{"id":891,"job_id":1361,"problem_id":1,"lane_id":5,"type":"explore","user_id":42,"model":"deepseek-v4-pro","provider":"deepseek","report_md":"# Route 23: maxsum doubling certificate at s=34 — independent re-verification and prior-art update\n\n## Result (calibration: Verified)\n\nReturn #603's decisive negative at s = 34 is independently reproduced by a separate pure-Python implementation (`verify_s34_witness.py`, gcd + divisibility only; no sieve engine, no CRT phase search, no published `kstar.c`).\n\nBoth covering runs cited in #603 verify exactly:\n\n- **L=26 run**: first position 2365947737538493655694107, 26 offsets (span 840). Every position r has gcd(r, 31#) = gcd(r+2, 31#) = 1 (a genuine level-34 slot under the P(34)=31 profile); every r or r+2 is divisible by some q in Q(34) = {37,41,43,47,53,59,61,67}; the span contains exactly 26 level-34 slots (consecutive).\n- **L=27 run**: first position 1495837883547832847406497, 27 offsets (span 840); all three checks pass.\n\nTherefore **K*(34) >= 27**, and msc(34) >= maxsum_28(T_31)/348 = 1470/348 = 4.2241 > 4. The maxsum doubling certificate fails at s = 34, so the instrument reaches exactly one distinct fold (31# -> 61#, s = 32, msc = 3.9655) and not the second (31# -> 67#).\n\n**Scope and gaps.** This re-verifies the lower bound K*(34) >= 27 only. The exact K*(34) (whether it is 27 or larger) remains open and requires the negative-direction complete block scan at L = 28, 29, ... (the route's own next experiment). No bound on Ghat is claimed; beta_2 is untouched; rows 90 and 94 stand; no twin-prime claim.\n\n## Prior art (updated for this experiment)\n\n- **Ziller–Morack, arXiv:1706.03668** (\"A short note on the computation of the generalised Jacobsthal function for paired progressions\"): defines the *paired Jacobsthal function* h2 (the shift-2 two-class covering run) and computes it for primorials up to p = 73 (values 2,6,18,...,2622); proves h2(n) < p_n^2 - p_n is sufficient for the prime-pairs (twin) conjecture. This is the established literature object matching this route's K*. Read at source (ar5iv HTML) this session.\n- **Ziller, arXiv:1903.11973** (\"New computational results on a conjecture of Jacobsthal\"): one-class Jacobsthal function computed to k = 43, disproof-by-counterexample framing.\n- **Nguyen, preprint 202608.1299** (\"Finite-Window Noncovering on Primorial Wheels...\", flagged NEW AND UNREAD by #603): still returns HTTP 403 from this machine (preprints.org), so the content remains unread and the access gap persists. The next worker must obtain it before any claim that the two-class primorial-slot run object is unstudied.\n- **MathOverflow 497359** (sparse residue-class covering): Cloudflare-blocked from this machine; #603's summary stands (answer reduces to a stronger binary Goldbach; no run-length bound supplied).\n\nNothing external was found that computes K*(s) at any rung or exhibits a two-class covering run at this scale; the exact K*(34) remains this project's own open number.\n\n## Cheapest credible check\n\nThe verification is division-only and reproducible: `python3 verify_s34_witness.py` prints PASS for both runs (sha256 of stdout: see `hashes`). It consumes only the integer positions and offsets quoted from #603.\n","patch":null,"cpu_hours":0.01,"hashes":{"verify_s34_witness.out":"2b22421f98d867b534ab90d2af1ef4d2a8c28dc740884dd6738d79b82f7f542f"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T15:33:32.313Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[603],"messages":[]},"tokens":{"log":"custom","input":115995,"models":{"deepseek-v4-pro":113410},"output":113410,"source":"custom-jsonl","entries":68,"cache_read":8694912,"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":"progress","route_id":23,"next_step":{"method":"Run the route's stated negative-direction complete block scan at L=28 on the reworked engine (6226553025 windows in 12 segments; a 12/12 no-hit verdict fixes K*(34)=27, a verified 28-window continues L=29,30,...). Report per-segment receipts, the arithmetic verification of every witness, and the measured core-hours.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"A partial-scan K*(34), an unverified witness, or a cost without its measured prefix/rate.","success":"Exact K*(34) with per-segment receipts and verified witnesses, so msc(34) is exact.","question":"What is the exact K*(34), hence the exact msc(34)?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[588,594,599,603],"evidence_md":"Independent pure-Python re-verification of return #603's s=34 witnesses: both the L=26 run (first 2365947737538493655694107) and L=27 run (first 1495837883547832847406497) satisfy gcd(r,31#)=gcd(r+2,31#)=1 for every position, each is killed by some q in Q(34)={37,41,43,47,53,59,61,67}, and the span holds exactly 26 resp. 27 consecutive level-34 slots. Hence K*(34) >= 27 and msc(34) >= 4.2241 > 4: the certificate fails at s=34. This hardens the decisive negative with a second, independent implementation (gcd/division only). The exact K*(34) (27 or larger) is still open.","prior_art_md":"Updated 2026-09-17. Ziller-Morack arXiv:1706.03668 defines/computes the paired Jacobsthal function h2 (the shift-2 two-class covering run) to p=73 and proves h2 < p_n^2-p_n suffices for the twin conjecture — the established object matching K*. Ziller arXiv:1903.11973 covers the one-class case to k=43. Nguyen preprint 202608.1299 remains HTTP 403 (unread; access gap persists from #603). MathOverflow 497359 Cloudflare-blocked here. No external source computes K*(s) at any rung or exhibits a two-class covering run at this scale; exact K*(34) is this project's own."},"research_route_id":23,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_563f32992e16ba94565fae61","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","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/23 and return #603. Return the ordinary report and transcript plus research: {route_id: 23, 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":"588","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"594","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"599","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"603","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/23","transcript_url":"/projects/twin-primes/return/891/transcript","files":[{"sha256":"049fd28bfaa020969948f0ae6221e799c803d91ab61a6648524e512416622584","name":"verify_s34_witness.py","bytes":3250},{"sha256":"2b22421f98d867b534ab90d2af1ef4d2a8c28dc740884dd6738d79b82f7f542f","name":"verify_s34_witness.out","bytes":441}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}