Precision-Graded Cohomology and Arithmetic Persistence for Network Sheaves

Persistent homology tracks topological features across geometric scales, encoding birth and death of cycles as barcodes. We develop a complementary theory where the filtration parameter is algebraic precision rather than geometric scale. Working over the $p$-adic integers $\mathbb{Z}_p$, we define \emph{arithmetic barcodes} that measure torsion in network sheaf cohomology: each bar records the precision threshold at which a cohomology class fails to lift through the valuation filtration $\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots$. Our central result -- the \emph{Digit-SNF Dictionary} -- establishes that hierarchical precision data from connecting homomorphisms between successive mod-$p^k$ cohomology levels encodes exactly the Smith normal form exponents of the coboundary operator. Bars of length $a$ correspond to $\mathbb{Z}_p/p^a\mathbb{Z}_p$ torsion summands. For rank-one sheaves, cycle holonomy (the product of edge scalings around loops) determines bar lengths explicitly via $p$-adic valuation, and threshold stability guarantees barcode invariance when perturbations respect precision. Smith normal form provides integral idempotents projecting onto canonical cohomology representatives without geometric structure. Results extend to arbitrary discrete valuation rings, with $p$-adic topology providing ultrametric geometry when available. Applications include distributed consensus protocols with quantized communication, sensor network synchronization, and systems where measurement precision creates natural hierarchical structure. The framework repositions torsion from computational obstacle to primary signal in settings where data stratifies by precision.