{"id":968,"job_id":1835,"problem_id":1,"lane_id":1,"type":"explore","user_id":42,"model":"deepseek-v4-pro","provider":"deepseek","report_md":"# Route 67 triage: the refined connection is exact; the loose-form support is the open question\n\nCalibration: **derived** (the correspondence is an exact identity from the quoted definitions); the loose-form support is open.\n\n## Result\n\nThe route's \"central uncertainty\" resolves **in favour of the connection** for the refined form. #159's dead-run length walks cumulative gap sums `S_m = G_m(i)` shifted by r into `K = {0, q-2}`; #161's `L(T_x,q)` is the longest run of slot residues in a 2-set `{a, a+2}` for any a. These coincide because `S_m = r_{i+m} - r_i (mod q)` **by definition**:\n\n```\nr + S_m in {0, q-2}   <=>   r_{i+m} in {r_i - r, r_i - r - 2} = {a+2, a},  a = r_i - r - 2\n```\n\nThe free shift r and the free anchor a are the **same freedom**. Hence a length-L dead run forces `L(T_x,q) >= L`, so the support of #159's refined correction is bounded by #161's maximum — a four-term (or shorter) sum wherever the table reads `<= 4`, and the producer's `LMAX = 8` truncation is idle there. This is the definitional step #965 flagged as the only place the connection could break; it is an identity, not a conjecture.\n\n## What remains open\n\nThe **loose form** `Q^loose_L` (per-gap condition `g mod q in {0, 2, q-2}`, no walk structure) is **not** covered by this correspondence. Its support — the quantity the producer actually truncates at 8 — is the real unknown, and asserting it stays below 8 without measurement would be the kind of overclaim this route must avoid.\n\n## Verdict\n\n**Promising.** The refined half is exact; the next experiment (a read-only comparison of #159's largest-L against #161's table, plus a loose-form support check on one small fold) is cheap, decisive, and directly targets the only remaining unknown.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T09:17:18.189Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[159,161,965],"messages":[]},"tokens":{"log":"custom","input":7043,"models":{"deepseek-v4-pro":19933},"output":19933,"source":"custom-jsonl","entries":10,"cache_read":4238976,"cache_write":0,"observed_models":["deepseek-v4-pro"]},"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,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"promising","route_id":67,"next_step":{"method":"Read-only: take #159's recorded largest-L field for T23 by 29/31/37 and T29 by 31/37 and compare against #161's table; then instrument tctRHS to emit argmax over L of Q^loose_L > 0 per (x,q,theta) and re-run exactly one small fold. No heavy computation.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"A shared pair where the two L disagree (one implementation wrong or the correspondence false), or a largest L > 8 (the producer's numbers were truncated).","success":"Every untruncated largest L <= L(T_x,q) <= 8, and the loose form's support <= 8: the refined correction is four-term or shorter, the producer's truncation is idle.","question":"Does #159's own largest untruncated L equal or stay below #161's L(T_x,q) at every shared (x,q) pair, and does the LOOSE form's support stay at or below LMAX = 8?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[159,161,965],"evidence_md":"#159's S_m = G_m(i) is the m-th cumulative gap, and #161's residues satisfy r_{i+m} = r_i + S_m (mod q) by definition, so 'r + S_m in {0,q-2}' <=> 'r_{i+m} in {r_i - r, r_i - r - 2}' = {a+2, a}. The free shift r and free anchor a are the same freedom, so a length-L dead run forces L(T_x,q) >= L: the refined correction's support is bounded by #161's maximum. The route's definitional uncertainty is therefore an exact identity for the refined form. The LOOSE form (per-gap, no walk) is not covered and remains open. No count was regenerated.","prior_art_md":"2026-09-17 (reusing #965's recorded search). One-class covering run is classical: Ford-Green-Konyagin-Maynard-Tao (large gaps, Annals 183); Jacobsthal j(n) (Kanold/Stevens/Hagedorn; Costello arXiv:1208.5342). The two-class variant (fold deletes v=0 and v=-2 mod q) has no external bound, so the coverage is internal: #159 (dead-run support, accepted/verified) and #161 (max-over-anchor table, accepted/verified) carry the two halves. The exact uncovered step is the LOOSE form's support, not the refined correspondence."},"research_route_id":67,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_a240901e1c256653d3e0ef23","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/67 and return #965. Return the ordinary report and transcript plus research: {route_id: 67, outcome: \"promising|progress|blocked|inconclusive|known|result\", 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>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"159","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"161","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"965","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/67","transcript_url":"/projects/twin-primes/return/968/transcript","files":[{"sha256":"883be7f21b9e71c287ef271e9c7e1635b0dfb5bcaae4e58cfdf55aed9d45bd84","name":"qa.md","bytes":31676}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}