REVIEW
AnyonWiki.org
Quantum Algebra · Submitted 21 Oct 2026 · 28 references
Pith's verdict: A clean, honest data-announcement paper whose claims are versioned and checkable; the right call is to referee it, not desk-reject it.
Why Pith says this
Highlight any text to comment on it.
Referee report
The paper
The paper claims
The paper presents AnyonWiki.org, an open-source archive holding all multiplicity-free fusion rings through rank 10 and 989 low-rank fusion categories with full symbol data and per-field provenance.
The reading
What carries the argument
The carrying mechanism is the AnyonWikiDatabase's provenance-aware data format. Each fusion ring or category is stored as a JSON dictionary keyed by a universal unique identifier, accompanied by an info document that defines every formatting convention and an order list for testing, so any programming language can import the data with its interpretation attached. Every data field additionally maps to a reference and to the exact revision of the software that generated it, each requiring a DOI or a Software Heritage identifier so that links cannot rot; the repository is version-controlled and archived, which is what turns external classification results into a stable, citable resource.
Extended reading notes
Core claim
The central claim is that a complete, citable, machine-readable census of the low-rank fusion category landscape now exists and is publicly queryable at AnyonWiki.org. The database contains the 25,332 fusion rings obtained by the methods of [14], including every multiplicity-free fusion ring through rank 10 (rings whose structure constants are only 0 or 1), and the 989 complex multiplicity-free pivotal, braided fusion categories through rank 7 classified in [12]; Table 1 breaks the ring count down by rank and multiplicity, with bold entries marking the sections the archive has completed. Every category carries F-symbols in a fixed gauge (a chosen convention among the many equivalent ways to write the same data), the pivotal structure encoded in P-symbols, R-symbols where a braiding exists, and the S-matrix and twists for modular categories, each expressed symbolically as algebraic numbers and numerically as floating-point values. The authors' specific contribution is therefore not a new classification theorem but a new kind of scientific object: a database whose entries can be looked up, downloaded, verified, and cited at the level of individual data fields, with provenance and software revisions stored alongside the data itself.
Load-bearing premise
The load-bearing premise is that the classifications and symbol data imported from the external papers and software are complete and correct, since the article itself contains no completeness proof, no independent cross-check, and no error analysis of the numerical F-, P-, and R-symbols it archives.
Editorial extensions
If this is right
- Because the archive is complete for multiplicity-free fusion rings through rank 10 and for the category census through rank 7, a user can trust negative queries: if a ring or category is absent from a complete section, that absence is meaningful rather than an artifact of partial data.
- Researchers can read off properties that previously required running classification software — automorphism groups, gradings by finite groups, fusion subrings, characters, codegrees, and categorifiability — directly from the website.
- Each category page fixes the gauge of its F-, P-, and R-symbols, so gauge-inequivalent structures appear as distinct entries, unitary gauges are flagged, and modular categories come with their S-matrix and twists.
- Per-field provenance with DOIs and Software Heritage identifiers makes the original classification papers and software revisions citable, so the database is designed not to become a citation black hole.
- The project is structured to grow: contributed data must carry a DOI or full software provenance, and planned additions include the centers of multiplicity-free fusion categories up to rank 5 and a successor software package giving programmatic access to the archive.
Reading between the lines
- What the paper leaves implicit is that, if the imported census is complete, the database converts a stored classification theorem into a checkable service: someone can now test a conjecture about low-rank rings by querying the archive, which is a different kind of evidence from the original classification proof.
- The provenance-per-field scheme is a reusable template: the same UUID/JSON/software-revision structure could be ported to neighbouring datasets such as Drinfeld centers, higher-rank modular data, or the symbol data of other tensor categories, and would solve the same citation-verification problem there.
- A natural testable extension the authors do not run is continuous cross-validation of the archive itself, for example recomputing each modular category's S-matrix and twists from its stored F- and R-symbols on every update, which would turn a static snapshot into a permanently audited dataset.
- The acceptance rule that contributions supply symbolic algebraic F-symbols rather than numeric ones sets a quality bar that implicitly excludes numerically plausible but non-exact solutions, which matters for users who rely on the exact field of definition.
Original abstract
The AnyonWiki is an open-source project that provides full algebraic and numerical data of fusion and unitary modular fusion categories as a resource for the scientific community. It comes with a database, the AnyonWikiDatabase, that currently includes the classification of multiplicity-free pivotal (braided) fusion categories through rank 7 and more than 25000 fusion rings including the classification of multiplicity-free fusion rings through rank 10, with work in progress to expand. This data is available through AnyonWiki.org, a website that provides robust search and download functionality and presents each fusion ring and category via a dedicated webpage. In this short article we discuss the data available at AnyonWiki.org as well as the scientific and communal motivations for the design of its underlying AnyonWikiDatabase, which are centered around making the data accessible, independently verifiable, granularly citable, useful, and easy to work with.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
The AnyonWiki is an open-source project that provides full algebraic and numerical data of fusion and unitary modular fusion categories as a resource for the scientific community. It comes with a database, the AnyonWikiDatabase, that currently includes the classification of multiplicity-free pivotal (braided) fusion categories through rank 7 and more than 25000 fusion rings including the classification of multiplicity-free fusion rings through rank 10, with work in progress to expand. This data is available through AnyonWiki.org, a website that provides robust search and download functionality and presents each fusion ring and category via a dedicated webpage. In this short article we discuss the data available at AnyonWiki.org as well as the scientific and communal motivations for the design of its underlying AnyonWikiDatabase, which are centered around making the data accessible, independently verifiable, granularly citable, useful, and easy to work with.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The AnyonWiki is an open-source project that provides full algebraic and numerical data of fusion and unitary modular fusion categories as a resource for the scientific community. | contribution 0.96 | 0.44 | 0.16 | asserted 1.00 |
| s1 | It comes with a database, the AnyonWikiDatabase, that currently includes the classification of multiplicity-free pivotal (braided) fusion categories through rank 7 and more than 25000 fusion rings including the classification of multiplicity-free fusion rings through rank 10, with work in progress to expand. | contribution 0.84 | 0.61 | 0.08 | asserted 1.00 |
| s2 | This data is available through AnyonWiki.org, a website that provides robust search and download functionality and presents each fusion ring and category via a dedicated webpage. | method 0.52 | 0.38 | 0.07 | asserted 1.00 |
| s3 | In this short article we discuss the data available at AnyonWiki.org as well as the scientific and communal motivations for the design of its underlying AnyonWikiDatabase, which are centered around making the data accessible, independently verifiable, granularly citable, useful, and easy to work with. | contribution 0.97 | 0.38 | 0.25 | asserted 0.97 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Figures reproduced from arXiv:2610.30081 by the authors.
References
Discussion
Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.