What was already known
Even-base measures were known to be spectral. For odd bases, failure of an orthogonal exponential basis did not rule out a Fourier frame.
AI involvement
De Dios Pont, Liehr and Taylor say GPT-5.5 identified the Bessel-bound obstruction that suggested their proof, helped simplify an expression and assisted formalization. They checked and rewrote all LLM-influenced paper content.
The authors identify the GPT-5.5-assisted failed-candidate obstruction as the seed of their final theorem, while they supplied and wrote the completed proof.
Importance
Resolves the middle-third Fourier-frame question and extends nonexistence to all odd bases.
Limitations
GPT-5.5 found an obstruction while testing a failed positive construction; the three human authors wrote the final theorem and proof. The Lean theorem and statement alignment are source-reported; human line-by-line review of generated Lean proof files is not claimed. Fu, Song and Wang later gave an independent proof, not a check of this argument.
Technical verification
The source reports a Lean formalization of Theorem 1.1 and an explicit human check and rewrite of all LLM-influenced mathematical content. The completed Palomar peer review documents the same theorem, statement alignment and reported formalization.
Formal checking
Formal NoFourierFrameExists theorem and reported statement/axiom alignment; source-reported.
- Checker
- Lean 4 kernel through the source formalization, as documented in the completed Palomar review
- Scope
- No Fourier frame for the fair {0,2} Cantor measure for every odd base b > 1
- Basis
- Reported by the linked checker
Formal checking
Authors checked and rewrote every LLM-influenced part of the paper; source-reported.
- Checker
- Jaume de Dios Pont, Lukas Liehr and Mitchell A. Taylor
- Scope
- The LLM-influenced obstruction and proof of odd-base Fourier-frame nonexistence
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The human authors explain the Bessel obstruction and full contradiction in Sections 1.5 and 4, with their own written proof.
Correction and publication history
Resolves the middle-third Fourier-frame question and extends nonexistence to all odd bases.
Read this published revision