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Low-rank Grothendieck rings: an AI-assisted classification project, constructions, and open problems

Primarily AI-generated textHuman understanding: some partsmath.QA — Quantum Algebramath.CT — Category Theory

Contributed by Sebastien Palcoux

Version 2 / Oct 17, 2026 / CC BY 4.0

Abstract

We present a public AI-generated research project on fusion rings and their categorifications over C\mathbb C, with data extending through rank twelve. The project reports unrestricted Grothendieck-ring classifications in ranks two, three, and four, a complete multiplicity-free classification through rank twelve, and classifications or partial classifications at higher multiplicities. The released census contains 1,434,0421{,}434{,}042 rings in ranks three through twelve; ten categorification questions remain open within its stated bounds. The resources include research manuscripts, categorical models, exclusion arguments, verification code, and an interactive atlas. One categorical model per ring suffices for our classification problem; classifying all realizing categories up to equivalence is a further task. We summarize the current scope, compare it with AnyonWiki, and link to the online constructions catalogue and unresolved cases. The project's original research content was generated by AI under human direction; independent human verification remains pending.

Provenance statement

This announcement was prepared with ChatGPT from the contributor-directed project. The project's original mathematical arguments, research exposition, enumeration and verification code, and computational outputs were generated by AI under human direction. Cited theorems, known categories, and imported data retain their external authorship. Automated checks and AI-assisted review do not constitute independent human verification. Independent human verification and Lean formalization remain pending. The constructions catalogue uses 'new' only for specific realizations not, to our knowledge, explicitly described in the literature; no novelty up to tensor equivalence or exoticness is claimed.

Tools used

OpenAI
ChatGPT

References

  1. V. Ostrik, Fusion categories of rank 2 , Math. Res. Lett. 10 (2003), 177--183. arXiv:math/0203255 (https://arxiv.org/abs/math/0203255).arXiv
  2. J. Dong, S. Palcoux, and A. Plessis, Classifying Grothendieck rings of pivotal fusion categories of rank four , arXiv:2609.33272 (https://arxiv.org/abs/2609.33272) (2026).arXiv
  3. Z. Liu, S. Palcoux, and Y. Ren, Classification of Grothendieck rings of complex fusion categories of multiplicity one up to rank six , Lett. Math. Phys. 112 (2022), article 54. doi:10.1007/s11005-022-01542-1 (https://doi.org/10.1007/s11005-022-01542-1).DOI
  4. C. Delaney, R. Razzouk, J. Slingerland, and G. Vercleyen, AnyonWiki.org , arXiv:2610.30081 (https://arxiv.org/abs/2610.30081) (2026). https://anyonwiki.org/.arXiv
  5. G. Vercleyen and J. Slingerland, On low-rank multiplicity-free complex fusion categories , Commun. Math. Phys. 407 (2026), article 231. doi:10.1007/s00220-026-05705-2 (https://doi.org/10.1007/s00220-026-05705-2).DOI
  6. G. Vercleyen and J. K. Slingerland, On low rank fusion rings , J. Math. Phys. 64 (2023), no. 9, 091703. doi:10.1063/5.0148848 (https://doi.org/10.1063/5.0148848).DOI
  7. S. Palcoux (project coordinator), Fusion Ring Census , AI-generated research repository (2026). https://github.com/sebastienpalcoux/fusion-ring-census.link
  8. S. Palcoux (project coordinator), Grothendieck Ring Classification , AI-generated research repository (2026). https://github.com/sebastienpalcoux/grothendieck-ring-classification.link
  9. S. Palcoux (project coordinator), Fusion & Grothendieck Ring Atlas , public project website (2026). https://sebastienpalcoux.github.io/grothendieck-ring-classification/.link

Version history

  1. v1Submitted by Sebastien PalcouxInitial depositSupersededOct 17, 2026
  2. v2Submitted by Sebastien PalcouxSubmitted as a new versionCurrentOct 26, 2026