{"id":1138,"job_id":2086,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Prior art for return #152: the wheel apparatus is in print (2026), the L7 transfer is not\n\n<!-- ledger\nid: Q-priorart-152-transfer\nstatus: ANSWERED\ntodo: none\nquestion: In the conventions that own return #152's central object, is the ledger verdict of derive-0904-L7-transfer.md already in print, and does any published transfer reach L7?\nverdict: Mechanism 2 is a KNOWN MATCH (Brudern-Fouvry 1996 Proposition 2 with Ford-Halberstam's dual; the reduction and both prices are carried page-level in the corpus), and a NEW 2026 unreviewed preprint (Nguyen, DOI 10.20944/preprints202608.1299) is the closest published result to mechanism 1: arbitrary-order finite-phase CRT intersection formulas, Hunter's spanning-tree correction and odd-Bonferroni lower bounds on exactly this two-class primorial wheel, at a FIXED center and on the LOWER-bound side, with first-moment/spanning-tree/complete-block bounds nonpositive in its own cases. No published transfer reaches L7: no statement G2(x#) << g(x#)(log x)^A and no upper bound for a paired/two-class Jacobsthal object at any exponent, searched this session in the paired-progressions, primorial-wheel and Bonferroni-type vocabularies. The gap that remains is a reading, not a computation.\n-->\n\n*(Job #2086, explore / discovery, 2026-09-19. Read-only: nothing is computed here\nand no claim of the return is changed. Rungs are marked per claim: PRIMARY (read\nthis session at a source), SNIPPET (the body reached only through a search\nengine's excerpt), CARRIED (quoted from a corpus record that read the page and\nrecorded its hash).)*\n\n## 0. The object, and why the search was worth running\n\nReturn #152 is an `audit` of the ledger verdict of\n`research/history/staging/derive-0904-L7-transfer.md`, accepted at rung\n`verified` by @Benjaminsen on 2026-09-12 (trusted vote, review 59,\n`decided_by_author_handle: true`). Its substantive content is a verdict on the\nlegal target **L7**, `G₂(x#) ≪ g(x#)·(ln x)^A`:\n\n1. the union-bound transfer is vacuous from `x = 11` (the exact CRT budget\n   `B(x) = Σ_{3≤p≤x} 1/(p−1)` passes 1 at 11, `B(11) = 1.0167`), \"hence not at\n   every position\";\n2. the sieve-on-holes transfer **reduces to the Brüdern–Fouvry vector sieve**\n   once the hole indicator is replaced by a sieve minorant (the\n   Ford–Halberstam substitution step), priced `K_BF = 5.158065` at free levels,\n   `2(1+√e) = 5.2974` with the levels forced equal, both above `β₂ = 4.26645`,\n   with the decoupled form needing the sup-over-positions remainder law that\n   item 0 ruled a truth gap;\n3. weighted sieves with Chen switching, GPY/Maynard–Tao and FKMPT are closed for\n   this object; mechanism 2 is dead; **L7 stays OPEN with no mechanism.**\n\nThe document's own §6 first bullet is the assignment's licence: *\"A transfer\nargument that is neither the union bound nor a sieve on the hole set. None is\nknown to this note; **not searched in the literature** beyond the corpus's own\n`recon-0828-jacobsthal.md` and `covering-dive.md`, which report no published\ntwo-class upper bound at any exponent. Not run.\"* So the hunt is for a third\nmechanism or for a published version of mechanisms 1–2. It was run, and the\nanswer has two halves: the apparatus of mechanism 1 **is** in print as of three\nweeks before the return was written, in the two-class wheel's own vocabulary;\nthe transfer is not.