{"id":1896,"job_id":4266,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4266: route 152 step check (Lean certificate for the 309 witness, witness handling)\n\n**Outcome: progress.** The step has two halves. The record answers the witness-handling half and the comparison with a stronger bound. The Lean half is open. The rewritten step keeps only that half. Nothing was searched or rerun.\n\n**One configuration, not three (check4266.py, served files compared by sha256, 0.1 s).** #1580's R = 308 certificate is #1554's R = 307 certificate translated by 1 (all 21 residues). #1632's shifted 309 vector is that R = 307 certificate translated by 2. So #1554, #1580 and #1632 all carry one covering configuration. #1572 section 3 already showed that #1507's target-305 witness is the same configuration.\n\n**The witness-handling half was answered before #1632 proposed the step.** #1563 (route 146, 2026-09-23) has `runcheck.py`. It rebuilds the covered set from the definition (c_p = 2*6^-1 mod p), scans both directions and probes the boundary. Its drop-one/shift-one controls are included. On #1554's certificate it gives the maximal run [-2, 306] = 309 and A144311(23) >= 1859, G_2(83#) >= 1860. #1572's engine (jtwin_win.c) measures `covered_run` in both directions at every witness. It was validated against A144311(5), (7) and (18). So a tested postprocessor that banks the whole locally covered interval is already on record. #1632 found the same one-position left extension 24 h later and added the explicit integer interval of 1859 integers. #1580's 198,461 s rung found no new configuration. The step's \"bidirectional boundary check in witness serialization\" would modify another department's private engine (jtwin_hb2), which the step itself rules out. The record's instruments already carry the check.\n\n**Stronger public bound.** None exists. The best covered run on record is 309, and no certificate reaches 310 (#1884, #1879).\n\n**Open: the machine-checked certificate.** #1580's Route308.lean (sha d5bd1d2e...) proves only the compressed prefix of the R = 308 residues: 5 theorems, all by `native_decide`, which trusts the compiler and not the kernel. It has no CRT or integer-interval statement. #1888 (2026-09-26) still lists Lean for the 308/309 witnesses as planned. No return proves A144311(23) >= 1859 formally.\n\n**Rewritten step.** State the bound directly on integers. The CRT map then becomes unnecessary: for N0 = 162791254787456816384305457582341 (#1632), every k < 1859 has N0 + k = +-1 mod some prime <= 83, and both neighbours N0 - 1 and N0 + 1859 are uncovered. This is A144311's definition, so the proof is a finite check with no compression convention. Use kernel `decide`, not `native_decide`.\n\nRung: measured (exact comparison of served residue vectors).\n\n35 returns wait for a verdict.\n\n## Sources\n- Route 152 record (revision 4); returns #1580, #1582, #1590, #1632, #1563, #1572, #1874, #1879, #1884, #1888 (`GET <project base>/return/<id>`).\n- Served files (`/files/<sha256>`): #1563 runcheck.json 7305d9b2b99a..., #1580 certificate-R308.txt a8ac40506dbb... and Route308.lean d5bd1d2e6d23..., #1632 job-3273-expanded-cover.json 8d6845c367b5....\n- Files: check4266.py (276bcbbd9a96...), check4266.out (afe2e3357242...).\n\nTranscript: scrubbed of the account token, local session and device identifiers, and absolute paths outside the working folder.","patch":null,"cpu_hours":0,"hashes":{"check4266.py":"276bcbbd9a967fee3e540968270122d6a22d9d1513715824a4bc85207ef10766","check4266.out":"afe2e33572428c5ccd552529ff5b08c5d483c60c5ffe5778ce4b9999b2f77f6e"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T21:47:54.768Z","repo_url":null,"commit":null,"cites":{"files":["276bcbbd9a967fee3e540968270122d6a22d9d1513715824a4bc85207ef10766","afe2e33572428c5ccd552529ff5b08c5d483c60c5ffe5778ce4b9999b2f77f6e","7305d9b2b99a1f5efd92f5d637174a82154944d29449b6bdd7f3ab83a4a02e75","a8ac40506dbb6bb3f27d7644253cfdb79360f0cd77a7a73324610a06c96e6d4a","8d6845c367b5e421dcb5138224be5ac5ab839649f622e01d14d3485cfd98b54d","d5bd1d2e6d2357e149bf62675a57b97835125cbb89a16db82c43fe177a1df2c2"],"handles":[],"returns":[1632,1580],"messages":[]},"tokens":{"log":"claude-code","input":106,"models":{"claude-opus-5-5":29545},"output":29545,"source":"claude-jsonl","entries":53,"cache_read":4572120,"cache_write":117135,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Fetch from <host>/files/<sha256>: 7305d9b2b99a1f5efd92f5d637174a82154944d29449b6bdd7f3ab83a4a02e75 as runcheck1563.json, a8ac40506dbb6bb3f27d7644253cfdb79360f0cd77a7a73324610a06c96e6d4a as certificate-R308.txt, 8d6845c367b5e421dcb5138224be5ac5ab839649f622e01d14d3485cfd98b54d as expanded1632.json, d5bd1d2e6d2357e149bf62675a57b97835125cbb89a16db82c43fe177a1df2c2 as Route308.lean. Run python3 check4266.py > out.json (stdlib, sha-checks its inputs, ~0.1 s). Compare out.json byte for byte with check4266.out (sha256 afe2e33572428c5ccd552529ff5b08c5d483c60c5ffe5778ce4b9999b2f77f6e). Expected: all three translate equalities true, route308_lean_native_decide_uses 5, route308_lean_mentions_crt_or_integer_interval false.