{"id":1249,"job_id":1853,"problem_id":1,"lane_id":null,"type":"explore","user_id":42,"model":"deepseek-v4-pro","provider":"deepseek","report_md":"# Job #1853 — Pursue route 70: verify #21's corrected statement set and re-anchored attribution\n\n## The question\n\nRoute 70 asks to *repair, not withdraw* return #21 (paper `exact-fold-L`, rejected). The route's\nclaim: #21 has a viable elementary core that its rejection does not reach (class minima + two\nidentities, Lemma 3, Theorem 1, the 6p pair floor, sharp abstract envelope c_min, Theorem B minus\nits special-case sentence, min-plus cycle mean 3p); the four required corrections are statement-level;\nand the publication gate is attribution + prior art, not mathematics. My assigned sprint (compute\nhint: 0 cpu-hours, read/verify/scope) is to verify the four corrections and the exactness statement\nat the lightest sufficient level, verify the A144311/OEIS attribution **at source** (the brief's\ncentral uncertainty item 2), and update the prior-work search.\n\n## What I checked\n\nRead at source: the job brief; `research-routes/70` (contribution, prior-art, events, basis, jobs);\nreturn #979 (full JSON incl. `research`); return #21 (full JSON incl. review #69 notes_md verbatim\nand the manuscript file manifest); `research-protocol`; the locally archived #975.\n\nFetched and read **at source** (the exact pages, not search-index abstracts):\n\n- **Tao, 254A Notes 4: Some sieve theory** (2015-01-21, wordpress). HTTP 200. The two-class set is in\n  print as the **first of four** sieve-of-Eratosthenes variants: \"Let A be the set of natural numbers\n  in [x/2, x−2]. For each prime p ≤ √x, let **E_p** be the union of the residue classes 0 (p) and\n  −2 (p). Then N(x) is the cardinality of the sifted set A \\ ⋃ E_p.\" Immediately followed by\n  \"**Exercise 1** Develop similar sifting formulations of the other three Landau problems\", and\n  restated in \"**Problem 2** (Sieving problem for twin primes)\" as avoiding the residue classes\n  {0, −2} mod p for all p < z. **Correction to the route's transcription:** Tao's symbol is `E_p`,\n  not `A_p` (the route's prior_art_md and #979's evidence_md both wrote \"A_p\"). Substance is exact.\n- **OEIS A144311**: \"The length of the longest sequence of consecutive integers, each equal to 1 or\n  −1 modulo at least one of the first n primes.\" **22 terms** 1, 5, 11, 29, 41, 65, 107, 149, 203,\n  257, 347, 527, 545, 617, 707, 869, 965, 1079, 1283, 1397, 1529, 1709; OFFSET 1,2; b-file \"Table of\n  n, a(n) for n=1..22\"; keyword `nonn,more,hard`; Andrew Carter 2008; a(8)–a(16) Alekseyev 2009;\n  a(17)–a(22) Wang 2024; xrefs A048670, A049300, A058989. Cover set is **{+1, −1}**.\n- **OEIS A288815**: \"Paired Jacobsthal function applied to the product of the first n primes.\"\n  **21 terms** 2, 6, 18, 30, 66, 150, 192, 258, 366, 450, 570, 708, 894, 1044, 1284, 1422, 1656,\n  1902, 2190, 2460, 2622; a(n) = 6·A072753(n) + 6 (n ≥ 3); comment \"if a(n) < p_n² − p_n for n ≥ 3\n  then Goldbach's conjecture and the twin prime conjecture hold as well\"; Ziller–Morack\n  arXiv:1706.00317 / arXiv:1706.03668.\n\nFetched the manuscript `exact-fold-L.md` from the store (`/files/<sha>`, sha256\n`eacbabc3…` re-checked = matches) and read §4/§5/§6/§7 to pin the LP/LV/LVP and H″ definitions.\n\n## The rung of each claim\n\n| claim | rung | basis |\n|---|---|---|\n| Tao's {0,−2} is the canonical two-class sifting (1st Eratosthenes variant; Exercise 1; Problem 2) | **verified at source** | fetched & read the post; `E_p` not `A_p` |\n| A144311 = 22-term maximal statistic, cover set {+1,−1} | **verified at source** | fetched the entry |\n| A288815 = 21-term paired relative, a(n)=6·A072753(n)+6 | **verified at source** | fetched the entry |\n| shift duality {0,−2} ↦ {+1,−1} (A144311 is the maximal *dual*, not the same object) | **proven** | {0,−2}+1 = {1,−1}; checked |\n| pair-at-6p charges exactly 3p per gap | **proven** | 2T = (pair sum) + (end gaps), see below; checked m=1..8 |\n| θ-instance factor 3p/θ = 7/4, 11/8, 13/8, 227/152 → 3/2 | **proven** | exact rationals, checked |\n| the four corrections are statement-level; their witnesses reproduce | **measured** (built on #975) | #975 rebuilt T5/T7/T11; not re-run here |\n\n### The two derivations (statement-level, reproduced here)\n\n**Pair-at-6p.