Diagonal Window Tests for Deferred Weighted Frequent Cauchy Sequences and Mean Coherence

This paper studies when a diagonal pairwise sampling condition of pre-Cauchy type can be upgraded to a genuine convergence criterion in the deferred weighted setting. Working with the window system determined by λ,μ, we introduce a diagonal pairwise framework and compare it with the corresponding two-parameter Pringsheim measure on N×N. This distinction makes it possible to separate within-window coherence from genuinely cross-window behavior. Within this setting, we formulate a deferred weighted diagonally statistically pre-Cauchy condition and identify additional assumptions under which it yields convergence. Our main results show that, once a mild stabilizing condition is imposed, either through the existence of a deferred weighted frequent limit point or through mean coherence of the window means, the diagonal pre-Cauchy property implies deferred weighted statistical convergence to a uniquely determined limit and consequently yields the deferred weighted frequent Cauchy property in the Pringsheim sense. For bounded sequences, we obtain computable characterizations in terms of the vanishing of diagonal double weighted window means of pairwise distances over the window squares, together with a bounded modulus refinement. We also give verifiable criteria based on adjacent-window coupling and dense subsequences of positive lower deferred weighted density. The examples and counterexamples show that diagonal pre-Cauchy alone does not rule out persistent oscillations and illustrate why an additional cross-window stabilization mechanism is needed in order to pass from diagonal tests to genuine two-parameter conclusions.