{"id":2601,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"O3 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 389–413:\n\n### 6.3 The smooth-number term\n\nSet $X=m+2$, $Z=z_1$. Then\n$$\n \\log X=L+O(L_2),\\qquad\n u=\\frac{\\log X}{\\log Z}\\sim\\frac{A L_2}{L_3}.\n \\tag{23}\n$$\nBoth $u$ and $Z$ tend to infinity. For example, with\n$\\varepsilon=1/2$, the range condition $u\\leq Z^{1-\\varepsilon}$\nholds eventually, because\n$\\log u=O(L_3)$ whereas\n$\\log Z=L L_3/(A L_2)$.\n\nMoreover $u\\log u=(A+o(1))L_2$. Input H therefore implies\n$$\n \\Psi(m+2,z_1)\n  =(m+2)L^{-A+o(1)}\n  =o(y/L)\\qquad(A>4).\n \\tag{24}\n$$\nFor the last step, the ratio to $y/L$ is bounded by a constant times\n$L^{4-A+o(1)}L_3^2/L_2^4$, which tends to zero.\nThis discharges the smooth-number range rather than assuming that a\nfixed-$u$ estimate applies while $u$ grows.\n\nDistinct remaining scope: No task for the full arbitrary-fixed-A smooth conclusion found. The parameter limits and fixed-A12 budget already proved must be reused.\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":"2026-10-09T14:49:19.970Z","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":"O3 OPEN: every fixed A>4 smooth equality and negligible-budget route","prior_art_md":"Accepted source2597 establishes the main theorem by fixed A=12 and proves relevant parameter limits; it does not establish the full Eqs.23-24 smooth equality/negligibility route for every A>4. The draft has not undertaken new literature discovery or proved an optimized Rankin estimate. Existing Q-kk-substitution and Q-two-class-lower-bounds are broad historical summaries, not this exact follow-up.","uncertainty_md":"The exact full statement is not established by the accepted three-claim package. No task for the full arbitrary-fixed-A smooth conclusion found. The parameter limits and fixed-A12 budget already proved must be reused. 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 O3 completion obligation from accepted source2597 as OPEN work. Full statement/domain: Original Eqs.23-24: fixed A>4, smooth equality and o(y/log y), not merely fixed A=12. Preserve the complete original statement reproduced in report_md and its locator `original-manuscript-verbatim.md`, lines 389–413. This is a proposed source/dependency/formalization path, not a novel theorem or a claim of completed source verification. No task for the full arbitrary-fixed-A smooth conclusion found. The parameter limits and fixed-A12 budget already proved must be reused."},"next_step":{"method":"Read the retained Eqs.23-24 and the checked SmoothLogLimit, SmoothParameterGeometry and SmoothBudget statements. Build an exact implication/hypothesis table for X=m+2, Z=z1, u=log X/log Z and fixed A>4. Preserve the original epsilon choice, floor+2, B and equality/o(y/log y) quantifiers. Map the original H input to every required range condition, reusing the existing parameter limits. Coordinate the missing full uniform equality with the distinct O2 route once its genuine IDs exist; do not invent a dependency ID. Distinguish a possible stronger finite upper-bound route for sufficient budgets from the still-open printed equality. Return one bounded missing-range or statement-binding subproblem; no proof execution.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The proposed implication needs A>=12, supplies only an upper bound, or leaves the original uniform H range unverified. Record that limitation rather than silently narrowing A>4 or promoting the fixed-A12 budget.","success":"An exact dependency/quantifier map identifies the earliest missing input for every fixed A>4 and a bounded next lemma, without repeating already checked limits. It explicitly leaves the full printed equality OPEN if only a sufficient upper bound is available.","question":"For every fixed A>4 with the original chooseB(A), which exact uniform smooth input and parameter implications are missing from Eqs.23-24 beyond the checked parameter limits and fixed-A12 sufficient budget?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2597],"evidence_md":"The published manuscript explicitly retains O3 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":237,"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":[237],"research_url":"/projects/twin-primes/research-routes/237","transcript_url":"/projects/twin-primes/return/2601/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}