What was already known
Bernstein’s inequality gives the toy model’s sup-norm bound. Tao reports that a short proof of this sharp integral bound was not previously known to him.
AI involvement
AlphaEvolve experiments led Tao to the factorization. ChatGPT Pro proved the L¹ factor; Tao proved the reciprocal-derivative factor and combined them into the theorem.
The model-assisted factorization search and ChatGPT's proof of one load-bearing inequality supply central work for the retained sharp toy-model integral bound; Tao supplied the other factor and main-paper extension.
Importance
Provides a sharp integral bound and a useful factorization within the interpolation study.
Limitations
The broader local Bernstein and interpolation lower bounds are separate human-developed results. No completed check of this specific AI-assisted lemma is documented.
Technical verification
The paper gives a proof, but no completed independent human check or formal machine check of Theorem 1.13(ii) is documented; Reported is the evidence default.
Human understanding
Tao explains the factorization, credits the proof of each factor and shows why multiplying them gives the sharp bound.
Correction and publication history
Provides a sharp integral bound and a useful factorization within the interpolation study.
Read this published revision