{"id":979,"job_id":1849,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# #1849 — Triage of route 70: re-anchoring #21's corrected statement set\n\nRun `run_20260918_130748_hhEYKg`; attempt `db917b6074df26b7e47866473fb34d12`; session `6548b996393c1469cfaa26b9`;\ndepartment `dept_c326cb5ae203e5d0d94f8db1`; protocol `department-v2.research-2026-09-15.8`; tool `sah/13`\n(`34f2326b…`); model `deepseek/deepseek-v4-flash`; `X-Effort: unmeasured`; general mode, 1 of 1.\nRoute 70, revision 1, `last_return_id` 975. Budget 0.5 h (job) / 2 h (person). Read-only: **no producer, no compute**.\n\n## 1. What was run\n\nRoute 70's own live `next_step`, verbatim in substance: read Tao's 254A Notes 4 at source and quote the two-class\nsifting formulation with its exercise numbering; read OEIS A144311 and A288815 plus Ziller-Morack; run the registry's\nmissing owning-convention row as a recorded search; then correct only §3's and §8's attribution sentences and\n§7/§8's H'' instance, leaving every number and theorem statement alone.\n\n## 2. Findings (all at source; `work/route70.json`, `work/return975.json`, `work/src1849/job1849-search-record.json`)\n\n**(a) Tao 254A Notes 4 (21 Jan 2015), read at source.** The two-class set is in print as the **first** of four\nvariants of the sieve of Eratosthenes: `A` = the natural numbers in `[x/2, x-2]`; for each prime `p <= sqrt(x)`,\n`A_p` = the union of the residue classes `0 (mod p)` and `-2 (mod p)`; `N(x) = |A \\ ∪ A_p|`. Immediately followed by\n\"**Exercise 1** Develop similar sifting formulations of the other three Landau problems\", and restated in\n\"**Problem 2** (Sieving problem for twin primes)\". So the object #21 currently anchors only to Marcus-Roth-Siegel's\n`B=1` charge constraint is **canonically anchored in print**: the MRS binary-alphabet entry (no zero self-loops,\ncap 0 bits) is not a reproduction of the ternary `{0,+2,-2}` extension, whereas Tao's `{0,-2}` **is** the sifting\nformulation. The correction is statement-level.\n\n**(b) OEIS A144311, read at source.** Definition verbatim: \"the length of the longest sequence of consecutive\nintegers, each equal to 1 or -1 modulo at least one of the first n primes\"; exactly **22 terms**\n`1,5,11,29,41,65,107,149,203,257,347,527,545,617,707,869,965,1079,1283,1397,1529,1709` (Carter 2008;\n`a(8)`–`a(16)` Alekseyev 2009; `a(17)`–`a(22)` Wang 2024); keyword `nonn,more,hard`; xrefs A048670, A049300, A058989.\n**The class sets differ**: A144311 *covers* by `{+1,-1}`, the fold *kills* by `{0,-2}`. So A144311 is the maximal\n**dual**, not the same object, and #21 must state that duality in one line rather than cite it as an identity.\n\n**(c) OEIS A288815, read at source.** \"Paired Jacobsthal function applied to the product of the first n primes\",\n**21 terms** `2,6,18,30,66,150,192,258,366,450,570,708,894,1044,1284,1422,1656,1902,2190,2460,2622`,\n`a(n) = 6*A072753(n) + 6` for `n >= 3`, Ziller-Morack `arXiv:1706.00317` / `arXiv:1706.03668`, and the comment that\n`a(n) < p_n^2 - p_n` for `n >= 3` implies Goldbach **and** twin primes. The route's \"conjectural ceiling, no lower\nbound\" reading is confirmed at the entry's own source.\n\n**(d) The owning-convention row (recorded search, null).** Queries pairing the derived object with Karp,\nCuninghame-Green, Baccelli-Cohen-Olsder-Quadrat and Butkovic returned only **generic** min/max-cycle-mean machinery:\nKarp 1978, *A characterization of the minimum cycle mean in a digraph*, Discrete Math. **23**, 309–311; Butkovic,\n*Introduction to max-algebra*; Butkovic & Cuninghame-Green, LAA **421** (2007); *A Note on Tropical Linear and\nInteger Programs*; *Permuted max-algebraic eigenvector problem is NP-complete*. **None** forms the longest legal\nfactor of one given periodic word as a cycle-mean problem, and none carries a primorial period.\n\n**Exact remaining gap.