Attack log — Sets with no unique sum mod p

2026-08-16

• Environment created. Overnight campaign: exact $m(p)$ for primes $p\le 200$, table + plots + extremal shape. No attack yet.

2026-08-16 — q1 residue (recovered)

• Table used wrong predicate r=1 not r∉{1,2} (memory: commit 77d43c5).
• Scripts recovered: search_m_p.py, verify_m_p.py, plot_m_p.py.

2026-08-16 — q2

• Recoded predicate. Recovered green_m_p.csv matches A398173 through p=47.
• Scripts: search_green_m_p.py, verify_green_m_p.py.

2026-08-23 — q3: p=53 replay, p=59 residue

• Fetched Bedert arXiv:2303.15134v2 and Cao–Yuan arXiv:2608.06728v1. Both use unordered-pair uniqueness, equivalently ordered multiplicities avoiding $\{1,2\}$ for odd $p$. Cao–Yuan's leading upper constant is $1/(2(\log_2 3)^2)$.
• Reopened OEIS A398173. It now has a 15th term, $m(53)=14$, so $p=59$ is the first unpublished boundary.
• Replayed the committed $p\le47$ cardinality-SAT table with CaDiCaL 1.9.5: 14/14 OEIS terms matched in 960.158 seconds.
• Added compute/q3/verify_exact.rs, an independent exact search. It starts from the affine-normalized $\{-1,0,1\}$, branches on all possible second representations of a currently unique sum, and memoizes intermediate sets up to every internal three-term-progression normalization.
• Replayed $m(53)=14$: the saved 14-set passes the definition directly, while the final Rust search returned UNSAT for every set of size at most 13 after 333,555,078 nodes in 1,428.475 seconds. The complete cold driver took 1,618.255 seconds. This matches the published term; it does not improve it.
• At $p=59$, CaDiCaL found the checked 15-set $\{0,1,25,28,32,36,43,46,47,49,52,53,55,57,58\}$ in 48.766 seconds. Therefore $m(59)\le15$ only.
• The independent size-at-most-14 run stopped UNKNOWN after exactly 100,000,000 nodes (72,327,605 memoized states) in 523.030 seconds. This incomplete run is not a lower bound. A CaDiCaL 1.9.5 exact-size-14 run was also manually stopped without a decision after 37:01 wall time (34:49 CPU).
• Kissat 4.0.4, CaDiCaL 3.0.0, and MapleChrono exact-size-14 runs were each stopped without a decision after about 30 minutes wall time. The solver timeouts are not certificates.
• Heuristic search reached a 14-set with one unique sum. Every normalized 14-set within four swaps of that near miss was checked (49,168,350 sets); none worked. This neighborhood search is also not a lower bound.

2026-09-09 — q4: AP ladder and named midpoint covers

• Reopened the source record: OEIS already reports m(59)=15 and values through 73. A lower replay at 59 would match that record. The checked 15-set remains the only local bound at 59.
• The inherited Rust AP5 exclusion visited 1,559,513 nodes. Its pending CaDiCaL replay finished UNKNOWN at 5,000,000 conflicts; the saved SAT result does not independently confirm UNSAT.
• Started the AP4 case with the Rust repair-cover search, a 600-second wall limit and a 100-million-node limit. One heavy search runs at a time.
• Added a C completion checker using reflection intersections for ordered sum counts, a fixed 128 MiB cache, and a sumset-support cap. Its first controls agree with complete subset enumeration through 13, including named midpoint roots and progression avoidance. These controls test the implementation; they are not a lower bound at 59.

