{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-08-05-hrt-conjecture-disproved/","md":"https://postcutoff.com/e/2026-08-05-hrt-conjecture-disproved/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-08-05-hrt-conjecture-disproved","date":"2026-08-05","date_precision":"day","short_title":"HRT conjecture (1996) disproved","deck":"12 time-frequency shifts of a Schwartz function are linearly dependent, found with ChatGPT-assisted guesswork","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.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2608.05044: Linear dependence of time-frequency shifts of a Schwartz function","url":"https://arxiv.org/abs/2608.05044","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Terence Tao: A partial digestion of the HRT counterexample","url":"https://terrytao.wordpress.com/2026/08/06/a-partial-digestion-of-the-hrt-counterexample/","type":"discussion","group":"reactions","domain":"terrytao.wordpress.com"}],"official":1,"filed":"2026-09-29","updated":"2026-10-19","orgs":["Faulhuber, Petersen, van Velthoven, Voigtlaender (academic mathematicians)"],"title":"HRT conjecture (1996) disproved: 12 time-frequency shifts of a Schwartz function are linearly dependent, found with ChatGPT-assisted guesswork","summary":"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"],"key_numbers":[],"tags":["math","harmonic-analysis","time-frequency-analysis","counterexample","ai-assisted"],"science":{"field":"mathematics","subfield":"harmonic analysis / time-frequency analysis","problem":"Heil–Ramanathan–Topiwala (HRT) conjecture on linear independence of time-frequency shifts","result":"Explicit counterexample: 12 time-frequency shifts of a Schwartz function are linearly dependent, disproving the HRT conjecture.","open_since":"1996","ai_system":["ChatGPT"],"human_role":"Human-led with AI assistance for strategy and parameter search; humans wrote and checked the proof","verification":{"key":"preprint","label":"Preprint only","text":"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"},"status":"pending","shock":"A well-known 30-year-old conjecture, widely believed true, turned out false, and the counterexample was found with a chatbot's help."},"body_md":"## What happened\n\nFour 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.\n\n## Why it matters\n\nIt 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.","disputed":[{"label":"Verification","text":"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","sources":[]}],"related":[{"id":"2026-08-05-sendov-conjecture-proved","url":"https://postcutoff.com/e/2026-08-05-sendov-conjecture-proved/","date":"2026-08-05","date_precision":"day","short_title":"Sendov's 1958 conjecture on polynomial roots proved with GPT-5.6 Pro","deck":"Tao simplifies and formalises it","takeaway":"It is a classic, well-known conjecture closed by AI, with the leading expert on the problem verifying and formalising the result.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":2,"official":1,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"]},{"id":"2026-07-20-jacobian-conjecture-counterexample","url":"https://postcutoff.com/e/2026-07-20-jacobian-conjecture-counterexample/","date":"2026-07-20","date_precision":"day","short_title":"Claude Fable 5 finds a counterexample to the Jacobian conjecture in dimension 3","deck":null,"takeaway":"The Jacobian conjecture is one of the most famous open problems in algebra.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":9,"official":6,"filed":"2026-09-29","updated":"2026-11-10","orgs":["Anthropic"]}],"people":[{"id":"terence-tao","name":"Terence Tao","url":"https://postcutoff.com/person/terence-tao/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-19","type":"edit","text":"Status confirmed → pending (unrefereed preprint; Tao's post is a partial digestion, not a full check)"},{"date":"2026-09-29","type":"filed","text":"Created"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-09-29","run":null,"sources_read":null,"updated":"2026-10-19","human_review":null,"version":{"date":"2026-10-19"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":97,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":36,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":127,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":66,"in_training_data":false}],"short_url":null}