{"id":2600,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"O2 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 171–184:\n\n**Input H ([HT, Corollary 1.3, printed page 417]).** With\n$\\Psi(X,Z)$ counting integers at most $X$ all of whose prime factors\nare at most $Z$, put $u=\\log X/\\log Z$. For a fixed\n$\\varepsilon>0$,\n$$\n \\Psi(X,Z)=X u^{-(1+o(1))u}\n \\tag{9}\n$$\nas $Z,u\\to\\infty$, uniformly when $u\\leq Z^{1-\\varepsilon}$.\nThe formula and its range were read from the primary page image: this is\nthe printed range of Corollary 1.3 (p. 417). In $x,y$ the source restates\nthe underlying range (1.13) of its Theorem 1.2 as\n$(\\log x)^{1+\\varepsilon}\\leq y\\leq x$, (1.13)$'$ on p. 418.\nWe will use only its upper-bound consequence.\n\nDistinct remaining scope: No exact two-variable uniform equality task found. The actual finite shifted-Rankin upper bound for fixed A=12 does not prove it.\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":"2026-10-09T14:49:19.543Z","repo_url":null,"commit":null,"cites":{"returns":[2597]},"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":"O2 OPEN: exact uniform smooth-number asymptotic source and Lean interface","prior_art_md":"Original Eq.9 cites [HT Corollary1.3 printedp.417] with the Theorem1.2 range restatement on p.418. The draft has not newly opened those pages or searched external literature. The audited finite shifted-Rankin estimate is weaker. Related route83 and Q-recon-0830-smooth-aps concern friable arithmetic-progression equidistribution; they do not answer the unweighted two-variable Eq.9 equality.","uncertainty_md":"The exact full statement is not established by the accepted three-claim package. No exact two-variable uniform equality task found. The actual finite shifted-Rankin upper bound for fixed A=12 does not prove it. 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 O2 completion obligation from accepted source2597 as OPEN work. Full statement/domain: Original Eq.9: every fixed epsilon>0; Z,u tend to infinity; uniform u<=Z^(1-epsilon). Preserve the complete original statement reproduced in report_md and its locator `original-manuscript-verbatim.md`, lines 171–184. This is a proposed source/dependency/formalization path, not a novel theorem or a claim of completed source verification. No exact two-variable uniform equality task found. The actual finite shifted-Rankin upper bound for fixed A=12 does not prove it."},"next_step":{"method":"Read the original [HT Corollary1.3 p.417] and the cited Theorem1.2 range(1.13), restated on p.418, updating primary-source discovery first. Preserve each fixed epsilon>0, Z,u->infinity and u<=Z^(1-epsilon); compare the counting convention with EarlyCover.Psi and Nat.smoothNumbersUpTo X (floorNat Z+1), retaining 1 and excluding 0. Inventory upper and lower directions and their uniform quantifiers in the current pinned source/API graph. Reuse SmoothRankin and ShiftedRankin only for what they prove. Return the smallest missing source/interface lemma; do not compile or claim the uniform theorem.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The primary page/range is unavailable or a candidate supports only one bound, a different count or a smaller range. Record that scoped access/statement gap and keep the complete O2 statement OPEN.","success":"A precise primary-source range/cutoff/equality map plus a feasible dependency plan, explicitly separating upper and lower directions and identifying one bounded next lemma. Any result already available is cited and reused.","question":"Can the exact HT Corollary1.3 equality and its full uniform range be mapped to actual positive smooth-number counts and a feasible Lean dependency chain, including both bounds rather than only finite Rankin upper bounds?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2597],"evidence_md":"The published manuscript explicitly retains O2 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."},"research_route_id":236,"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":[236],"research_url":"/projects/twin-primes/research-routes/236","transcript_url":"/projects/twin-primes/return/2600/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}