Mesoscopic Blindness of Hypersingular Riesz Polarization: Defect Budgets, Tangent Coercivity, and Gap-Moment Thresholds

We study quantitative stability and instability for one-dimensional hypersingular Riesz polarization. For the periodized Riesz kernel of exponent s > 1, equally spaced points have optimal polarization with exact lattice constant Cs = 2(2s − 1) ζ(s). We construct explicit near-maximizers with persistent mesoscopic density defects. If RM = o(M) consecutive gaps are compressed by a fixed factor c > 1 and the missing length is spread over the remaining gaps, then the relative polarization deficit is s(1 − 1/c)RM/M up to lower-order terms, while every growing subblock of the compressed region has relative discrepancy c − 1. Thus a single near-optimal configuration carries a whole cone of bad mesoscopic scales. A localized defect-budget theorem shows that, to leading order, the polarization loss depends only on the total amount of compression inside an o(M) defect zone. Consequently no deficit-only modulus can control discrepancy on all growing submacroscopic scales, and no universal recovery threshold o(εM) is possible. We also develop a positive tangent theory at the equally spaced configuration. For the inverse-square case, the exact first-variation Fourier symbol and a finite sine-sum Green kernel give the sharp finite-dimensional tangent constant Si(π)/π. For every s > 1, the first variation factors through a positive probability convolution of the gap deformation. Its Fourier multiplier has a dimension-free positive spectral floor, while its long-wave limit yields the sharp continuum ratio 1/s, exactly matching the compressed-block obstruction. Finally, we construct a cluster-compensated hole of mesh length G whose polarization deficit satisfies εM = Os(Gs). This implies that no universal gap-moment estimate Σ|gj − 1|p = O(εM) can hold for p > s; in particular, global quadratic gap stability fails for 1 < s < 2. The full nonlinear recovery problem on scales r ≫ εM remains open.