A uniform Tits alternative for endomorphisms of the projective line

The paper proves precisely the j = 1 statement: if f1 and f2 are polarized by the same ample line bundle and PrePer(f1) ≠ PrePer(f2), then f1 and f2 themselves generate a free semigroup of rank 2. This is Theorem 1.1, proved via a refined ping–pong argument on the complete metric space of heights HL using a new ping–pong lemma for injective contractions (Proposition 2.1) and the canonical-height fixed points of αf (the pullback contraction) . The candidate solution claims the j = 1 case is likely open and only obtains freeness after passing to powers, contradicting the paper’s main theorem and proof .