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Assuming the existence of a measurable cardinal, we produce generic extensions V₁ V₂ of the universe with the same cardinals, cofinalities, and bounded subsets of _ω, such that V₂ is generated over V₁ by a cofinal $ω$-sequence in _ω, and V₂ is a minimal generic extension over V₁, i.e., every set of ordinals in V₂ V₁ generates V₂ over V₁.
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Gitik et al. (2026) studied this question.
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