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

Abstract Let $${\mathbb{I}}$$ be a fixed admissible ideal on $${\mathbb{N}}$$ and let $$(\lambda_n,\mu_n)$$ be a deferred window scheme equipped with positive weights $$w=(w_k)$$ . Using the associated window proportions $$D_n(\cdot)$$ and the square/rectangular pair counts $$Q_n(\cdot)$$ and $$D_{m,n}(\cdot)$$ , we define $${\mathbb{I}}$$ -deferred weighted statistical convergence, its diagonal statistically pre-Cauchy variant, and the corresponding $${\mathbb{I}}_2$$ -deferred weighted frequent Cauchy property on $${\mathbb{N}}\times{\mathbb{N}}$$ . We first prove that $${\mathbb{I}}$$ -deferred weighted statistical convergence always yields $${\mathbb{I}}_2$$ -deferred weighted frequent Cauchy behavior. Our main upgrade theorem (Theorem 4.4) shows that, for bounded sequences, diagonal pairwise control can be promoted to $${\mathbb{I}}$$ -deferred weighted statistical convergence provided that the deferred weighted window means form an $${\mathbb{I}}$$ -convergent sequence; in particular, it suffices that the mean sequence is $${\mathbb{I}}$$ -Cauchy in $$\mathbb{R}$$ . Examples show that diagonal testing alone cannot replace the full rectangular scheme and that the mean-coherence assumption is essential.