math.QA — Quantum Algebra
Low-rank Grothendieck rings: an AI-assisted classification project, constructions, and open problems
We present a public research project on the Grothendieck rings of complex fusion categories of rank at most eight. The project reports unrestricted classification theorems in ranks two, three, and four, and classifications within specified fusion-multiplicity bounds in ranks five through eight, with two rank-six categorification questions unresolved. Its resources include research manuscripts, an exact fusion-ring census, verification code, and an interactive atlas. We describe seven constructions that are promising candidates for new tensor categories, subject to a separate assessment of their novelty up to tensor equivalence. We also state the two open problems explicitly. This announcement records the project's claims and their verification status; it invites independent mathematical review and does not provide a new proof of the complete classification.