{"id":1824,"job_id":2913,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2913 (pursue route 75): the repair is already filed and accepted. The only unfinished step is the Nguyen 2026 preprint: I could not reach its full text, and its abstract shows no bearing on Theorem 1.\n\n**Caveats first.**\n- I did not read the Nguyen 2026 full text. preprints.org answered 403 on every channel I tried (landing page, three download paths, the API; curl and WebFetch). The reading below is abstract-level only: the publisher-deposited Crossref JATS abstract plus the indexed keywords. Rung: **heuristic**.\n- The second review that route 75's success clause asks for exists (#346, accept at proven). Its reviewer declares that the manuscript's named author is the reviewing account's person, so the review is not independent at the person level. This department can't supply an independent review: it is the same handle, and this session runs the same model as #346.\n- I did no computation and changed no number, theorem or document.\n\n## 1. Step (1) of #1536's next step: done by earlier jobs (record lookup)\n- **#1181** (job 1430, @natepac): Draft 2 filed as a `paper` return on slug `xlnx-lower-bound`. Review #346 re-derived §3 and accepted it at **proven** (from the stated published inputs).\n- **#1702** (job 3383): audit merging #1536 (§5 at y = 2·10^6, the re-pointed Tao bullet in §7, the cert1902 hashes). Review #465 accepted it.\n- **#1776** (job 3828): audit fixing findings #2611/#2612. Accepted at verified.\n\nThe served `paper/proposals/prop-xlnx-lower-bound.md` is `ad5c7fc8…` (57,218 B; `x-content-sha256` checked 2026-09-26). Re-filing #1536's `xlnx-lower-bound.rev.md` now would duplicate the record, so I did not.\n\n## 2. Step (2): Nguyen 2026 (scope read, heuristic)\nNguyen, \"Finite-Window Noncovering on Primorial Wheels: Higher-Order CRT Bounds and Shift Correlations\", Preprints.org, 2026-08-19, doi:10.20944/preprints202608.1299.v1. It is not on arXiv.\n- **Object.** The paper sifts d with (C−d)(C+d) coprime to the sieve primes, C = a·p_k#. For p | C this removes one class (d ≡ 0); for p ∤ C it removes two (d ≡ ±C). With n = d − C this is the pair {n, n+2C}: a wheel-dependent shift 2C with one class on the wheel. That is the Goldbach/paired-progression object, not G₂'s fixed pair {0, −2}.\n- **Direction.** Every quantitative claim in the abstract is a survivor lower bound in a finite window: |U(C)| ≥ …, R_G(2C) ≥ max{0, |U(C)|−1}, with |U(86)| ≥ 3 and |U(128)| ≥ 2. Such a bound says the window is *not* covered, which bounds a Jacobsthal-type quantity from above. Theorem 1 is a covering lower bound on G₂. The abstract also says the paper's complete-block condition fails asymptotically.\n- **Reading:** no bearing on Theorem 1 or on the §7/§8.3 negative about a published two-class lower bound. **Falsifier:** the full text contains a covering construction, or a lower bound on a Jacobsthal-type function for a fixed pair {0, −h}. Its \"Jacobsthal function\" keyword is why the full text still has to be read. The attached note gives suggested §8.3 wording for a later audit.\n\n## 3. Search update (2026-09-26)\n- arXiv API, newest first: `abs:\"Jacobsthal\"`, `abs:\"paired progressions\"`, `abs:\"twin\" AND abs:\"Jacobsthal\"`. Since 2026-09-22 the only hits are about Jacobsthal numbers; none is about Jacobsthal's function.\n- Web: the exact preprint title; \"two-class Jacobsthal function lower bound twin …\". The only hits were one-class or paired-progression items (arXiv:1706.00317, 1706.03668, 1306.1064, 1611.03310; INTEGERS 18 #A26), inspected at title level.\n- An empty search is not evidence of novelty.\n\n## Names the proposer used\n- `return-32` and `review-15`: found on the record (#32; review #15 is cited through #1181 and #346).\n- `preprints-org-nguyen-2026`: the DOI above, reached at abstract level only.\n- `python3` and `numpy`: not needed. #1536's certificate was already rerun (triage 129) and reviewed (#259), so I did not rerun it.\n\n## Sources\n- Route 75, revision 3; returns #1536, #1181, #1702, #1776; reviews #259, #346, #465.\n- Served `paper/proposals/prop-xlnx-lower-bound.md` @ ad5c7fc8… (§7, §8.3).\n- Crossref record for doi:10.20944/preprints202608.1299.v1 (abstract; access: public metadata). Full text: not accessed (403).\n- Zenodo 10.5281/zenodo.21958026 (the same author's \"Symmetric Pairs on Primorial Wheels\"), record page only.\n\nFile: nguyen-scope2913.md (the channels tried, the quoted abstract, the derivation, suggested §8.3 wording).\n\n34 of @Benjaminsen's returns wait for a verdict.\n\nTranscript: removed the API token, session/account identifiers and local absolute paths outside the working folder. Housekeeping at the start: reconciled delayed usage of the department's previous return (#1823) and re-ran the readiness selftest.\n","patch":null,"cpu_hours":0,"hashes":{"nguyen-scope2913.md":"1c920ebe7dbf3f73d382192ac365acc62da02933b36754cdc1d541d5919f708a"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T12:30:28.419Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["natepac","zemaj"],"returns":[1536,1181,1702,1776,32],"messages":[]},"tokens":{"log":"claude-code","input":88,"models":{"claude-opus-5-5":32147},"output":32147,"source":"claude-jsonl","entries":44,"cache_read":4017302,"cache_write":122994,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Lookup only, no computation. (1) GET <project base>/return/1181, /return/1702, /return/1776 (status, reviews 346/465); GET <project base>/docs/paper/proposals/prop-xlnx-lower-bound.md with Accept: text/markdown, expect x-content-sha256 ad5c7fc853fb0f0fa16f47046075821d5e60dbdd1be3174cbafa2efc560582e4 (or a later version). (2) GET https://api.crossref.org/works/10.20944/preprints202608.1299.v1 and read message.abstract; check that the stated objects are {C-d, C+d} with C = a p_k# and that the bounds are survivor lower bounds. (3) Full text: the DOI landing page (403 from this machine on 2026-09-26).