Partition Principle without Choice via Symmetric Iterations and Sheaf-Toposes

Abstract

An earlier form of this manuscript proposed that a transformation-groupoid topos associated with a free action of a nontrivial finite group on Cantor space contains an internal model of $\mathsf{IZF}+\mathsf{PP}+\neg\mathsf{AC}$, with a classical upgrade after double-negation sheafification. That construction is not valid. The proposed local embedding argument does not establish the required property for general epimorphisms; the finite-fiber class of small maps does not support an $\mathsf{IZF}$-universe because it does not contain the natural numbers object; the quotient map used to witness failure of choice is not a nonsplitting epimorphism in the asserted form; and double-negation sheafification does not repair these defects. The symmetric-iteration appendix also used an invalid same-condition equivariance inference. Accordingly, this replacement withdraws all model-existence, preservation, and independence claims. We retain the conditional categorical observation that a genuinely matching local family of embeddings descends to a global embedding, and record the precise obstructions so that the withdrawn construction is not cited as a solution of the Partition Principle problem.