Tables of practical invariants for distinguishing multiplicity-free fusion categories up to rank 7

Gert Vercleyen

REVIEW 3 major objections 6 minor 10 references

Tables of practical invariants for distinguishing multiplicity-free fusion categories up to rank 7

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Small tables of invariants distinguish every multiplicity-free fusion category up to rank 7 in the Anyonica census, under the assumption that the census is complete.

desk verdict Practical invariant tables that separate the Anyonica census up to rank 7 — honest, useful, and conditional on the still-unverified completeness of that census. read the letter →

arxiv 2507.00652 v3 pith:FV5ZONKJ submitted 2025-07-01 math-ph math.CTmath.MPmath.QA

Gert Vercleyen This is my paper

classification math-phmath.CTmath.MPmath.QA MSC 18M2018M15

keywords multiplicity-freefusioncategoriessystemsgauge-splitbasesinvariantspentagonequationsbraidedpivotalstructureclassification

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether the Anyonica census of multiplicity-free pivotal fusion categories up to rank 7 is a genuine classification, and answers the uniqueness half of that question. Working under the assumption that every listed set of skeletal data really is a category, it provides tables of small sets of invariants--rational expressions built from F-symbols, R-symbols, and left quantum dimensions--such that two categories with the same fusion ring are equivalent exactly when their invariant values coincide after applying a fusion-ring automorphism. The tables are arranged so the comparison can be done by hand, and they also let a researcher identify which entry on the anyonwiki corresponds to a known set of skeletal data. The paper is explicit that completeness of the census--no missing categories--remains the harder, unproven step.

What carries the argument

The load-bearing construction is the gauge-split basis of a fusion system: a tuple $(I, D)$ of rational monomials in the formal F-symbols, R-symbols, and left quantum dimensions, where the elements of $I$ are gauge-invariant, the elements of $D$ are gauge-dependent and can be set to arbitrary nonzero values by a gauge transform, and every formal monomial factors uniquely as a product of powers of these elements. With such a basis, the orbit $S(I)$ of the invariant part under fusion-ring automorphisms, together with its values $[S(I)]_\Phi$, forms a complete invariant of the system $\Phi$, since the original skeletal data can be recovered from it. The paper then prunes redundant entries from this complete invariant to obtain the small tables, choosing invariants that avoid zero-valued F-symbols when possible, separate F-, R-, and pivotal data, and keep the tuples short enough for hand computation.

What would settle it

Run a fresh, independent enumeration of multiplicity-free pivotal fusion categories of rank up to 7--using different software and equation-solving strategies--and find either a category missing from the census or two listed categories that are inequivalent yet produce identical values for every invariant in their table.

Watch

Extended reading notes

Core claim

The paper's central claim is that, given the skeletal data proposed by Anyonica is correct, the tables in Section 4 completely distinguish all inequivalent multiplicity-free pivotal braided and non-braided fusion categories (MFPBFCs and MFPNBFCs) up to rank 7. For each multiplicity-free Grotendieck ring of rank at most 7, the paper lists a small tuple of gauge-invariant rational monomials, possibly organised as an orbit under the ring's automorphism group, whose value is different for every inequivalent category in the census. Categories with the same fusion ring are equivalent as pivotal braided fusion categories exactly when the indices $n_F$, $n_R$, $n_P$ in the naming $[\mathrm{RingName}]_{n_F,n_R,n_P}$ agree, and the invariant tables provide the same information directly. The author frames the result as resolving the inequivalence statement of the census under the correctness assumption, turning the 'each category listed only once' assertion into something any reader can check manually.

Load-bearing premise

The census of multiplicity-free pivotal braided and non-braided fusion categories up to rank 7 is complete, in the sense that no category is missing; if any one is absent, the invariant tables cannot distinguish it from a listed entry and the classification claim collapses.