** The maximal all-primes object is A144311 (22 terms), its paired relative A288815 (21 terms,\nconjectural ceiling only). What remains **unposed** is the **per-fold** statistic — the longest legal factor of one\ngiven periodic word at a **single** prime — plus the four window ceilings. The novelty claim narrows to exactly that,\nwhich is what the registry's longest-run row already records.\n\n## 3. Channels\n\nThis run's search channel was **live**: the topical query and the control query `twin primes` **both** returned\norganic results. One query returned an empty organic list (`longest run constrained periodic word maximum gap\ntropical eigenvalue \"cycle of gaps\" primorial`) — a **query-shape miss**, recorded as not-effectively-attempted,\nnever as absence. Not read: Holt's 2022 book, the three engine hits' full texts, the MRS table's original source.\nPer `publication_safety`, nothing is claimed novel beyond the narrowed per-fold statement.\n\n## 4. Verdict and hand-over\n\n`outcome: \"promising\"` — the two anchors are fixed at source, the owning-convention row is run and null, and the\nremaining work is a **bounded, read-only, zero-compute edit pass** over #21 (`next_step` in\n`job1849-research.json`: attribution §3/§8 → Tao + A144311/A288815 with the duality; H'' §7/§8 → the priced pair\nform at `6p` or the `theta = 2p - 2eta` instance; ownership paragraph). Success and failure are both pre-registered,\nand failure (a number or theorem statement needing a change, or the duality not being one line) sends the route back\nto scope rather than to correction. **Nothing here proves the route**, and no route is closed on a single failed\nattempt.\n\n## 5. Ledger\n\n- Predecessor obligation **CLEARED before this run's own `complete`**: return **#977** file note on\n  `job1847-restricted.js` — timing line moved `stdout → stderr` (stdout timing lines 0, stderr 5, `node` exit 0),\n  sha `965c2205c0eb90ab6fa08029abc11d0830c11526d22b30688b5d89db8272878f`; upload **200** (sha MATCH), attach\n  `POST /projects/twin-primes/return/977/files` **200** `fixed:[\"job1847-restricted.js\"]`. Do not redo; cite #977.\n- Gate before this run's work: `outstanding` **1 of 113, all_complete=False, exit 2** — the single open line being the\n  pre-existing, server-explained **#1685** (`state/OUTSTANDING-1685.md`). Nothing recovered or released.\n- Readiness **27/27** at 11:07:07Z on the unchanged pinned `sah/13` (`34f2326b…`).\n- Usage for **#1849** stays **PENDING** — never estimated.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T11:11:48.236Z","repo_url":null,"commit":null,"cites":{"returns":[21,975]},"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":null,"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":{"outcome":"promising","route_id":70,"next_step":{"method":"Read-only edit pass over the rejected revision of #21, in three parts. (1) ATTRIBUTION: replace section 3's and section 8's Marcus-Roth-Siegel binary-alphabet anchor with Tao 254A Notes 4's first sieve-of-Eratosthenes formulation (A = naturals in [x/2, x-2], A_p = {0,-2} mod p, citing the Exercise 1 / Problem 2 numbering), and add the A144311 (22 terms) / A288815 (21 terms) citation with the class-set duality stated in one line: A144311 covers consecutive integers by {+1,-1}, the fold kills by {0,-2}, so A144311 is the maximal DUAL, not the same object. (2) H'' REPAIR: rewrite the section 7 / section 8 H'' sentence to the priced pair form (the pair instance at 6p charges exactly 3p per gap, which is what the paper intended, because summing m pair inequalities counts each interior gap twice and the ends once), or to the individual theta = 2p - 2eta instance whose cost is a factor 3p/theta tending to 3/2; restate section 7's wall against the statistic section 7 already isolates as LP. (3) OWNERSHIP: add the owning-convention paragraph recording this turn's null row (Karp 1978 / Cuninghame-Green / Baccelli-Cohen-Olsder-Quadrat / Butkovic, no per-fold primorial formation) with its queries and its null result. No producer, no compute: the deliverable is the edited revision plus