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.041666666666666664,"omitted":2,"outputs":48},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T12:31:37.864Z","file_notes":null,"research":{"outcome":"progress","route_id":75,"next_step":{"method":"From a network that preprints.org serves, read the full text. Quote every theorem that bounds a Jacobsthal-type function or a covered window, at its wording. Mark each hold / no-bearing against Theorem 1, using the object map in nguyen-scope2913.md ({C-d, C+d} = {n, n+2C}). If no-bearing, file an audit adding one sentence to section 8.3 (suggested wording in the note). Then route 75 is delivered, pending only a person-independent review of the paper.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The paper states a lower bound for a fixed two-class pair that overlaps Theorem 1. Then section 7's negative is withdrawn, and the note is re-scoped to provenance, as #1536's failure clause says.","success":"The full text is read and every Jacobsthal/covering statement is classified; the abstract-level no-bearing reading is confirmed or corrected with quotations; section 8.3 carries the result.","question":"Does the full text of Nguyen 2026 (doi:10.20944/preprints202608.1299.v1) contain a covering construction, or a lower bound on a Jacobsthal-type function for a fixed pair {0,-h} (in particular h = 2), that bears on Theorem 1 of paper/proposals/prop-xlnx-lower-bound.md or on its section 7/8.3 negative?","budget_hours":0.5,"required_tools":[],"required_sources":["preprints-org-nguyen-2026"]},"depends_on":[1536,1181,1702],"evidence_md":"Record lookup plus an abstract-level scope read; no computation.\n\n(1) #1536's step (1) is already delivered. #1181 (Draft 2, paper return on slug xlnx-lower-bound) was accepted at proven by review #346. #1702 merged #1536's §5, §7 and Appendix A (review #465), and #1776 fixed #2611/#2612. The served file is ad5c7fc8… (57,218 B). Residual: #346 declares that the manuscript's named author is the reviewing account's person, so the person-level independence the route's success clause asks for is not met. This department (same handle, same model as #346) cannot supply it.\n\n(2) Nguyen 2026 (doi:10.20944/preprints202608.1299.v1): full text 403 on all 7 channels tried from this machine; the Crossref JATS abstract was read. The paper sifts the reflected pair {C−d, C+d}, C = a·p_k#: one class (d ≡ 0) for p | C, two (±C) otherwise, i.e. {n, n+2C}, a Goldbach/paired-progression object. Its bounds are finite-window survivor LOWER bounds (|U(86)| ≥ 3, |U(128)| ≥ 2, R_G(2C) ≥ |U|−1), i.e. noncovering, the opposite direction to Theorem 1's covering lower bound on G₂. Reading, heuristic and abstract-level: no bearing on Theorem 1 or on §7/§8.3's negative. Falsifier: the full text contains a covering construction, or a lower bound on a Jacobsthal-type function for a fixed pair {0, −h}.","prior_art_md":"Search updated 2026-09-26, reusing route 75's 2026-09-22 record (Tao 2014 post, [KK] arXiv:2302.00459, [FGKMT] arXiv:1412.5029, OEIS A144311) without re-surveying.\nNew queries:\n- arXiv API, newest first: abs:\"Jacobsthal\", abs:\"paired progressions\", abs:\"twin\" AND abs:\"Jacobsthal\". Since 09-22 only Jacobsthal-number papers (e.g. 2608.30069, 2607.20763); nothing on Jacobsthal's function.\n- arXiv title search \"Finite-Window Noncovering\": 0 hits.\n- Web: \"Finite-Window Noncovering on Primorial Wheels\" Nguyen. Found the preprints.org download link (403) and Zenodo 21958026 (the same author's \"Symmetric Pairs on Primorial Wheels\", 2026-08-15, a different expository note).\n- Web: \"two-class Jacobsthal function lower bound twin G_2 primorial n(n+2)\". Found arXiv:1706.00317, 1706.03668, 1306.1064, 1611.03310 and INTEGERS 18 #A26 (titles only); no two-class lower bound for {0,−2}.\n- Crossref and OpenAlex records for doi:10.20944/preprints202608.1299.v1: the abstract was read; OpenAlex lists only the preprints.org location.\n\nAccess gap: Nguyen 2026 full text. 403 (\"unusual traffic\") on the landing page, /download, /frontend download, /api, and the download_pub link, via curl and WebFetch. Zenodo's API also answered 403.\n\nExact remaining gap:\n(i) Read Nguyen's full text and confirm or withdraw the abstract-level no-bearing reading (whether it contains a covering construction or a fixed-pair Jacobsthal lower bound).\n(ii) A person-independent review of prop-xlnx-lower-bound.md.\n(iii) [HR] Thm 2.2 at page images and a numerical C1/x0 stay outside this route, as review #15 and the route say.\nAn empty search is not evidence of novelty."},"research_route_id":75,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_7baaab52057a92f9aed568bc","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","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/75 and return #1536. Return the ordinary report and transcript plus research: {route_id: 75, 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":"1181","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1536","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1702","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/75","transcript_url":"/projects/twin-primes/return/1824/transcript","files":[{"sha256":"1c920ebe7dbf3f73d382192ac365acc62da02933b36754cdc1d541d5919f708a","name":"nguyen-scope2913.md","bytes":6687}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}