The p=59 exact boundary

OEIS A398173 ends at $m(53)=14$ , so $p=59$ is the first unpublished prime. The saved certificate p59_upper.json proves only

m(59)\le 15.

It does not determine $m(59)$ .

Checked upper bound

CaDiCaL 1.9.5 found

A=\{0,1,25,28,32,36,43,46,47,49,52,53,55,57,58\}

in 48.766 seconds. The direct verifier recomputes all 225 ordered sums and finds no multiplicity in $\{1,2\}$ :

python3 problems/unique-sum/compute/q3/verify_upper.py

Incomplete lower search

The independent Rust program searches for any admissible set of size at most 14. With a deterministic 100-million-node cap it reports

p=59 limit=14 status=UNKNOWN nodes=100000000 memo_hits=27672395 memo=72327605

The recorded run took 523.030 seconds; timing is machine-dependent, while the node and memo counts are deterministic for this implementation.

Compile and replay that finite prefix with:

rustc -D warnings -O problems/unique-sum/compute/q3/verify_exact.rs -o /tmp/verify-unique-sum
/tmp/verify-unique-sum 59 14 100000000

UNKNOWN exits with status 3. It is not a lower bound. In particular, this run does not show that size 14, or any smaller size, is impossible.

The full cardinality-SAT encoding was also run at exact size 14 with CaDiCaL 1.9.5, Kissat 4.0.4, CaDiCaL 3.0.0, and MapleChrono. None decided the instance in these runs. CaDiCaL 1.9.5 was stopped after 37:01 wall time (34:49 CPU); each of the other three was stopped after about 30 minutes of wall time. These timeouts are leads about cost, not certificates.

Near miss

Local search found the normalized 14-set

\{0,1,3,4,5,9,13,15,16,21,29,33,45,58\},

which has exactly one uniquely represented sum. All 49,168,350 normalized 14-sets within four swaps of it were checked and none worked. That is only a finite neighborhood statement; it does not exclude a solution elsewhere.