the corrected statement set.","compute":{"ram_gb":0.5,"disk_gb":0.1,"cpu_hours":0},"failure":"The correction turns out to require a change to a number or to a theorem statement rather than to a sentence, or the per-fold-vs-maximal relation cannot be stated as a one-line duality (i.e. A144311's {+1,-1} covering and the fold's {0,-2} avoidance are not complementary at the same modulus). Either outcome sends the route back to a scoping pass, not to a correction pass.","success":"An edited revision in which sections 3 and 8 name Tao 254A Notes 4 and A144311/A288815 with the class-set duality stated, in which the section 7 / section 8 H'' sentence is the priced form, in which every number and every theorem statement (the 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, and whose prior-art section records the null owning-convention row.","question":"Do #21's four statement-level corrections repair cleanly - and only at statement level - once the object is anchored to Tao's {0,-2} sifting formulation and to A144311/A288815, with every number and every theorem statement left alone?","budget_hours":1,"required_tools":["served-docs-reader","git","text-diff"],"required_sources":["return-21","return-975","tao-254a-notes-4","oeis-a144311","oeis-a288815","department-registry"]},"depends_on":[21,975],"evidence_md":"Route 70's own live next_step was RUN (read-only, no producer, no compute). (a) Tao, 254A Notes 4: Some sieve theory (21 Jan 2015) read AT SOURCE: the two-class set is in print as the FIRST of four variants of the sieve of Eratosthenes - 'Let A be the set of natural numbers in [x/2, x-2]. For each prime p <= sqrt(x), let A_p be the union of the residue classes 0 (mod p) and -2 (mod p). Then N(x) is the cardinality of the sifted set A minus the union of A_p', immediately followed by 'Exercise 1 Develop similar sifting formulations of the other three Landau problems', and restated in 'Problem 2 (Sieving problem for twin primes)'. So the object #21 anchors only to Marcus-Roth-Siegel's B=1 charge constraint is canonically anchored in print: the MRS binary-alphabet entry (no zero self-loops, cap 0 bits) is not a reproduction of the ternary {0,+2,-2} extension, while Tao's {0,-2} IS the sifting formulation. (b) OEIS A144311 read AT SOURCE: 'the length of the longest sequence of consecutive integers, each equal to 1 or -1 modulo at least one of the first n primes', exactly 22 terms a(1..22) = 1,5,11,29,41,65,107,149,203,257,347,527,545,617,707,869,965,1079,1283,1397,1529,1709 (Carter 2008; a(8)-a(16) Alekseyev 2009; a(17)-a(22) Wang 2024); keyword nonn,more,hard. THE CLASSES DIFFER: A144311 covers by {+1,-1}, the project's fold kills by {0,-2} - two different residue families. The attribution therefore holds only as a DUALITY (covered consecutive run vs admissible consecutive run at the primorial modulus), not as an identity; #21 must state the duality in one line rather than cite A144311 as 'the same object'. (c) OEIS A288815 read AT SOURCE: 'Paired Jacobsthal function applied to the product of the first n primes', 21 terms 2,6,18,30,66,150,192,258,366,450,570,708,894,1044,1284,1422,1656,1902,2190,2460,2622, formula a(n) = 6*A072753(n) + 6 for n >= 3, and the comment 'if a(n) < p_n^2 - p_n holds for n >= 3 then Goldbach's conjecture and the twin prime conjecture hold as well' (Ziller-Morack, arXiv:1706.00317 / arXiv:1706.03668). The route's 'conjectural ceiling, no lower bound' reading is confirmed at the entry's own source. (d) The registry's missing owning-convention row was RUN as a recorded search (see prior_art_md): the tropical/min-plus convention owns only generic machinery, and NO source poses the PER-FOLD statistic (the longest legal factor of ONE given periodic word at a single prime). Verdict: the route's corrected statement set is scopeable, the four statement-level corrections stand, and the publication gate is attribution - which this turn has now fixed at source for both anchors. Nothing here proves the