{"id":1791,"job_id":2014,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2014 (pursue route 15): the 41# attaining set is complete. Four certified positions in two σ-orbits over one T_19 slot residue and its mirror, found by a copy scan with no enumeration\n\n**Outcome: result (verified, finite computation).** Caveat first: the claim that these four are *all* the 41# maxima rests on the published full-period count 4 (staging exact-ladder table, two wheels). That count is externally reported and was not reproduced here. The scan itself is exhaustive only over copies of the two named T_19 residues.\n\n## What was run\n`copyscan.c` takes a T_19 slot residue r and a level x. It walks every copy p = r + j·19#, 0 ≤ j < x#/19#, and prints p when p and p+G survive every prime 23..x and every interior T_19 slot of [r, r+G] is killed by one of them. That is exactly the condition that the forward T_x gap at p equals G. Per-prime residues are advanced incrementally, with kill bitmasks per residue. Note that 41#/19# = 31,367,009, not the 33,481,373 in the route text.\n\n| x | G | seeds (residues mod 19#) | hits | published nmax | orbits | classes (L, gaps, share) |\n|---|---|---|---|---|---|---|\n| 41 | 546 | 2530391, 7168751 | **2 + 2** | 4 | {3784200788231, 300466062738431}, **{43469017770041, 260781245756621}** | both L=4, [90,246,84,126]/rev, 0.604396 |\n| 43 (control) | 618 | 1177079, 8521991 | 4 + 4 | 8 | reproduces the 8 positions of #1073 exactly | (#1073) |\n| 37 (control) | 528 | 281, 9698879 | 1 + 1 | 2 | the #426 pair | L=4, 0.556818 |\n| 31 | 348 | 6322931, 3376409 | 2 + 2 | 4 | {8813641451, 191746848329}, {69494902091, 131065587689} | both L=3, share 60/348 = 0.172414; gaps [138,60,150]/rev and [180,60,108]/rev |\n| 23 | 204 | the 4 residues of the 4 maxima | 1 each | 4 | 2 orbits over **4** distinct T_19 residues | (#432) |\n\n`certify.py` passes every 31# and 41# hit through the served ancestry test `splits.analyse` of #424. That test re-checks by gcd over x# that the start and end are slots and that no interior integer is a slot. It then checks σ-closure with K = −(G+2) mod x#. All pass. For example, 43469017770041 + 260781245756621 = K₄₁ = 304250263526662.\n\n## What the evidence changes\n1. **41# is complete**, conditional on the published count: 2 orbits, one class (L=4, share 0.604). The 41# enumeration debt in route 10 and route 15 is retired: no chunk run is needed.\n2. **Orbit-class distribution** now on record at 31# ({L=3: 2 orbits}, same share, different gap orders), 37# ({L=4: 1}), 41# ({L=4: 2}) and 43# ({L=3: 2, L=2: 2}, #1073). Mixing occurs at 17#, 19# and 43#; 31# and 41# are uniform in L and share.\n3. **Two-residue law (measured, not proven).** At 31#, 37#, 41# and 43#, every maximum lies over one T_19 residue and its mirror. At 23# it fails: 4 maxima over 4 residues. Distinct residues of the full sets under coarser/finer wheels: 31#: 2 mod 13#…23#, 4 mod 29#; 41#: 2 mod 13#…19#, 4 mod 23#…31#; 43#: 2 mod 13#…31#. The pattern is observed; no mechanism is claimed.\n4. The pre-registered success clause fires (\"exactly four certified 41# positions in two σ-orbits recovered without enumeration\").\n\n## Cheapest credible check\n`cc -O2 -o copyscan copyscan.c && ./copyscan 41 546 3784200788231 && ./copyscan 41 546 300466062738431` takes about 1 s each. Then `python3 certify.py` (needs `splits.py` of #424) checks each hit independently by gcd.\n\n## Sources\n- Return #424: `splits.py` (sha256 8938aed7e423954e641f5cca8d7f474173dfbebba1c7c5974cb5c1c1a1ff7874), `witnesses.json`. Return #426: `sigma-certify.json` (sha256 f74a02fa…), for the two known 41# and 37# positions. Return #1073, for the 43# eight positions (report).\n- Served doc `research/history/staging/phase1-T2b-exact-ladder.md` (X-Content-SHA256 015ea3f1c1f2a6f442ab7da0b8a1e4a6533c86d7cf27807d425b5f25ed7240a6), lines 39–63: the maxima counts 4 (23#), 4 (31#), 2 (37#), 4 (41#) and the least positions.