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Abstract

We determine the worst polynomial degree needed to detect a negative atom under local variation and weak exponential-moment bounds. For a tail scale H(x)=x/W(x) with a controlled growth ratio, put Ψ(t)=1+∫₁ᵗdx/[xW(x)]. When Ψ diverges, the worst detecting degree satisfies Ψ(max{1,D_W(R)})=Θ(R^(1/q)) for every fixed local exponent q>0. The same estimates yield the full-line origin Christoffel order λ_n≍Ψ(n)^(-q) for the canonical density proportional to |x|^(q-1)e^(-bH(|x|)). Finite-clock masking and logarithmic critical scales follow within this framework. We also construct one smooth nondecreasing profile with H(x)/log x→∞ and divergent Ψ, for which the canonical density masks the atom at every degree for every fixed local exponent q>0 and density rate b>0. Thus divergent clock and existence of all moments do not replace growth control. The proofs combine classical quasianalytic propagation and localization with explicit moment and interpolation estimates, retaining signed and singular backgrounds and precise fixed-parameter quantifiers.