Symmetric Iterations with Countable and $<κ$-Support: A Framework for Choiceless ZF Extensions

Abstract

This replacement corrects the earlier manuscript. Its claimed general countable- and $<κ$-support iteration framework does not establish the advertised preservation or application theorems. The proposed limit filters can be improper; the dependent-choice arguments pass incorrectly from ground-model sequences of hereditarily symmetric names to arbitrary sequences in a generic extension; the class-length union argument does not verify Separation, Replacement, or Power Set; the Partition Principle argument uses an invalid same-generic compatibility inference; and the singular-support fusion argument assumes that a coherent inverse-limit family is a condition even when its support may have size $κ$. Accordingly, the main preservation, construction, class-length, dependent-choice, Partition Principle, and singular-cardinal claims are withdrawn. We retain only several conditional algebraic observations: pullback and completion operations on subgroup filters, closure of ground-model families of hereditarily symmetric names under canonical tupling, the standard ZF theorem for a genuine symmetric system, and ordinary conditional tail-closure facts. The remainder is preserved as a historical record of the superseded approach and is not asserted.