Precision-Graded Cohomology and Arithmetic Persistence for Network Sheaves

TL;DR

This work develops a theory of arithmetic persistence for network sheaves over discrete valuation rings, using the $\pi$-adic filtration to organize cohomology by algebraic precision and to extract torsion via mod $\pi^k$ reductions. The central tool, the Digit-SNF Dictionary, equates the growth of digit-map images $\mathrm{im}(\partial_k)$ with the Smith normal form exponents $\{a_j\}$ of the coboundary, turning torsion into a computable barcode where bars have lengths $a_j$. It further provides Saturation Splitting to yield canonical integral idempotents and establishes Truncated Stability ensuring barcode invariance under high-precision perturbations, justifying numerical robustness for measured data. The cyclotomic interpretation via cycle holonomy shows how simple loop measurements determine bar lengths in rank-one cases, and the framework extends to multi-dimensional holonomy and multiparameter persistence, with concrete applications to distributed consensus and sensor networks where precision naturally induces hierarchical structure. Overall, the paper recasts torsion as the primary signal in precision-stratified data, delivering both theoretical insight and practical algorithms grounded in Smith normal form and Bockstein theory.

Abstract

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.

Theorems & Definitions (86)

• Remark 1.1
• Example 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Remark 2.6
• Definition 3.1
• Lemma 3.2
• proof