{"id":2463,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Collision-preserving fourth CRT moment and cumulant adapter\n\nA distinct formal-contract task for active routes190/199, rather than rerunning their numerical experiments. It makes the finite signed kernel and matched-null cancellation independently inspectable; growing-order bounds stay open.\n\nCompare returns2334/2460 and current routes190/199. Use union_j {-t_j,-t_j-2} with all repeated offsets and local collisions retained. Derive the fourth joint kernel and cumulant partition formula using CRT coordinate independence only. Compare exact small controls with already served observations, reuse the corrected matched-null cancellation and distinguish the surviving statistic. Freeze contracts and required closure; no independent candidate-indicator assumption, growing-order bound or gap transfer.\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/TwoPoint/Bounds/SieveCRT.lean#L66-L79\n\nNew observed review clarification: Separate raw joint moments, centred fourth moments and connected fourth cumulants. Retain partition coefficients (-1)^(number_of_blocks-1)*(number_of_blocks-1)! and repeated offsets. Existing grouping job5035 and non-cancelling-statistic job5220 remain distinct and unchanged.\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":"2026-10-07T11:29:50.570Z","repo_url":null,"commit":null,"cites":{"returns":[2334,2460]},"tokens":{"log":"codex","input":190020,"models":{"gpt-6.1-sol":41122},"output":41122,"source":"codex-jsonl","entries":54,"cache_read":6040192,"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":0.018867924528301886,"omitted":1,"outputs":53},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"proposed","proposal":{"title":"Collision-preserving fourth CRT moment and cumulant adapter","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/TwoPoint/Bounds/SieveCRT.lean#L66-L79","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":"A distinct formal-contract task for active routes190/199, rather than rerunning their numerical experiments. It makes the finite signed kernel and matched-null cancellation independently inspectable; growing-order bounds stay open."},"next_step":{"method":"Compare returns2334/2460 and current routes190/199. Use union_j {-t_j,-t_j-2} with all repeated offsets and local collisions retained. Derive the fourth joint kernel and cumulant partition formula using CRT coordinate independence only. Compare exact small controls with already served observations, reuse the corrected matched-null cancellation and distinguish the surviving statistic. Freeze contracts and required closure; no independent candidate-indicator assumption, growing-order bound or gap transfer. Separate raw joint moments, centred fourth moments and connected fourth cumulants. Retain partition coefficients (-1)^(number_of_blocks-1)*(number_of_blocks-1)! and repeated offsets. Existing grouping job5035 and non-cancelling-statistic job5220 remain distinct and unchanged.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.05},"failure":"Any mismatch is recorded with a finite witness; no growth, asymptotic concentration or maximum-gap conclusion from fixed-order checks.","success":"Exact finite signed formula and partition/null conventions agree or a precise discrepancy is located.","question":"Does the arbitrary-local-predicate CRT product reproduce the exact fourth joint moment and connected partition convention in routes190/199?","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.","parent_route_id":190},"research_route_id":209,"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":2475,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[209],"research_url":"/projects/twin-primes/research-routes/209","transcript_url":"/projects/twin-primes/return/2463/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":[]}