bibkey: jenkin2003compositecovering
authors: "Scott Jenkin; Jamie Simpson"
year: 2003
title: "Composite covering systems of minimum cardinality"
doi: 10.5281/zenodo.7607541
url: https://zenodo.org/api/records/7607541/files/d13.pdf/content
claim: "Containing-divisor replacement and restriction to a prime branch preserve covering; the branch reduction explicitly permits repeated residual moduli."
strata_touched:
  - D5/S3/Arith/Congruence/TwoOddPrimeUncoveredDensity
license: citation-only
triage: anchor

Divisor replacement and prime-branch restriction

The inspected primary scan is Integers 3 (2003), A13, 11 pages, published 24 September 2003. The DOI above identifies the Zenodo archive record. The abstract and introduction define covering by union equal to the integers; disjointness is not required. Page 1 explicitly notes that the moduli form a multiset and may repeat.

Lemma 1, page 3, replaces a class modulo m by its containing class modulo any divisor mu>1. Lemma 2 on the same page compresses the ordered support primes to the first primes, retaining exponent patterns. The paper’s canonical composite incongruent systems therefore contain every composite divisor of every modulus. For odd-only systems admitting prime moduli, the analogous admissible divisor descent includes all divisors greater than one; this is an application of the replacement argument, not the paper’s literal composite-only canonical definition.

Theorem 8, pages 7–8, partitions the p-divisible original classes by their residue modulo p. Every resulting branch is covered by the p-free moduli together with d/p for that branch’s p-divisible labels. Its proof uses the full reduction d/gcd(d,p), without flatness or square-free hypotheses. It does not assert distinctness of the resulting moduli or provide a covering choice among duplicate classes.

Lemma 7, pages 6–7, groups moduli by their p-adic heights and gives a test for unhelpful moduli. Its proof also supplies a fixed-residue counting argument. For a fixed covering and 1<=j<=alpha=max_m v_p(m), if the classes of heights below j miss an integer x, fix x’s complete p-free coordinate and vary the higher p digits above its depth j-1 prefix. The classes of heights at least j must cover all p^(alpha-j+1) such points. Writing M_i for the multiset of moduli of height i gives sum_(i=j,…,alpha) p^(-i)*|M_i| >= p^(1-j). At j=1 this implies at least p p-bearing classes whenever the p-free subfamily does not cover. An actual private point of a p-bearing class supplies that premise; this application of the proof does not identify private essentiality with the paper’s modulus-set definition of helpfulness.

Theorem 3, page 3, cites Simpson’s bound n >= 1+sum_p H_p(p−1) for an irredundant whole cover with full period product_p p^H_p. That bound is already retained in report 343. No source text or scan is vendored by this citation note.

Report 350 applies these reductions to a hypothetical distinct odd cover chosen first by minimum cardinality, then by minimum modulus sum. It identifies the literal parent-child residual collisions and their private-point constraints. The source-support deductions there are separately proved; they are not attributed as statements of this paper or as Lean results.