HRT conjecture (1996) disproved: 12 time-frequency shifts of a Schwartz function are linearly dependent, found with ChatGPT-assisted guesswork
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. This disproves the Heil–Ramanathan–Topiwala (HRT) conjecture with an explicit 12-point example. ChatGPT helped with the initial strategy and parameter guesswork. The proof was written by hand and certified numerically, not in Lean.
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
Science result
- Field
- mathematics / harmonic analysis / time-frequency analysis
- Problem
- Heil–Ramanathan–Topiwala (HRT) conjecture on linear independence of time-frequency shifts (open since 1996)
- Result
- Explicit counterexample: 12 time-frequency shifts of a Schwartz function are linearly dependent, disproving the HRT conjecture.
- AI system
- ChatGPT
- Human role
- Human-led with AI assistance for strategy and parameter search; humans wrote and checked the proof
- Verification
- Handwritten proof with certified numerics; expert-checked (Tao digest); not formalised in Lean; peer review pending
- Status
- confirmed
- Why surprising
- A well-known 30-year-old conjecture, widely believed true, turned out false, and the counterexample was found with a chatbot's help.
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.
Changelog
- 2026-09-29: created (lead from data/leads.md)
Related events
- Claude Fable 5 finds a counterexample to the Jacobian conjecture in dimension 3 ★★★★★
- Sendov's 1958 conjecture on polynomial roots proved with GPT-5.6 Pro; Tao simplifies and formalises it ★★★★
Sources (2)
- paperarXiv 2608.05044: Linear dependence of time-frequency shifts of a Schwartz function
- discussionTerence Tao: A partial digestion of the HRT counterexample
id: 2026-08-05-hrt-conjecture-disproved · updated 2026-09-29 · open in the interactive timeline