What was already known
Green asked for the threshold. Earlier amenability constructions and an upper restriction K<p^(1/100) left a wide gap.
AI involvement
Ma, Tang and Xu credit ChatGPT 5.4 with the retained q-adic valuation formulation. They developed the final proof; the amenability idea was already known to human researchers.
The first-person disclosure identifies the retained, central valuation step as mostly AI-origin, while the surrounding proof and previously known amenability route have substantial human input.
Importance
The valuation step helps identify the sharp logarithmic threshold.
Limitations
The lower construction uses a previously known amenability idea. The authors corrected faulty model drafts and developed the surrounding Fourier and probabilistic arguments. No completed check of the final valuation proof is documented.
Technical verification
The paper presents proofs and notes iterative correction of AI errors, but no completed human or machine check of the final q-adic upper-bound argument is expressly documented. Green's comments improved the writing and prior-work context, not a stated proof audit.
Human understanding
The human authors explain Fourier concentration, approximate dilation invariance, the valuation sum and the lower Følner construction, including where early model arguments had gaps.
Correction and publication history
The valuation step helps identify the sharp logarithmic threshold.
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