Digit-SNF Dictionary Framework

• Digit-SNF Dictionary is a comprehensive framework linking network sheaf cohomology with SNF torsion invariants, defining arithmetic barcodes that track precision decay.
• The methodology leverages valuation filtrations and connecting homomorphisms to convert torsion profiles into actionable precision thresholds for consensus in networked systems.
• The theory extends to rank-one and vector-valued sheaves, providing critical insights for synchronization, sensor networks, and adaptive resource allocation in quantized data analysis.

The Digit-SNF Dictionary is a comprehensive framework linking hierarchical precision data arising from network sheaf cohomology over discrete valuation rings to the algebraic structure encoded by the Smith normal form (SNF) of coboundary operators. It provides a precise correspondence between arithmetic barcodes—interval invariants tracking the "decay" of cohomology classes under decreasing precision—and the SNF exponents that classify torsion submodules. This theory reframes torsion not merely as a computational artifact but as a refined signal quantifying the precision threshold at which global consistency in networked systems is attained or lost. The results are formulated and proven for network sheaves of free modules over $\mathbb{Z}_p$ and extend to arbitrary discrete valuation rings (DVRs), supporting robust applications in synchronization, distributed consensus, and precision-sensitive data analysis (Ghrist et al., 1 Nov 2025).

1. Algebraic Foundations: Smith Normal Form and Cohomology Torsion

Let $G$ denote a finite graph and $\Fsheaf$ a network sheaf of free $\Z_p$-modules. The cohomology complex is given by

$$C^0 = \Z_p^{n_0},\qquad C^1 = \Z_p^{n_1},\qquad d: C^0 \longrightarrow C^1.$$

Over the principal ideal domain (PID) $\Z_p$, SNF theory guarantees the existence of invertible matrices $U \in \mathrm{GL}_{n_1}(\Z_p)$ and $V \in \mathrm{GL}_{n_0}(\Z_p)$ such that

$$U d V = \operatorname{diag}(p^{a_1},\,\dots,p^{a_r},\, 0,\dots,0),\qquad 0 \le a_1 \le \dots \le a_r.$$

The cohomology module

$H^1(G;\Fsheaf) = \operatorname{coker}(d) \cong \Z_p^{n_1 - r} \oplus \bigoplus_{j=1}^r \Z_p / p^{a_j} \Z_p$

has torsion decomposed as $\bigoplus_{j=1}^r \Z_p / p^{a_j} \Z_p$. The sequence $\{a_j\}$, the positive SNF exponents, completely determines the torsion profile. This algebraic perspective is crucial for extracting precision-graded invariants from network data (Ghrist et al., 1 Nov 2025).

2. Valuation Filtration and the Arithmetic Barcode

The valuation filtration on the coefficient ring induces associated filtrations on sheaves and their cohomology: $\Fsheaf \supseteq p \Fsheaf \supseteq p^2 \Fsheaf \supseteq \cdots,$

$H^i(G;\Fsheaf) \supseteq p H^i \supseteq p^2 H^i \supseteq \cdots.$

Each graded piece $p^k H^i / p^{k+1} H^i \cong H^i(G;\Fsheaf/p\Fsheaf)$ is an $\F_p$-vector space, yielding a persistence module indexed by $k$. The structure theorem for finitely generated $\Z_p$-modules gives: $H^i(G;\Fsheaf) \cong \Z_p^{b_i} \oplus \bigoplus_{j=1}^{r_i} \Z_p / p^{a_{i,j}} \Z_p,$ from which the arithmetic barcode is defined: $\Bar^i(G;\Fsheaf) = \underbrace{[0,\infty) \;\sqcup\;\dots\;\sqcup\;[0,\infty)}_{b_i} \sqcup \bigcup_{j=1}^{r_i} [0,a_{i,j}).$ Infinite bars $[0,\infty)$ correspond to free summands, while finite bars $[0,a_{i,j})$ correspond to $\Z_p / p^{a_{i,j}} \Z_p$ torsion (Ghrist et al., 1 Nov 2025).

3. Digit–SNF Dictionary: Precision Profiles from Connecting Homomorphisms

The core result, the Digit–SNF Dictionary, equates digit profiles of connecting homomorphisms arising from the valuation filtration with the Smith exponents:

• For each $k \geq 1$, form the connecting map

$\delta^i_k: H^i(G; \Fsheaf/p^k \Fsheaf) \longrightarrow H^{i+1}(G; p^k\Fsheaf/p^{k+1}\Fsheaf) \cong H^{i+1}(G; \Fsheaf/p\Fsheaf)$

and write $d_k = \dim_{\F_p} \mathrm{im} \, \delta^i_k$.

