Statistical and ideal convergence of double sequences via 𝔅-summability in 2-normed spaces

We study statistical and ideal convergence of double sequences in 2-normed spaces under nonnegative RH-regular matrix families. For a family of averages over adjacent row pairs, we give a necessary and sufficient condition for ideal convergence of the transforms to recover convergence of the original sequence. The condition involves the ideal and the lower level sets of the coefficient that separates the two entries of each pair. It applies to all double sequences. We characterize separately when a finite quantitative bound exists and determine its optimal constant. An explicit two-member family recovers statistical convergence although neither member does so alone and no finite quantitative bound is available. The same family fails to recover Pringsheim convergence. We also prove a general diagonal Tauberian estimate for bounded sequences, separate statistical convergence of transforms from exceptional-set matrix mass, and establish density-one, Cauchy and filter-set results under explicit hypotheses. Double Cesàro examples illustrate the distinctions between the convergence notions.