ASTP PROTOCOL. ARTICLE #2: MATHEMATICAL FORMALISM OF HIERARCHICAL DIMENSIONAL DEGRADATION

Historical Note and Version Status This document is a **mathematical supplement** to the preprint "ASTP v6.5: Discrete 3‑sphere as an Alternative to ΛCDM" (DOI: 10.5281/zenodo.20055206). It supersedes the earlier mathematical appendix for ASTP v6.1 (DOI: 10.5281/zenodo.19522462), which was based on a phenomenological parametrisation with an effective parameter x_eff. The present version derives all geometric and kinematic relations from the single kinematic axiom A1 (linear growth of the 4‑ball radius R(t) = R₀ + c t) and the assumption of a 3‑sphere topology. No phenomenological parameters are required. While the main publication focuses on the comparison with supernova and BAO data, the present text provides a **self‑contained, rigorous derivation** of the cosmological distance formulas, the sound horizon prediction, and the associated mathematical framework. It can be used independently for further theoretical development or numerical implementation. Thus, this Mathematical Appendix for ASTP v6.5 should be cited together with the main preprint when referring to the ASTP interface.