Editorial extensions

If this is right

Two skeletal fusion systems in the census with the same Grotendieck ring are the same category if and only if their reduced invariant values agree, so the uniqueness question for rank up to 7 is settled once the correctness assumption is granted.
A researcher who has computed the F-, R-, and pivotal data of an unknown rank-7 category can identify its Anyonica and anyonwiki entry against these tables without solving any polynomial equations.
The three indices $n_F$, $n_R$, $n_P$ provide a bookkeeping-level equivalence invariant: matching triples mean equivalent pivotal braided categories, and fixing only some indices compares the category at the level of fusion, braiding, or pivotal structure alone.
Because the correctness of the census data has been double-checked with independent symbolic computations, the 'no duplicates' part of the classification is as rigorous as the trust placed in those computer algebra systems.
Completeness remains the only gap: nothing in the tables can detect a missing category, and the paper identifies this as the step that needs an independent reimplementation of the search.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If gauge-split bases behave as well for higher ranks, the same reduction recipe could produce hand-checkable invariant tables for future censuses; the paper notes it is unknown whether the convenient property 'no zero F-symbol is needed' persists beyond rank 7.
The invariant tuples could double as stable fingerprints for a fusion-category database, letting different research groups compare entries without exchanging full skeletal data.
The existence of four duplicate Rep(D7) entries in the census shows that completeness failures are concrete, so a targeted independent search of rank-7 fusion rings would be a meaningful test of the completeness assumption.
The tables implicitly define a normal form for census entries under gauge and automorphism equivalence, which could be used to canonicalize newly found categories before adding them to any catalogue.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. The paper delivers lookup tables: for each multiplicity-free pivotal fusion ring up to rank 7, a small tuple of F-symbol, R-symbol, and quantum-dimension invariants that separates every braided and non-braided category in the Anyonica census. And the separation claim is explicitly conditional on that census being correct. The author says so plainly, and completeness of the census is not established.

What's genuinely new is the reduction work. The complete invariant (S(I), [S(I)]) comes from the author's earlier gauge-split basis machinery, but turning it into hand-checkable tables — choosing which invariants to keep, handling orbits under fusion-ring automorphisms, explaining the three table formats — is a real service. Anyone with skeletal data who wants to identify the corresponding anyonwiki entry will use these tables directly.

The paper earns credit for honesty. It lists the three statements a classification needs, marks correctness as checked (including by the independent TensorCategories.jl package), and admits completeness rests on the absence of silent bugs. It even reports the four Rep(D7) duplicates caused by a software bug. The invariant direction is sound: different tuples imply inequivalent categories, so the uniqueness claim goes through if the skeletal data is correct.

The soft spots are the ones the author names, and they add up. Completeness is load-bearing, and it is the premise that already failed once in practice. A missing category would have no row in the tables, and the reduction from the full invariant to the small tuples was validated only against census entries — so a missing category could slip past the reduced invariants even when the full invariant would catch it. On top of that, the pipeline depends on proprietary Mathematica, and no machine-readable census or completeness-check script is provided. The conditional framing is honest and limits the damage, but the "classification up to rank 7" wording should not be read as more than a conjecture plus strong evidence.

Who benefits: anyone working on anyon models, topological quantum computation, or fusion category classification who needs to distinguish or identify these categories. It deserves a serious referee, accepted conditionally: the tables are the contribution, and the author should be asked to release the machine-readable census and an independent completeness check, or at least specify what evidence would be convincing.

Referee Report

3 major / 6 minor

Summary. This paper addresses whether the Anyonica census of multiplicity-free pivotal braided and non-braided fusion categories up to rank 7 can be regarded as a proper classification. The author discusses correctness, completeness, and uniqueness, and the main technical contribution is a method for producing small sets of gauge-invariant combinations of F-symbols, R-symbols, and quantum dimensions that are also invariant under fusion-ring automorphisms. The paper then presents extensive tables in Section 4 listing these invariants for every category in the census. Under the explicit assumption that the census data are correct, the tables distinguish all listed categories; the paper is candid that completeness of the census is not verified and that the reduction to small invariants is validated only on the census itself.

Significance. The tables are a useful practical resource: they provide a manual-checkable way to identify a skeletal category against the anyonwiki data and to verify inequivalence of two listed categories, and the one-way implication 'same category implies same invariant tuple' is rigorous. The paper also gives appropriate credit to computational infrastructure: the Anyonica code is open-source, correctness of the skeletal data has been independently checked with TensorCategories.jl/Oscar, and the author clearly states the limits of computational evidence. The main weakness is that the central classification statement is conditional on the completeness and correctness of the Anyonica census, and because the invariant reduction is optimized on that census, the tables cannot detect missing categories.

major comments (3)

