Rigorous stability theory for arithmetic barcodes in the ultrametric setting

Develop a rigorous stability theory for arithmetic barcodes associated to valuation persistence modules of network sheaves over discrete valuation rings, including formal definitions of bottleneck distance and interleaving for ultrametric persistence modules and the derivation of corresponding stability bounds.

Background

The paper introduces arithmetic barcodes that encode torsion in network sheaf cohomology over discrete valuation rings, relating precision-graded digit maps to Smith normal form exponents. While truncated stability under p-adic perturbations is established (Theorem C), a full stability framework analogous to classical bottleneck and interleaving theories for field-based persistence is not developed.

Establishing a comprehensive stability theory in the ultrametric context would provide robust quantitative guarantees for comparing arithmetic barcodes across datasets or perturbations, paralleling the role of bottleneck distance and interleavings in standard persistent homology.