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Claim. Let $D=\{p-1:p\text{ prime},\ 5\le p\le877\}$, a set of $149$ moduli. For every choice of residues $a_d$ ($d\in D$), the classes $a_d\pmod d$ do not cover the integers. Equivalently, every covering system whose moduli are pairwise distinct numbers $p-1$ with $p\ge5$ prime uses a modulus $p-1$ with $p>877$: a covering system using only some moduli of $D$ would stay a cover after one class is added for each missing modulus.

Covers. The case of Problem 273 in which every modulus is $p-1$ with $p\le877$: no such covering system exists. This is a finite statement and a partial step toward a negative answer. Whether a covering system with distinct moduli $p-1$, $p\ge5$ prime, exists remains open.

Credit. The proof runs the distortion sieve of Balister, Bollobás, Morris, Sahasrabudhe and Tiba, in the sieve setup of their square-free paper with its Lemma 3.3 class-mass recursion, along base-$p$ digit coordinates that follow the prime-adic box correspondence of Berger, Felzenbaum and Fraenkel. The digit form of the sieve, its loss bound over sets of completed classes, the digit order and thresholds, and the exact certificate are Plasma AI's. The same statement is Theorem 1.1 of the earlier research note of 2026-07-11 recorded on the ideal_ombrer claim page, a pending claim. That note, its certificates and its proof were not used or consulted, and this derivation is not a review of it.

Standing. Author-recorded at tier 0, with no independent review. The first $18$ digit coordinates complete the $49$ moduli of $D$ that are $7$-smooth and lose at most $115/126$ in total; every later coordinate completes fewer moduli than its size and loses nothing, so a positive proportion of the digit box stays uncovered for every residue choice. The lead's evidence program rebuilds $D$, reads the retained digit order and thresholds, recomputes the total exactly in rational arithmetic, and checks that each later digit completes fewer moduli than its size. The loss bound it applies is a written argument on the lead's digit page. Catalog acceptance is pending.

Depends on. The odd-covering distortion lead (tier 0), through its calibration on Problem 273 and its digit form of the sieve.