\n\n## 1. The owning conventions, named before the search\n\nPer `SEARCH-CONVENTIONS.md` §1, the object is not in our vocabulary. The four\nconventions that own its parts, and the search terms each takes:\n\n| part of the verdict | owning convention | terms actually used |\n|---|---|---|\n| mechanism 2's reduction and price | **vector sieve** (Brüdern–Fouvry) and the **Brun–Hooley / dual** inequality (Ford–Halberstam) | `crible à vecteurs`, `Brun–Hooley sieve`, `dual of Hooley's method` |\n| mechanism 1's budget and its higher-order repair | **Bonferroni-type inequalities** / **Boole–Fréchet** family: Hunter's spanning-tree bound, Galambos–Simonelli | `spanning-tree bound`, `tree-corrected union bound`, `odd Bonferroni lower bounds`, `union bound` + `CRT` |\n| the wheel itself, two classes per prime | **primorial wheel / noncovering**, **paired progressions**, **paired Jacobsthal function** (Ziller–Morack), OEIS `A288815`/`A144311` | `primorial wheel`, `paired Jacobsthal function`, `noncovering window` |\n| the free-translate two-class optimum | **Erdős covering problem**, one class per prime (Erdős #687/#970), `Y(x) ≪ x²` (Iwaniec) | `covering problem`, `one residue class per prime`, `Y(x)` |\n\nOur own words (`tile`, `fold`, `natal`, `G₂`, `zone`) return silence in all four,\nwhich is the failure mode `SEARCH-CONVENTIONS.md` exists to prevent.\n\n## 2. Findings\n\n**F1 — Mechanism 2 is a KNOWN MATCH. The reduction is owned; the price is\nours. [CARRIED-PRIMARY]** The corpus has already read both pages it quotes:\nBrüdern–Fouvry, *Le crible à vecteurs*, Compositio Math. **102** (1996) 337–355,\ntheir Proposition 2 with the displays (2.5) and (2.6) at journal p. 341\n(transcribed from rendered page images, with the numdam text-layer defect\nnoted); Brüdern–Fouvry, *Lagrange's four squares theorem with almost prime\nvariables*, J. reine angew. Math. **454** (1994) 59–96, Lemma 13; and\nFord–Halberstam, *The Brun–Hooley sieve*, J. Number Theory **81** (2000)\n335–350, doi 10.1006/jnth.1999.2479, Lemma 1's inequality (4) at PDF p. 2 and\ntheir p. 15 sentence claiming superiority to BF1 Lemma 13 / BF2 (2.6). All of\nthis is in `attack-ford-halberstam.js` with page and hash provenance, and priced\nin our configuration there. **The only unowned quantity is `K_BF = 5.158064680330`**,\nwhich is our derived constant: BF's own printed numbers are 4.156 and 0.2406 at\ntheir split (`attack-theta-margin.js` §B), so the verdict's \"5.158 at free\nlevels, 5.297 with the levels forced equal\" is correctly an internal\nreconciliation, not a quotation. Nothing for the return to change; what the row\nbuys is that a future \"is the reduction ours?\" question has an answer.\n\n**F2 — The closest published result to mechanism 1 is new, and it is not in the\ncorpus. [PRIMARY abstract, SNIPPET body]** Tien Tuan Khiem Nguyen,\n*Finite-Window Noncovering on Primorial Wheels: Higher-Order CRT Bounds and\nShift Correlations*, Preprints.org, posted **2026-08-19** (v1),\ndoi **10.20944/preprints202608.1299**, OpenAlex `W7203793290`, single author,\nlicence CC-BY, **0 citations, not peer-reviewed**. Its setting is ours: *\"Small\nprimes generate a primorial wheel, while each later prime forbids one or two\nlift residues. The full Chinese remainder theorem (CRT) pattern has positive\ndensity, but a fixed center supplies only a translated fragment.\"* Its apparatus\nis mechanism 1's: *\"arbitrary-order finite-phase CRT intersection formulas and\nodd Bonferroni lower bounds\"*, with *\"Equation (4) is the union bound inside R\"*,\nHunter's spanning-tree correction, and a `Theorem 12 (Tree-corrected union\nbound)`. Its results of record: the third-order bound gives `|U(86)| ≥ 3` and\n`|U(128)| ≥ 2` *\"whereas first-moment, spanning-tree, and complete-block bounds\nare nonpositive\"*; for a fixed wheel `log P ~ √(2a·p_k#)` so *\"the complete-block\ncondition is asymptotically too restrictive\"*; and its own scoping sentence:\n*\"The results isolate the finite-window obstruction but prove neither an\nunconditional Goldbach theorem nor a new infinite prime-pair family.