  A fundamental challenge in multiparameter persistent homology is the absence
of a complete and discrete invariant. To address this issue, we propose an
enhanced framework that realizes a holistic understanding of a fully
commutative quiver's representation via synthesizing interpretations obtained
from intervals. Additionally, it provides a mechanism to tune the balance
between approximation resolution and computational complexity. This framework
is evaluated on commutative ladders of both finite-type and infinite-type. For
the former, we discover an efficient method for the indecomposable
decomposition leveraging solely one-parameter persistent homology. For the
latter, we introduce a new invariant that reveals persistence in the second
parameter by connecting two standard persistence diagrams using interval
approximations. We subsequently present several models for constructing
commutative ladder filtrations, offering fresh insights into random filtrations
and demonstrating our toolkit's effectiveness in analyzing the topology of
materials.
