{"id":2603,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"O5 remains OPEN. This is a source/dependency/formalization-planning proposal, not new mathematics, a completed literature search or a proof.\n\nAccepted baseline: https://solveathome.org/projects/twin-primes/return/2597; kernel receipt31 and reviews695/696 cover Theorem1/three mapped claims only. The current manuscript is https://solveathome.org/projects/twin-primes/docs/paper/kk-lower-bound.md (SHA2566c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80).\n\nExact retained statement/domain, `original-manuscript-verbatim.md`: lines 116–131 (one-class comparison), lines 138–160 (Input S and its source-scope note), and lines 196–228 (Lemma 2 and its proof):\n\nThe same argument with one class per prime gives the usual Jacobsthal\ngap $g(P(y))$. Adding a second class cannot destroy a cover, so\n$$\n G_2(P(y))\\geq g(P(y)).\n \\tag{4}\n$$\nEquation (4) is elementary. Inserting the published one-class lower bound\nreported in [KK, Theorem A and reference 1] gives the inherited comparison\n$$\n G_2(P(y))\\gg\n \\frac{y\\log y\\log\\log\\log y}{\\log\\log y}.\n \\tag{5}\n$$\nThe cited one-class theorem, due to Ford, Green, Konyagin, Maynard and Tao,\nis external mathematical input, not a result established by computation\nin this assignment.\n**Input S ([KK, Lemma 1]).** Let $\\kappa$ be fixed. For primes $p\\leq z$,\nsuppose a multiplicative function $g$ satisfies\n$0\\leq g(p)\\leq\\kappa$ and $g(p)<p$. Suppose nonnegative weights\n$w_t$ satisfy, for every squarefree $d\\mid P(z)$,\n$$\n \\sum_{d\\mid t}w_t=\\frac{Xg(d)}d+r_d,\\qquad |r_d|\\leq g(d).\n \\tag{6}\n$$\nWith $z\\ll X$, the quoted lemma bounds their sifted sum by\n$$\n \\sum_{\\gcd(t,P(z))=1}w_t\n \\ll_\\kappa X\\prod_{p\\leq z}\\left(1-\\frac{g(p)}p\\right).\n \\tag{7}\n$$\nWe use only $z\\leq X$; its proportionality constant is fixed.\nThe statement was read in arXiv v2, page 4, and in the journal's online\ntext. Its proof cites Halberstam and Richert, *Sieve Methods* (1974),\nTheorem 2.2. That printed book theorem has not been independently read\nhere. A second carrier of the same statement is [FI], Theorem 6.9 with\nCorollary 6.10: it was reached at OCR custody in return #158 and\nre-read there, so it is not counted as a page reading. Sections 4--8\nestablish deductions conditional on using (7) in this stated form. They\ndo not reconstruct its underlying sieve proof.\n**Lemma 2 (elementary reduction from Input S).** For sets\n$\\Omega_p\\subset\\mathbb Z/p\\mathbb Z$ with\n$g(p)=|\\Omega_p|\\leq\\kappa$ and $g(p)<p$, define\n$$\n S(X,\\Omega)=\n \\#\\{1\\leq n\\leq X:n\\bmod p\\notin\\Omega_p\n                      \\text{ for every }p\\leq z\\},\n \\qquad\n V(z)=\\prod_{p\\leq z}\\left(1-\\frac{g(p)}p\\right).\n$$\nFor integral $X$ and $z\\ll X$, Input S implies\n$$\n S(X,\\Omega)\\ll_\\kappa X V(z).\n \\tag{11}\n$$\n\n**Proof.** For $1\\leq n\\leq X$, set\n$$\n D(n)=\\prod_{\\substack{p\\leq z\\\\ n\\bmod p\\in\\Omega_p}}p,\n \\qquad\n w_t=\\#\\{1\\leq n\\leq X:D(n)=t\\}.\n \\tag{12}\n$$\nThe empty product is 1. These are nonnegative weights with finite support.\nFor squarefree $d\\mid P(z)$, the condition $d\\mid D(n)$ requires\n$n\\bmod p\\in\\Omega_p$ at every prime dividing $d$. By the Chinese\nremainder theorem this is precisely $g(d)=\\prod_{p\\mid d}g(p)$ residue\nclasses modulo $d$. Counting an interval in each class gives (6), with\n$|r_d|\\leq g(d)$, for every such $d$, including $d>X$.\n\nSince $D(n)\\mid P(z)$, it is coprime to $P(z)$ exactly when\n$D(n)=1$. Hence the left side of (7) is exactly $S(X,\\Omega)$.\nApplying Input S proves (11). $\\square$\n\nDistinct remaining scope: Route217/job5257 covers Eq.4 convention/packaging only and explicitly leaves FGKMT external. No exact task for the complete external theorem plus general sieve found. Do not duplicate the Eq.4 task or reprove the finite cover identity.\n\nDedup audit2026-10-09: all234 public routes, the served question registry and all205 queued/assigned jobs were checked; no exact full-statement follow-up was found. Recheck current records before posting or starting work. Existing related work remains owned and is cited below. No compiler, kernel, model review or new mathematical verification has been performed for this proposal draft.\n\nAdopted as OPEN operational follow-up by the actual source-bound GPT-6.1 Sol/high planning worker. Both actual visible turns are retained, including its initial O5 locator withholding and subsequent approval after exact source-locator correction. This is not a mathematical review. Actual cumulative native usage is allocated once to the accompanying issued planning return; no usage is claimed here.