Exact Geometric-Failure Tails for Period-Doubling Hénon-like Maps

This preprint studies exact geometric-failure probabilities in infinitely period-doubling Hénon-like dynamics.For every fixed map in a sufficiently small native Hénon-like class with positive Jacobian on its Cantor attractor, the paper establishes an exact leading asymptotic for the tail of the geometric precision-failure observable. The asymptotic identifies the universal power-law exponent, the leading coefficient, the dependence on the average Jacobian, and the correction induced by source generation and observation time.The proof combines a coefficient-one comparison between physical parent-child geometry and an exact fixed-point defect with finite-word renewal asymptotics and observation-time counting. Uniform versions are also obtained for specified scaled analytic stable families. Further consequences include a conditional Pareto law for failure size, an asymptotically independent uniform law for the canonical tower offset, sharp positive-moment thresholds, and a critical logarithmic size-weighted law.This record is Preprint Version 2. It is a substantially revised presentation of an earlier Zenodo preprint from October 2026, with the exact moving-threshold tail law and its geometric and probabilistic consequences presented as the central results. The marked renewal structure developed in this preprint also admits a generation-weighted extension. A companion manuscript studies exponential weights in the canonical source generation, leading to Gaussian generation fluctuations and corresponding joint and deviation laws. --- Author's Note: Artificial intelligence (AI) tools were utilized in the course of this research, including language translation, drafting assistance, and code verification. (이 연구는 번역, 원고 검토 및 코드 검증 등에 AI를 활용하여 수행되었습니다.) + 2026/11/04Supplementary numerical investigations suggest that oscillations in the fixed-point defect count at finite thresholds are consistent with a correction associated with a candidate resonance of the transfer operator. This observation is exploratory and separate from the proved results of the preprint. Rigorous certification of the resonance, quantitative bounds on the remaining error, and any extension to secondary asymptotics for geometric-failure tails of actual Hénon-like maps require further work.