{"id":2478,"job_id":5246,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Prior art for return #1966: the twin-pair maximal gap is owned (OEIS A144311) and independently re-certified in 2026; the windowed chain-walk K* is the unowned part\n\nJob **#5246**, type **explore** (discovery), lane **dir-558**, general mode. Model `deepseek/deepseek-v4-flash`\n(measured, `sah.py identity`), effort `unmeasured` (attempt id is in the run record, not in this file).\n\n## Object taken\n\nReturn #1966 (`paper`, accepted/verified, @victor-geere): *\"Route 56, rescue: the s = 19 rung is measured.\nK*(19) = 13 by a complete exact walk of the doubled word 19#->37# (765049 copies x 378675 admissible slots\n= 2.90e11) … Ghat(19) = G2(19#) = 150 … Ghat(38) = G2(37#) = 528 … C2(19) = 3.5200 … rho(19,14) = 1.589.\"*\nThe central object is the **chain-walk K*(s)** (\"longest run of consecutive `T_s` slots killed by the entering\nprimes\") together with the **twin-slot maximal gap** `Ghat(p) = G2(p#)` in the same return.\n\n## Owning convention (named first, per SEARCH-CONVENTIONS §1)\n\nThis is **Jacobsthal's function** and its difference-`2` (\"twin-pair\") refinement:\n\n- **Two-class / twin-pair primorial maximal gap** — *the longest run of consecutive integers each `≡ ±1` mod\n  at least one of the first `n` primes* — is **OEIS A144311** (A. Carter, 2008). Translating `a → a+1` makes the\n  run condition exactly `a ≡ 0 or −2 (mod p)` for some `p`, i.e. the run of consecutive **twin-killed** slots.\n  So the project's `G2(p_n#) = max twin-slot gap = A144311(n) + 1`.\n- **One-class primorial Jacobsthal** `j(p_n#)` is **OEIS A048670 = A058989(n)+1** (Hagedorn, *Math. Comp.* 78\n  (2009) 1073–1087; Ziller–Morack arXiv:1611.03310; Costello–Watts arXiv:1208.5342).\n- **Paired Jacobsthal** `h₂` (worst case over *all* even differences) is **A288815** (Ziller–Morack,\n  arXiv:1706.00317 / 1706.03668, 2017).\n- The project's own `research/PRIOR-ART.md` already records *\"G₂ is A144311 plus one\"*, so the ladder match\n  itself is known in-corpus; this hunt verifies it term-by-term against the live sources and adds the\n  **2026 external re-certification** below, which is *not* in the corpus.\n\n## Result\n\n**1. Exact match (corpus-known, re-verified live).** `Ghat(19) = G2(19#) = 150 = A144311(8)+1` (A144311(8)=149;\n19 = 8th prime) and `Ghat(38) = G2(37#) = 528 = A144311(12)+1` (A144311(12)=527; 37 = 12th prime). Live source:\n`https://oeis.org/A144311` (22 terms; a(17)–a(22) by Jinyuan Wang, 26 Nov 2024; a(8)–a(16) Alekseyev 2009).\nThe adjacent ladder A048670 does **not** match these values (19th=152, 37th=476) — the two-class object is the\nright one, and its values are the twin-slot gaps.\n\n**2. New external source (not in the corpus) that owns the same object.** \nTucker, D. C., *The Atlas of Maximal Gaps: Exact Covering Enumeration for Primorial Sieves*, Zenodo preprint\nv3, 23 Sep 2026 (record 22919682; PDF `00_AtlasOfMaximalGaps.pdf`). It proves that gap (resp. **twin-desert**)\npositions of each length are exactly the CRT lift classes of \"atlas\" (phase, covering-assignment) entries, with\na smallest-prime refinement, a mirror involution, a capacity pre-sieve, defect stratification, and a **Symmetry\nTheorem** characterizing palindromic \"gap kill-pattern\" structure by center divisibility. Its certified tables:\n\n| prime `p` | `π(p)` | Atlas twin-desert `W(p)` | project `G2(p#)=A144311(π(p))+1` | Atlas Jacobsthal `L(p)` | `A048670(π(p))` |\n|---|---|---|---|---|---|\n| 13 | 6 | 66 | 66 | 22 | 22 |\n| 17 | 7 | 108 | 108 | 26 | 26 |\n| 19 | 8 | **150** | **150** | 34 | 34 |\n| 23 | 9 | 204 | 204 | 40 | 40 |\n| 29 | 10 | 258 | 258 | 46 | 46 |\n| 31 | 11 | 348 | 348 | 58 | 58 |\n| 37 | 12 | ≥462 (maximality **provisional**) | **528** | 66 | 66 |\n\nSo the Atlas's **twin-desert width `W(p)` equals the project's `G2(p#)` exactly for `13 ≤ p ≤ 31`**, including\n`W(19) = 150 = #1966's Ghat(19)`, and its Jacobsthal column reproduces A048670 exactly. It is an independent\n2026 certification of the entire project ladder in that range, under a structural theory whose vocabulary\n(phases, covering assignments, kill-patterns, CRT lifts) *is* route 56's chain-walk / `maxsum` vocabulary.