Two Counterexamples in the Geometry of Numbers
The paper gives counterexamples in dimensions eight and nine to two problems: the Cartesian-product problem posed by Cassels for critical determinants and formulated by Zong for lattice packings, and a question raised by Sarnak, formulated as a conjecture by Chiu, on whether height among unit-volume flat tori is minimized by a lattice maximizing its shortest nonzero vector.
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Geometry of Numbers, Lattices
- Posed by
- J. W. S. Cassels and Chuanming Zong; Peter Sarnak, formulated by Chiu
- Year posed
- —
- Years open
- —
- Solved
- 2026-07-13
- Model
- ChatGPT 5 + Codex
- Vendor
- OpenAI
- Collaborators
- Nihar Gargava
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 20 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
The acknowledgements state the counterexamples were found using ChatGPT, and that the final paper is partly the author's own writing and partly written with the help of Codex.
Verification
No independent check, but the results are explicit counterexamples in dimensions eight and nine and are therefore checkable by inspection. Preprint, not refereed.