---
id: "2026-08-05-hrt-conjecture-disproved"
url: "https://postcutoff.com/e/2026-08-05-hrt-conjecture-disproved/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-08-05"
date_precision: day
category: science
importance: 3
confidence: high
status: [Event confirmed, Awaiting review]
verification: Preprint only
sources: 2
editor: human editor
human_review: null
version: "2026-10-19"
---

As of: 2026-11-13 13:49 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-08-05-hrt-conjecture-disproved/

# HRT conjecture (1996) disproved

Full title: 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

## 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

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 97 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 36 days after its cutoff
- Gemini 4 Hydrogen (training cutoff March 2026): 127 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 66 days after its cutoff

## Sources

1. [arXiv 2608.05044: Linear dependence of time-frequency shifts of a Schwartz function](https://arxiv.org/abs/2608.05044) (arxiv.org, paper)
2. [Terence Tao: A partial digestion of the HRT counterexample](https://terrytao.wordpress.com/2026/08/06/a-partial-digestion-of-the-hrt-counterexample/) (terrytao.wordpress.com, discussion)

## Changes

- 2026-10-19: Status confirmed → pending (unrefereed preprint; Tao's post is a partial digestion, not a full check)
- 2026-09-29 (filed): Created

## Related

- 2026-08-05: [Sendov's 1958 conjecture on polynomial roots proved with GPT-5.6 Pro](https://postcutoff.com/e/2026-08-05-sendov-conjecture-proved/index.md)
- 2026-07-20: [Claude Fable 5 finds a counterexample to the Jacobian conjecture in dimension 3](https://postcutoff.com/e/2026-07-20-jacobian-conjecture-counterexample/index.md)
- People: [Terence Tao](https://postcutoff.com/person/terence-tao/)