Environment-Activated Cosmic-Dawn Coupling Window in a Single-Field S3 Ontology: Halo-Only Demonstration with Press–Schechter and Sheth–Tormen

This note defines a minimal, reproducible “cosmic-dawn coupling window” module in a single-field (S^{3}) ontology and computes its halo-only imprint on nonlinear collapse and rare-halo abundances. The environmental activation is encoded by a bounded bump gate in virial potential depth together with a redshift envelope localized near (z\sim 15)–25, while the background expansion and linear growth are kept GR-like by construction. Using a collapse-threshold surrogate and standard Press–Schechter and Sheth–Tormen mass functions, the paper reports mass- and redshift-dependent abundance-ratio predictions and provides the full script, figures, and JSON outputs for exact reproducibility.