CRITICAL RECURRENCE IN THE REAL QUADRATIC FAMILY

The paper proves that if (δ_n) is admissible and the series ∑ δ_n/(log n) with an additional log* loss diverges, then Leb(Λ(δ_n) ∩ CE) = 0, via a Benedicks–Carleson–type exclusion on start-up intervals, a phase–parameter distortion lemma, and a careful accounting of complete returns, including a 1/log n separation and an explicit log* loss coming from unbounded distortion. These elements are all present and internally consistent in the manuscript (Theorem A/B; return taxonomy; distortion; Schlömilch step) . By contrast, the model changes the loss to an exponential form τ^{(log* n)^3} (not in the paper) and misstates the globalization step by invoking PR as a full-measure property of CE via Avila–Moreira. The paper actually avoids relying on such typicality and uses the inclusion Λ(δ_n) ∩ CE ⊂ PR (for admissible (δ_n)) to localize the argument to PR and then cover PR by start-up intervals . The model also claims a stronger measure-zero statement on NR (not proved in the paper), and misreads the placement of the log* loss factor that the manuscript attributes to unbounded distortion and to the block structure between complete returns .