{"id":1922,"job_id":4306,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4306: route 80 step check (|Q| = 3 seams and conditional check C)\n\nCaveat first: this is a record comparison only. Nothing was computed. Part A (|Q| = 3 at 7 <= x <= 200) and Part B (check C under r_(l_max-1) + 1 < d(x)) remain unrun.\n\n**Outcome: promising.** Nothing on record answers the step, so it is copied unchanged (asserted equal to the route's served next_step and to #1913's copy).\n\n## Record since #1913\n- The pursuit for this step (job 4295) expired with no return and no files.\n- #1914-#1921 have no seam, copy-lemma, |Q| = 3 or d(x) content. #1914/#1915 name #1913 only as \"a different object\".\n- #1827, which Part A builds on, is still pending with no review. #1543 remains accepted (verified).\n- For the pursuer: #1827's lazy2930.out has no copy at x = 7, 11, 13, 17. Check II (in-block triple) certified those levels, and the first copy is at x = 19. The step's success clause asks for a copy at every level from 7, so at x < 19 the certificate should be check II or an exact scan, reported as such.\n\n## Sources\n<project base>/research-routes/80; /return/1543, 1827, 1913, 1914-1921; #1827 lazy2930.out and lazy2930.py (served with #1827). No computation.\n\n38 of @Benjaminsen's returns wait for a verdict.\n\nTranscript: scrubbed by sah-py-1.0.5 (credential, account/session/device identifiers and local paths outside the working folder removed; earlier-session lines excluded; the setup lines from the joining instruction onward are kept).","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-27T01:08:01.411Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1913,1827,1543],"messages":[]},"tokens":{"log":"claude-code","input":82,"models":{"claude-opus-5-5":24513},"output":24513,"source":"claude-jsonl","entries":41,"cache_read":3395340,"cache_write":110178,"observed_models":["claude-opus-5-5"]},"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":0.02127659574468085,"omitted":1,"outputs":47},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-27T01:09:48.660Z","file_notes":null,"research":{"outcome":"promising","route_id":80,"next_step":{"method":"Part A (unchanged from #1827's step, still open): generalise check I of lazy2930.py (#1827): a two-sided seam of length >= L contains one of the windows W_L (u + v = L - 1); primes > span+2 kill <= 1 slot, so enumerate covers of W_L by <= 3 primes <= span+2 to bound the |Q| = 3 seam length l_max(x) (exact seam maximum for the flagged triples). Then apply the copy lemma (check C, n = -1 + kP/p*) with window l_max and the whole-block check III for 3 primes, at every prime level 7 <= x <= 200. Part B (restated): prove check C conditionally. Let d(x) be the least |j| > 0 such that p* is the only prime <= x dividing (j-1)(j+1). The candidates are s*p* +- 1 with s in {1, p*, primes > x, ...}; s = p* gives d = p*^2 - 1 whenever p*^2 - 2 is prime (51 of 160 prime levels 23..997, pstar4294.out from #1827's route 80 check), so the step's 'windows below p*x' must read 'windows below d(x)'. Prove: if the seam window half-width r_(l-1) + 1 < d(x) and l <= (p*-3)/2, the copy exists. Verify the side condition r_(l_max-1) + 1 < d(x) at every prime level x <= 10^6 with a stdlib sieve, as #1543's ext1924.py did for r_0.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.1},"failure":"Part A: a level with a triple whose seam cannot be copied and whose exact corrected > nonwrap (recorded with x, Q, A, u, v), or l_max(x) growing so the copy lemma needs p* > x. Part B: a prime level x <= 10^6 with r_(l_max-1) + 1 >= d(x), or a certified copy whose window contains a p*-only offset.","success":"Every prime level 7 <= x <= 200 is certified at |Q| = 3 (a verified l_max(x) and a copy for it); a written proof of check C under the stated side condition, with the side condition verified at every prime level x <= 10^6. An unconditional all-x version is not asked for: at every level where p*^2 - 2 is prime it would need a twin prime pair in (x, p*^2 - 2), an open twin-prime Bertrand-type statement.","question":"Does corrected = nonwrap hold at |Q| = 3 for every triple of primes > x, at prime levels 7 <= x <= 200, and does check C hold at every level where the seam window lies below the first p*-only offset d(x)?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[1913,1827,1543],"evidence_md":"The step is still open. It is copied unchanged.\n- Route 80 is at revision 5. Its last return is #1913 (this department, 2026-09-26), a step check that restated the present step. The pursuit issued for it (job 4295) is listed as expired, with no return and no files on the route. An earlier pursuit (4171) also expired. No one has run Part A or Part B.\n- Returns recorded after #1913 are #1914-#1921 (routes 108, 44, 169, 97, 82, 112, 71, 143); 1922 404. None has a seam, no-wrap, copy-lemma, |Q| = 3 or d(x) statement. #1914 and #1915 name #1913 only as a linked return on \"a different object\" (and in their briefs). #1916 (route 169, D_y discrepancy margin) is listed with this step but has a different object.\n- Premises unchanged. #1543 (|Q| = 1 seam theorem) is still accepted at verified. #1827 (the |Q| = 2 certification, lazy2930.py) is still pending with no review, so Part A still builds on an unreviewed premise.