Hierarchical SB Determinants on Multi-Dimensional Arrays

We introduce a novel algebraic structure called the Hierarchical SB Determinant (HSD), which extends classical determinants to multi-dimensional arrays (tensors).The construction recursively combines determinants of lower-dimensional slices through 2×2 block operations, producing a compact representation of high-dimensional tensor volumes.HSD preserves key properties such as scaling, antisymmetry, and degeneracy conditions, and admits a natural geometric interpretation as a discrete accumulation of oriented cell volumes.This framework provides a potential tool for discrete geometry, lattice-based physics, and tensor analysis, and opens avenues for further exploration in multi-linear algebra and numerical applications.