We compute every cohomology-annihilator ideal and its least nonnegative stabilization index for S[x]/(x^n), where n >= 2 and S is a polynomial or formal power-series ring over any field. In base dimension zero, the eventual ideal is (x^floor(n/2)) and the index is 1. In base dimension one, the ideal is (x^(n-1)) and the index is 2 in every characteristic; this formula holds over every nonfield principal ideal domain. In base dimension d >= 2, the ideal is (x^(n-1)), with index d+1, when n is invertible in the field; it is zero, with index 0, when the characteristic divides n. Maps through projectives are described by an explicit transfer. A classical symmetric-power nilpotent operator gives a trace and colon-ideal obstruction in positive characteristic, and finite-free induction treats every divisible truncation degree. The field comparison also yields strict failures of equality under polynomial extension and localization.

Version 1.1 extends version 1.0 to every characteristic dividing n and to formal power-series bases in every dimension. It is a revision of the same manuscript, not a separate work. The broad noether-algebra question in OWR-12868-001 remains unresolved. The preprint is AI-assisted, self-audited and unrefereed; no independent review, proof-assistant certification, or absolute priority claim is asserted.