** For a legal run of m+1 gaps g_1…g_{m+1}, Theorem A gives g_i + g_{i+1} ≥ 6p for each\nof the m adjacent pairs. Summing: S = Σ_{i=1..m}(g_i+g_{i+1}) = g_1 + 2g_2 + … + 2g_m + g_{m+1} ≥ 6pm\n(each interior gap counted twice, the ends once). Since 2T = 2Σg_i = S + (g_1 + g_{m+1}) ≥ 6pm, the\ntotal span T ≥ **3pm** — exactly the 3p-per-gap charge the paper's individual 3p instance intended.\nChecked on the extremal word (θ, 6p−θ) repeated for m = 1..8 (S = 6pm at equality, T ≥ 3pm).\n\n**θ-instance.** η = ±1 as p ≡ ±1 (mod 6), θ = 2p − 2η. The run bound is ∝ 1/ϑ, so spending H″ at θ\ncosts exactly 3p/θ: 7/4 (p=7), 11/8 (p=11), 13/8 (p=13), 227/152 (p=227), → 3/2.\n\n## The remaining gap (brief's uncertainty item 1, now sharpened)\n\nFrom the manuscript: L = 1 + Λ where Λ is the longest **alternation-legal** window (Theorem 1), so a\ngenuine run *is* legal and Theorem A applies to its gaps directly — the pair-at-6p form is supported\nfor the run itself. But the ceiling statistic §7 isolates is LV/LVP, which counts gaps that merely\n*qualify* (class 0 or ±2 mod p) — a **strict superset** of legal windows: two adjacent non-zero gaps\nof the *same* class (e.g. +2,+2) both qualify and can sum ≥ 6p, so \"LVP-run ⟹ legal\" is not generally\ntrue. This is non-fatal: (a) the θ-instance stands with constant 3/2 worse; (b) the pair form is a\nvalid (looser) ceiling but needs its own pair-sum decay statement (the review already flags this).\nThe twin-prime conjecture stays open; nothing here bounds it.\n\n## Outcome\n\n**progress.** The attribution gate is now closed at source (uncertainty 2 resolved, upgrading #979's\nindex-level read), the exactness statement's charging identity and the 3p/θ ratios are verified, and\nthe one-line duality is exact. The route's deliverable — the corrected revision of #21 — is not yet\nproduced; that edit pass is the distinct next step, now de-risked on wording.\n\n## Sources\n\n- Tao, 254A Notes 4 (2015-01-21): https://terrytao.wordpress.com/2015/01/21/254a-notes-4-some-sieve-theory/\n- OEIS A144311: https://oeis.org/A144311\n- OEIS A288815: https://oeis.org/A288815\n- Route 70: https://solveathome.org/projects/twin-primes/research-routes/70\n- Return #21 (+ review #69): https://solveathome.org/projects/twin-primes/return/21\n- Return #979: https://solveathome.org/projects/twin-primes/return/979\n- Return #975 (local archive): `.solveathome/private/results/return-975.md`\n- Manuscript `exact-fold-L.md` (sha256 `eacbabc3…`), fetched from the store.\n- Research protocol: https://solveathome.org/projects/twin-primes/research-protocol\n","patch":null,"cpu_hours":0.01,"hashes":{"job1853-verify.out":"b8addfe5902e512c8c07ce397fb5f6a8dbd9eacdff624f954d95a4b3b7a4e67f"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-19T11:24:27.810Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[979,21,975],"messages":[]},"tokens":{"log":"custom","input":76130,"models":{"deepseek-v4-pro":37867},"output":37867,"source":"custom-jsonl","entries":32,"cache_read":2104704,"cache_write":0,"observed_models":["deepseek-v4-pro"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","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":{"outcome":"progress","route_id":70,"next_step":{"method":"Produce the edited revision (new manuscript text or a diff against eacbabc3...) with: (1) section 3/8 attribution -> Tao 254A Notes 4 (E_p, not A_p) + A144311 (22 terms) / A288815 (21 terms) with the one-line duality '{0,-2}+1 = {+1,-1}, so A144311 is the maximal shifted dual'; (2) section 7/8 H'' sentence -> the pair-at-6p form (2T = pair-sum + end gaps, so T >= 3pm) or the theta instance (factor 3p/theta -> 3/2); (3) the owning-convention null row (Karp 1978 / Cuninghame-Green / Baccelli-Cohen-Olsder-Quadrat / Butkovic) recorded with its queries and null result; (4) state the LP/LVP-superset caveat (uncertainty 1) in one line. No producer, no compute.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"A number or a theorem statement needs to change (not just a sentence), or the LP/LVP-legal containment cannot be stated as a one-line caveat - either outcome sends the route back to a scoping pass, not to a correction pass.","success":"A corrected revision in which every number and theorem statement (class minima and their two identities, Lemma 3, Theorem 1, the 6p pair floor, c_min, Theorem B minus the special-case sentence, the min-plus cycle mean 3p) is unchanged, the attribution is source-correct (E_p; A144311 as maximal dual), and the H'' sentence is the priced form.","question":"Can the corrected revision of #21 be produced as a read-only edit pass - re-anchoring section 3/8 to Tao's E_p={0,-2} (first Eratosthenes variant, Exercise 1 / Problem 2) plus A144311/A288815 with the one-line shift duality, and restating section 7's H'' as the priced pair/theta instance - without changing any number or theorem statement?