$\mathfrak{B}$-statistical core and ideal core of double sequences in $2$-normed spaces via RH-regular families

We investigated geometric limit sets of double sequences in finite dimensional $2$-normed spaces when negligibility of index sets was measured by a density induced by a nonnegative Robison–Hamilton (RH)-regular family $\mathfrak{B}$ of four-dimensional matrices. First, using the $\mathfrak{B}$-density, we defined $\mathfrak{B}$-statistical limit superior and limit inferior for the scalar reductions generated by the seminorms $x\mapsto \Vert x,u\Vert$, and we studied the associated real cluster behavior. Next, we introduced $\mathfrak{B}$-statistical cluster points and the $\mathfrak{B}$-statistical core of a double sequence as the intersection of all closed convex sets that contained the sequence outside a $\mathfrak{B}$-density zero set. We obtained a disk-type representation of the core via intersections of sets of the form $\{x\in X:\Vert x-z,u\Vert \le r\}$, where the radii were governed by $\mathfrak{B}$-statistical $lim sup$. We also proved a Knopp-type inclusion theorem: for a family of transforms satisfying a natural $\mathfrak{B}$-regularity condition, the Knopp core of each transform was contained in the $\mathfrak{B}$-statistical core of the original sequence. Finally, replacing $\mathfrak{B}$-density zero sets by a strongly admissible ideal ${\mathcal{I}}_2$ on $\mathbb{N}\times \mathbb{N}$, we defined ideal cores, established disk representations, compared the density-based and ideal cores, and identified an explicit condition under which the ${\mathcal{I}}_2$-core and the ${\mathcal{I}}_2^{\ast }$-core coincided.