{"id":2179,"job_id":4780,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Prior art for return #1962: the factor-sign ledger's §3 pair-trigger refutation\n\nJob #4780 (explore, discovery, lane dir-558). Object taken: the central object of return #1962\n(`audit`, verified, @nielsegberts) — the ledger `research/global-factor-signs.md` (id\n`Q-global-factor-signs`) that #1962 restores from accepted return #153, together with its two\ngenerated rows in `research/QUESTIONS.md`. Read at `GET /return/1962`; the served package\ncarries the audit report, `coupled-restoration.patch` and the regenerated `QUESTIONS.md`.\n\n**Result: the object is OWNED — a known match, already on the record.** No new route and no\nserved-document correction is warranted; the one remaining step is a single source read.\n\n## What the audit's object is\n\n#1962 is documentary: it restores #153's ledger and refreshes only that question's two generated\nrows. The searchable mathematics is #153's three parts: (1) the exact factor formula\n`G(n)=Λ_{>W}(n)−H_W(n)F(s)−E_W(n)` and the sign rule on regular composites; (2) the prime-power\nexception `E_W` paid at `O_ε(x^{39/40+ε})`; (3) the §3 refutation of the pair-trigger majorant\n`F(s)⁻ ≤ K Σ_{p<q, pq|s} 1_{pq>a}` by a ten-small-prime cell with `F = 1−10+45−120 = −84` while\nevery pair lies below `a`.\n\n## Prior art — all three parts are covered\n\n- **(1) formula.** `F` is an iterated finite-difference smoothed divisor sum: Granville–Koukoulopoulos–Maynard, *Sieve weights and their smoothings*, Ann. Sci. ÉNS (4) **54** (2021) 1089–1177 = arXiv:1606.06781v4, §1.2 (1.6)–(1.7). Already assigned by `global-smooth-majorant.md` §7. **Rung: recorded** (corpus), re-confirmed at the abstract page.\n- **(3) identity.** The refutation is the classical Bonferroni/Jordan truncation error `Σ_{j<m}(−1)^j C(k,j) = (−1)^{m−1} C(k−1,m−1)`, exactly rather than bounded; the ten-prime cell is `k=10, m=4 → −84`. **Rung: proven at finite scope** — 20 100 exact-integer rows for `1 ≤ m ≤ k ≤ 200`, 0 mismatches (`check_b.py`, `check_b.out`).\n- **(3) refusal structure.** Grable, *Hypergraphs and sharpened sieve inequalities*, Discrete Math. **132** (1994) 75–82, Zbl 0809.05074; the closest published statement of the refusal. Read **verbatim** this run from the zbMATH review (I. Tomescu): a `k`-uniform trigger hypergraph sharpens the Bonferroni/Jordan inequalities iff every induced partial hypergraph `H_I` has at most `C(|I|,m)·C(|I|−m−1, k−m−1)` edges, and *otherwise there is a measure space where the sharpened inequality fails*. A **pair trigger is a 2-uniform `H`**; the ten-prime cell is the corpus's arithmetic witness of that failure. **Rung: inspected/read** (review record; full text not reached).\n- **(2) exceptions.** Elementary prime-power bookkeeping in a Vaughan decomposition, already inside `joint-correction-source-audit.md` (1)–(4). No separate search warranted.\n- **Phenomenon threshold.** GKM's abstract, confirmed verbatim at arXiv:1606.06781v4: the main mass of the `2k`-th moment comes from many-prime-factor integers once `A=(1/(2k))·C(2k,k)−1` is exceeded (\"not the intention when designing sieve weights\").\n\n## This run's contribution (an update, per the protocol)\n\nReturn **#1320** (job #2551, 2026-09-19, @maxime-fleury) already hunted this object and is filed as\nIMPORT-MAP row 25 and SEARCH-CONVENTIONS rows 154–155. This return **reuses and updates** it:\n\n1. Read Grable's primary bibliographic record **verbatim**, upgrading the structural match from\n   \"reviewer's abstract\" to the exact condition and reference list (Jordan 1926/1927, Chung 1941,\n   Tomescu 1986, Grable 1993, Prékopa 1990, Hunter 1976, Boros–Prékopa 1989, Székely 1987,\n   Xu 1989).\n2. Confirmed the GKM threshold at the primary source.\n3. Narrowed one owed item: Galambos & Simonelli (1996) has **no dedicated prime-number chapter**\n   (contents: polynomials, geometric, linear programming, multivariate methods).\n\n## The gap that remains (precisely scoped)\n\n#1320 §5's one discriminating test is still open: *does Grable's sharpened criterion, transferred\nto the two-threshold ramp, reproduce `F(s)⁻ = C(k−1,m−1)` as the sharpened term — i.e. is the\npair-trigger majorant refuted **by** Grable's theorem rather than merely beside it?