We prove that no covering system with distinct odd moduli can have its least common multiple supported on at most four distinct odd primes (for arbitrary exponents). Equivalently, any odd covering system---if one exists---must use moduli involving at least five distinct odd primes. The proof introduces a weight function method. Moduli are partitioned into prime-power towers and composites; the towers define a "weight region" W in Z/LZ via CRT, and a union bound shows the composites cover at most an R-fraction of W with R = 41/45 < 1 for the worst-case prime set {3,5,7,11}. This leaves at least L/40 integers provably uncovered. The same method yields R = 2/3 for three primes (a short self-contained proof) and extends to five primes via a three-level refinement---weight function, Bonferroni correction, and pigeonhole-forced collisions at prime 3---which proves the impossibility unconditionally for 98.2% of exponent configurations. The remaining 1.8% reduce to a CRT coverage maximality conjecture (NC ≤ 0), for which we provide an analytical proof at k ≤ 3 primes and exhaustive computational verification over 9,000,000+ exact configurations at k = 4 with zero violations.