Finite–Radical Obstructions to Distinct Covering Systems: Exact Dual Certificates in CRT QuotientsOk

We establish an unconditional, finite obstruction to odd distinct covering systems supported on a fixed prime radical. Working in the finite CRT quotient \mathbb{Z}/(3^2\cdot5^2\cdot7^2)\mathbb{Z}, we formulate the covering problem as a set–cover linear program with a distinct–modulus constraint and prove infeasibility by exhibiting an exact dual (Farkas) certificate. Despite the total harmonic volume exceeding one, rigid overlap in the subgroup lattice prevents full coverage. The result is finite, exact, and purely arithmetic—requiring no entropy arguments, asymptotics, or medium assumptions. This paper supplies a concrete arithmetic witness supporting the impossibility of “zero-novelty” completion mechanisms and provides the finite obstruction layer underlying broader structural and informational reductions of the Erdős odd distinct covering systems problem.