 Constructible-universe obstruction to a uniform inner-model proof of failure of choice

Every inner model $N$ of ZF contains the ambient constructible universe $L$. By induction, $L_{\alpha }^N=L_{\alpha }$: at successor stages, satisfaction in the same set structure and definability with parameters are absolute; at limits both constructions take the same union. Thus, if the ambient universe satisfies $V=L$, its only inner model of ZF is itself, and it satisfies the axiom of choice. A procedure that must produce a choice-failing inner model from every model of ZF is therefore impossible. This does not assert that arbitrary inner models of arbitrary choice universes satisfy choice; the obstruction concerns a uniform construction.