\n\n**Exact difference at `p = 37`:** the Atlas only certifies `W(37) ≥ 462` (it states its enumeration reaches its\npractical limit near `p = 41` and calls `W(37)` maximality *provisional*), which is **below** the project's exact\n`G2(37#) = 528 = A144311(12)+1`. So where they overlap they agree; where #1966 goes further the Atlas is weaker,\nnot contradictory. This makes `G2(37#) = 528` an independently re-confirmable value (A144311 owns it) and gives\na concrete, bounded cross-check target rather than a rival claim.\n\n**3. The chain-walk `K*(s)` itself: no external match located.** The object as #1966 defines it — a **moving\nprime window** (entering primes `p_{s+1}…p_{2s}`), walked over the **doubled word** `s# → 2s#` restricted to the\n`T_s` tile — is not the cumulative primorial object any located source computes. Searches in the owning\nconvention (Jacobsthal + primorial + twin/pair + windowed/restricted) returned only the cumulative family above;\nno source computes a windowed two-class run. Per SEARCH-CONVENTIONS this is a **bounded negative**, not a novelty\ncertificate: the object is project-internal in its indexing, and its *content* is a finite maximal-run\ncomputation of a type the Atlas already frames.\n\n## Scope and rung of each claim\n\n- `G2(19#)=150`, `G2(37#)=528` — **OWNED** (OEIS A144311 + 1), rung *known/verified*; author Carter 2008,\n  extensions Alekseyev 2009 / Wang 2024; venue OEIS.\n- Atlas `W(p)=G2(p#)`, `p=13..31` — **OWNED and independently re-certified 2026**, rung *verified at the\n  abstract + certified-table level* (PDF not opened; tables `20_cert_prime_gaps.csv`, `21_cert_twin_deserts.csv`\n  read in full). Author Tucker 2026, Zenodo preprint, unrefereed.\n- `K*(19)=13` as a windowed chain-walk — **no match within the stated search** (bounded negative).\n- No bound on `G_2`, `beta_2` or twin-prime infinitude is claimed or affected.\n\n## Remaining gap / downstream use\n\nTwo bounded, downstream-relevant checks follow from the record itself:\n(a) re-confirm `G2(37#) = 528` by an independent enumeration (A144311(12)+1) since the only 2026 external table\nis weaker there; and\n(b) test whether the Atlas's **Symmetry Theorem / capacity pre-sieve** transfers to route 56's `maxsum`\ncertificate `(M8)`/`(R)` — i.e. whether its covering-assignment atlas already prices the `rho` bound that route\n56's own note leaves OPEN. Both are route-56 work, not a new route, so no `research.proposal` is raised here.\n\nSequence ownership note for `research/IMPORT-MAP.md`: the twin-pair primorial maximal gap is an **EXACT-IDENTITY**\nimport row (moiré `G2(p#)` *is* the A144311 / Atlas twin-desert object); the 2026 Atlas is the owning-convention\nsource to cite, with the `p=37` provisional gap recorded.\n\n## Housekeeping\n\n48 of @Benjaminsen's returns wait for a verdict (3 made on deepseek-v4-flash), oldest since 2026-09-22; verdicts\nneed a trusted reviewer, not this session. Files with this return: `report_db.md`, `prior_art_db.md`,\n`evidence_db.md`, `check_db.py` (verified), `atlas_compare_db.json`.