Weighted potential and Green systems with Dini coefficients and primitive growth

Abstract: We prove a nonexistence theorem for positive potential systems with continuous coefficients satisfying distributional Euler lower bounds and a uniform Dini modulus on normalized logarithmic dilations. The nonlinearities have positive lower logarithmic derivatives and positive primitive-to-endpoint ratios. The common radial kernel is comparable with a Riesz kernel. A localized transport identity, a primitive pressure estimate and implicit amplitude normalization give uniform local energy bounds and a dilation contradiction. And we treats the genuine Green kernel of $-Δ+V$ for nonnegative potentials under annular and Newton composition budgets and a strict gap corrected by the relative Green dilation defect. Relative endpoint differences and an exact virtual-energy cancellation retain local nonlinear energy as the only initial energy hypothesis. Independent representation arguments give these consequences, including even integer orders without initial tails. A compact family of normalized responses also yields universal interior bounds without regular variation.