Arithmetic Barcodes: Algebraic Precision

• Arithmetic barcodes are defined using algebraic precision via valuation filtrations, replacing geometric scale to capture torsion phenomena in persistent homology.
• They leverage digit maps and the Smith normal form to identify lifting obstructions and compute bar lengths through torsion summands over discrete valuation rings.
• The framework connects persistent homology, quiver representations, and combinatorial counting via inverse problem formulations and threshold stability for robust analysis.

Arithmetic barcodes are barcodes in which the filtration parameter is algebraic precision rather than geometric scale. In the formulation developed over a discrete valuation ring, especially $\mathbb{Z}_p$, they measure torsion in network sheaf cohomology: each bar records the precision threshold at which a cohomology class fails to lift through the valuation filtration

$$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$

and bars of length $a$ correspond to $\mathbb{Z}_p/p^a\mathbb{Z}_p$ torsion summands (Ghrist et al., 1 Nov 2025). The same phrase also appears in an inverse-problem viewpoint in persistent homology: given a Betti curve $\beta$, one asks how many distinct barcodes produce that same curve, equivalently how many decompositions of $\beta$ into interval indicator functions exist; that count is identified with the Kostant partition function and with magic juggling sequences (Ashley et al., 9 Feb 2026).

1. Precision filtrations and arithmetic persistence

The arithmetic-barcode framework replaces the scale filtration of ordinary persistent homology by the valuation filtration of a discrete valuation ring. In the general DVR setting one works with

$$R \supseteq \mathfrak{p} \supseteq \mathfrak{p}^2 \supseteq \cdots,$$

and in the $p$-adic case with

$$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots.$$

For cochains on a graph $G$ with a sheaf $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$0, the filtered pieces are defined by

$$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$1

so filtration depth is interpreted as divisibility or precision level (Ghrist et al., 1 Nov 2025).

This formalism treats torsion as the primary signal rather than as a computational inconvenience. A cohomology class can be lifted through the tower

$$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$2

and its “death” occurs at the precision level where lifting fails. The associated exact sequences

$$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$3

produce connecting homomorphisms

$$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$4

called digit maps. These digit maps record the obstructions to lifting mod-$$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$5 classes to mod-$$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$6 and ultimately to the full $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$7-adic module (Ghrist et al., 1 Nov 2025).

The successive quotients of the filtration are residue-field valued: $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$8 as $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$9-vector spaces, where $a$0. This produces a persistence module whose intervals encode precision lifetimes rather than scale lifetimes. A plausible implication is that the usual persistence vocabulary of birth, death, and survival can be transferred to settings where the controlling parameter is arithmetic rather than geometric.

2. The Digit-SNF Dictionary and interval decomposition

The central structural result is the Digit-SNF Dictionary, which identifies stagewise lifting obstructions with the Smith normal form exponents of the coboundary operator. If

$a$1

is the coboundary, then there exist unimodular matrices $a$2 such that

$a$3

with

$a$4

and $a$5. For each $a$6, the digit connecting homomorphism satisfies

$a$7

and consequently

$a$8

Thus the image dimensions of the digit maps are cumulative counts of torsion bars, while first differences recover exact multiplicities (Ghrist et al., 1 Nov 2025).

The barcode itself is obtained from the invariant factor decomposition of cohomology. Each free summand contributes an infinite bar $a$9, and each torsion summand $\mathbb{Z}_p/p^a\mathbb{Z}_p$0 contributes a finite bar $\mathbb{Z}_p/p^a\mathbb{Z}_p$1. In the paper’s “Barcode Decomposition” formulation,

$\mathbb{Z}_p/p^a\mathbb{Z}_p$2

where $\mathbb{Z}_p/p^a\mathbb{Z}_p$3. This gives an interval-module decomposition formally parallel to classical persistence, but with intervals indexed by precision thresholds (Ghrist et al., 1 Nov 2025).

The paper also notes that the digit maps factor through the Bockstein,

$\mathbb{Z}_p/p^a\mathbb{Z}_p$4

and that the Bockstein spectral sequence collapses at $\mathbb{Z}_p/p^a\mathbb{Z}_p$5 for graphs because the complex only has degrees $\mathbb{Z}_p/p^a\mathbb{Z}_p$6 and $\mathbb{Z}_p/p^a\mathbb{Z}_p$7. This places arithmetic barcodes within a standard homological-algebraic framework rather than outside the ordinary apparatus of persistence theory.

