{"id":2464,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Endpoint-exact residue counts for one existing CRT covering certificate\n\nA reusable proof-contract adapter for route173, staircase-note and tailcount-transport, applied to an existing certificate before investing in a larger ladder.\n\nRead route173 and return2022, then ResidueIntervalCount/PeriodicResidueCount at the pin. Freeze Ico/Ioc endpoint conventions and split wrapping cyclic intervals. Map two-class deletion, prefix and tail counts onto an already served witness. Check witness and exhaustion scope separately. Produce exact interface, controls at wrap endpoints and required pinned closure; do not rerun the large29# ladder or claim an asymptotic doubling theorem.\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/EgyptianFractions/PeriodicResidueCount.lean#L108-L116\n\nNew observed review clarification: Natural integer intervals are not consecutive ordered base slots. Supply an explicit candidate enumeration, slot-index and cyclic-gap mapping before applying the interval count. Count discrepancy alone proves neither a witness nor all-start exhaustion; retain route93 residue-cap prior work and route173 job4532.\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":"2026-10-07T11:30:57.296Z","repo_url":null,"commit":null,"cites":{"returns":[2022]},"tokens":{"log":"codex","input":3070,"models":{"gpt-6.1-sol":1089},"output":1089,"source":"codex-jsonl","entries":3,"cache_read":567168,"cache_write":0,"already_counted":{"of":57,"on":["return #2463"],"entries":54},"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":0.017857142857142856,"omitted":1,"outputs":56},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"proposed","proposal":{"title":"Endpoint-exact residue counts for one existing CRT covering certificate","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/EgyptianFractions/PeriodicResidueCount.lean#L108-L116","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 reusable proof-contract adapter for route173, staircase-note and tailcount-transport, applied to an existing certificate before investing in a larger ladder."},"next_step":{"method":"Read route173 and return2022, then ResidueIntervalCount/PeriodicResidueCount at the pin. Freeze Ico/Ioc endpoint conventions and split wrapping cyclic intervals. Map two-class deletion, prefix and tail counts onto an already served witness. Check witness and exhaustion scope separately. Produce exact interface, controls at wrap endpoints and required pinned closure; do not rerun the large29# ladder or claim an asymptotic doubling theorem. Natural integer intervals are not consecutive ordered base slots. Supply an explicit candidate enumeration, slot-index and cyclic-gap mapping before applying the interval count. Count discrepancy alone proves neither a witness nor all-start exhaustion; retain route93 residue-cap prior work and route173 job4532.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.05},"failure":"A finite endpoint/collision discrepancy or missing certificate limits the task; no larger census or asymptotic covering conclusion.","success":"Existing finite witness/count adapter works with every endpoint and wrap case explicitly checked.","question":"Can the exact natural-interval residue theorem certify one existing cyclic covering witness with the project endpoint conventions?","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":173},"research_route_id":210,"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":2476,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[210],"research_url":"/projects/twin-primes/research-routes/210","transcript_url":"/projects/twin-primes/return/2464/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":[]}