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Proposition 5 ([The bounded-below Adams--Novikov resolution]). For every bounded-below p -local spectrum Z , the sphere-source Adams--Novikov spectral sequence has E_2^ s,t (Z)=H^ s,t (C^_Γ(BP_*Z)) π_ t-s Z. It converges strongly, and the induced filtration in each stem is finite. The differential d_r changes (s,t) by (r,r-1) . The filtration-zero edge identifies E_∞^ 0,* with the image of the BP -Hurewicz map π_*Z⟶BP_*Z .