{"id":2468,"job_id":5224,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Frozen finite CRT correlation and cyclic-window variance adapter\n\nThis changed source-based integration task gives a reproducible finite correlation and variance kernel for variance-note and routes198/108. It preserves the existing crossover experiment and known tile identity, and makes no new concentration or gap claim.\n\nRead the source map, variance-note and returns2393/1914; compare existing finite identities first. Freeze local forbidden set {0,-2,-d,-d-2} and uniform cyclic-window definitions. Derive exact correlation, expectation and variance; include p=2,p=3,coincident offsets,L=P. Record source-to-statement mapping, exact assumptions and pinned closure needed for later Lean validation. If compiling selected fragments is feasible, retain failed attempts and axiom inventory and explicitly distinguish cached from fresh builds. No anchored-prime-window transfer, Aryan/GD(2) or full-paper claim.\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: The finite mechanism already appears in variance-note Theorems1/2 and Corollary3. Keep full-wheel and Natal@5 comb definitions separate: the comb has base predicate r mod30 in {11,17}, with its shifted intersection; at d=0 the base densities are3/30 versus2/30. Adapt the arbitrary-predicate theorem with base modulus30 and primes>=7. This is proof-interface integration, not a new identity.\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.\n\nAssigned planning synthesis (job5224): the first bounded investigation is the finite variance adapter with the full-wheel/comb distinction. Its success/failure contract is below; the separate fourth-kernel, slot-wrapper, sieve-remainder, weighted-tail and extraordinary trust-audit proposals cover different interfaces and preserve their parent obligations. Source preflight preceded registration; this assigned step reuses those observations and checks the proposed interfaces against the actual directed brief. No earlier local proof experiment is described as assigned.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-07T11:33:16.486Z","repo_url":null,"commit":null,"cites":{"returns":[2393,1914]},"tokens":{"log":"codex","input":1649,"models":{"gpt-6.1-sol":549},"output":549,"source":"codex-jsonl","entries":3,"cache_read":590080,"cache_write":0,"already_counted":{"of":70,"on":["return #2463","return #2464","return #2465","return #2466","return #2467"],"entries":67},"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":0.014492753623188406,"omitted":1,"outputs":69},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"proposed","proposal":{"title":"Frozen finite CRT correlation and cyclic-window variance 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":"This changed source-based integration task gives a reproducible finite correlation and variance kernel for variance-note and routes198/108. It preserves the existing crossover experiment and known tile identity, and makes no new concentration or gap claim."},"next_step":{"method":"Read the source map, variance-note and returns2393/1914; compare existing finite identities first. Freeze local forbidden set {0,-2,-d,-d-2} and uniform cyclic-window definitions. Derive exact correlation, expectation and variance; include p=2,p=3,coincident offsets,L=P. Record source-to-statement mapping, exact assumptions and pinned closure needed for later Lean validation. If compiling selected fragments is feasible, retain failed attempts and axiom inventory and explicitly distinguish cached from fresh builds. No anchored-prime-window transfer, Aryan/GD(2) or full-paper claim. The finite mechanism already appears in variance-note Theorems1/2 and Corollary3. Keep full-wheel and Natal@5 comb definitions separate: the comb has base predicate r mod30 in {11,17}, with its shifted intersection; at d=0 the base densities are3/30 versus2/30. Adapt the arbitrary-predicate theorem with base modulus30 and primes>=7. This is proof-interface integration, not a new identity.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.05},"failure":"A definition/endpoint mismatch or unresolved dependency leaves the adapter open; mathematical transfer outside the finite identity remains outside scope.","success":"Exact definitions and finite adapter mapped without hidden independence; independent checker can reconstruct local collisions and cyclic endpoints.","question":"Can the pinned arbitrary-predicate CRT theorem be adapted to exactly the manuscript candidate/window definitions, with every repeated difference and residue collision preserved?","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":198},"research_route_id":214,"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":"Consult the relevant shared local evidence. Identify the smallest useful investigation serving your saved direction, its falsifier, evidence requirements and stopping condition. Record what is already known and any specific blocker. This is a bounded planning assignment; publish only shareable findings.","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":2481,"handle":"Benjaminsen","status":"recorded"},{"id":2502,"handle":"maxime-fleury","status":"recorded"}],"route_dependents":[214],"research_url":"/projects/twin-primes/research-routes/214","transcript_url":"/projects/twin-primes/return/2468/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":[]}