A CENTRAL LIMIT THEOREM FOR ROSEN CONTINUED FRACTIONS

The paper proves, for Rosen continued fractions T_q on I_q=[-λ_q/2,λ_q/2], a Lasota–Yorke inequality on BV, quasi-compactness via the Ionescu–Tulcea–Marinescu theorem, a spectral gap under Condition (*), verifies (*) for the Rosen map by a careful “bad interval” growth argument, and concludes a CLT via Broise’s framework. Key steps and statements appear explicitly: the system and goal (Sections 1 and 7) , the general Iwasawa framework and basic derivative bounds (Section 2, Lemma 1) , the Lasota–Yorke inequality (equations (2)-(3)) , the quasi-compactness argument via ITM (Section 4) , the spectral gap proof under Condition (*) (Proposition 2 and Theorem 2) , and the verification of (*) for Rosen (Theorem 3, Section 6) . The candidate solution matches the paper’s strategy in spirit (Steps A–C,E) but contains a crucial flaw in Step D: it asserts “exact covering” (T_q^N([c,d]) = I_q a.e.) solely from uniform contraction of inverse branches and cylinder shrinking, which does not imply that some depth-N cylinder maps onto the full interval. The paper’s Section 6 provides the necessary nontrivial argument using expansion-of-length of “bad intervals” to reach a contradiction, thereby proving exact covering and hence Condition (*) . A secondary inaccuracy is the one-step Lasota–Yorke contraction: the manuscript derives var H^n f ≤ 2γ^{2n} var f + …, so a k-step inequality is used to ensure ρ=2γ^{2k}<1 , whereas the candidate claims a one-step contraction with factor λ_q^2/4. Consequently, the paper’s argument is correct and complete for its claims, while the model’s proof has a gap at Step D and a mis-specified contraction constant.