A fundamental difficulty in multiparameter persistent homology is the absence of a complete and discrete invariant. To address this challenge, we propose an enhanced framework that not only achieves a holistic understanding of a fully commutative quiver's representation via synthesizing interpretations obtained from intervals but also can tune the balance between approximation resolution and computational complexity. This framework is evaluated on commutative ladders of both finite-type and infinite-type. In the former, we discover an efficient method for the indecomposable decomposition leveraging solely one-parameter persistent homology. In the latter, we introduce a new invariant that reveals partial persistence in the second parameter by connecting two standard persistence diagrams using interval approximations. We then introduce several models for constructing commutative ladder filtrations, offering new insights into random filtrations and demonstrating our toolkit's effectiveness by analyzing the topology of materials.