Han's Conjecture
For a finite-dimensional algebra , finite global dimension forces for all large . Han conjectured the converse: eventual vanishing of Hochschild homology should detect homological smoothness. Disproved by an explicit finite-dimensional -algebra with for every and , built by transporting Krah's phantom into a singularity category via one-periodic folding.
- Result
- Disproved(see note)
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Construction
- Field
- Homological algebra
- Posed by
- Yang Han
- Year posed
- 2006
- Years open
- 20y
- Solved
- 2026-07-31
- Model
- GPT-5.6 Sol Ultra
- Vendor
- OpenAI
- Collaborators
- Bochao Kong, Yeqin Liu, Yu Shen
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 25 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
The counterexample is an ordinary algebra concentrated in degree zero, with the strongest possible vanishing in positive degrees, so the phenomenon needs no grading or differential. Liu and Shen had already disproved the differential-graded version in December 2025 without any AI involvement; the classical case is the one that fell with a model in the loop.
What the AI did
The paper's acknowledgment in full: "The counterexample presented in this paper was discovered with the assistance of OpenAI's GPT-5.6 Sol Ultra model. All mathematical arguments and references were independently verified by the authors." In a counterexample paper the algebra is the whole result, so crediting the model with its discovery is a claim about the central object, not about support work. The hedge "with the assistance of" keeps this below the top tier.
Verification
A preprint days old with no independent review. The surrounding evidence is unusually strong for something this new: two of the three authors disproved the differential-graded analogue of the same conjecture in December 2025, this paper extends that program, and the construction runs on named recent machinery (Krah's phantom from Inventiones 2024, Chen's partial-resolution theorem, the Wang-Arunachalam-Keller identification) rather than novel unpublished tools. That is provenance, not verification, and the tier reflects the difference.
Sources
Submitted by FrostyBadger576 on