Isomorphism criteria for the dark zone below the bricks

Develop effective isomorphism criteria or invariants that control, in general, the information coming from the "dark zone" (the area below the bricks) in matrices in first reduced echelon form, to decide isomorphism A(T) ≅ A(S) beyond Wall and Measure sequence data.

Background

The authors define two invariants—the Wall and the Measure sequence—that classify aspects of matrices in reduced echelon forms and provide powerful tests to separate isomorphism classes. However, they note that matrices can share the same Wall and bricks yet differ in the 'dark zone' (entries below the brick blocks), and current criteria do not generally resolve such cases.

They explicitly acknowledge lacking a general criterion for the dark zone and indicate plans to address this in future work.