The size dichotomy cannot close: the controlled-interval horn delivers log g(k) = o(k) if and only if max(A) = rad(S)^{o(1)}, pinned from both sides by Rankin above and simplex volume below, and the other horn cannot supply that because after gcd-normalization the height of a primitive admissible set of sub-extremal size is unbounded.
hunts/support_95bb5cb7 (Record 78 of 98 in chronological sequence)
Guiding Question
Does a size dichotomy (large-gap descent vs smooth-number counting in a controlled interval) yield log g(k) = o(k) for Erdos 126, and if not, where exactly does it fail?
Method & Verification
The counting horn was pinned from both sides (Rankin upper, simplex-volume lower), and height witnesses for primitive admissible triples were enumerated by full trial division up to 5.6e14 for k=2, |A|=3.
Lineage & Relationships
Primary Sources (at pin 8fa46e134)
Editorial Notes
Question filled from the HuntSpec. Disposition maps the arm is settled negative; the conjecture is untouched to completed for the arm, not for Erdos #126. The arity paragraph cites Evertse-Schlickewei-Schmidt and Erdos-Stewart-Tijdeman from literature and was not re-derived here. The audit correction that r_186989's 'every optimal witness has elements < 50' fails is reported here and was not edited into r_186989.
Date Provenance
commit a9e9f8219d2d6a4fbc3440b3cc249f2deabd07c4, hunts/support_95bb5cb7/RESULTS.md, author 2026-08-24T20:55:56-05:00