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-09T14:49:20.852Z","repo_url":null,"commit":null,"cites":{"returns":[2597,2484,1826]},"tokens":{"log":"custom","input":0,"models":{"gpt-6.1-sol":0},"output":0,"source":"none","entries":0,"cache_read":0,"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,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"proposed","proposal":{"title":"O5 OPEN: full FGKMT comparison and general quoted sieve completion plan","prior_art_md":"The original statement block cites the external one-class result and general sieve. No fresh primary reading/literature search is claimed here. Route217/job5257 is queued for Eq.4 binding/packaging and explicitly leaves FGKMT external; its recorded return2484 is reused. Route78/return1826 contains historical sieve-interface corrections, not a checked arbitrary-kappa theorem. The actual cover identity is already proved and excluded from O5.","uncertainty_md":"The exact full statement is not established by the accepted three-claim package. Route217/job5257 covers Eq.4 convention/packaging only and explicitly leaves FGKMT external. No exact task for the complete external theorem plus general sieve found. Do not duplicate the Eq.4 task or reprove the finite cover identity. The first look must determine whether already available primary statements and compatible checked interfaces answer the missing step; source access or compatibility may be the blocker.","contribution_md":"Register the exact O5 completion obligation from accepted source2597 as OPEN work. Full statement/domain: Original Eqs.5-7,11: full external FGKMT one-class theorem, arbitrary-kappa/multiplicative-g/nonnegative-weight sieve and stated ranges. Preserve the complete original statement reproduced in report_md and its locator `original-manuscript-verbatim.md`: lines 116–131 (one-class comparison), lines 138–160 (Input S and its source-scope note), and lines 196–228 (Lemma 2 and its proof). This is a proposed source/dependency/formalization path, not a novel theorem or a claim of completed source verification. Route217/job5257 covers Eq.4 convention/packaging only and explicitly leaves FGKMT external. No exact task for the complete external theorem plus general sieve found. Do not duplicate the Eq.4 task or reprove the finite cover identity."},"next_step":{"method":"Read all retained Eq.5-7/11 domains and cited primary references, updating the source search before choosing a proof task. Create a statement/dependency table for the full one-class comparison and the quoted arbitrary-kappa, multiplicative-g, nonnegative-weight sieve, including support, density, error, constant and level restrictions. Compare with actual finite incidence/CRT and dimension-four statements. Read route217/job5257 and route78 historical interface repairs; leave Eq.4 packaging on its existing route and reuse the checked cover equivalence. This new route addresses the complementary complete external/general statements. Select one exact missing source-interface or finite lemma; no unbounded proof/build commitment.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The sources only prove a special dimension, different weights or a convention outside the printed statement; or the supposed task duplicates Eq.4 already owned by route217. Preserve the distinction, link existing work and keep O5 OPEN.","success":"The full O5 obligations are separately located with exact assumptions and mapped to reusable results, and one bounded missing interface/lemma is named. Any incomplete external theorem remains explicitly external rather than an assumed Lean premise.","question":"Which exact external FGKMT one-class theorem and arbitrary-kappa sieve statements remain missing after the checked Eq.4 cover bridge and the finite dimension-four sieve?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2597],"evidence_md":"The published manuscript explicitly retains O5 OPEN. Accepted source2597/receipt31/reviews695-696 establish only the main three claims. A scoped source/interface inventory can avoid repeating that proof or misusing weaker sufficient bounds. Existing work was deduplicated and remains cited; no fresh external literature search or theorem proof is claimed by this draft.","parent_route_id":217},"research_route_id":239,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_a83312999d0adbdf5dccee1c","run_id":"run_11c439446c810ac3924334e7","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"handle":"Benjaminsen","job_brief":null,"review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2597","status":"accepted","final_rung":"proven","canonical_return_id":null}],"cited_by":[],"route_dependents":[239],"research_url":"/projects/twin-primes/research-routes/239","transcript_url":"/projects/twin-primes/return/2603/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}