[§3.2 and §4] The claim that the Section 4 tables 'completely distinguish all MFPBFCs and MFPNBFCs up to rank 7' is load-bearing on the completeness of the Anyonica census, which the author explicitly leaves unverified. A missing category would have no row in the tables, and the reduction from the complete invariant (S(I), [S(I)]_Φ) to the small practical tuples in §3.3.2 is justified by checking which gauge-invariants are constant or redundant per fusion ring 'for all categories', a check that in practice is performed only on the census entries. Thus the tables could fail to separate a missing category from a listed one even if the full invariant would separate them. The paper should either state the theorem only over the census, as a precisely scoped conditional statement, or provide an independent completeness verification such as a second enumeration, machine-readable census data, and rerunnable scripts.
[§3.3.2 and §4.5] The tables are the main result, but the computation that produces them is not reproducible from the manuscript alone. The paper states that 'it turns out' every Gröthendieck ring up to rank 7 admits a single S(I) that distinguishes all fusion systems, and that certain invariants are redundant, but neither the script nor the machine-readable values of all F- and R-symbols are included. Since the distinguishing-power claim rests on these computations, I ask the author to supply, as supplementary material or through a linked repository, the Anyonica census data, the exact invariant-selection code, and an output file that regenerates Tables 5–116, so that a reader can check the distinctness of the printed tuples rather than take them on faith.
[§3.3.1] The statement that (S(I), [S(I)]_Φ) is a complete invariant is asserted rather than proved. Property 4 of Definition 3.4 gives an expression for each monomial, but the recovery of the full skeletal data from the tuple and the statement that this tuple is invariant under arbitrary gauge transforms and fusion-ring automorphisms need a precise lemma. If this lemma is already in the companion work [Ver24b], it should be stated or reproduced here, because the reduction in §3.3.2 relies on it; if it is new, it needs a proof.

minor comments (6)

[Introduction, item 3] 'MFPBCFs' should be 'MFPBFCs'.
[§3.1] 'is could be regarded' is a grammatical typo.
[§3.3.2] 'de gauge-invariants' should read 'the gauge-invariants'.
[Table 52] The row for [FR6,1,0_1]_{3,5,1} is missing its opening parenthesis; it reads '{ 1, −0.707, ...' instead of '{ (1, −0.707, ...'.
[§4.5.54] The section heading 'FR7,1,2 3' lacks a colon, unlike neighboring headings such as 'FR7,1,2 4 ∶ Rep(SD16)'.
[§4.1] The shorthand for roots of unity, e.g. '𝜁 5 16', is printed with a space between the numerator and denominator; defining it as a single symbol such as ζ_16^5 would reduce ambiguity in the tables.

Circularity Check

2 steps flagged · score 4.0 of 10

The distinguishing-power claim is fitted to and checked against the author's own Anyonica census, whose completeness is explicitly assumed; the four Rep(D7) duplicates show the census has had bug-induced miss-classifications.

fitted input called prediction [Section 3.3.2 (Reduction of invariants) and Section 4 (Tables of invariants)]
"After removing the elements of each tuple in S(I) whose values are equal for all categories, we obtain a smaller invariant. ... It turns out that, for MFP(N)BFCs up to rank7, every Gröthendieck ring ℛ provides a list of tuples of formal symbols S(I) that ... distinguishes between all fusion systems with Gröthendieck ring ℛ."

The reduced invariants are pruned using the census-wide value distribution ('equal for all categories'), so the surviving tuple separates the census rows by construction. The paper then presents this in-sample separation as an unconditional classification claim: the tables 'distinguish between all MFPBFCs and MFPNBFCs up to rank 7'. That claim closes over categories not in the census only if the census is complete, and Section 3.2 says completeness is assumed: 'the completeness of the data relies on the assumption that there are no silent critical bugs in Anyonica and/or Mathematica and that no human error was made while processing the data.' The author's reported discovery of four equivalent Rep(D7) entries produced by a software bug shows the census has concrete failure modes.
self citation load bearing [Section 3.3.1, paragraph following Definition 3.4]
"In [Ver24b], it is shown that any fusion system (pivotal or non-pivotal, braided or non-braided) has an infinite number of gauge-split bases, and there exists an efficient method for constructing such bases."

The complete invariant (S(I), [S(I)]_Phi) on which the entire reduction rests requires a gauge-split basis, and both the existence theorem and the construction method are imported from the author's own prior work [Ver24b]. The census being distinguished is also the author's own (Anyonica / [Ver24b]), and the naming scheme n_F, n_R, n_p is likewise from [Ver24b]. No independent source establishes the completeness of the census or the adequacy of the construction for the closure claim; the only external check cited (TensorCategories.jl) verifies correctness of skeletal data, not that the reduced invariants separate every category up to rank 7.