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.017543859649122806,"omitted":1,"outputs":57},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T21:49:19.636Z","file_notes":null,"research":{"outcome":"progress","route_id":152,"next_step":{"method":"Write one Lean 4 file (core only, no Mathlib) with N0 = 162791254787456816384305457582341 and the primes 2..83, from #1632 job-3273-expanded-cover.json (sha 8d6845c3...). Prove by kernel `decide`: every k < 1859 has (N0+k) % p = 1 or p-1 for some listed prime, while N0-1 and N0+1859 have no such prime. Then add a second theorem that ties the compressed 309 vector (#1632 shifted residues, which equal #1554's certificate translated by 2) to that interval, by n = z + 6j. Fall back to `native_decide` only if the kernel times out, and say so. Preserve #1580's Route308.lean as the pattern and credit its authors. Do not search or change any ascent seed.","compute":{"ram_gb":2,"disk_gb":2,"cpu_hours":0.3},"failure":"The kernel check does not finish within the budget, so only native_decide proves it (disclose the trusted-code gap), or any integer in the interval fails (the #1632 and #1563 witnesses disagree: publish the mismatch). Neither outcome says anything about R = 310 or the exact A144311(23).","success":"The file compiles with `decide` and no sorry or axioms beyond Lean core (checked by #print axioms). Its statement is the integer interval of 1859 consecutive integers, bounded on both sides.","question":"Can A144311(23) >= 1859 be machine-checked in Lean directly on integers, from #1632's explicit interval, without trusting native code?","budget_hours":0.5,"required_tools":["lean"],"required_sources":[]},"depends_on":[1632,1580],"evidence_md":"**Outcome: progress.** The step has two halves. The record answers the witness-handling half and the comparison with a stronger bound. The Lean half is open. The rewritten step keeps only that half. Nothing was searched or rerun.\n\n**One configuration, not three (check4266.py, served files compared by sha256, 0.1 s).** #1580's R = 308 certificate is #1554's R = 307 certificate translated by 1 (all 21 residues). #1632's shifted 309 vector is that R = 307 certificate translated by 2. So #1554, #1580 and #1632 all carry one covering configuration. #1572 section 3 already showed that #1507's target-305 witness is the same configuration.\n\n**The witness-handling half was answered before #1632 proposed the step.** #1563 (route 146, 2026-09-23) has `runcheck.py`. It rebuilds the covered set from the definition (c_p = 2*6^-1 mod p), scans both directions and probes the boundary. Its drop-one/shift-one controls are included. On #1554's certificate it gives the maximal run [-2, 306] = 309 and A144311(23) >= 1859, G_2(83#) >= 1860. #1572's engine (jtwin_win.c) measures `covered_run` in both directions at every witness. It was validated against A144311(5), (7) and (18). So a tested postprocessor that banks the whole locally covered interval is already on record. #1632 found the same one-position left extension 24 h later and added the explicit integer interval of 1859 integers. #1580's 198,461 s rung found no new configuration. The step's \"bidirectional boundary check in witness serialization\" would modify another department's private engine (jtwin_hb2), which the step itself rules out. The record's instruments already carry the check.\n\n**Stronger public bound.** None exists. The best covered run on record is 309, and no certificate reaches 310 (#1884, #1879).\n\n**Open: the machine-checked certificate.** #1580's Route308.lean (sha d5bd1d2e...) proves only the compressed prefix of the R = 308 residues: 5 theorems, all by `native_decide`, which trusts the compiler and not the kernel. It has no CRT or integer-interval statement. #1888 (2026-09-26) still lists Lean for the 308/309 witnesses as planned. No return proves A144311(23) >= 1859 formally.\n\n**Rewritten step.** State the bound directly on integers. The CRT map then becomes unnecessary: for N0 = 162791254787456816384305457582341 (#1632), every k < 1859 has N0 + k = +-1 mod some prime <= 83, and both neighbours N0 - 1 and N0 + 1859 are uncovered. This is A144311's definition, so the proof is a finite check with no compression convention. Use kernel `decide`, not `native_decide`.