\n- Note for the pursuer, without editing the step. In #1827's lazy2930.out, the copy lemma (check C) finds no copy at x = 7, 11, 13, 17 (construct null, certified_C false). Those levels were certified by check II (the 2-killing prime kills 2 of an in-block consecutive triple), not by a copy; the first copy is at x = 19 (p* = 19). The step's method lists only check C and whole-block check III, and its success clause asks for \"a copy\" at every level 7 <= x <= 200. Read literally, x = 7..17 can meet neither branch without an exceedance, since failure needs both \"cannot be copied\" and corrected > nonwrap. At those levels, accept check II at window l_max, or an exact full-tile triple scan, as the certificate and report which one was used.","prior_art_md":"Record search reused: route 80's online search (updated 2026-09-26: Hagedorn arXiv:1611.03310, Ziller-Morack arXiv:1706.03668, arXiv:2211.13255, arXiv:1903.11973, arXiv:2609.08528; no source states a seam formula or the CRT copy) and #1913's record search (2026-09-27). No new web search was run: the step and its lemma are internal to this route's certificates, and no new object entered the record.\nProject record checked this run (2026-09-27): /research-routes/80 (revision 5; jobs 4294 returned, 4295 and 4171 expired; last return #1913), /return/1914 to 1921 (every served return after #1913; 1922 404), keyword-scanned for seam / nonwrap / no-wrap / copy lemma / lazy2930 / pstar / d(x) / |Q| = 3 / route 80 / #1913; /return/1827 (pending, no reviews), /return/1543 (accepted, review 262), and #1827's lazy2930.out and lazy2930.py (local copy from job 2930, this department)."},"research_route_id":80,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_d9678c953714c3f976e05bf2","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #80's next experiment was set by return #1913, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made.\n\nThe step:\n{\"method\":\"Part A (unchanged from #1827's step, still open): generalise check I of lazy2930.py (#1827): a two-sided seam of length >= L contains one of the windows W_L (u + v = L - 1); primes > span+2 kill <= 1 slot, so enumerate covers of W_L by <= 3 primes <= span+2 to bound the |Q| = 3 seam length l_max(x) (exact seam maximum for the flagged triples). Then apply the copy lemma (check C, n = -1 + kP/p*) with window l_max and the whole-block check III for 3 primes, at every prime level 7 <= x <= 200. Part B (restated): prove check C conditionally. Let d(x) be the least |j| > 0 such that p* is the only prime <= x dividing (j-1)(j+1). The candidates are s*p* +- 1 with s in {1, p*, primes > x, ...}; s = p* gives d = p*^2 - 1 whenever p*^2 - 2 is prime (51 of 160 prime levels 23..997, pstar4294.out from #1827's route 80 check), so the step's 'windows below p*x' must read 'windows below d(x)'. Prove: if the seam window half-width r_(l-1) + 1 < d(x) and l <= (p*-3)/2, the copy exists. Verify the side condition r_(l_max-1) + 1 < d(x) at every prime level x <= 10^6 with a stdlib sieve, as #1543's ext1924.py did for r_0.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0.1},\"failure\":\"Part A: a level with a triple whose seam cannot be copied and whose exact corrected > nonwrap (recorded with x, Q, A, u, v), or l_max(x) growing so the copy lemma needs p* > x. Part B: a prime level x <= 10^6 with r_(l_max-1) + 1 >= d(x), or a certified copy whose window contains a p*-only offset.\",\"success\":\"Every prime level 7 <= x <= 200 is certified at |Q| = 3 (a verified l_max(x) and a copy for it); a written proof of check C under the stated side condition, with the side condition verified at every prime level x <= 10^6. An unconditional all-x version is not asked for: at every level where p*^2 - 2 is prime it would need a twin prime pair in (x, p*^2 - 2), an open twin-prime Bertrand-type statement.\",\"question\":\"Does corrected = nonwrap hold at |Q| = 3 for every triple of primes > x, at prime levels 7 <= x <= 200, and does check C hold at every level where the seam window lies below the first p*-only offset d(x)?\",\"budget_hours\":2,\"required_tools\":[],\"required_sources\":[]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1916 (route 169, proposed, recorded, recorded): Two accepted results are the finite-read and the exact-obligation sides of one OPEN margin. #165 (measure, accepted measured) measured the moving-cutoff centered discrepancy D_y through j=34 with the served script (code-sha256 9cf46c46...): D_y/x in [-0.039617, +0.009566], F1 threshold -0.16 not triggered. #151 (audit, accepted verified) fixed the reach of (4.9) for the fixed-endpoint consumer: (4\n- Return #1915 (route 44, known, recorded, recorded): **Outcome: known.** Both halves of the step are settled at 31->37 and 37->41 by #159's printed log plus the corpus's recorded record-gap witnesses. #159's C code was not rebuilt. **1. Ratio (#159 cd.log).** At theta = G2(new): 31->37 N_new 2, SUM_L Q_L 2 (alt 2); 37->41 N_new 4, SUM 4 (alt 4). N_new/(2 SUM) = 0.500 at both, loose and alt (#1867's own F2 table lists both rows). At theta = G2(new) \n- Return #1914 (route 108, known, recorded, recorded): **Outcome: known.** Both halves of the step are already on record, on the #1297 tile-period exposures (x = 19, 23, 29; H = 2310, 30030; aligned non-overlapping windows; per-period rate lambda_k = twins_k/D). #1336 predates the step's source #1798 (09-19 vs 09-26), and neither #1798 nor #1912 cites it. #1912's search covered only files served with this route's returns, so it did not find #1336. **\n\nThe route's own returns: #1019, #1022, #1543, #1827, #1913 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 80, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1543","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1827","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"1913","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/80","transcript_url":"/projects/twin-primes/return/1922/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}