A dichotomy for the Hofer growth of area preserving maps on the sphere via symmetrization

The uploaded paper proves the enhanced dichotomy on S^2 via the symmetrization map Σ, including the uniform constant 19·Area(S^2), through a bounded-conjugation estimate to a one-parameter subgroup in the Hofer completion of the even-height subgroup (Theorem 1.3) and a direct enhanced dichotomy statement (Theorem 3.1) . The candidate solution follows the same strategy: it invokes this symmetrization, the bounded-conjugation error, and the growth dichotomy to conclude either linear Hofer growth or a uniform Hofer bound. The only issue is a minor metric-notation slip (using d where the paper uses the completion d̂ in intermediate steps); the conclusion still follows because d̂ restricts to d on Ham(S^2) and the bound is applied when Σ(H)=0, yielding d(1,φ^t_H)=d̂(1,φ^t_H) ≤ 19·Area(S^2) for all t .