full rationale

This paper is partially circular in its classification-strength claim, but not in its core invariant construction. The self-contained part is the complete invariant (S(I), [S(I)]_Phi): given any gauge-split basis, property 4 of Definition 3.4 lets the paper argue in-paper that the skeletal data can be recovered from the tuple, so the complete invariant genuinely separates equivalence classes. The circularity enters at the reduction step (Section 3.3.2): invariant elements are pruned by checking which values are constant 'for all categories' — in practice, for all rows of the author's own Anyonica census — and the surviving tuple is then presented as distinguishing 'between all fusion systems with Gröthendieck ring ℛ' and, in Section 4, 'all MFPBFCs and MFPNBFCs up to rank 7'. The separating power of the practical tables is therefore only as strong as the census's unproven completeness, which Section 3.2 explicitly flags: 'the completeness of the data relies on the assumption that there are no silent critical bugs in Anyonica and/or Mathematica and that no human error was made while processing the data.' The author's discovery of four equivalent Rep(D7) entries caused by a software bug (Section 3) shows this assumption has already failed once. The framework is also built on a self-citation chain: the census, the gauge-split basis existence and construction method, and the naming scheme all come from [Ver24b]/[Ver24a], and no external benchmark validates the closure claim. However, the author is transparent about the conditionality — Section 1 states that 'the correctness of the proposed classification becomes a matter of faith', and the inequivalence claim is explicitly 'under the assumption that all proposed data is correct' — and the in-sample row distinctness is visibly verified in the tables, so the formal direction (different invariants imply inequivalent census entries) is sound. The partial circularity is real but bounded, warranting a score of 4.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central contribution depends on the correctness and completeness of the Anyonica census, on the gauge-split basis theorem from the author's prior work, and on the reliability of the exact computer algebra used to generate the tables. No new entities or fitted numerical parameters are introduced.

assumptions (6)

ad hoc to paper The Anyonica census contains correct skeletal data for all listed multiplicity-free pivotal braided and non-braided fusion categories up to rank 7.
The paper verifies correctness computationally in Section 3.1 but does not prove it. The inequivalence claim in the abstract is made explicitly under this assumption.
ad hoc to paper The Anyonica census is complete: no category of rank at most 7 is missing.
Stated in Section 3.2 to be hard to verify. The classification framing requires completeness, and the Rep(D7) duplicate incident shows this is not trivial.
domain assumption Every fusion system admits a gauge-split basis, and the complete invariant built from it reconstructs the skeletal data.
Takes the construction and proof from the author's prior work [Ver24b]; used in Section 3.3.1 to justify the invariant method.
domain assumption Exact symbolic computations in Mathematica and TensorCategories.jl used for verification are error-free.
The correctness check in Section 3.1 relies on RootReduce and high-precision numerical evaluation; no formal machine-checked proof is provided.
domain assumption All categories are considered in a gauge where vacuum F-symbols and R-symbols equal 1.
Section 4.4 states that the tables assume this normalization and that users must check that their data satisfies it.
ad hoc to paper The tables in Section 4 were computed without transcription errors.
No code or data files are shipped to regenerate the 59 tables; the values are asserted in the text.

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Cite this review

Pith. "Pith review of Tables of practical invariants for distinguishing multiplicity-free fusion categories up to rank 7." pith.science (2026). https://pith.science/paper/FV5ZONKJ

@misc{pith2026250700652,
  author       = {Pith},
  title        = {Pith review of: Tables of practical invariants for distinguishing multiplicity-free fusion categories up to rank 7},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FV5ZONKJ}},
  note         = {Machine review of arXiv:2507.00652}
}

read the original abstract

This paper discusses to what extent the census of multiplicity-free fusion categories up to rank 7, proposed by the software package Anyonica and the anyonwiki website, can be regarded as a proper classification. The questions of correctness and completeness are briefly discussed, and the question of inequivalence of the provided categories is resolved under the assumption that all proposed data is correct. This is done by providing tables of small sets of invariants for these categories, for which (a) their invariance under automorphisms can be checked manually, and (b) the (in)equivalence of two skeletal fusion categories can be checked manually. These invariants can also be used to identify the category on the anyonwiki that corresponds to one for which the skeletal data is known.

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Works this paper leans on

10 extracted references · 6 canonical work pages

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An Invitation to the Mathematics of Topological Quantum Computation

[Row16] E C Rowell. “An Invitation to the Mathematics of Topological Quantum Computation”. In:Journal of Physics: Conference Series698 (Mar. 2016), p. 012012.doi: 10.1088/1742-6596/698/1/012012. [RT90] N. Y. Reshetikhin and V. G. Turaev. “Ribbon Graphs and Their Invaraints Derived from Quantum Groups”. In:Communications in Mathematical Physics127.1 (Jan. ...

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[Tur92] Vladimir G. Turaev. “Modular Categories and 3-Manifold Invariants”. In:International Journal of Modern Physics B06.11n12 (June 1992), pp. 1807–1824.doi: 10.1142/S0217979292000876. [TV17] Vladimir Turaev and Alexis Virelizier.Monoidal Categories and Topological Field Theory. Vol

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