Odd Distinct Covering Systems: Arithmetic Renormalization and the Structural Dichotomy

This paper completes a multi-part structural analysis of the Odd Distinct Covering System (ODCS) problem. Building on exact finite arithmetic obstructions, stabilization dynamics, and an arithmetic renormalization framework developed in earlier papers, we establish a rigorous dichotomy: any ODCS must either involve infinitely many distinct prime factors or exploit a presently unknown arithmetic mechanism capable of defeating obstruction amplification under scale lifting. The key contribution is the synthesis of finite dual (Farkas) obstructions with renormalization dynamics, showing that bounded-radical constructions cannot escape obstruction by forming compatible towers over increasing prime powers. This reframes the Erdős ODCS problem as a problem of structural emergence rather than density alone, and isolates the precise frontier at which new arithmetic phenomena would be required. This work is unconditional except where explicitly stated, and is designed to serve as a conceptual and structural capstone for the accompanying Papers D–G.