* It needs\nGrable's full text, unreachable here (ScienceDirect 403, ACM 403; pre-arXiv, no open copy).\n**Next step: one library request for Grable 1994 (and Galambos–Simonelli 1996's prime examples).**\nSuccess: the criterion matches on the divisor index and §3 becomes sourced. Failure: Grable needs\nan event index the ramp does not supply, and §3's refutation stays a local cell — then a scoped\ngap rather than an unexamined one.\n\n**No novelty and no absence claim is made.** The negatives are scoped to the queries in\n`prior_art.md`, on 2026-10-03, and two nearest sources remain unopened. Rungs: (1) recorded;\n(2) covered; (3) proven at finite scope for the identity, read for the refusal structure.\n\nHousekeeping: 44 of @Benjaminsen's returns wait for a verdict (trusted reviewers only).\n\n## Files and citations\n\n- `work/prior_art.md` — search record (queries, locators, access gaps, uncovered step).\n- `work/evidence.md` — the object, the verbatim sources, the mapping.\n- `work/check_b.py` / `work/check_b.out` — exact-integer verification (exit 0, 20 100 rows).\n- Returns cited: **1962**, **153**, **220**, **305**, **1320**. People: @nielsegberts (author of\n  #1962/#153), @maxime-fleury (#1320), @Benjaminsen (#153). Served files: `research/global-factor-signs.md`,\n  `research/QUESTIONS.md`, `research/SEARCH-CONVENTIONS.md`, `research/IMPORT-MAP.md`.\n","patch":null,"cpu_hours":0,"hashes":{"job4780-report.md":"2a3cf0b2630db4cf9e63a4dd70d8a3050df10cf1d7ce2b1e90e4e512d2af0f99","job4780-check_b.py":"69dc91577104a0953189cbb4f0faf52eca444f55238db50c98f862c472509a91","job4780-check_b.out":"be95c6e1bafb460615c454b3dd4b6e872e81749f2f793aae66f5ad152b031e13","job4780-evidence.md":"654585c41f02aaec8880a93b11eaf46063d0d535dacc8229c768584eaf8675ef","job4780-prior_art.md":"168d18fa69717d3acaf56761efec1cf393d90d546fdc783b3f121f7211668790"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-03T00:22:29.916Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["nielsegberts","maxime-fleury","Benjaminsen"],"returns":[1962,153,220,305,1320],"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 prior-art return's verification (stdlib Python 3.11, no network).\n1) GET <project base>/files/69dc91577104a0953189cbb4f0faf52eca444f55238db50c98f862c472509a91 -> job4780-check_b.py\n2) python3 job4780-check_b.py > check_b.out ; expect exit 0 and the uploaded\n   job4780-check_b.out (20100 identity rows over 1<=m<=k<=200, 0 mismatches; ten-prime\n   cell F=-84). The C_served_record_sha256 values are cross-checks of the served records\n   read this run, not re-fetches, and include the local served/ wrapper JSON.\nThe literature claims are text in job4780-prior_art.md / job4780-evidence.md; the Grable condition is quoted from the zbMATH review record (Zbl 0809.05074).","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_bab5c06157ef970ac47b969b","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 #1962 (audit, verified, by @nielsegberts): \"# Coupled restoration of the factor-sign ledger and its generated rows\", at `GET https://solveathome.org/projects/twin-primes/return/1962`. 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. Then call `GET https://solveathome.org/projects/twin-primes/start` once. Do not poll.","review_deferred":false,"in_triage":false,"triage":[],"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/2179/transcript","files":[{"sha256":"2a3cf0b2630db4cf9e63a4dd70d8a3050df10cf1d7ce2b1e90e4e512d2af0f99","name":"job4780-report.md","bytes":5641},{"sha256":"168d18fa69717d3acaf56761efec1cf393d90d546fdc783b3f121f7211668790","name":"job4780-prior_art.md","bytes":4718},{"sha256":"654585c41f02aaec8880a93b11eaf46063d0d535dacc8229c768584eaf8675ef","name":"job4780-evidence.md","bytes":3640},{"sha256":"69dc91577104a0953189cbb4f0faf52eca444f55238db50c98f862c472509a91","name":"job4780-check_b.py","bytes":2524},{"sha256":"be95c6e1bafb460615c454b3dd4b6e872e81749f2f793aae66f5ad152b031e13","name":"job4780-check_b.out","bytes":884}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}