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

Abstract

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 $P^*_{M,s}=C_sh^{-s}$, where $$C_s=2(2^s-1)\zeta(s),\qquad h=L/M.$$ We construct explicit near-maximizers with persistent mesoscopic density defects. Compressing $R_M=o(M)$ consecutive gaps by a fixed factor $c>1$ and spreading the missing length over the remaining gaps gives $$\varepsilon_{M,s} =s\left(1-\frac1c\right)\frac{R_M}{M} +O_s\!\left(\frac{R_M^2}{M^2}+M^{1-s}\right),$$ 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, and no deficit-only modulus can control all growing submacroscopic intervals. In the sharp regime the largest bad scale is of order $\varepsilon M$, so no recovery threshold $o(\varepsilon M)$ is possible. A more general localized construction shows that the leading polarization cost depends only on the total compression budget. At the equally spaced configuration we identify the first variation exactly. For $s=2$ its circulant Fourier symbol is $$\lambda_q=-i e^{\pi iq/M}\frac{4\pi^3}{M^2}q(M-q),$$ and Green-kernel inversion yields the sharp finite-dimensional tangent constant $\operatorname{Si}(\pi)/\pi$. For every $s>1$ the first variation factors through a positive probability convolution of the gap deformation. The convolution multiplier has a dimension-free spectral floor $$c_s=\frac{2^s\beta_{\rm D}(s+1)}{(2^s-1)\zeta(s)}>0,$$ where $\beta_{\rm D}$ is the Dirichlet beta function. Its low-frequency limit gives the sharp continuum ratio $1/s$, exactly matching the compressed-block obstruction. Finally we exhibit a second nonlinear defect mechanism: a large empty gap screened by endpoint clusters. If the hole has mesh length $G$, it can remain near-optimal at cost $\varepsilon M=O_s(G^s)$. Consequently no universal estimate of the form $\sum_j\vert{}g_j-1\vert{}^p\lesssim_{s,p}\varepsilon M$ can hold for $p>s$; in particular, global quadratic gap stability fails throughout $1<s<2$. The full nonlinear recovery problem above the scale $\varepsilon M$ remains open.