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Abstract

We study the degree at which polynomial square integrals recover all isolated negative atoms of a real signed measure. Fix the number of atoms k≥2, their separation from the remainder d>0, and a Gaussian moment rate c>0. For normalized atomic weights 1/k, remainder variation at most R, and a fixed Gaussian moment budget, the worst degree for recovering k negative directions is asymptotic to (1)/(8cd^2)[log(R/Δ^(2(k-1)))]^2 as the minimum atom separation Δ tends to zero, allowing arbitrary jointly varying R≥1. The upper bound uses classical Laguerre estimates and Lagrange cardinals; a vanishing-jet subspace gives the matching lower bound. For an exponential rather than Gaussian budget, the same obstruction shows that the collision power in an earlier sufficient bound is necessary on the logarithmic scale, for every fixed k. For every weaker tail e^(c|x|^p), 0<p<1, we give a fixed-budget positive background that hides an isolated negative atom from every polynomial square above an explicit contrast, and extend this construction to compact negative measures. The classical kernel, orthogonal-polynomial and Stieltjes-class ingredients are credited explicitly; the conclusions concern degree and negative index, not computational complexity.