaps between corrected inner and outer distance offsets of volume-constrained minimizers of an anisotropic fractional perimeter. For small even, mean-zero angular perturbations of the isotropic kernel, we bound the measure of bad directions by the actual directed radial inversion energy at a fixed center. The estimate retains both the offset distance and the gap threshold, with constants uniform for fractional orders in a fixed interval bounded away from zero. Its proof propagates ordered density blocks to every finer distance scale: a binary phase chain gives a quantitative lower bound for actual directed pairs, and disjoint distance bands can then be summed in the original energy. Together with a spherical-foliation defect budget, this yields uniform decay of the bad-direction measure at fixed geometric ratios. No graph or convexity assumption is imposed on the original minimizer.