Claims

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1966_01_01_rogers: Rogers' theorem, first published in Halberstam and Roth's Sequences (1966): for fixed moduli the density of integers in none of the classes is largest when every residue is zero, so equal residues cover least; a book result.

1986_04_01_simpson: Simpson's 1986 bound: the density covered by one class per modulus is at least the alternating sum of reciprocal least common multiples, attained when all residues agree; accepted on the refereed paper.

2025_08_25_cambie: Stijn Cambie's arXiv note (v2 crediting GPT Pro Sol 5.6 with one observation): exact balanced-partition formulas for structured moduli, a Chvátal-hard knapsack order, and NP-hardness of zero density for lists.

2026_09_10_onishi: Yoshiharu Onishi's 2026 manuscript (with GPT-5.6 Sol and GPT-6 Astra): the maximum covered density is the largest clique inclusion-exclusion value over the realizable graphs a gcd dynamic program enumerates; no review recorded.

2026_09_22_schroeder: Michael Schroeder's 2026 Zenodo manuscript on the first question: an exact formula when noncoprime classes can be made disjoint, #P- and NP-hardness results, and fixed-parameter tractability in treewidth; no review recorded.

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