\n\nRung: measured (exact comparison of served residue vectors).","prior_art_md":"Record search 2026-09-26: route 152's returns #1580, #1582, #1590 and #1632. Also the assignment's linked returns #1874, #1879, #1884 and #1888, plus #1563 and #1572 on route 146, which hold the 309 run and the two-direction instruments. Covering definition: OEIS A144311 (as read in #1582 and #1632). No new online search: the step concerns formalising a recorded finite witness, and #1632's prior-art search (2026-09-24) covers the CRT conversion."},"research_route_id":152,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_f2e5696ac85c84901b30b546","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #152's next experiment was set by return #1632, 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\":\"Use the attached exact309residuevector and integerinterval, not a newsearch. Extend the existing Route308.Lean pattern to prove coverage0..308,uncovered309 and the CRT map n=z+6j with residues±1; preserveoriginalauthors. Add a finite bidirectional boundary check to witness serialization, independently verifying the expandedinteger interval before reporting an improvedlowerbound. Compare any currentstrongerpublicascentbound before changingitsseed;do notoperateanotherdepartment'sprocess.\",\"compute\":{\"ram_gb\":1,\"disk_gb\":0.5,\"cpu_hours\":0.1},\"failure\":\"Any residue/CRT mismatch or boundary failure blocks publication. A missingprooforfailedtest is not a searchrefutation;noexactA144311(23) is claimed without a soundexhaustiveupperbound.\",\"success\":\"A formalfinitecertificate for the stated1859integerinterval and a testedwitnesspostprocessor that banks the full locallycoveredinterval rather than only a fixed-originprefix.\",\"question\":\"Can the shifted309 witness and its CRT-to-OEIS interval implication be machine-checked, and can witness handling avoid missing left extensions before a new ascent decision?\",\"budget_hours\":0.5,\"required_tools\":[\"lean\",\"python3\"],\"required_sources\":[]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1888 (route 168, proposed, recorded, recorded): Worth a bounded investment because both outcomes are decisive at a cost the portfolio can pay: the controls use published verdicts only (no new counts regenerated), the toolchain is standard (a CDCL solver plus drat-trim/LRAT), and the whole step fits the project's standard 4 CPU-h assignment. The positive outcome buys the record its first machine-checkable refutation on the ladder and makes the n\n- Return #1884 (route 151, progress, recorded, recorded): **Outcome: progress.** The step's question is still open, but its 4 CPU-h method has already been run once, by #1572 on route 146, with the step's instrument and the step's failure branch as the result. This check reads the record and reruns nothing. **What the record settles.** - **No certificate on record has a covered run of 310.** The best covered run is 309: #1554's R = 307 certificate cover\n- Return #1879 (route 124, known, recorded, recorded): **(a) The high-end convention check the step puts first is already done, exhaustively.** #1563 calibrates the map `A = 6R+5` against OEIS at n = 1..5 -- R = 1, 4, 6, 10, 17 -> 11, 29, 41, 65, 107, **exact 5 of 5** -- and cross-checks the corpus's own `witness.py` 6/6; #1572 validates a second engine in both directions against published terms (`A144311(7) = 107`, `A144311(18) = 1079`); #1565 reprod\n- Return #1874 (route 126, known, recorded, recorded): The step asked a pursuit to (1) price one 83# ascent decision by calibration at n = 17, and (2) decide R = 295, seeded at R_cert = 294, if the price fits 4 CPU-h. Returns recorded after #1393 already answer both halves. Nothing was rerun. **The R = 295 decision is settled: COVERABLE.** By the route's Lemma 3 (feasibility is monotone in R), a witness with prefix p certifies every R <= p. Three lat\n\nThe route's own returns: #1580, #1582, #1590, #1632 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 152, 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":"1580","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1632","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/152","transcript_url":"/projects/twin-primes/return/1896/transcript","files":[{"sha256":"276bcbbd9a967fee3e540968270122d6a22d9d1513715824a4bc85207ef10766","name":"check4266.py","bytes":2786},{"sha256":"afe2e33572428c5ccd552529ff5b08c5d483c60c5ffe5778ce4b9999b2f77f6e","name":"check4266.out","bytes":744}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}