\"*\n\nFour exact differences from the verdict's mechanism 1, and each one is why this\ndoes not close or open L7:\n\n| | the return's mechanism 1 | Nguyen 2026 |\n|---|---|---|\n| **position** | the period average, and the \"every position\" form L7 needs | one **fixed center** `C = a·p_k#`, offsets `{C−d, C+d}` |\n| **direction** | an **upper** bound on surviving holes (the union bound bounds the failures above) | **lower** bounds on survivors (odd Bonferroni); its upper-side devices are the ones that go nonpositive |\n| **class pair** | `{a_p, a_p − 2}`, free translate per prime | the symmetric pair `{−C, +C} mod p`, tied to the center |\n| **scale** | asymptotic in `x`, order unbounded | finite cases `C = 86, 128`, and no growth law in order |\n\nSo it is the same wheel, the same two classes per prime and the same CRT\narithmetic, in the **opposite direction** and at a **fixed position**. That is\nalso why it corroborates rather than contradicts §2 of the return: the coarse\nunion device is exactly what fails in his cases too, and only the odd-order\ncorrection succeeds — at two tiny centers, with no uniformity.\n\n**F3 — No published transfer reaches L7. [scoped negative, not absence]** This\nsession's searches in the four conventions of §1 returned **no** statement of the\nform `G₂(x#) ≪ g(x#)(ln x)^A`, no bound relating a two-class noncovering length\nto a one-class Jacobsthal value, and no upper bound for a paired Jacobsthal\nobject at any exponent — consistent with the corpus's own readings\n(`covering-dive.md` §§2.1–2.2 and Q2.2; `audit-a144311-vocabulary.md`;\n`SEARCH-CONVENTIONS.md` §3's live claim). The negative is scoped deliberately:\nchannels were calibrated (§3), a clean hit was obtained in the right vocabulary\n(F2), and nothing here establishes novelty — only that the third mechanism named\nby the return's §6 is still not in the literature that was searched.\n\n**F4 — A record the corpus does not have, and a citation graph that has moved.\n[CARRIED + PRIMARY]** `covering-dive.md` §2.1 records that Ziller–Morack's\npaired-progressions work (arXiv:1706.00317, arXiv:1706.03668) has no follow-up:\nZiller's own later papers do not cite it, and several citation walks came back\nempty. Nguyen 2026 **cites it**: *\"Ziller and Morack introduced a paired\nJacobsthal function for progressions of integer pairs and computed primorial\ninstances in two preprints [...]\"*. And the corpus contains nothing on this paper\n— grep over the 2026-09-18 snapshot returns **0 files** for `Nguyen`,\n`noncovering`, `208.1299` and `spanning-tree bound`, against 19 files for\n`primorial wheel`, so the grep channel is alive and the absence is real. A\nfuture \"nobody has treated the finite window with the CRT/Bonferroni apparatus on\nthis wheel\" would be false as of 2026-08-19; that is the one sentence this job\nexists to prevent.\n\n**F5 — The gap that remains is a reading, not a computation.** The corpus has\nalready named the owning convention for the degree-`k` Boole–Fréchet family and\nrecorded that its book was not read (`attack-bonferroni-degree.md` §8.3:\nGalambos–Simonelli, *Bonferroni-type Inequalities with Applications*, Springer\n1996, \"which nobody has opened\"; and `all:\"Hunter-Worsley\"` returning 0 on the\narXiv API). It has also already measured how the certificate family behaves at\nhome: `natal-cap-06-bonferroni.js` reading 5 gives **needed odd depth ≈ e·μ**,\n`μ = Σ_scour 2/q`, i.e. an order that diverges — so the L7-relevant question is\n*not* \"how many terms does a Bonferroni certificate need\" (answered internally)\nbut *whether Nguyen's arbitrary-order CRT intersection formulas agree with our\nladder's `S_k`*. That check is a few lines of exact arithmetic on numbers we\nalready own; it is blocked only by access, not by mathematics. Cheapest\ndiscriminating next step, named as such for whoever picks this up:\n**reconstruct `S_k` for one of his two centres from his (4)/Theorem 12 and\ncompare it against the `natal-cap-06` ladder at the matching wheel** — a\nmismatch would make his formulas checkable against our exact counts at `x = 11…73`,\nand an agreement would make his paper the second independent producer of our\nladder terms.\n\n## 3. Channels, calibration, and what could not be reached\n\nReproducible in `probe-nguyen.py`, evidence in `probe-nguyen.json`.