Precision-Graded Cohomology and Arithmetic Persistence for Network Sheaves (2511.00677v1)

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.

• The paper introduces a novel framework that leverages algebraic precision based on DVR filtrations to compute arithmetic barcodes.
• It employs the Digit-SNF Dictionary and saturation splitting to connect digit maps with Smith normal form exponents for robust network sheaf cohomology.
• The study applies these methods to distributed systems, demonstrating numerical stability and effective resource allocation in quantized communication.

Precision-Graded Cohomology and Arithmetic Persistence for Network Sheaves

Introduction and Motivation

This paper develops a theory of cohomology for network sheaves over discrete valuation rings (DVRs), with a focus on the $p$-adic integers $\mathbb{Z}_p$. Unlike classical persistent homology, which tracks topological features across geometric scales, the filtration parameter here is algebraic precision: the hierarchy of powers of the uniformizer (e.g., $p$ in $\mathbb{Z}_p$) stratifies cohomology by the ability to lift classes through increasing levels of precision. The central objects are arithmetic barcodes, which encode torsion in network sheaf cohomology as intervals whose lengths correspond to precision thresholds.

The framework is motivated by applications in distributed systems, sensor networks, and synchronization problems, where communication and measurement are quantized and precision is a natural stratification. The theory reinterprets torsion, often a computational obstacle in integral topology, as a primary signal in settings where data is inherently stratified by precision.

Algebraic Foundations: Network Sheaves over DVRs

Network sheaves are cellular sheaves on graphs, with stalks given by free modules over a DVR $R$ (e.g., $\mathbb{Z}_p$). The cohomology groups $H^0$ and $H^1$ measure global sections and obstructions, respectively. Over a DVR, $H^1$ generically carries torsion, distinguishing integral from rational cohomology. The valuation filtration $$R \supseteq \pi R \supseteq \pi^2 R \supseteq \cdots$$ (where $\pi$ is the uniformizer) stratifies modules by precision, and the associated graded structure encodes data at each level of refinement.

The cochain complex is $$C^0(G; \mathcal{F}) \xrightarrow{d} C^1(G; \mathcal{F})$$, with $d$ represented as a matrix over $R$. The structure theorem for modules over a PID gives $H^1 \cong R^b \oplus \bigoplus_j R/\pi^{a_j}$, where the exponents $a_j$ are the invariant factors and correspond to torsion summands.

Main Results: Digit-SNF Dictionary, Saturation Splitting, and Stability

Digit-SNF Dictionary

The central result is the Digit-SNF Dictionary, which establishes a precise correspondence between hierarchical precision data (digit connecting homomorphisms between mod-$\pi^k$ cohomology levels) and the Smith normal form exponents of the coboundary operator. Specifically, the dimension of the image of the digit map at level $k$ equals the number of torsion summands with exponent at most $k$:

$$\dim_{R/\pi} \operatorname{im}(\delta_k) = \#\{j : 1 \leq a_j \leq k\}$$

where $\delta_k$ is the connecting homomorphism in the long exact sequence associated to the short exact sequence of sheaves $$0 \to \mathcal{F} \xrightarrow{\pi^k} \mathcal{F} \to \mathcal{F}/\pi^k \mathcal{F} \to 0$$.

This correspondence enables two computational routes to the barcode: via Smith normal form or via digit map ranks at successive precision levels.

Saturation Splitting

The saturation splitting theorem constructs explicit integral idempotents projecting onto canonical representatives for cohomology classes. The saturation of the image of $d$ is the minimal direct summand containing $\operatorname{im} d$, and the quotient $sat(\operatorname{im} d)/\operatorname{im} d$ is isomorphic to the torsion part of $H^1$. The Smith normal form provides integral idempotents $\Pi_{sat}$ and $\Pi_{free}$, which commute with reduction modulo $\pi^k$ and yield canonical bases for cohomology.

Truncated Stability of Arithmetic Barcodes

The truncated stability theorem ensures that arithmetic barcodes are robust under high-precision perturbations: if two coboundaries $d$ and $d'$ agree modulo $\pi^m$, then their barcodes coincide for all bars of length less than $m$. This makes the invariants numerically stable and suitable for applications involving measured or quantized data.

Arithmetic Barcodes and Valuation Persistence

The valuation filtration on cohomology defines a persistence module indexed by precision level $k$. The barcode decomposition is:

• Each free summand $R$ yields an infinite bar $[0, \infty)$.
• Each torsion summand $R/\pi^a$ yields a finite bar $[0, a)$.

The arithmetic barcode thus encodes the precision thresholds at which cohomology classes fail to lift. For rank-one sheaves, cycle holonomy determines bar lengths: the $p$-adic valuation $val(h(C) - 1)$ of the holonomy around a cycle $C$ gives the bar length.

Geometric Interpretation: Cycle Holonomy

For unit sheaves (rank-one sheaves with edge restrictions given by units), the holonomy around a cycle $C$ is $h(C) = \prod_{e \in C} m_e$. The bar length is $val(h(C) - 1)$, quantifying the precision to which the cycle is consistent. In the case of edge-disjoint cycles, the torsion splits as a direct sum, and the barcode is determined cyclewise.

For higher-rank sheaves, matrix holonomy generalizes this: the Smith normal form of $H(C) - I$ (where $H(C)$ is the product of edge transformations around $C$) yields bar lengths for each principal direction.

Computational and Algorithmic Implications

The theory provides practical algorithms for computing arithmetic barcodes:

• Smith normal form over $R$ via elementary row/column operations or Hensel lifting.
• Digit map ranks via cohomology computations over $R/\pi^k$.

Threshold stability enables adaptive precision algorithms: compute digit ranks up to the available precision, and refine only where long bars are detected. The saturation splitting identifies which components of the edge-cochain space require high precision, guiding resource allocation in distributed systems.

Applications: Distributed Consensus with Quantized Communication

In distributed consensus protocols (e.g., clock synchronization, federated learning, formation control), agents communicate quantized data over a network. The arithmetic barcode provides a topology-dependent lower bound on the bit complexity required to detect cycle inconsistencies. For scalar consensus, the maximal bar length among cycles determines the necessary communication precision. For vector-valued consensus, the Smith exponents of matrix holonomies yield anisotropic precision requirements.

The framework extends to dynamic and multiparameter settings, where both geometric and algebraic filtrations interact.

Theoretical and Practical Implications

The results reposition torsion from a computational nuisance to a central invariant in settings where data is stratified by precision. The arithmetic barcode provides a new class of persistent invariants for networked systems over DVRs, with robust stability properties and direct computational accessibility. The theory suggests further development of ultrametric persistence modules, multiparameter invariants, and algorithmic implementations for large-scale systems.

Conclusion

This work establishes a comprehensive theory of precision-graded cohomology and arithmetic persistence for network sheaves over DVRs. The Digit-SNF Dictionary, saturation splitting, and stability theorems provide a robust algebraic infrastructure for analyzing torsion as a persistent feature stratified by precision. Applications in distributed systems, sensor networks, and synchronization problems demonstrate the practical relevance of the framework. Future directions include rigorous stability theory, algorithmic complexity analysis, extension to higher-dimensional sheaves, and development of multiparameter persistence invariants. The algebraic and topological tools developed here are poised to impact a broad range of systems where ultrametric precision hierarchies are intrinsic.