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Abstract

We determine the worst polynomial degree needed to detect an isolated negative atom in a signed measure under coarse local-mass and weak exponential-moment bounds. For a tail scale H(x)=x/W(x), with W continuous, nondecreasing and subject to a sublinear power-ratio bound, let Ψ(t)=1+∫_1^td x/[xW(x)]. If this clock diverges, the minimax detecting degree satisfies Ψ(D_W(R))=Θ(R^(1/q)) for every fixed local exponent q>0. If it converges, one fixed atomless reference law, scaled with R, masks the negative atom from every polynomial square. No differentiability or regular variation is required. A finite positive moment realization and classical Bang propagation give the uniform signed-measure upper bound. A compact polynomial multiplier and singular Jacobi evaluation give matching fixed-law lower bounds and masking. Logarithmic and iterated-logarithmic tail regimes follow as applications, including double- and triple-exponential critical degrees. We distinguish the original signed-class recovery theorem from the classical determinacy thresholds and approximation ingredients used in its proof.