Deferred weighted diagonal upgrade principles for windowed statistical convergence: external ideals and frequent cauchy behavior

Let I be a fixed admissible ideal on N and let (λₙ,μₙ) be a deferred window scheme equipped with positive weights w=(wₖ) . Using the associated window proportions Dₙ(·) and the square/rectangular pair counts Qₙ(·) and Dm,n(·) , we define I -deferred weighted statistical convergence, its diagonal statistically pre-Cauchy variant, and the corresponding I₂ -deferred weighted frequent Cauchy property on N×N . We first prove that I -deferred weighted statistical convergence always yields I₂ -deferred weighted frequent Cauchy behavior. Our main upgrade theorem (Theorem 4.4) shows that, for bounded sequences, diagonal pairwise control can be promoted to I -deferred weighted statistical convergence provided that the deferred weighted window means form an I -convergent sequence; in particular, it suffices that the mean sequence is I -Cauchy in R . Examples show that diagonal testing alone cannot replace the full rectangular scheme and that the mean-coherence assumption is essential.