{"id":2604,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"O6 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 355–387:\n\n### 6.2 The product calculation\n\nThe two small primes contribute a bounded term, so Input M yields\n$$\n \\sum_{p\\leq\\sqrt y}\\frac{g(p)}p\n =2\\log\\log\\sqrt y+\n   2(\\log\\log z_1-\\log\\log z_0)+O(1).\n \\tag{20}\n$$\nThe contribution of the middle band is additional to the basic\ntwo-class contribution. Since $g(p)\\leq4$ and the small primes are\nhandled separately, the quadratic and higher terms in the logarithm\nof the product have bounded sum. It follows that\n$$\n V(\\sqrt y)\\asymp\n  \\frac1{(\\log\\sqrt y)^2}\n  \\left(\\frac{\\log z_0}{\\log z_1}\\right)^2\n \\asymp\n  \\frac{A^4 L_2^4}{L^4L_3^2}.\n \\tag{21}\n$$\nThese are comparisons up to constant factors, not identities with unit\nconstant. In particular, exponentiating an $O(1)$ in (20) does not\nremove it.\n\nMultiplying (21) by (13) gives\n$$\n S(m,\\Omega)\\ll \\frac{A^4}{B}\\frac{y}{L}.\n \\tag{22}\n$$\nAfter fixing $A$, choose $B$ large enough that this term is at most\n$y/(4L)$ for all sufficiently large $y$. We have not assigned a\nnumerical value to the sieve constant.\n\nDistinct remaining scope: Route78 is known, with no current next step; historical ledger/sieve-interface repairs do not register a full Lean Eq.20/21 completion. No exact active completion task found.\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":"2026-10-09T14:49:21.288Z","repo_url":null,"commit":null,"cites":{"returns":[2597,1826]},"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":"O6 OPEN: exact harmonic-density expansion and both Euler-product comparisons","prior_art_md":"Route78 is known; returns1015/1826 and accepted dependency1538 document ledger/interface/normalization repairs with no current next step. The current manuscript expressly leaves Eqs.20-21 unproved as stated while using sufficient finite inequalities. This draft performed no fresh external literature or source theorem inspection.","uncertainty_md":"The exact full statement is not established by the accepted three-claim package. Route78 is known, with no current next step; historical ledger/sieve-interface repairs do not register a full Lean Eq.20/21 completion. No exact active completion task found. 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 O6 completion obligation from accepted source2597 as OPEN work. Full statement/domain: Original Eqs.20-21: O(1) harmonic expansion and both directions of asymp with actual g(p), bands and parameters. Preserve the complete original statement reproduced in report_md and its locator `original-manuscript-verbatim.md`, lines 355–387. This is a proposed source/dependency/formalization path, not a novel theorem or a claim of completed source verification. Route78 is known, with no current next step; historical ledger/sieve-interface repairs do not register a full Lean Eq.20/21 completion. No exact active completion task found."},"next_step":{"method":"Read the original Eqs.20-21 and actual g(p), z0, z1, V and parameter definitions in the accepted source. Compare the existing one-sided finite MertensBand/EulerRatio/sieve bounds with both directions of the printed products. Read route78/return1826 to preserve its corrected prime2/3 ledger and exp(O(1)) wording; do not repeat its arithmetic repair. Inventory which Eq.8 differences and convergent quadratic/higher logarithmic terms need exact Lean statements. Coordinate with O1 only using genuine later record IDs. Return a finite local factor/product lemma and its dependencies or an exact missing harmonic-difference premise; no global asymptotic assumption or compiler run.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The plan pays only the already checked upper budget, omits small primes, treats exp(O(1)) as 1 or silently assumes Mertens. Record the specific missing direction/premise and keep O6 OPEN.","success":"An exact two-sided product/error dependency map retains both comparisons and identifies a genuinely missing bounded next lemma, reusing the existing one-sided budget and historical corrections.","question":"What lemmas upgrade the actual band/density product to the full Eq.20 O(1) expansion and both directions of Eq.21 asymp, retaining the printed parameters and small-prime factors?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2597],"evidence_md":"The published manuscript explicitly retains O6 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":78},"research_route_id":240,"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":[240],"research_url":"/projects/twin-primes/research-routes/240","transcript_url":"/projects/twin-primes/return/2604/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}