• The sequence $d_0 \le d_1 \le \cdots$ is nondecreasing, and

$$d_k = \#\{j : 1 \le a_j \le k\}, \quad d_k - d_{k-1} = \#\{j : a_j = k\}.$$

Thus, each jump in $d_k$ directly records the multiplicity of the torsion summand $\Z_p/p^k\Z_p$. The entire barcode structure is determined by the sequence $(d_k)$, making the filtration dynamics a proxy for the underlying SNF invariants (Ghrist et al., 1 Nov 2025).

4. Explicit Barcodes: Holonomy Formulae for Rank-One Sheaves

For rank-one "unit" sheaves (each stalk $\cong \Z_p$; restrictions given by multiplication by $\lambda_e \in \Z_p^\times$), each graph cycle $\gamma$ carries a well-defined holonomy

$$h(\gamma) = \prod_{e \in \gamma} \lambda_e \in \Z_p^\times.$$

The barcode interval associated with $\gamma$ has length given by the $p$-adic valuation: $$\text{length} = v_p(h(\gamma) - 1),\qquad \text{interval }[0, v_p(h(\gamma) - 1)).$$ This provides a direct cycle-wise method for computing the precise threshold at which cocycle incompatibility becomes visible under precision reduction. For vector-valued sheaves, cycle holonomy generalizes to products of edge matrices, and the torsion profile is extracted from the SNF of $H(\gamma) - I_d$ (Ghrist et al., 1 Nov 2025).

5. Threshold Stability and Universality Over DVRs

A salient stability property is that the digit sequence $(d_k)$ and thus all bars of length $< m$ are determined by the reduction of the coboundary $d$ modulo $p^m$: $d \equiv d' \pmod{p^m} \implies \Bar^i(G;\Fsheaf)\cap [0,m) = \Bar^i(G;\Fsheaf')\cap [0,m).$ If $m > \max_j a_{i, j}$, the entire barcode remains stable under such perturbations. All constructions and results transfer verbatim to an arbitrary DVR $(R, \pi)$, replacing $p$ with $\pi$ and $v_p$ with the $\pi$-adic valuation $v_\pi$ (Ghrist et al., 1 Nov 2025).

6. Applications in Network Synchronization and Quantized Systems

The Digit–SNF Dictionary framework yields practical tools across a variety of engineering and data science contexts:

• Clock synchronization (scalar consensus): Agents' comparisons of local clock-rates modulo $2^b$ correspond to sheaf restrictions with finite precision. The arithmetic barcode quantifies for each cycle the minimal number of bits $b \ge \max_\gamma v_2(h(\gamma) - 1)$ needed to detect inconsistency, thereby specifying the consensus threshold.
• Sensor network synchronization (unit-valued data): The precision barcode expresses the $p$-adic depth to which phases can be globally synchronized.
• Vector-valued consensus and formation control: Edge-wise transformations given by $T_e \in \GL_d(\Z_p)$ lead to matrix-valued holonomies, with the Smith exponents of $H(\gamma) - I_d$ determining bar lengths for vector data quantization.
• Bit allocation and saturation splitting: The saturation splitting, given by SNF-derived idempotents, identifies subspaces of data most sensitive to torsion and thus most demanding in bit-precision requirements, enabling adaptive resource allocation (Ghrist et al., 1 Nov 2025).

7. Broader Context and Expansions

The Digit–SNF Dictionary establishes a deep connection between algebraic precision (via discrete valuation, filtrations, and torsion) and structural network properties in a manner analogous but orthogonal to classical persistent homology, where geometric filtering is replaced by arithmetic thresholds. Through this lens, torsion encodes "loss of liftability" across precision strata, with barcodes serving as a primary invariant classifying system behavior under limited resolution. These principles extend to composite settings (e.g., consensus with quantized communication, federated learning with compression). The theory is robust to sheaf and field generalizations so long as the coefficient ring is a DVR; $p$-adic topology provides an ultrametric geometric underpinning when available, but purely algebraic statements remain valid in the absence of topological structure (Ghrist et al., 1 Nov 2025).