Research log — Sets with no unique sum mod p

2026-08-16

OEIS A398173 then had 14 terms through p=47: 3,4,5,7,7,8,9,10,11,11,12,13,13,13.
Bedert, arXiv:2303.15134; Cao–Yuan, arXiv:2608.06728.
The first recovered run used the wrong predicate ( $r=1$ rather than $r\notin\{1,2\}$ ); memory commit 77d43c5.
The recoded green_m_p.csv matched A398173 through p=47.

2026-08-23

Fetched and read Bedert, arXiv:2303.15134v2. Definition 1 uses unordered-pair uniqueness; in ordered notation the paper states that a unique sum is exactly $1\le r_A(g)\le2$ . Theorems 3 and 5 give $m(p)\ge\omega(p)\log p$ , with $\omega(p)\gg\sqrt{\log^{(3)}p}/\log^{(4)}p$ , and $m(p)\ll(\log p)^2$ .
Fetched and read Cao–Yuan, arXiv:2608.06728v1. Theorem 1.1 gives $m(p)\gg\log p\log\log p$ . Theorem 1.3 gives the independent upper construction with leading constant $1/(2(\log_2 3)^2)=0.1990361769\ldots$ ; the paper explicitly says all unsubscripted logarithms are natural.
Reopened OEIS A398173. It now has 15 terms through $p=53$ , with $m(53)=14$ added by Paweł Kwaczyński on 2026-07-29. Thus $p=59$ is the first prime at which a finite computation here could extend the published table.

2026-09-05 — q4 source check

Fetched Bedert v2 with scripts/arxiv_fetch.py and read Definition 1 and the ordered representation convention in section 2. Both multiplicities 1 and 2 are forbidden. Opened the abstract and HTML as well.
Fetched Cao–Yuan v1 with the same script and read the introduction and Theorems 1.1–1.3. The explicit asymptotic lower constant does not decide the boundary at 59. Opened the abstract and HTML. No asymptotic improvement is attempted here.
OEIS A398173: the browser fetch timed out; scripts/oeis_lookup.py A398173 succeeded after network authorization, showing the sequence definition and its first 12 terms (the helper truncates output). The 15-term baseline supplied for this run is also in the notebook; all 15 saved witnesses passed the direct arithmetic verifier.
The full OEIS JSON entry subsequently succeeded via curl. It has 20, not 15, terms, through 73: 3,4,5,7,7,8,9,10,11,11,12,13,13,13,14,15,15,16,16,16. The extension is attributed to Paweł Kwaczyński on 2026-08-08. compute/q4/published_oeis.json saves the numerical data and attribution.
Opened unique-sum-free-cert. Its README says two independent searches established the values at 59, 61, 67, 71, and 73, but retained lower proof objects are not supplied there for those rows. The 53 ledger records hashes of proofs deleted after checking. This is a published claim, not a lower proof replayed by q4. It credits Scheinerman's 2019 thesis for witnesses through 59 and links Watson for retained lower proofs.
Opening the linked Zenodo DOI failed in the browser; a GitHub API directory request returned HTTP 403.

2026-09-09 — resumed q4 record check

The full OEIS JSON entry again returns 20 terms through 73, unchanged from the saved September 5 snapshot. The notebook's local replay through 53 is not the current publication boundary. In particular, OEIS reports m(59)=15.
Reopened Kwaczyński's certificate repository and Watson's notes. Both acknowledge the later finite values. Their verification and retained proof claims are source reports, not proof objects replayed in this run.
Fetched Bedert v2 with scripts/arxiv_fetch.py 2303.15134v2 --text pdf and read Definition 1. Both ordered multiplicities 1 and 2 are forbidden. No new asymptotic statement is used in this continuation. The browser's OEIS page request timed out; curl of the JSON entry succeeded.
During the same-day continuation, reopened the Bedert abstract and Cao–Yuan abstract, and reread the previously fetched Bedert PDF text. A fresh curl of the OEIS JSON entry exactly matches the saved 20-term record. The ordinary OEIS page again timed out in the browser. The work below concerns finite replay only.

2026-09-09 — q5 record check

Opened the Bedert abstract and the HTML of v2. Definition 1 is the unordered unique-sum predicate; section 2 states that a unique sum is exactly $1\le r_A(g)\le 2$ . Theorems 3 and 5 are the $\omega(p)\log p$ lower bound and the $(\log p)^2$ upper bound. No finite table.
Opened the Cao–Yuan abstract and the HTML of v1. Theorem 1.1 is $m(p)\gg\log p\log\log p$ . Theorem 1.3 is the ternary symmetric-square upper bound with leading constant $1/(2(\log_2 3)^2)$ . Unsubscripted logs are natural. The paper does not decide $m(59)$ .
scripts/arxiv_fetch.py on both ids (metadata only) matched those abstracts. scripts/oeis_lookup.py A398173 returned the sequence name and a truncated prefix. A curl of the OEIS JSON entry again has 20 terms through 73: 3,4,5,7,7,8,9,10,11,11,12,13,13,13,14,15,15,16,16,16. compute/q5/published_oeis.json stores that snapshot.
Opened unique-sum-free-cert. Its README claims $m(59)=15$ with hash-bound CaDiCaL LRAT ledgers for $k=2..14$ . That is a published-adjacent lead, not a proof object replayed in q5. The notebook's lower half is the named-cover search.

2026-09-09 — q6 record check

Reopened the Bedert abstract and the Cao–Yuan abstract. No finite table; versions unchanged.
scripts/arxiv_fetch.py on both ids and scripts/oeis_lookup.py A398173 again. Curl of the OEIS JSON entry still has 20 terms through 73, including $m(61)=15$ with example $\{0,1,2,3,4,6,15,21,22,24,42,49,55,56,58\}$ as a(17). compute/q6/published_oeis.json stores that snapshot. The example is a lead until the local ordered-count check; that check passed.