Classifying Persistence Modules over a PID via Structure Directed Acyclic Graphs

This manuscript develops a complete classification of one-parameter persistencemodules over a principal ideal domain via structure directed acyclic graphs (DAGs). Three main contributions: (1) Classification: The structure DAG is a complete isomorphism invariant; therealizability theorem establishes a bijection between DAG isomorphism classesand persistence module isomorphism classes. Krull-Schmidt follows unconditionallyvia DAG connected components. The category with fixed parameters isrepresentation-finite. (2) Computational framework: The local-global principle reduces the isomorphismproblem over Z to a finite family of barcode computations over small Artinianprincipal ideal rings, realizable with existing persistent homology software.DAG construction and isomorphism testing run in polynomial time. (3) Applications: The ephemeral dependency graph Γ links graph topology tohomological invariants (Ext-nilpotency, Yoneda lengths, minimal generators).The framework unifies Gabriel's barcode, the Luo-Henselman-Petrusek criterion,and the Ghrist-Ding arithmetic barcode as special cases. A self-contained excerpt isolating the key counterexample (stepwise SNF is not a complete invariant) is available at https://doi.org/10.5281/zenodo.20252298.