{"id":"61ea018a-ba51-4679-8a10-3f9c43a6b14e","arxiv_id":"2507.00652","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every multiplicity-free fusion ring up to rank 7, a table of small invariants distinguishes all inequivalent pivotal braided and non-braided fusion categories in the Anyonica census.","lead":"This paper provides tables of small, manually checkable invariants that distinguish between the multiplicity-free fusion categories collected in the Anyonica census up to rank 7. The tables give researchers a practical way to tell whether two such categories are equivalent and to identify a category from its skeletal data.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The unverified completeness of the Anyonica census is load-bearing: the invariant reduction is checked only against the census, and the census has already had bug-induced duplicates.","rationale":"The reader's weakest-assumption diagnosis is correct and is the most load-bearing issue. The invariant construction itself (Section 3.3.1) is mathematically sound: the complete invariant (S(I), [S(I)]) is gauge/automorphism invariant and complete by the gauge-split basis properties. The practical tables are obtained by a reduction that is validated only against the census. Therefore the same unverified premise—completeness of the census—underpins both the coverage of the tables and the completeness of the reduced invariants. The paper is transparent about this limitation and even documents a software-bug-induced duplicate, which strengthens the concern. A missing category would not merely add a row; it could change the equivalence relation that the reduced invariants are supposed to decide. The proposed test—re-running the enumeration for one ring with an independent solver—would directly settle whether the census is complete, and hence whether the classification claim holds. Until such a test is done (or the complete log files are independently verified), conditional acceptance is the right verdict.","tokens_in":117631,"tokens_out":15589,"duration_ms":179883,"concrete_test":"Independently re-enumerate all fusion systems for at least one ring with a large solution count (e.g., FR4,1,0_1 = Z2⊗Z2 or FR7,1,6_1 = Z7) using an independent implementation (e.g., TensorCategories.jl/Oscar) that solves the pentagon, hexagon, and pivotal equations from the fusion ring, and compare the resulting gauge/automorphism-equivalence classes with the number of rows in the paper's table for that ring. If the independent enumeration yields any additional class, the census is incomplete and the classification claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's claim that the Section 4 tables completely distinguish all multiplicity-free pivotal braided and non-braided fusion categories up to rank 7 is a classification claim: it requires that every such category appears in the Anyonica census. Section 3.2 explicitly states that completeness is much harder to verify than correctness and relies on the absence of silent critical bugs in Anyonica and Mathematica, plus no human error, and notes a concrete past failure: four Rep(D7) categories were duplicates produced by a software bug. Completeness is doubly load-bearing here. First, a missing category would have no row in the tables, so the 'classification' would be incomplete. Second, the reduction from the complete invariant (S(I), [S(I)]) in Section 3.3.2 to the small practical tuples is justified by checking which gauge-invariants are constant or redundant 'for all categories'; in practice this check is performed on the census entries only. If a category is missing, the reduced invariants could fail to separate it from a listed one even though the full invariant would do so. The paper supplies no machine-readable census, no completeness-check script, and no independent re-enumeration; the dependence on proprietary Mathematica makes the completeness premise non-reproducible.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":117872,"tokens_out":4433,"duration_ms":53127,"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":[{"comment":"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.","section":"§3.2 and §4"},{"comment":"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.","section":"§3.3.2 and §4.5"},{"comment":"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.","section":"§3.3.1"}],"minor_comments":[{"comment":"'MFPBCFs' should be 'MFPBFCs'.","section":"Introduction, item 3"},{"comment":"'is could be regarded' is a grammatical typo.","section":"§3.1"},{"comment":"'de gauge-invariants' should read 'the gauge-invariants'.","section":"§3.3.2"},{"comment":"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, ...'.","section":"Table 52"},{"comment":"The section heading 'FR7,1,2 3' lacks a colon, unlike neighboring headings such as 'FR7,1,2 4 ∶ Rep(SD16)'.","section":"§4.5.54"},{"comment":"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.","section":"§4.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is honest and carefully hedged, and I do not see a deliberate attempt to overclaim. The main concern is that the deliverable is effectively an appendix to the author's own computational census, and the lack of machine-readable census data and generation scripts is a substantial reproducibility gap for a data-heavy paper. If the journal accepts computational classification results, I would strongly encourage requiring the data and code as supplementary material before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat'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.\n\nThe 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.\n\nThe 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.\n\nWho 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.","headline":"Practical invariant tables that separate the Anyonica census up to rank 7 — honest, useful, and conditional on the still-unverified completeness of that census.","tokens_in":118328,"tokens_out":3859,"would_cite":true,"duration_ms":46147,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18M20","18M15"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["multiplicity-free fusion categories","fusion systems","gauge-split bases","invariants","pentagon equations","braided fusion categories","pivotal structure","classification"],"falsifier":"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.","tokens_in":117404,"feed_emoji":"🧮","tokens_out":7764,"duration_ms":83500,"temperature":0.7,"pith_summary":"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.","feed_headline":"Invariant tables tell every rank-7 fusion category apart","feed_subtitle":"Two census entries match exactly when their few invariant values agree, so equivalence can be checked by hand.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the census of skeletal data and the gauge-split basis method from which the invariant tables are built.","marker":"[Ver24b]"},{"why":"Introduces fusion systems and the equivalence of skeletal data under gauge transforms and fusion-ring permutations, the relation the invariants must detect.","marker":"[DHW13]"},{"why":"The Anyonica software package whose dataset is the object of the classification claim.","marker":"[Ver24a]"},{"why":"Independent verification of the census data's correctness, used to support the assumption that each listed skeletal system is a genuine category.","marker":"[MT]"},{"why":"Provides the Oscar symbolic algebra backend for that independent verification.","marker":"[25]"},{"why":"The anyonwiki website the tables are designed to look up against.","marker":"[VS]"}],"fun_headline_variants":["Few numbers tell rank-7 fusion categories apart","Tiny invariant sets separate every rank-7 fusion category","Hand-checkable invariants identify every rank-7 fusion category","Small invariant tuples distinguish all rank-7 fusion categories"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Few numbers tell rank-7 fusion categories apart","Tiny invariant sets separate every rank-7 fusion category","Hand-checkable invariants identify every rank-7 fusion category","Small invariant tuples distinguish all rank-7 fusion categories"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001568,"raw_usage":{"total_tokens":6228,"prompt_tokens":879,"completion_tokens":5349,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":495,"completion_tokens_details":{"reasoning_tokens":5284}},"tokens_in":495,"tokens_out":5349,"duration_ms":37188,"temperature":1.0,"reasoning_tokens":5284,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:10:23.787631+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}