---

Claim. Let $\mathcal F$ be a finite family of congruence classes with pairwise distinct odd moduli greater than one that contains a class $b_p\pmod p$ for each prime $3\le p\le73$. A family without such a class may be given one: the moduli stay distinct and odd, and a cover stays a cover. Let $M$ be the largest divisor of the least common multiple of the moduli whose prime factors are at most $73$, let $Q_s$ be the set of residues modulo $M$ that avoid every class $b_p\pmod p$, and let $R$ be the set of residues in $Q_s$ that lie in no class of $\mathcal F$ whose modulus has no prime factor above $73$. If $|R|\ge|Q_s|/4$, then $\mathcal F$ does not cover the integers, whatever its classes whose moduli have a prime factor above $73$ and whatever the exponents in those moduli. Equivalently, in a distinct covering system with odd moduli, completed in this way, the classes with $73$-smooth moduli cover more than three quarters of $Q_s$.

Covers. A class of hypothetical distinct odd coverings on Problem 7: those whose classes with $73$-smooth moduli leave at least a quarter of $Q_s$ uncovered. For these, no classes with a prime factor above $73$ in their moduli can complete a cover. The class contains the lead's explicit concentration family, whose uncovered proportion is at least $773/1920$ and on whose uncovered set every measure has system constant above $152.14$, beyond the threshold $138.877$ of the published square-free tail interface. One explicit configuration with uncovered proportion below a quarter, the $16$-class extension on the lead's concentration page, is excluded in the same way through an adapted product measure. Families whose $73$-smooth classes leave less than a quarter of $Q_s$ uncovered are otherwise not addressed, and the unrestricted question remains open.

Credit. The distortion sieve is that of Balister, Bollobás, Morris, Sahasrabudhe and Tiba, from the sieve setup of their square-free paper (Algebra & Number Theory 15 (2021)). The tail recurrence of their Lemma 6.2 and the termination criterion of their Theorem 6.1 are from their density paper (Invent. Math. 228 (2022)), used at their published statements. The digit coordinates follow the prime-adic box correspondence of Berger, Felzenbaum and Fraenkel. The digit form of the sieve, the large-prime tail with actual class systems, the convex-order bound for exact large-prime losses, and the quarter-density criterion with its outward rational certificate are Plasma AI's; the lead asserts no novelty for them.

Standing. Author-recorded at tier 0, with no independent review. The quarter-density criterion rests on an outward-rounded exact certificate, computed by the lead's evidence program in integer arithmetic. It bounds the summed large-prime loss of the uniform measure over the primes from $79$ through $19997$ by $201/1000$, and the resulting tail constant by $94000$, below the certified lower bound $101000$ for the termination threshold of Theorem 6.1. The convex-order bound, the tail theorems and the reduction to digit coordinates are written arguments. Catalog acceptance is pending.

Depends on. The odd-covering distortion lead (tier 0), through its exact-loss tail, its system interface and its digit form of the sieve.