A countable-support symmetric iteration separating PP from AC

Abstract

We study a countable-support Cohen symmetric seed model and a proposed symmetric-iteration approach to separating the Partition Principle $\mathsf{PP}$ from the Axiom of Choice. The previously claimed final construction is withdrawn. Its package forcing targets right inverses, and therefore a localized splitting principle stronger than the ordinary localized Partition Principle. For the fixed seed parameter $S=A^ω$, that splitting principle together with $\mathsf{SVC}(S)$ implies $\mathsf{AC}$, contrary to the intended preservation of a non-well-orderable Cohen set. Independently, the final non-well-orderability argument confuses stabilization of forcing names with an action inside one fixed generic extension, and the limit-stage $\mathsf{SVC}(S)$ argument relies on an invalid truncation lemma. The retained positive result is the Cohen symmetric seed [ \mathcal{N}\models\mathsf{ZF}+\mathsf{DC}+\mathsf{SVC}(S)+\neg\mathsf{AC}, \qquad S=(A^ω)^{\mathcal{N}}. ] The package and iteration sections are preserved only as a record of the superseded approach. No model of $\mathsf{ZF}+\mathsf{DC}+\mathsf{PP}+\mathsf{AC}_{\mathsf{WO}}+\neg\mathsf{AC}$ is claimed here.