Moving Physical Finite Clock Laws and Integer Resonance revised

This preprint is the third paper in a series on geometric failures in period-doubling Hénon-like maps. It studies the finite-clock regime, where the logarithmic Jacobian scale grows proportionally to observation time, for two specified scaled families.The main result approximates the unconditioned physical marked measure by a moving finite Gaussian-selector measure. Within the stated admissible parameter range, this yields relative total mass asymptotics and bounded Lipschitz control of the conditional joint law of failure size, observation offset, and source generation. The approximation accommodates oscillating or divergent standardized boundary parameters without requiring convergence of the threshold-to-time ratio.At noninteger limiting clocks, the offset has a finite discrete uniform law and is independent of Pareto size and Gaussian source generation. Integer resonance introduces dependence between offset and generation through a Gaussian-selected boundary slot. The manuscript proves the required family-specific geometric admission, physical recovery, and sourcewise transfer, and includes counterexamples clarifying the limits of stronger distributional and relative Gaussian rare-tail interpretations.--- 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.)