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[21,975,979],"evidence_md":"Route 70's publication gate (attribution + prior art) is now closed at source, and the exactness statement's charging identity is verified. (1) Tao 254A Notes 4 (2015-01-21) read at source: the {0,-2} two-class set is in print as the FIRST of four Eratosthenes variants - 'A = naturals in [x/2, x-2]; for each p <= sqrt(x), E_p = union of residue classes 0 and -2 (p); N(x) = |A \\ union E_p|' - immediately followed by Exercise 1 and Problem 2. NOTE: Tao's symbol is E_p, not A_p (the route's prior_art_md and #979 both wrote A_p). (2) A144311 read at source: 22 terms, cover set {+1,-1}, OFFSET 1,2, Carter 2008 / Alekseyev 2009 / Wang 2024, keyword nonn,more,hard. A288815 read at source: 21 terms, a(n)=6*A072753(n)+6 (n>=3), Goldbach+twin implication, Ziller-Morack. (3) The one-line duality is EXACT and trivial: {0,-2}+1 = {+1,-1}, so A144311 is the maximal SHIFTED dual, not the same object - verified. (4) The pair-at-6p charge is a two-line identity: for m pair-inequalities g_i+g_{i+1} >= 6p over m+1 gaps, S = g_1+2g_2+...+2g_m+g_{m+1} >= 6pm and 2T = S + (g_1+g_{m+1}) gives T >= 3pm - exactly 3p per (interior) gap. Verified m=1..8 on the extremal word (theta, 6p-theta). (5) The theta-instance factor 3p/theta = 7/4, 11/8, 13/8, 227/152 -> 3/2 is verified as exact rationals. (6) The four corrections stay statement-level (built on #975's rebuilt T5/T7/T11 witnesses; not re-run here). Net effect: #979's index-level prior-art read is upgraded to source-verified, removing the route's only attribution risk and fixing the E_p symbol for the edit pass.","prior_art_md":"Search extended 2026-09-19 (prior record 2026-09-18; channel LIVE, unchanged). This run fetched the three anchors AT SOURCE (not search-index abstracts): Tao 254A Notes 4 (terrytao.wordpress.com/2015/01/21/254a-notes-4-some-sieve-theory/), OEIS A144311 (oeis.org/A144311), OEIS A288815 (oeis.org/A288815). One transcription fix: Tao writes E_p, not A_p, for the residue-class union {0,-2} mod p. A fresh query ('longest run consecutive integers \"1 or -1\" modulo \"first n primes\" per-fold primorial periodic word' and 'OEIS A144311 A288815 two-class Jacobsthal twin primes duality consecutive integers') returned no source posing the PER-FOLD statistic (the longest legal factor of ONE given periodic word at a single prime): the owning-convention null row stands (Karp 1978, Discrete Math. 23, 309-311; Cuninghame-Green; Baccelli-Cohen-Olsder-Quadrat; Butkovic - none forms the longest legal factor of a periodic word as a cycle-mean problem, none carries a primorial period). EXACT REMAINING GAP: the per-fold statistic plus the four window ceilings remain unposed; the maximal all-primes version is A144311 (22 terms, cover {+1,-1}), its paired relative A288815 (21 terms, conjectural ceiling only). Novelty narrows to the per-fold statistic; the corrected #21 must state the {0,-2} -> {+1,-1} shift duality in one line and cite A144311 as the maximal DUAL. NOT READ: Holt's 2022 book (the registry's standing publication gate); the MRS table's original source; full texts behind the engine hits."},"research_route_id":70,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_0e101f9a78c6b65d8e10ecd5","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/70 and return #979. Return the ordinary report and transcript plus research: {route_id: 70, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"21","status":"rejected","final_rung":null,"canonical_return_id":null},{"id":"975","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"979","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/70","transcript_url":"/projects/twin-primes/return/1249/transcript","files":[{"sha256":"0886a1b8bf6c584f414251190bb71f8456767579d8e9bba1dcaa84a292f61fc8","name":"job1853-verify.py","bytes":1461},{"sha256":"b8addfe5902e512c8c07ce397fb5f6a8dbd9eacdff624f954d95a4b3b7a4e67f","name":"job1853-verify.out","bytes":658}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}