# zonegap-witnesses — second witnesses adopted, the due OEIS flip-search run, and one loud hit *(Follow-up to `zonegap-prior-art.md`'s NOT-REACHED list, 2026-08-21. Own new files only: `research/twin-gap-witnesses-01.js` and this note. Nothing committed, nothing pushed.)* ## 1. The witness verdicts: AGREE / AGREE, no extension, no conflict Adoption artifact: **`research/twin-gap-witnesses-01.js`** (embedded, guards throw on any cross-source mismatch). Two witnesses fetched and machine-parsed this session — no hand transcription anywhere; the A113274/A113275 arrays were copied by `awk` from the adopted `research/a113274-gap-records.js`. - **Luhn, pzktupel.de `RecordGaps/GAP02.php`** (HTML sha256 `d12779fb37c5113bcceb40f238bcff6cc4e6908f2048e42ac85049b42e570028`): 82 rows. Gap and start EXACT against A113274/A113275 82/82; `next = start+gap` row-exact, so the lesser-to-lesser convention is verified, not assumed. His **merit column is `1.32032363 · ∫_start^next dt/ln²t`** (the HL expected twin-pair count across the gap; 1.32032363 = 2·C₂ = KW2019's `C_{2,H}` — the ZONE-POSTULATE §4 notation collision applies), reproduced on all 82 rows, worst deviation 0.0115% — adopted as a second transcription guard. **No record beyond 82**; the page certifies "Maximal gaps below 1.00·10¹⁷", so the trusted envelope does NOT extend. - **Kourbatov, arXiv:1309.4053v1 Table 2** (PDF sha256 `733f8e77aba31d7f2bd5dee7da6e42b73f07be49a2bc1f442d3a658b10d108df`, pdftotext extraction with a per-row `next − start = gap` structural check): exactly **72 rows = A113274 records 1..72, EXACT 72/72** — the middle-range second witness the recon flagged, matching A113274's own "terms up to a(72) from Kourbatov". - **The discoverer ledger** (new data OEIS does not carry): Rathbun 1–41 (1998), Wolf 42–57 (1999), Fischer 58–73 (2008), Oliveira e Silva 74–75 (2013), Smith 76 / Ritschel 77 (2019), Raab 78–82 (2019–2021). OEIS credits CONTRIBUTORS (Kourbatov → a(72), TOS a(73)–a(75), Raab a(76)–a(82)); the ledgers differ at 73, 76, 77 as roles, not numbers. - **What "second witness" honestly means here:** Luhn's page and A113274 cite each other (shared upstream, mersenneforum t=24303). Records 1..72 now have three carriers (+ TOS's starred rows to 75 via zonegap-01 CUSTODY 2); 73..82 remain transcription-custody only — independent recomputation stays a paper-phase deliverable. ## 2. The due OEIS check: one loud HIT (the pair-count object is owned), the gap objects absent Channel: `oeis.org/search?fmt=text`, calibrated in-session (known positives `2,6,18,30,66,150,192,258` → A288815 and `2,4,6,10,14,22,26,34,40,46,58,66` → A048670, both returned first query). Sequence values generated by a 15-line mirror of zonegap-01's conventions and cross-checked against its embedded output (envelope-step p's, band headmax, min count) before searching. Full query log: session scratchpad `oeis-batch-results.txt`, 33 queries. **THE HIT — the pair-count-per-zone sequence is in OEIS, one index over.