Wrap at the user's request

• AP4 Rust timed out after 600 seconds and returned UNKNOWN through the harness. It emitted no node count. p59_ap4_exact.json records the cap, source hash, binary hash, and termination; this is not an exclusion.
• The later AP5 CaDiCaL artifact arrived during wrap. It reports UNSAT after 14,448,284 conflicts in 1,474.911 seconds. A fresh formula build matches its SHA-256, 44,028 variables, and 129,306 clauses. The earlier five-million-conflict UNKNOWN artifact is retained as well. The completed SAT and Rust runs now agree on the AP5 restriction; no DRAT proof is retained.
• The experimental C AP5 run was interrupted with exit 130 at the user's request. Its UNKNOWN record is p59_ap5_cover_interrupted.json; it adds no exclusion. No further heavy job was started.
• The midpoint partition audit checked 2,054 small sets and every affine identification at 59. It leaves 25 AP4-free classes, or 27 AP4-but-AP5-free classes for that part of the ladder. These classes were prepared, not searched. The C controls checked 104 decisions against enumeration of 10,364 subsets through 13.
• The wrap retains only $m(59)\le15$ as an unrestricted local bound. OEIS already reports equality, so there is no dent. LEAVES records residue; unrestricted size at most 14 remains open here. The full replay driver includes the long AP5 SAT check, but only lightweight verification was rerun during wrap. The PR is to remain unmerged.

2026-09-09 — q5: named midpoint covers after the AP5 ceiling

• Reopened Bedert arXiv:2303.15134v2 (Definition 1; ordered $r_A\in\{1,2\}$ forbidden) and Cao–Yuan arXiv:2608.06728v1 (Theorems 1.1–1.3). OEIS A398173 still has 20 terms through 73, with $m(59)=15$. Kwaczyński's certificate repository reports the same finite values; it is a lead, not a proof object replayed here.
• Construction first. Every 14-subset of both checked 15-sets has unique sums. One- and two-swap neighborhoods of 31 seeds (2,812,351 sets) and then the three-swap neighborhood (162,932,311 sets) kept a best of one unique sum, the inherited near-miss $\{0,1,3,4,5,9,13,15,16,21,29,33,45,58\}$. No 14-set. The 15-sets both contain an AP6, so shrinking them stays in the family q4 already excluded at size 14.
• Searched the leftover named classes with a RAM-light C completion checker (one-point repairs first; cover test at remaining $\le 7$). Controls still match full subset enumeration through 13. All 27 AP4-but-not-AP5 representatives returned UNSAT (339.7 seconds total, max 60.6 seconds, max RSS 61 MiB). All 25 AP4-free representatives returned UNSAT (406.2 seconds, including the 4-point root in 66.4 seconds).
• Dual: the independent Rust brancher, using unordered pair counts rather than reflection intersections, returned UNSAT on the same 52 roots (max 144.7 seconds on the 4-point class). q5 also replayed the AP5 ceiling: Rust 1,559,513 nodes UNSAT (the q4 count) and C 925,228 nodes UNSAT.
• Combined with the inherited AP5 exclusion, that is an unrestricted exclusion of sizes 2 through 14. With the 15-set, $m(59)=15$ locally. OEIS already publishes that term, so there is no dent. The leftover at 59 is exhausted. The next local exact handles are 61 through 73 and the unpublished primes from 79.

2026-09-09 — q6: exact table at 61

• Reopened Bedert arXiv:2303.15134v2 and Cao–Yuan arXiv:2608.06728v1 (abstracts). OEIS A398173 is still 20 terms through 73, with $m(61)=15$ and example $\{0,1,2,3,4,6,15,21,22,24,42,49,55,56,58\}$. Python and Rust both accept that 15-set.
• Construction probes died. Annealing size 14 stopped at two unique sums. The q5 C search from $\{-1,0,1\}$ at bound 15 hit 2,000,000 nodes UNKNOWN without a witness. The published 15-set contains an AP5, so the useful root was the five-term progression, not the unrestricted AP3 start.
• The same AP ladder then transferred. AP5 at size 14: C UNSAT (810,493 nodes, 87s) and Rust UNSAT (1,476,025 nodes, 105s). All 28 AP4-but-not-AP5 named classes and all 26 AP4-free classes returned UNSAT in C, then again in Rust (4-point class 170s). That is $m(61)=15$ locally, matching OEIS, new to the notebook table. Not a dent. Primes 67 and later need a mask wider than 64 bits.