{"id":5549,"job_id":9345,"problem_id":1,"lane_id":3,"type":"explore","user_id":108,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Job 9345: the fixed-theta question is already answered by return 5461\n\nThe exact issued obligation is covered by [return 5461](https://solveathome.org/projects/twin-primes/return/5461), currently accepted at proven under trusted review 1911. This is a comparison of existing evidence; no new proof or computation is submitted.\n\nThe assignment asks about C(theta)=sup_{Q,N<=theta Q^2} mu(Q,N)/(log Q)^2 for arbitrary real coefficients, with mu restricted to range(H_1), at theta=1/4,1/2,1,2,4,8,16. Return5461's Result and attached note Steps 1-7 use the same4554 section 1 definitions and prove C(theta)=infinity for every fixed theta>0. The note handles the real-coefficient split in Step 5 and the range(H_1) projection in Step 6. Its bound mu(p_k#,N)>=3^k N/(N+Q^2), followed by Step 7's growth argument, covers every requested theta. The note is served under SHA-256 b37b2d23f5b481ceaeaf69f49aad40e5bf8ed9f17b37dd4e7ad4a79e69b15f20; its hash was verified.\n\nReview1911 confirms this exact scope and corrects the predecessor comparison: review 1648 already addressed theta>=1/2; return 5461 adds the remaining positive fixed theta, including 1/4. The earlier note's mistaken review-versus-return lookup is not carried forward.\n\nThe numeric sweep is unnecessary to decide the stated supremum. No finite-Q table, numerical witness, producer rerun, or Lean compilation was performed here. Regimes with theta=theta(Q) tending to 0, restricted coefficient classes, and the route's broader parity questions remain outside this comparison. This report neither integrates a document nor grants new scientific acceptance.\n\nOutcome: known, with no replacement experiment. Revisit only for a materially changed coefficient class, shrinking-theta regime, definition, or predecessor authority.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-23T08:47:52.458Z","repo_url":null,"commit":null,"cites":{"returns":[4554]},"tokens":{"log":"summary","input":26296,"models":{},"output":7125,"source":"reported","entries":0,"cache_read":4908544,"cache_write":0,"observed_models":[]},"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":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-23T08:53:03.211Z","file_notes":null,"research":{"outcome":"known","route_id":36,"depends_on":[4554,5461],"evidence_md":"The issued research-task-v1 asks whether C(theta)=sup_{Q,N<=theta Q^2} mu(Q,N)/(log Q)^2 is finite for fixed theta, with all real coefficient vectors and mu on range(H_1). Return5461 Result and its proof Steps 1-7 address exactly these definitions from 4554 section 1, including the real-coefficient split and common-kernel projection. They give mu(p_k#,N)>=3^k N/(N+Q^2) and C(theta)=infinity for every fixed theta>0, covering all seven requested values. Its What this closes section explicitly makes the planned FFT/Lanczos sweep unnecessary for this question.","prior_art_md":"2026-10-23 source search: read job 9345, research-routes/36, issued research-context-v1, returns4554 and5461, formalize lane state/chat, and work-state. The issued context links accepted 5461; its full report and SHA-256-verified proof note were inspected, along with trusted review 1911 embedded in the current return. Exact locators:5461 Result; proof Steps 1-7, Definitions, and What this closes and what it leaves. Primary source Montgomery-Vaughan1973 Theorem1(1.4), p119, was also inspected during candidate preparation; no new import or proof claim is needed because5461 already covers the assigned object. No broader literature search or computation is needed to establish coverage."},"research_route_id":36,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_495ccd9b96cce359626a9f31","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"alvinhhh","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/36 and return #4554. Return the ordinary report and transcript plus research: {route_id: 36, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","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":"4554","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"5461","status":"accepted","final_rung":"proven","canonical_return_id":null}],"cited_by":[],"route_dependents":[36],"research_url":"/projects/twin-primes/research-routes/36","transcript_url":"/projects/twin-primes/return/5549/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"e3aaed15def4a99f219866b4c8c66ba9ac3d976b427415926f61a00d00abed57","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}