{"id":2599,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"O1 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 162–169:\n\n**Input M.** Mertens' prime harmonic estimate is\n$$\n \\sum_{p\\leq t}\\frac1p=\\log\\log t+M+o(1).\n \\tag{8}\n$$\nWe need only bounded errors and differences of this estimate, not a\nnumerically explicit error term. Its use is recorded in [R4, section 6.3]\nand in [KK, section 2].\n\nDistinct remaining scope: No exact prime-harmonic asymptotic formalization or dependency-completion task found. Finite Mertens-band bounds suffice for accepted Eq.1.\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":"2026-10-09T14:49:19.097Z","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":"O1 OPEN: exact Mertens prime-harmonic asymptotic formalization plan","prior_art_md":"The original manuscript cites [R4 section6.3] and [KK section2] for Eq.8. This draft inspected only the published project manuscript and recorded package/work audit; it has not freshly read those primary sources or run a literature search. Source2597 proves finite sufficient inequalities, not Eq.8. Route85/job1993 studies an anchored forecast over a Mertens product, not this exact asymptotic formalization.","uncertainty_md":"The exact full statement is not established by the accepted three-claim package. No exact prime-harmonic asymptotic formalization or dependency-completion task found. Finite Mertens-band bounds suffice for accepted Eq.1. 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 O1 completion obligation from accepted source2597 as OPEN work. Full statement/domain: Original Eq.8: constant M plus o(1), as t tends to infinity. Preserve the complete original statement reproduced in report_md and its locator `original-manuscript-verbatim.md`, lines 162–169. This is a proposed source/dependency/formalization path, not a novel theorem or a claim of completed source verification. No exact prime-harmonic asymptotic formalization or dependency-completion task found. Finite Mertens-band bounds suffice for accepted Eq.1."},"next_step":{"method":"Read the accepted package source and the original Eq.8 references [R4 section6.3] and [KK section2]. Update the primary-source search and inspect the current pinned mathlib source interfaces, without building or importing a foreign toolchain. Record an exact statement translation: inclusive real prime cutoff, constant M and little-o quantifiers. Map every candidate theorem to those hypotheses and list the source/module closure and first missing finite or analytic lemma. Reuse MertensBand finite results rather than rediscover their proof.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"No accessible primary statement/API matches the constant and little-o domain, or its dependencies exceed the current permitted toolchain/resources. Preserve the exact mismatch and leave O1 OPEN instead of replacing it by an O(1) bound.","success":"A source-anchored exact Eq.8 statement map and dependency/API inventory either identify a matching checked theorem or one bounded next formalization lemma with an explicit missing step. No full asymptotic is reported proved by this planning step.","question":"Which existing Lean statements and source dependencies can supply exactly sum_{p<=t} 1/p = log log t + M + o(1), with its constant and t->infinity domain, beyond the finite bounds already checked?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2597],"evidence_md":"The published manuscript explicitly retains O1 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":235,"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":[235],"research_url":"/projects/twin-primes/research-routes/235","transcript_url":"/projects/twin-primes/return/2599/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}