Spectral Coherence on the Developmental Attractor

This preprint develops the spectral structure of the developmental flow introduced in the companion paper A Global Attractor for the Developmental Flow. Once the movement operator of developmental geometry is restricted to its global attractor, it induces a Koopman semigroup on the Banach space of continuous functions over the attractor. The main result of this paper is the Spectral Coherence Theorem, which shows that the number of isolated spectral components of the Koopman semigroup is invariant under the flow. This coherence index provides a stable spectral skeleton for the long‑time behavior of the developmental system and establishes a persistent analytic structure underlying the geometric framework of M1–M6. This work is part of the Developmental Geometry Series, which develops the analytic, geometric, and dynamical foundations of the developmental program. It builds directly on the author’s earlier paper on the geometric origin of Benford’s Law (DOI: 10.5281/zenodo.18765897).