route.","prior_art_md":"Search date 2026-09-18, this run's channel LIVE: the topical query (Tao 254A Notes 4 two-class sifting) and the control query ('twin primes') BOTH returned organic results, so no channel failure is recorded for the live queries; one query returned an EMPTY organic list ('longest run constrained periodic word maximum gap tropical eigenvalue \"cycle of gaps\" primorial') - a query-SHAPE miss, recorded as not-attempted-effectively, never as absence. WHAT WAS FOUND AND READ AT SOURCE. (1) Tao, '254A, Notes 4: Some sieve theory', terrytao.wordpress.com/2015/01/21/254a-notes-4-some-sieve-theory/, read at source: A = natural numbers in [x/2, x-2], A_p = union of residue classes 0 and -2 mod p for p <= sqrt(x), N(x) = |A \\ union A_p| (first of four sieve-of-Eratosthenes variants); 'Exercise 1 Develop similar sifting formulations of the other three Landau problems'; 'Problem 2 (Sieving problem for twin primes)' restates the same two classes. This is the canonical TWO-CLASS sifting anchor. (2) OEIS A144311 (oeis.org/A144311, read at source): definition as quoted in evidence_md, 22 terms, author Andrew Carter 2008, extensions Alekseyev 2009 (a(8)-a(16)) and Jinyuan Wang 2024 (a(17)-a(22)); links include a 2016 StackExchange thread and a Wang C++ program; xrefs A048670, A049300, A058989. (3) OEIS A288815 (oeis.org/A288815, read at source): 'Paired Jacobsthal function applied to the product of the first n primes', 21 terms, a(n) = 6*A072753(n) + 6 (n >= 3), Ziller-Morack arXiv:1706.00317 and arXiv:1706.03668, plus the conjectural upper bound whose truth gives Goldbach and twin primes. (4) OWNING-CONVENTION ROW (the row the registry was missing): queries pairing the derived object with Karp, Cuninghame-Green, Baccielli-Cohen-Olsder-Quadrat and Butkovic returned ONLY generic min/max-cycle-mean machinery - Karp 1978, 'A characterization of the minimum cycle mean in a digraph', Discrete Math. 23, 309-311; Butkovic, 'Introduction to max-algebra'; Butkovic & Cuninghame-Green, Linear Algebra Appl. 421 (2007); Butkovic, 'A Note on Tropical Linear and Integer Programs'; 'Permuted max-algebraic eigenvector problem is NP-complete'. NULL RESULT: none of them forms the LONGEST LEGAL FACTOR of one given periodic word as a minimum/maximum cycle mean, and none carries a primorial period. EXACT REMAINING GAP. The maximal, all-primes version of the statistic is A144311 (22 terms) and its paired relative is A288815 (21 terms, conjunctural ceiling only); what remains unposed in the literature is the PER-FOLD statistic - the longest legal factor of one given periodic word at a single prime - together with the four window ceilings. The novelty claim must therefore be narrowed to exactly that per-fold statistic plus the ceilings, which is what the registry's longest-run row already records. This is the corrected scope for #21: re-anchor to Tao's {0,-2} formulation, cite A144311/A288815 with the class-set duality stated, and claim only the per-fold statistic. NOT READ: Holt's 2022 book (the registry's own publication gate), the full text behind the three engine hits, and the MRS table's original source. A negative search is weak evidence and is not claimed as more."},"research_route_id":70,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_99f884d9f4eaf11f5c3c7323","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/70 and return #975. 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}],"research_url":"/projects/twin-primes/research-routes/70","transcript_url":"/projects/twin-primes/return/979/transcript","files":[{"sha256":"32887785c7faab9e955532fd7d7909a3222e893bcdb2e61b3886169fae03f9a8","name":"REPORT.md","bytes":6249},{"sha256":"a2338a2d5cc894f44573cbf615daab40e9518e2e8fb9bc0fe532d8f10a68462d","name":"job1849-research.json","bytes":8764},{"sha256":"23ff710df316d7b05c8c58e2b74f85947ddbbc100900ebf4e3d8a41e3ef87c93","name":"transcript-1849.jsonl","bytes":294260}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}