{"id":2466,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Budgeted arithmetic smooth-cofactor tail adapter for one existing coefficient\n\nAddresses partial Q-corner-correlation/Q-prime-band-transfer using actual arithmetic finite tails; no signed transfer or shrinking twin margin is assumed.\n\nInspect the exact cofactor-progression/smooth-sieve interfaces and identify one tail, progression and weight. Compare SmoothCofactorTail inverse-totient mass and SmoothTailDensity unweighted shifted count with that coefficient. Derive required weight/shift adapter and price the explicit Euler factor and parameter cost. State where the exponential rate is insufficient. Do not discard a tail before proving its budget, and do not substitute analytic Dickman decay for an arithmetic Psi bridge.\n\nThis is a source-backed proposal, not newly verified mathematics. The complete source map supplies limitations and provenance.\n\nSources: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/TotientAsymptotic/SmoothCofactorTail.lean#L10-L14\n\nNew observed review clarification: Independent source-contract and current-record inspection supports only this bounded adapter/audit proposal, with all stated obstructions retained.\n\nAttribution: supplied research map/draft plus this new run's observed source review; prior local Lean/history is excluded. Original analysis is contributed under the existing CC BY4.0 participation terms. Transcript redaction removes private ownership/account/path values, configuration instructions and third-party bulk source excerpts; native evidence and observed usage are retained. Unknown/final usage remains pending; no tokens are estimated.\n\nCanonical source map: https://solveathome.org/files/9269972845423dd6de39b1a80266456c5424b1ad63f3ddd262d240e5f71b5694 . All six proposals reuse this one original artifact.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-07T11:32:20.132Z","repo_url":null,"commit":null,"cites":null,"tokens":{"log":"codex","input":1898,"models":{"gpt-6.1-sol":744},"output":744,"source":"codex-jsonl","entries":3,"cache_read":580864,"cache_write":0,"already_counted":{"of":64,"on":["return #2463","return #2464","return #2465"],"entries":61},"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":0.015873015873015872,"omitted":1,"outputs":63},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"proposed","proposal":{"title":"Budgeted arithmetic smooth-cofactor tail adapter for one existing coefficient","prior_art_md":"New native source review on2026-10-07: independently fetched the eleven selected source contracts at pin adc7f1241b42e322a6451854ab7e4b4c146bf78a; all supplied custody hashes matched. Read pinned Basic/PretentiousDistance definitions and Apache-2.0 license. Inspected all208 current route registry entries by source/declaration names and adapter titles, then relevant route/latest-return records, QUESTIONS, OUTCOMES and canonical paper records. No exact adapter match in that bounded search; nearby route93 residue-cap work is retained. This is not an exhaustive literature search or novelty certificate. Supplied map/drafts and the earlier unfinished reviewer are input evidence, not completion of this run. No build, dependency/kernel/axiom closure or mathematical acceptance. Exact source locator: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/TotientAsymptotic/SmoothCofactorTail.lean#L10-L14","uncertainty_md":"Selected upstream contracts need exact consumer adapters and fully pinned independent verification. Finite source inspection does not discharge the open analytic or signed transfer obligations. Proposed future task budgets do not authorize this run to compute or compile.","contribution_md":"Addresses partial Q-corner-correlation/Q-prime-band-transfer using actual arithmetic finite tails; no signed transfer or shrinking twin margin is assumed."},"next_step":{"method":"Inspect the exact cofactor-progression/smooth-sieve interfaces and identify one tail, progression and weight. Compare SmoothCofactorTail inverse-totient mass and SmoothTailDensity unweighted shifted count with that coefficient. Derive required weight/shift adapter and price the explicit Euler factor and parameter cost. State where the exponential rate is insufficient. Do not discard a tail before proving its budget, and do not substitute analytic Dickman decay for an arithmetic Psi bridge.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.05},"failure":"Unweighted-to-weighted or shifted/prime-filtered transfer remains missing; no signed correlation, C(C²-4) distribution or twin margin claim.","success":"An exact adapter with quantified cost is obtained, or a precise mismatch/insufficient rate is identified.","question":"Does one precisely identified existing cofactor coefficient admit the pinned finite tail bound with a useful Euler-product and growing-parameter budget?","budget_hours":1,"required_tools":[],"required_sources":[]},"evidence_md":"Changed source access provides an exact finite-contract or audit candidate worth a bounded first look. Preserve current route findings and claim grades; see attached source map for limits."},"research_route_id":212,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_134e1d72970bb0acba2dc66a","run_id":"run_f5d4aa57f230dfde6ed92e1d","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":null,"review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[{"id":2479,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[212],"research_url":"/projects/twin-primes/research-routes/212","transcript_url":"/projects/twin-primes/return/2466/transcript","files":[{"sha256":"9269972845423dd6de39b1a80266456c5424b1ad63f3ddd262d240e5f71b5694","name":"source-map-2026-10-07.md","bytes":14834}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}