** Query `4,8,9,16,17,21,29,30,41,48,50,61,74,87` (our per-zone counts from p = 5) returns **A273257, "Number of twin primes between prime(n) and prime(n)^2"** (Jesse H. Crotts, Aug 2016; b-file to n = 10000 by Greathouse). The match is no coincidence: since (p, p′) contains no prime, the twin openers in our zone (p, p′²) are exactly those in [p′, p′²), so **our per-zone pair count at p_n equals A273257(n+1) for every n ≥ 2** (only the p = 2 zone differs, via their disjoint-pairs rule at (3,5),(5,7); upper endpoints cannot differ because a+2 = p′² is impossible). The family around it, all read in full internal format: - **A057767** (Nomoto, Oct **2000**): twin pairs between P(n)² and P(n+1)² — the count companion of A091591/A091592, with Hasler's 2019 comments: "a(n) > 0 for infinitely many n" ⟺ TPC, a conjectured record-low ladder each term of which would imply TPC, and the empirical lower bound a(n)/n ≳ 1/11 (quoted into A273257 as "about n/11"). - **A308777** (Michel Marcus, Jun 2019): twin PRIMES (members, not pairs) between p and p², inclusive; links Birdsey–Schay, arXiv:1906.09220, "A Sieve for Twin Primes" (2019) — a paper attached to exactly this window. - **A143738** (2008): twin pairs between n and n² for ALL n (openers counted); A079047: plain primes between prime(n) and prime(n)². **Novelty-position change, stated loudly:** the zone's **pair-count** object is OWNED — at window (p, p²) it has been in OEIS since 2016 and our per-zone count sequence is A273257 shifted by one index; at squares-to- squares windows since 2000, with the TPC-equivalence and a linear-growth observation already recorded there by Hasler in 2019. Any zone-count claim must cite this family. What remains outside OEIS is the **per-zone MAX-GAP object Z₂ and the boundary slacks**, searched as follows (every query 0 hits; denominators per `SEARCH-CONVENTIONS.md` §2): | object | base sequence (zonegap-01 conventions) | variants searched, all 0 hits | |---|---|---| | per-zone Z₂ ladder | `2,6,12,30,30,30,36,72,150,150,150,150` | base, ±1, ±2, /2, offsets 1 and 3 (8 queries) | | Z₂ envelope record values (A113274 minus the never-attained records 4 and 16) | `2,6,12,30,36,72,150,168,210,282,372,498,630,924` | base, ±1, ±2, /2, offset 2 (7 queries) | | envelope-step p's | `2,3,5,7,17,19,23,47,73,113,137,157,241,433` | base, offsets 2 and 4 (3 queries) | | head slack (firstOpener − p) | `1,2,6,4,6,4,12,10,6,12,10,4,18,16,12,6` | base, ±1, ±2, offset 2 (6 queries) | | head as absolute opener (smallest twin opener > p) | `3,5,11,11,17,17,29,29,29,41,41,41,59,59,59,59,71,71,71` | base, offset 1 (2 queries) | | tail slack (p′² − lastOpener) | `4,8,8,14,20,8,14,8,14,80,50,14,62,68` | base, −2, offset 2 (3 queries) | | pair count per zone | `2,3,4,8,9,16,17,21,29,30,41,48,50,61` | base, ±1 → 0 hits; **offset 2 → A273257** (4 queries) | These absences carry the §2 flip discipline (±1, ±2, /2, offsets) on a channel calibrated the same minutes; they do not cover indexings not listed (e.g. Z₂ read at record points only in a first-occurrence convention — that inverse indexing is the owned A113274/TOS ladder itself, recon §3). ## 3. The remaining NOT-REACHED literature: one settled by proxy, one still walled - **Korevaar, Indag. Math. 23 (2012) 269–299** = "The prime-pair conjectures of Hardy and Littlewood", doi 10.1016/j.indag.2011.12.001 (identified from A091592's %H this session). ScienceDirect still refuses (bot wall on the article and its `/pdf`, 403 on WebFetch; Elsevier API: metadata only, `openaccessArticle=true` bronze — the PDF endpoint needs a key; CORE's five stored copies are all dead blobs, `BlobNotFound`). **Proxy read instead:** its arXiv precursor Korevaar, "Prime pairs and Zeta's zeros", arXiv:0806.0934v1 (sha256 `2506a027f6841fd8da7681614efa1fc119b2500132fe49fdef997db106f4c13c`, 30 pp, read via pdftotext): the programme is Wiener–Ikehara/Tauberian equivalences for the HL pair-correlation conjecture — `Legendre` occurs 0 times, `square` only as "square partial sums", no interval-between-squares or maximal-gap statement anywhere. A091592 links the survey as general prime-pair background. **Verdict: page-level read of the Indag text still NOT REACHED, but the twin-zone relevance is now bounded: nothing in this programme owns or neighbors the zone object.