{"id":2602,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"O4 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 186–192:\n\n**Input P.** The prime number theorem gives\n$$\n \\pi(y)-\\pi(y/2)\\sim\\frac{y}{2\\log y}.\n \\tag{10}\n$$\nNo explicit prime-count threshold is required. This is the final counting\ninput in [KK, section 2] and [R4, section 7].\n\nDistinct remaining scope: No exact dyadic count asymptotic task found. The checked eventual cardinality lower bound y/(3 log y) does not imply this asymptotic.\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":"2026-10-09T14:49:20.412Z","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":"O4 OPEN: exact dyadic prime-count asymptotic dependency plan","prior_art_md":"Original InputP is the dyadic count asymptotic. Accepted source2597 supplies an elementary eventual lower budget sufficient for Eq.1, not a PNT asymptotic. The current audit found no exact queued O4 task. No new primary-source or PNT library inspection has been performed for this draft.","uncertainty_md":"The exact full statement is not established by the accepted three-claim package. No exact dyadic count asymptotic task found. The checked eventual cardinality lower bound y/(3 log y) does not imply this asymptotic. 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 O4 completion obligation from accepted source2597 as OPEN work. Full statement/domain: Original Eq.10: pi(y)-pi(y/2)~y/(2 log y). Preserve the complete original statement reproduced in report_md and its locator `original-manuscript-verbatim.md`, lines 186–192. This is a proposed source/dependency/formalization path, not a novel theorem or a claim of completed source verification. No exact dyadic count asymptotic task found. The checked eventual cardinality lower bound y/(3 log y) does not imply this asymptotic."},"next_step":{"method":"Read retained InputP/Eq.10 and PrimeReserveBound. Update primary-source discovery for the exact asymptotic and inspect compatible pinned Lean source interfaces. Translate pi at real inclusive cutoffs and the strict dyadic difference before considering Nat floors; record the floor/end-point error and all eventual quantifiers. Map a candidate prime-number theorem or equivalent input to the ratio limit using genuine checked dependencies. Reuse the actual y/(3log y) reserve proof; do not infer an asymptotic from it, migrate toolchains or start a large source build. Return a minimal exact interface/dependency plan.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Only a Chebyshev lower bound, foreign incompatible library, unchecked PNT premise or differently scoped interval is available. Keep O4 OPEN and record the precise statement or compatibility blocker.","success":"A pinned-source plan for the exact dyadic asymptotic and cutoff conversion identifies matching APIs or a first bounded missing lemma, with compatibility/import/resource requirements explicit.","question":"What source-compatible Lean chain proves pi(y)-pi(y/2)~y/(2 log y) with the literal real endpoints, beyond the checked reserved-prime lower budget?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2597],"evidence_md":"The published manuscript explicitly retains O4 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":238,"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":[238],"research_url":"/projects/twin-primes/research-routes/238","transcript_url":"/projects/twin-primes/return/2602/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}