Blow-up of multipliers in meromorphic families of rational maps

The paper’s Theorem A is correctly stated and proved using a non-Archimedean strategy: when the non-Archimedean Lyapunov exponent λ(f_na)=0, uniform boundedness of multipliers follows from prior work; when λ(f_na)>0, Theorem 1.2 gives that, for each ε>0 and all large n, a (1−ε)-fraction of rigid period‑n points are repelling with multiplier ≥Ae^{nλ(f_na)/2}; then Lemma 1.7 transfers this to the complex family, yielding the required blow‑up for most cycles as t→0 . By contrast, the model’s Phase‑2 argument contains two critical errors: (i) it conflates λ=0 with potential good reduction and claims holomorphic extension across t=0 (which need not hold in degenerating families; the paper instead cites a uniform multiplier bound at λ=0 without requiring extension) ; and (ii) its “counting lemma” is invalid as stated: from an average lower bound (slope λ in log|t|^{-1}) and a pointwise upper bound (slope σ≥0), the fixed-threshold proportion estimate “≥1−ε” cannot be made uniform in t when σ>λ, since the ratio used tends to 1 as log|t|^{-1}→∞. The paper’s proof avoids this by using non-Archimedean repelling periodic point counts and a branch-by-branch transfer via Lemma 1.7, which yields the (1−ε) proportion uniformly for small |t| .