** - **Guy, UPINT §A8/§A9 — still NOT REACHED at page level.** Both archive.org copies (`unsolvedproblems0000guyr` 1994, `unsolvedproblems0003guyr` 2004) are lending-restricted: direct `_djvu.txt` answers 401, the node's `fulltext/inside.php` answers "Item not available", `api.archivelab.org` is dead (connect timeout), and Google Books was 429 for the prior session. The recon's caveat stands: whether §A8 records a named twin-between-squares remark is unverified. ## 4. Proposed PRIOR-ART.md additions (not applied — for the orchestrator) Carry recon §6's three bullets (Kourbatov–Wolf 2019; the Legendre-type twin conjectures bullet; the data-carriers line) and amend/add as follows: 1. **Amend the data-carriers line** (recon §6.3): "Luhn's pzktupel.de GAP02 is A113274's live continuation channel, **cross-checked record-exact 82/82 against A113274/A113275 with its merit column reproduced as 2C₂·∫dt/ln²t (`research/twin-gap-witnesses-01.js`); certifies 'below 1.00·10¹⁷'; carries the discoverer ledger Rathbun/Wolf/Fischer/Oliveira e Silva/Smith/Ritschel/Raab 1998–2021**. Kourbatov's tables paper arXiv:1309.4053 Table 2 is a second witness for records 1..72, exact." 2. **NEW bullet — the OEIS twin-count-per-window family owns the zone's count object.** A057767 (2000): twin pairs between consecutive prime squares, with Hasler's 2019 TPC-equivalence comment, a conjectured record-low ladder, and the empirical a(n)/n ≳ 1/11; A273257 (2016): twin pairs in (prime(n), prime(n)²) — **equal to the zone (p, p′²) pair count up to one index shift** (no prime lies in (p, p′)), b-file to n = 10⁴; A308777 (2019, members not pairs), A143738 (2008, all n), and Birdsey–Schay arXiv:1906.09220 attached to the (p, p²) window. The per-zone MAX-GAP object Z₂, the head/tail slacks, and the envelope-step ladder remain absent from OEIS at the flip-search discipline (§2 above, 33 calibrated queries). 3. **NEW line under the Kourbatov bullet:** the 2013 heuristic ceiling's constant appears on Luhn's live page as the merit normalization 1.32032363 = 2C₂, i.e. the record-tracking community operates inside the same HL normalization the corpus adopted (notation collision noted at ZONE-POSTULATE §4). ## 5. Artifacts this session | artifact | source | sha256 | |---|---|---| | Luhn GAP02 HTML | `pzktupel.de/RecordGaps/GAP02.php` | `d12779fb37c5113bcceb40f238bcff6cc4e6908f2048e42ac85049b42e570028` | | Kourbatov tables, arXiv:1309.4053v1 | `export.arxiv.org/pdf/1309.4053` | `733f8e77aba31d7f2bd5dee7da6e42b73f07be49a2bc1f442d3a658b10d108df` | | Korevaar precursor, arXiv:0806.0934v1 | `export.arxiv.org/pdf/0806.0934v1` | `2506a027f6841fd8da7681614efa1fc119b2500132fe49fdef997db106f4c13c` | OEIS entries read in full internal format: A273257, A308777, A057767, A079047, A143738, A091592 (%H only, for the Korevaar identification). Query log: session scratchpad `oeis-batch-results.txt` (33 queries, two calibration positives passing in the same minutes). --- *History layer: process record, staging. See `research/history/CHANGELOG.md` for the corpus rule.*