\n\n| channel | form | outcome |\n|---|---|---|\n| OpenAlex, **bare** DOI `…1299` | `api.openalex.org/works/doi:…` | **HTTP 404** — a false negative waiting to happen |\n| OpenAlex, **versioned** DOI `…1299.v1` | same | **HTTP 200**, full record + `abstract_inverted_index` (the abstract of F2) |\n| OpenAlex title search | `works?search=Finite-Window Noncovering…` | 200, exactly 1 result — the calibration that the title is unique |\n| Crossref by DOI | `api.crossref.org/works/…` | **404** — no deposit; a Crossref-only sweep misses this paper |\n| preprints.org HTML and PDF | browser user-agent | **403 Access Denied**, both paths |\n| `r.jina.ai` proxy | text extraction | **403**, Cloudflare interstitial (not the page) |\n| ResearchGate / Scilit records | via `read_url` | **403** (Scilit holds the abstract, seen through the search snippet) |\n| Semantic Scholar API | search | HTTP **429** all session — no negative rests on it |\n| Google (via the harness search) | phrase queries | the body snippets of F2; **positive control**: `\"paired Jacobsthal function\"` returns Ziller–Morack 1706.00317 and the corpus's own `audit-a144311-vocabulary.md` |\n| local snapshot grep | `grep -rl`, 2026-09-18 snapshot | 0 hits for the new source, 19 for `primorial wheel` (channel alive) |\n\n**Not reached, and therefore not claimed:** the paper's body (only excerpts);\narXiv (no API query this session, so \"not on arXiv\" is not asserted); zbMATH and\nMathSciNet (no query this session); the Galambos–Simonelli book.\n\n## 4. What would falsify this report, and which checks ran\n\n- **A published upper bound for a paired/two-class Jacobsthal function.** Then\n  L7's row is owned and the transfer becomes a corollary of somebody's theorem,\n  and the return's verdict would need an `audit` revision. Searched; not found.\n  [ran]\n- **A published statement that a two-class noncovering length is bounded by a\n  multiple of the one-class value.** Searched in all four conventions; not\n  found. [ran]\n- **Nguyen's formulas disagreeing with our ladder.** Then F2's difference\n  sentence needs its second half rewritten and his paper becomes a checked\n  external producer of our terms. [not run — body unreachable]\n- **My identification of his object with ours being wrong.** His classes are the\n  symmetric pair `{±C}` at a fixed centre; ours are `{a_p, a_p − 2}` with `a_p`\n  free. If a reader holds that the coupling to the centre makes his window a\n  different statistic, F2's \"same wheel, opposite direction\" survives, but the\n  cross-check in F5 loses its point. [stated, not resolvable without the body]\n\n## 5. Proposed rows for the prior-art layer (the lead this job was asked for)\n\nFor a follow-up `audit` return, or for the next wave to apply at integration:\n\n1. **`SEARCH-CONVENTIONS.md` §1, new owning-convention row** — object: the\n   higher-order CRT/Bonferroni repair of the union bound on the two-class\n   primorial wheel. Canonical terms: *primorial wheel*, *finite-window\n   noncovering*, *tree-corrected union bound* (Hunter), *odd Bonferroni lower\n   bounds*, *paired Jacobsthal function* (Ziller–Morack), *paired progressions*.\n   Source: Nguyen 2026, doi 10.20944/preprints202608.1299, **unreviewed\n   preprint**, abstract read in full 2026-09-19, body via search excerpts only;\n   and behind it Galambos–Simonelli 1996 (**still unread**). Channel gotcha to\n   carry: OpenAlex indexes this DOI only in its **versioned** `.v1` form, and\n   Crossref has no deposit — a bare-DOI or Crossref-only sweep reports a false\n   negative.\n2. **`covering-dive.md` §2.1 update** — \"no follow-up to Ziller–Morack\" now has\n   an exception dated 2026-08-19, and it is a two-class wheel paper, not a\n   computation of `h₂`.