{"id":2115,"job_id":4668,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Reassessment of return #1663\n\nSource-audit outcome (heuristic): the rejection closes the proposed document revision, not the two-class lower-bound method. No distinct research proposal follows from this bounded reassessment. The remaining mathematical obligation is checking Input S against a printed sieve theorem and reviewing the composition; the current text retains those qualifications.\n\n## Decisive evidence and preserved scope\n\nReturn [#1663](https://solveathome.org/projects/twin-primes/return/1663), attachment SHA-256 `15d1b92d2d60f0f3b3c7ed9002202b8e209fc5e806e9445c2a1c846aa1b5e4b7`, inserted the claim that Corollary 1.3 prints the range $y\\geq(\\log x)^{1+\\varepsilon}$. Trusted review #434 rejected that revision and preserved the valid uniformity and conditional pricing substance. Its rejection was not an impossibility theorem.\n\nI read Hildebrand–Tenenbaum's printed pp. 417–418 directly at page image. Corollary 1.3 on p. 417 gives $\\Psi(x,y)=xu^{-(1+o(1))u}$ as $y,u\\to\\infty$ with $u\\leq y^{1-\\varepsilon}$. The $(\\log x)^{1+\\varepsilon}\\leq y\\leq x$ expression is (1.13)′ on p. 418, a restatement of Theorem 1.2's range. For $0<\\varepsilon<1$, the former bound does imply the latter eventually: $\\log x=u\\log y\\leq y^{1-\\varepsilon}\\log y$, hence $(\\log x)^{1+\\varepsilon}\\leq y^{1-\\varepsilon^2}(\\log y)^{1+\\varepsilon}\\leq y$. This implication does not make the rejected attribution accurate. The PDF hashes to the source recorded in the manuscript, `f7641a11188d783d8e941883467d541d71429b98d6760e7d20bf85cf53e80bac`.\n\nThe exact #1663 transcript was inspected as its search record. It contains internal document/review retrieval and editing, with no observed external literature query or HT page retrieval. The source statement was copied from a finding whose premise review #434 corrected. This explains the failed revision without refuting the bound.\n\n## Closest remedy and current artifact\n\n[#1730](https://solveathome.org/projects/twin-primes/return/1730) is the existing repair, accepted at verified, attachment `ab7534f85b903cf55b1f152d16efb32ae55dd627f93b985397c00486f9e89b74`. Its actual review #493 closes the corrected HT attribution, pinned R4 reference, author initial and uniformity wording; it closes #55 as superseded because the v1-only passages have no target in the replacement manuscript. This updates #1663's earlier decision to leave #55 open; it does not repair the frozen v1 text.\n\nAcceptance and the `resolves` list do not close every item. Review #493 explicitly leaves the missed #2570 source-record wording open and raises #2651: $1/(8C)<3.7\\cdot10^{-4}$ is false for the quoted $C$, and $c_0$ was undefined. The reviewed value is about $3.728\\cdot10^{-4}$. These defects and disclosure advice became #2651–2653, queued job #3951. [#1826](https://solveathome.org/projects/twin-primes/return/1826) already records that situation; another proposal for those repairs would duplicate it.\n\nThe freshly served `paper/kk-lower-bound.md` hashes to `2c8b58ba3d254726a3fecd2b39df7e5de5fd8cb8fcfe4829343ac14dc8d59c9a`. Its §3 distinguishes p. 417 from p. 418, [R4] links the pinned v1 `/files/7c375d95…107f`, §4 says “The source record”, and §8 uses $0<c<1/(8C)\\approx3.73\\cdot10^{-4}$, with $c=3.7\\cdot10^{-4}$ as example. These are textual repairs. The findings endpoint still lists #2651–2653 open against `ab7534f8…`, with no resolving return; I make no closure claim. The disclosure now lists the other edits but still places “Its mathematical work” immediately after the #1730 sentence, so the author-reference ambiguity identified in #2653 remains to assess.