What was already known
The prime-power bound fℓ(n)≤n was sharp. Babai and Frankl asked whether it persisted for composite moduli, even ℓ=6.
AI involvement
Bukh, Chao and Zheng report that GPT-5.6 Sol first proposed the retained lower-bound construction in response to their prompts. The upper-bound theorems are their work without AI assistance.
A named model supplied the retained central lower-bound construction, but the paper does not document AI leadership of the whole proof or project.
Importance
Answers the composite-modulus question and matches the leading coefficient of a known upper bound.
Limitations
Bukh, Chao and Zheng developed the proof and independently obtained the paper’s improved upper bounds. No completed check of the AI-origin lower construction is documented; the lower-order term remains open.
Technical verification
Section 2 provides a proof, but no completed human or machine checking of the construction is documented. Authorship and proof availability are not themselves a completed check.
Human understanding
The human authors explain the two ingredients: low inner rank in the other prime-power characteristics and CRT-based cancellation of mixed Gram blocks.
Correction and publication history
Answers the composite-modulus question and matches the leading coefficient of a known upper bound.
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