HRT conjecture (1996) disproved
12 time-frequency shifts of a Schwartz function are linearly dependent, found with ChatGPT-assisted guesswork
Event confirmedAwaiting review
The takeaway
arXiv 2608.05044 (5 Aug 2026), by Markus Faulhuber, Philipp Petersen, Jordy Timo van Velthoven and Felix Voigtlaender, shows that finitely many time-frequency shifts of a Schwartz function can be linearly dependent.
Status
- Claim
Event confirmedAwaiting review
- Our reporting
- High confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 19 October 2026
Your AI and this story
- GPT-6 Astra97 days after its cutoff
- Claude Opus 5.536 days after its cutoff
- Gemini 4 Hydrogen127 days after its cutoff
- Grok 4.766 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 36 days before it.
Key facts
- HRT conjecture (Heil, Ramanathan, Topiwala, 1996): any finite set of distinct time-frequency shifts of a nonzero L² function is linearly independent
- Counterexample: 12 time-frequency shifts of a nonzero Schwartz function with a nontrivial vanishing linear combination
- Key certified numerical step: an operator-norm distance below the 1/3 threshold (value 0.333032 per Tao’s digest)
- AI role (per Tao): ChatGPT assisted with the initial proof strategy and ‘AI-assisted guesswork’ to choose parameters; final arguments handwritten with a readable overview
- v2 adds a separate, purely analytic proof of a qualitative counterexample; Python code in the arXiv ancillary files
- Follow-ups: Vignon Oussa proposed a four-point counterexample with Arb (interval arithmetic) verification
What happened
Four time-frequency analysts posted a counterexample to the HRT conjecture. Tao’s next-day digest explains that ChatGPT helped them find a workable strategy and good parameter choices. The decisive estimate was then certified by traditional numerical computation, and the paper itself was written by hand.
Why it matters
It is a clean example of the “AI-assisted, human-written” mode of discovery. It sits alongside the autonomous, Lean-verified results of summer 2026 and settles a conjecture that had resisted proof for three decades.
What is disputed or not yet verified
| Verification | Unrefereed preprint by specialists (Faulhuber, Petersen, van Velthoven, Voigtlaender) with certified numerics; v2 adds a purely analytic proof of a qualitative counterexample; Terence Tao posted a partial digestion; not formalised in Lean |
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Sources
2 sources from 2 sites. Numbers match the chips in the text.
2 sources: 1 primary, 1 reaction
Primary
Reactions
- Terence Tao: A partial digestion of the HRT counterexampleterrytao.wordpress.com, discussion
Changes
- Status confirmed → pending (unrefereed preprint; Tao’s post is a partial digestion, not a full check)
- Filed