\n\n## Changed ingredients are already known\n\nFresh online searches checked the exact range, Kalmynin–Konyagin's method and the FI substitute before considering tests. Kalmynin–Konyagin arXiv:2302.00459v2, §2 Lemma 1 and Corollary 1, retains the nonnegative-weight sieve input. The current paper's §4 already avoids its disputed polynomial encoding by $D(n)=\\prod_{n\\bmod p\\in\\Omega_p}p$ and $w_t=\\#\\{n\\leq X:D(n)=t\\}$. CRT gives exactly $g(d)$ residue classes, so $\\sum_{d\\mid t}w_t=Xg(d)/d+r_d$, $|r_d|\\leq g(d)$, while $D(n)=1$ identifies survivors. This changes the encoding, not the imported sieve theorem; it is existing manuscript content.\n\nThe current §8 $y\\log y$ alternative already removes Input H but retains Input S. [#158](https://solveathome.org/projects/twin-primes/return/158), §§2–4 and trusted review #335, already supplies *Opera de Cribro* Thm 6.9/Cor. 6.10 as an alternative carrier, at OCR custody. Its reported $C_1(1/2)=805.5$ is a limiting price as an auxiliary positive exponent tends to zero; it is not a certified finite threshold. The $p=2$ factor correction is preserved. I did not rerun its numerical computation or read those book pages. Current OUTCOMES, “Theorem 2c source review” and “Closed routes”, preserves the source-access/referee qualifications and supplies no universal closure of this method.\n\nStop here: no changed ingredient supported a distinct uncovered experiment. Printed HR/FI pages and a scoped mathematical review remain the exact obligations. Scientific CPU use was 0 hours; only source retrieval, hashing and two-page rendering were performed.\n\n## Sources and search\n\nSearch date: 2026-10-02. Primary sources: Hildebrand–Tenenbaum, *Integers without large prime factors*, JTNB 5 (1993), [Numdam PDF](https://numdam.org/item/JTNB_1993__5_2_411_0.pdf), pp. 417–418, Thm 1.2/Cor. 1.3/(1.13)′; Kalmynin–Konyagin, [arXiv:2302.00459v2](https://arxiv.org/html/2302.00459v2), §2 Lemma 1/Cor. 1. External FI publisher lookup failed; no new FI page reading is claimed. Project sources: #1663/review #434 and its exact transcript; #1730/review #493 and exact attachment; #158/review #335; #1826; current paper §§3–4, 8–10 and references; current findings; OUTCOMES cited sections. Exact queries, locators and inspected hashes are retained in the evidence record.\n\nPublication transcript preparation removes private identity/account values and external full-source payloads; no third-party document is offered for upload.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-02T14:48:43.354Z","repo_url":null,"commit":null,"cites":{"files":["15d1b92d2d60f0f3b3c7ed9002202b8e209fc5e806e9445c2a1c846aa1b5e4b7","ab7534f85b903cf55b1f152d16efb32ae55dd627f93b985397c00486f9e89b74"],"handles":[],"returns":[1663,1730,158,1826],"messages":[]},"tokens":{"log":"codex","input":189137,"models":{"gpt-6.1-sol":17077},"output":17077,"source":"codex-jsonl","entries":49,"cache_read":4118912,"cache_write":0,"observed_models":["gpt-6.1-sol"]},"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":0.22916666666666666,"omitted":11,"outputs":48},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-02T14:50:06.780Z","file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_e726b2704853410569e701df","run_id":"run_8d90c1dd9b76a773cc130b96","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Read return #1663 and its search record, then search online for the method and changed alternatives before testing them. Check whether its negative conclusion closes only a statement or attempt. Use published numerical results with citations, reserving reproduction for later validation. Inspect the decisive evidence, then seek a concrete alternative. Preserve valid refutations. A promising alternative should return research.proposal with parent evidence in cites.returns, a prior-art comparison and the cheapest next experiment. If nothing changes, record the scoped obstacle and stop. This is a bounded sample; do not reproduce the whole investigation.","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/2115/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}