3. Cycle holonomy, threshold stability, and canonical representatives

For rank-one sheaves, the arithmetic-barcode formalism becomes explicitly computable. In a rank-one unit sheaf on a graph $\mathbb{Z}_p/p^a\mathbb{Z}_p$8, each vertex and edge stalk is $\mathbb{Z}_p/p^a\mathbb{Z}_p$9, and each edge has a unit scaling $\beta$0. For an oriented edge $\beta$1,

$\beta$2

For a cycle $\beta$3, the holonomy is

$\beta$4

On the cycle graph $\beta$5, the barcode is determined by $\beta$6: if $\beta$7, then

$\beta$8

so there is an infinite bar; if $\beta$9 is a unit, then

$\beta$0

so the barcode is empty; and if

$\beta$1

then

$\beta$2

so the barcode is a single finite bar $\beta$3 (Ghrist et al., 1 Nov 2025).

This criterion is reflected in the coboundary determinant. For the cycle graph,

$\beta$4

and

$\beta$5

for some unit $\beta$6, hence

$\beta$7

The bar length is therefore exactly the valuation of the holonomy defect.

Arithmetic barcodes also satisfy a threshold stability statement. If two coboundary operators satisfy

$\beta$8

then for every $\beta$9,

$$R \supseteq \mathfrak{p} \supseteq \mathfrak{p}^2 \supseteq \cdots,$$0

and hence the bar multiplicities for lengths $$R \supseteq \mathfrak{p} \supseteq \mathfrak{p}^2 \supseteq \cdots,$$1 agree: $$R \supseteq \mathfrak{p} \supseteq \mathfrak{p}^2 \supseteq \cdots,$$2 Equivalently,

$$R \supseteq \mathfrak{p} \supseteq \mathfrak{p}^2 \supseteq \cdots,$$3

If

$$R \supseteq \mathfrak{p} \supseteq \mathfrak{p}^2 \supseteq \cdots,$$4

then the entire barcode is preserved. The paper also introduces integral idempotents projecting onto the kernel, the saturated image of the coboundary, and the free part of cohomology, with reduction modulo $$R \supseteq \mathfrak{p} \supseteq \mathfrak{p}^2 \supseteq \cdots,$$5 commuting with these projectors for all $$R \supseteq \mathfrak{p} \supseteq \mathfrak{p}^2 \supseteq \cdots,$$6 (Ghrist et al., 1 Nov 2025). This provides canonical cohomology representatives without requiring an inner product or Hodge theory.

4. Barcode spaces, coordinates, and stratifications

Arithmetic barcodes remain part of the broader theory of barcode spaces. In that broader setting, a persistence barcode is a finite multiset of intervals. If a barcode has exactly $$R \supseteq \mathfrak{p} \supseteq \mathfrak{p}^2 \supseteq \cdots,$$7 intervals, it can be encoded as

$$R \supseteq \mathfrak{p} \supseteq \mathfrak{p}^2 \supseteq \cdots,$$8

where $$R \supseteq \mathfrak{p} \supseteq \mathfrak{p}^2 \supseteq \cdots,$$9 is the birth time and $p$0 is the length of the $p$1-th interval; because intervals are unordered, one passes to an orbit space under the action of the symmetric group $p$2, obtaining $p$3, and then the full barcode space

$p$4

where $p$5 identifies barcodes differing only by deletion of zero-length bars (Verovsek, 2016).

Within this framework, tropical algebra supplies stable numerical coordinates on barcode space. In max-plus arithmetic,

$p$6

The resulting tropical rational functions are built from finitely many $p$7, $p$8, and $p$9 operations applied to linear forms. The paper constructs 2-symmetric functions compatible with permutation invariance and proves Lipschitz stability with respect to both bottleneck and Wasserstein distances. A countable family of tropical rational coordinates separates points in barcode space, providing a stable feature map from persistent homology to standard machine-learning pipelines (Verovsek, 2016).

