Arithmetic Renormalization: The Mechanism of Non-Liftability in Odd Distinct Covering Systems

We introduce an arithmetic renormalization framework to analyze liftability in the Odd Distinct Covering System (ODCS) problem. Treating dual certificates in finite quotients as dynamical state vectors rather than static obstructions, we define a scale-dependent invariant, the Coverage Fraction βₑ, which governs feasibility under lifting along prime-power towers. We prove the Fiber Damping Lemma, showing that new distinct moduli introduced at higher scales capture only a uniformly bounded fraction of lifted obstruction mass. As a consequence, we establish the Obstruction Amplification Theorem: finite dual obstructions are repulsive under lifting. This resolves the Non-Liftability Conjecture posed in Paper F, showing that no bounded-radical odd distinct covering can escape finite arithmetic constraints via tower lifting.