Global Nonexistence of Odd Distinct Covering Systems

We prove the unconditional nonexistence of Odd Distinct Covering Systems (ODCS). The argument synthesizes a certified finite obstruction in the quotient ring \mathbb{Z}/11025\mathbb{Z} with a structural constraint governing radical extensions. The key new result is the Distinctness Allocation Lemma, which shows that extending a covering system by a new prime q strictly increases the covering deficit. A finite uncovered set lifts to q disjoint fibers, while the newly available distinct moduli can eliminate at most one copy. The remaining fibers must be covered by composite moduli formed from the original kernel divisors; however, distinctness forces these divisors to be partitioned among the fibers. Since the full kernel divisor set is already insufficient to cover the base deficit, any such partition is a fortiori insufficient. This yields a structural induction excluding all bounded-radical and unbounded-radical constructions alike. The result closes the final remaining escape route in the Erdős Odd Distinct Covering Systems problem.