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.
