HRT Conjecture
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Statement
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.
Context
The central open problem of time-frequency and Gabor analysis for thirty years, with a continuous partial-results literature, but little known outside harmonic analysis.
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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.
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0 human reviews · 0 machine checksNo person has reviewed this attempt. It has not been checked at all.
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