A complementary geometric description views the space $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots.$$0 of barcodes with $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots.$$1 bars as a quotient

$$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots.$$2

with strict barcodes classified by a permutation $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots.$$3 comparing the birth-ordering and death-ordering. Using the Coxeter complex of $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots.$$4, this is extended to arbitrary barcodes by marked double cosets of parabolic subgroups. The resulting “Coxeter coordinates”

$$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots.$$5

record the means and standard deviations of births and deaths together with an angular Coxeter-complex component. The strata consist of barcodes with the same averages and standard deviations of birth and death times and the same permutation type, and the quotient description produces modified bottleneck and Wasserstein-type metrics in which bars are not matched to the diagonal (Brück et al., 2021).

These classical constructions do not define arithmetic barcodes, but they supply the ambient language in which barcode-valued invariants are represented, compared, and converted into coordinates.

5. The inverse problem: counting barcodes with a fixed Betti curve

A distinct use of the phrase “Arithmetic Barcodes” concerns an inverse problem for persistent homology. A barcode $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots.$$6 on $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots.$$7 is a multiset of intervals $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots.$$8, with multiplicities allowed,

$$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots.$$9

and its Betti curve $G$0 counts how many intervals contain each index: $G$1 Equivalently,

$G$2

where $G$3 is the multiplicity of $G$4. The inverse problem is therefore: given $G$5, count all nonnegative integer families $G$6 such that

$G$7

The corresponding fiber is denoted

$G$8

(Ashley et al., 9 Feb 2026).

Representation-theoretically, a persistence module $G$9 is interpreted as a type $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$00 quiver representation. By the structure theorem for finite-dimensional representations, every such module decomposes uniquely into indecomposables, and for type $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$01 the indecomposables are precisely interval modules $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$02. The Betti curve is the dimension vector,

$$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$03

so counting barcodes with fixed $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$04 is counting decompositions of a dimension vector into interval indecomposables (Ashley et al., 9 Feb 2026).

The first main theorem identifies this count with the Kostant partition function: $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$05 where $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$06 are the simple roots of type $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$07. In this setting the positive roots

$$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$08

correspond combinatorially to intervals $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$09. The paper also gives the recursive formula

$$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$10

where $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$11 means that $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$12 is non-increasing, $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$13, and $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$14 for all $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$15. This recursion groups barcodes according to the multiset of intervals that begin at $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$16 (Ashley et al., 9 Feb 2026).

A second main theorem gives a bijection with magic juggling sequences. For

$$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$17

the map

$$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$18

is a bijection. Combined with the juggling identity

$$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$19

this yields the barcode–Kostant identity again. The statistical significance is that the count $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$20 quantifies how lossy the passage from barcodes to Betti curves is: the examples $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$21 and $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$22 come from $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$23 and $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$24 different barcodes, respectively (Ashley et al., 9 Feb 2026).

6. Conceptual significance and research directions

Arithmetic barcodes recast persistence in terms of precision hierarchies. In the network-sheaf formulation, the filtration parameter is algebraic precision; torsion classes $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$25 become finite bars; digit maps recover Smith exponents; cycle holonomy provides explicit bar lengths in rank-one cases; and threshold stability gives barcode invariance under perturbations that respect precision (Ghrist et al., 1 Nov 2025). The applications named in that framework include distributed consensus protocols with quantized communication, sensor network synchronization, and systems where measurement precision creates natural hierarchical structure.

The inverse-problem formulation addresses a different question: not how a barcode is produced from a filtered object, but how much information is lost when one replaces a barcode by a Betti curve. There the count of the fiber $$\mathbb{Z}_p \supseteq p\mathbb{Z}_p \supseteq p^2\mathbb{Z}_p \supseteq \cdots,$$26 becomes a quantitative measure of compression, and the equivalence with the Kostant partition function and with magic juggling sequences places the question simultaneously in persistent homology, representation theory, and enumerative combinatorics (Ashley et al., 9 Feb 2026).

Taken together, these usages show that “arithmetic barcodes” names a family of ideas rather than a single construction. In one direction, arithmetic data become barcodes through valuation filtrations and Smith normal form. In another, barcodes themselves become arithmetic objects through exact counting problems. A plausible implication is that both directions enlarge the role of barcodes beyond descriptive summaries: they become interfaces between topological data analysis, discrete valuation algebra, quiver representation theory, and combinatorial models of counting.