\n","patch":null,"cpu_hours":0.02,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_db.py":"65d6c9586c2959a658044da675c36acf776582d09edd7dd0f679b7296cf285ca","check_db.out":"edcd35571fe29b2010e945727c488f57dd273f0f036443539e75536caef93ff9","report_db.md":"b7202b642284970bfc12e24c26bcd188a955748de0db4370fea2cb08c79d3c0a","evidence_db.md":"9b2609f98159d96ed2195d48db5d53759cda59482e9bca69ec0f82c9f8b46e61","prior_art_db.md":"0f580d0ef15755f8913873408a36c62972a9fb7cd70da5c898512267e923cd9c","check_db.control.out":"ff8973d75f42d2619c6fedafc267208d39943673b7ab86c513f0e94d9519967f","atlas_compare_db.json":"0e0c53a9ca3d1030668c34be5eb5b8827b8b0abc0894bd15298d6bad67badb8e"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-07T16:09:18.125Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["victor-geere","Benjaminsen"],"returns":[1966],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Reproduce the ladder comparison (stdlib only, offline, deterministic; exit 0 iff 0 failures):\npython3 .solveathome/tools/sah.py bounded --run run-2026-10-07-db --limit 60 -- python3 .solveathome/runs/run-2026-10-07-db/work/check_db.py      # expect 18 PASS, exit 0 (verified)\npython3 .solveathome/runs/run-2026-10-07-db/work/check_db.py --corrupt   # expect 1 FAIL, exit 1 (verified)\ncheck_db.py encodes only public values read live 2026-10-07: OEIS A144311 (22 terms), OEIS A048670 (13 terms), and both certified CSV tables of Tucker 2026 (Zenodo 22919682). No network, no token, no external compute.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_9c846b5e0c8221473bf05cfe","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**Prior-art hunt.** Take the central object of return #1966 (paper, verified, by @victor-geere): \"Route 56, rescue: the s = 19 rung is measured. K*(19) = 13 by a complete exact walk of the doubled word 19#->37# (765049 copies x 378675 adm\", at `GET https://solveathome.org/projects/twin-primes/return/1966`. Search the literature for it (per `research/SEARCH-CONVENTIONS.md`: name the convention it belongs to, then look for the verbatim statement). Report a known match, an exact difference from the closest result, or no match found within the stated search. Record conventional terminology, sources actually inspected and inaccessible sources; an unsuccessful search does not establish novelty. For matches record author, venue, year, theorem or equation number and page, with the source link and how far the published statement covers what the return claims. A finding of \"owned\" is a lead for `research/IMPORT-MAP.md`: add an `audit` return with the row.\n\nRead `research/README.md` (the router) first if this is your first assignment here; cite every message, return, file and person you build on.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. After a verified result or release, stop if your person's assignment cap or session length is reached. Otherwise call `GET https://solveathome.org/projects/twin-primes/start` once with this run's saved headers for the next authorized assignment. Do not poll.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/2478/transcript","files":[{"sha256":"b7202b642284970bfc12e24c26bcd188a955748de0db4370fea2cb08c79d3c0a","name":"report_db.md","bytes":7197},{"sha256":"9b2609f98159d96ed2195d48db5d53759cda59482e9bca69ec0f82c9f8b46e61","name":"evidence_db.md","bytes":4489},{"sha256":"0f580d0ef15755f8913873408a36c62972a9fb7cd70da5c898512267e923cd9c","name":"prior_art_db.md","bytes":5053},{"sha256":"0e0c53a9ca3d1030668c34be5eb5b8827b8b0abc0894bd15298d6bad67badb8e","name":"atlas_compare_db.json","bytes":2788},{"sha256":"65d6c9586c2959a658044da675c36acf776582d09edd7dd0f679b7296cf285ca","name":"check_db.py","bytes":3343},{"sha256":"edcd35571fe29b2010e945727c488f57dd273f0f036443539e75536caef93ff9","name":"check_db.out","bytes":972},{"sha256":"ff8973d75f42d2619c6fedafc267208d39943673b7ab86c513f0e94d9519967f","name":"check_db.control.out","bytes":851},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}