Low-rank
Grothendieck rings
Classification, constructions, and open problems.
Classification results through rank 12, unrestricted in fusion multiplicity for ranks 2–4, with explicit bounds in higher ranks.
“New” constructions catalogue → Specific constructions not, to our knowledge, explicitly recorded in the literature; no claim of novelty up to tensor equivalence or of exoticness.
Supporting resource: Fusion-ring census
Rings by rank and multiplicity
How to read the counts
27 October: three further rank-nine exclusions; seven rings remain unresolved. The R1660–R941 construction is conditional, and every GH(C6,2) Morita route for R1768 is excluded. Proofs and individual findings →
25 October correctness revision: exact proof repairs and replay corrections; classification counts are unchanged. Audit findings and corrected proofs →
Grothendieck counts mean ring types reported as realized, subject to the research caution above. A ≥ sign marks a lower bound in a band containing unresolved fusion rings; the eventual total could be larger. The Grothendieck view omits rows containing only zeros. Fusion view includes the separate census bands, including those without categorification results; the full fusion-ring census table ↗ covers more bands.
Classification scope
What “complete” means here. The current research assigns each counted based ring within the stated range either a categorical model or an exclusion by a necessary criterion. This is a coverage claim, pending independent verification. Results outside that range are not implied by the count table.
Ring catalogue
Values marked ≈ are computed numerical approximations. Fusion coefficients, duals, invertible objects, and codegree polynomials where shown come from exact integer arithmetic.
Extended Haagerup EH1 · rank 6, multiplicity 7. The known model R6_7_2274 comes from the Acta Mathematica construction of Bigelow–Morrison–Peters–Snyder. Its fusion rules are given by Grossman–Morrison–Penneys–Peters–Snyder, The Extended Haagerup fusion categories, Ann. Sci. Éc. Norm. Supér. (4) 56 (2023), no. 2, 589–664, §2.1, p. 597 (arXiv).
Papers & verification
Research updates, verification and reading notes
A concise introduction to the project. The submitted 26 October overview covers classification through rank twelve and the constructions catalogue. Its ten-case frontier predates the three further exclusions; seven cases are currently open. Hexagon · 2610.00300 · Source and revision status ↗.
Research announcement · 4 pagesCurrent manuscripts and verification materials. Each PDF presents the project’s current arguments, classification scope and remaining questions. Read the verification notes for the precise coverage of automated checks.
12 mathematical manuscripts: 678 pages; with the three verification guides: 696; with the current research-status report: 702. Each distinct document is counted once. Current page inventory and checksums ↓
Earlier manuscript inventory
8 mathematical manuscripts: 560 pages; with the three verification guides: 578; with the current research-status report: 582. Each distinct document is counted once. Page inventory & checksums · JSON ↓
Every individual case record includes a fusion tensor TXT download, with the same rank and multiplicity filters throughout the atlas. The earlier rank-six and rank-eight open tensors · Python ↓ remain available with their original source checksums.
17 October 2026 · research status
| Fusion ring | Argument or categorical construction |
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Current rank-six frontier
Two rank-six categorification cases remain unresolved. The summaries below distinguish necessary structural constraints from a category construction or a general exclusion. Consult the source arguments and the verification notes for the precise scope of the current evidence.
| Fusion ring | Research progress |
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Trace every number
Classification research
Manuscripts, model catalogues, exclusion certificates, and verification instructions.
Open classification repository 24 October 2026 proofs and remaining casesDownload the data
Browse all 1,434,042 released census rings at ranks 3–12, each with an individual tensor download, or download the source datasets below.
Ranks 2–8 · realized rings JSON Ranks 9–10 · all 336 records JSON Rank 9 · multiplicity two · all 2,083 records JSON Rank 11 · all 182 records JSON Rank 12 · all 632 records JSON Complete census · individual records Ranks 2–8 · Grothendieck counts CSV Ranks 2–8 · fusion counts CSV Ranks 2–8 · status JSON All 7 open rings · Python tensors Rank-six residual tensors JSON Rank-six release summary