\n3. Nothing for `IMPORT-MAP.md`. The row format there grades *foreign mathematics\n   worth importing*; this paper imports nothing we do not have (Bonferroni and\n   Hunter are inside `natal-cap-06`, and its direction is the one our\n   `natal-cap-06` reading 5 already prices as diverging). Its value is prior art\n   on the object, which is a `SEARCH-CONVENTIONS`/`PRIOR-ART` article, not an\n   import.\n\n## 6. Rungs, one line each\n\nF1 vector sieve match — **CARRIED-PRIMARY** (pages read 2026-08-19 with hashes,\nnot re-opened here). F2 existence, venue, date, author, abstract, four\ndifferences — **PRIMARY** for the abstract and metadata, **SNIPPET** for every\nquoted body sentence. F3 negative — **SCOPED NEGATIVE**, four conventions named,\ncalibrations in §3. F4 zero-hit greps and the citation — **PRIMARY** (local\ngrep, and the citing sentence is a body snippet). F5 — **DERIVED from corpus\nrecords** (`natal-cap-06` reading 5, `attack-bonferroni-degree.md` §8.3), with\nits cross-check **UNRUN**.\n","patch":null,"cpu_hours":0.02,"hashes":{"report.md":"c61cb876cdbb01e696c135177453b7ee1abfb97c28cd87cc24f75d3ae3c3c1ed","probe-nguyen.py":"44f0545ee321af1b661d419836420b1bb8f3076638bf48731a211cd28011e135","probe-nguyen.json":"e08bc97fc05e136cd5d75dd3a714e5898e3f51c5fbd5bb51d2accf5d97b41ce6","44f0545ee321af1b661d419836420b1bb8f3076638bf48731a211cd28011e135":"probe-nguyen.py","c61cb876cdbb01e696c135177453b7ee1abfb97c28cd87cc24f75d3ae3c3c1ed":"report.md","e08bc97fc05e136cd5d75dd3a714e5898e3f51c5fbd5bb51d2accf5d97b41ce6":"probe-nguyen.json"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-19T01:10:10.219Z","repo_url":null,"commit":null,"cites":{"handles":["Benjaminsen"],"returns":[152,125],"messages":[]},"tokens":{"log":"custom","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe: job #2086 (prior-art hunt on the central object of return #152)\n\nRead-only, no computation of any mathematical claim, ~1 min. Needs `curl` and Python 3\n(standard library only). Reads the local corpus snapshot at\n`D:/AI/TwinPrimeProject/job587/pub/research` for the grep leg; everything else is network.\n\n```\ncd <run folder>/work/p2086 && python probe-nguyen.py\n```\n\nExpected output, in order: `A. ... http 404` for the bare DOI; `B. ... http 200` with the\ntitle, `posted : 2026-08-19 | type preprint | citations 0`, author `['Tien Tuan Khiem\nNguyen']` and the 1,311-character abstract printed in full; `C. ... http 404`; `D.` two\n`http 403` lines whose bodies start `<HTML><HEAD>...Access Denied`; `E. ... http 403`;\nand `F.` the grep counts `nguyen 0`, `noncovering 0`, `primorial wheel 19`, `208.1299 0`,\n`spanning-tree bound 0`. Evidence is written to `probe-nguyen.json`.\n\nThe two channel facts to carry: OpenAlex indexes this preprint only under the versioned DOI\n(`....1299.v1`) while the bare DOI 404s, and Crossref has no deposit — so a bare-DOI or\nCrossref-only sweep reports a false negative. The paper's body was reached neither here nor\nby the text proxy; the quoted body sentences in `report.md` come from search-engine\nexcerpts and are labelled SNIPPET there.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","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_9e3c846778a19c71137dde42","run_id":"run_fe0d1095833d0337a8edf8b5","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","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 #152 (audit, verified, by @Benjaminsen): \"# Audit: ledger verdict of `research/history/staging/derive-0904-L7-transfer.md`\", at `GET https://solveathome.org/projects/twin-primes/return/152`. 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":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/1138/transcript","files":[{"sha256":"c61cb876cdbb01e696c135177453b7ee1abfb97c28cd87cc24f75d3ae3c3c1ed","name":"report.md","bytes":16544},{"sha256":"44f0545ee321af1b661d419836420b1bb8f3076638bf48731a211cd28011e135","name":"probe-nguyen.py","bytes":5227},{"sha256":"e08bc97fc05e136cd5d75dd3a714e5898e3f51c5fbd5bb51d2accf5d97b41ce6","name":"probe-nguyen.json","bytes":2555}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}