HRT Conjecture
Heil, Ramanathan and Topiwala conjectured in 1996 that any finite set of time-frequency shifts of a nonzero square-integrable function is linearly independent. This refutes it: there is a Schwartz function admitting 12 linearly dependent time-frequency shifts.
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Construction
- Field
- Harmonic Analysis
- Posed by
- Christopher Heil, Jayakumar Ramanathan, Pankaj Topiwala
- Year posed
- 1996
- Years open
- 30y
- Solved
- 2026-08-05
- Model
- GPT-5.6 Pro
- Vendor
- OpenAI
- Collaborators
- Markus Faulhuber, Philipp Petersen, Jordy Timo van Velthoven, Felix Voigtlaender
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 35 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
ChatGPT (GPT-5.6 Pro) developed the counterexample and its proof strategy in dialogue with the authors. They initiated the search, told it to enlarge the phase-space configuration when it stalled, steered it from abstract methods toward elementary ones, and had it turn an existential construction into a fixed finite one. The authors then replaced several of the model's arguments with elementary proofs and closed gaps they call occasionally substantial: the whole of Section 6.2 and Appendix A was originally two sentences.
Verification
A preprint one day old with no external confirmation. The authors certify their numerical estimates with interval arithmetic, but that is their own rigor rather than third-party review.
Source
Submitted by QuietMarten105