We present an argument excluding finite covering systems with distinct odd moduli whose least common multiple has at most six distinct prime factors, with unrestricted prime exponents and integer residues. The argument refines capped reweighting by comparing the positive-part expectations of every auxiliary divisor count with those of a product of independent finite geometric-tail factors. For the first six odd primes the resulting rational budget is approximately 0.91015158 < 1. A digit-embedding prime-replacement lemma transfers the fixed-prime exclusion to arbitrary prime supports. The probability argument is finite: truncation preserves the small product atoms needed for the calculation, and a finite geometric sum bounds the mean. Both full statements have been verified in Lean and Aristotle, with independent checks of the returned source and unrestricted exponents. Historical priority is not asserted.