Higher Cotorsion Pairs on Image-Nilpotent $\eta$-Extensions

Let $\text{cB}$ be an abelian category with enough projective and injective objects, and let $\text{cA}=\text{cB}{⋉}_{\eta }\text{sF}$ be an $\eta$-extension induced by a right exact endofunctor $\text{sF}$. We introduce uniform image-nilpotence for $\eta$-extensions and characterize it in terms of the nilpotence of the associative natural transformation $\eta :{\text{sF}}^2\to \text{sF}$. Using the associated $\eta$-Loewy filtration, we prove a lifting theorem for right $n$-cotorsion pairs over image-nilpotent $\eta$-extensions and establish a corresponding completeness theorem. As applications, our results recover and unify related constructions for split nilpotent ring extensions, comma categories, formal triangular matrix rings, Morita context rings and classical trivial extensions.