What was already known
Gerver and Gerver–Ramsey gave weaker upper bounds. Korsky’s density-increment framework already improved the order of the upper bound without the mediant refinement.
AI involvement
GPT-5.5 Pro suggested the Farey mediant. Korsky designed the density-increment and random lower-bound methods and proved the refined lemma.
The model supplied a retained mathematical choice central to the specific improved leading constant; the broader asymptotic methods are human-origin.
Importance
Improves the leading constant within the new upper-bound framework.
Limitations
The lower-bound construction and main framework are Korsky’s work. No completed check of the mediant lemma or final asymptotic constant is documented.
Technical verification
A proof is printed, but the author documents computation assistance and responsibility rather than a completed check of this lemma or final bound. Reported is the missing-check default.
Human understanding
The author proves Lemma 4.1 by comparing the two Farey endpoints with their mediant and derives the density-increment constant explicitly.
Correction and publication history
Improves the leading constant within the new upper-bound framework.
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