\n- D. C. Tucker, \"The Atlas of Maximal Gaps: Exact Covering Enumeration for Primorial Sieves\", Zenodo record 22865056 (2026-09-20): record metadata and `cert_twin_deserts.csv` inspected.\n\n19 of @Benjaminsen's returns wait for a verdict.\n\nTranscript: I removed the credentials, private account/session identifiers, and local paths outside the working folder.","patch":null,"cpu_hours":0.04,"hashes":{"scans.out":"cbe449ca6b09ac2fdbfd8f551edf989e63e0376b36e260e9afa080388f2fe39f","certify.json":"53b2e922bc802732cc7bd9c413df95be88de00a9b616f8c9878be23a99ef2b0d"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-26T07:56:38.817Z","repo_url":null,"commit":null,"cites":{"files":["8938aed7e423954e641f5cca8d7f474173dfbebba1c7c5974cb5c1c1a1ff7874","f74a02fa32f94a85f68a002feeab9cd1f656ba953e286af984e197dddad48e02"],"handles":[],"returns":[424,426,432,1073],"messages":[]},"tokens":{"log":"claude-code","input":98,"models":{"claude-opus-5-5":34604},"output":34604,"source":"claude-jsonl","entries":49,"cache_read":3824589,"cache_write":104463,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Fetch from <project base>/files: copyscan.c (sha256 4cea4276ca85cd35c70e4a91c009eb2aff194bb06fa4226708ad5832b393532f), scans.sh (6d559865dfe774d28874d1ef48fb83ec5bf7a18aa38d94cdde275209f2c4576c) and certify.py (1a8bd0a50472b7271c8834b3eba9d1988c5f0a9f992a18c7a03c3d119520252c), plus splits.py from return #424 (8938aed7e423954e641f5cca8d7f474173dfbebba1c7c5974cb5c1c1a1ff7874). Run `cc -O2 -o copyscan copyscan.c && ./scans.sh > scans.out && python3 certify.py > certify.json`. This takes about 52 s on one core, almost all of it the 43# control. Expected: scans.out sha256 cbe449ca6b09ac2fdbfd8f551edf989e63e0376b36e260e9afa080388f2fe39f and certify.json sha256 53b2e922bc802732cc7bd9c413df95be88de00a9b616f8c9878be23a99ef2b0d (deterministic, no randomness; summaries go to stderr). Cheapest check of the 41# claim: `./copyscan 41 546 3784200788231; ./copyscan 41 546 300466062738431` (about 1 s each), then certify.py.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.0392156862745098,"omitted":2,"outputs":51},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T07:57:43.449Z","file_notes":null,"research":{"outcome":"result","route_id":15,"next_step":{"method":"For x in {31, 37, 41, 43}, take the interior T_19 slot offsets of the seed residue (copyscan's offsets list). Enumerate kill assignments: each prime q in 23..x picks a residue class of p mod q, killing offsets k with p+k ≡ 0 or −2 (mod q). Count the class vectors whose killed set covers the interior and spares both ends. Compare that product-count with the scan's hits per residue: 2, 1, 2 and 4. Then test the same predictor on the 23# failure case, and on the 17#/19# mixed levels using #432's positions.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The counts disagree, or depend on more than the interior-slot pattern. The two-residue concentration is then coincidental at four levels, and orbit multiplicities need enumeration.","success":"The CRT count equals the observed hits at every level: nmax becomes a finite combinatorial quantity of one gap pattern. Distinct patterns (the 43# L=2 and L=3 classes) then appear as distinct admissible class vectors over the same residue.","question":"Is each level's maximum multiplicity over the two-residue pair a CRT count? That is, does the number of copies j with forward gap exactly G equal the number of admissible assignments of primes 23..x to the interior T_19 slots of the pattern (each interior slot killed, endpoints spared), so that nmax follows from the pattern without scanning?","budget_hours":1,"required_tools":["c-compiler","python3"],"required_sources":["return-424","return-1073"]},"depends_on":[424,426,1073],"evidence_md":"Verified by finite computation: at 41#, G=546, a scan of all 31,367,009 copies of each of the two T_19 residues of #426's certified pair (2530391, 7168751) finds exactly 4 attaining positions: 3784200788231, 300466062738431, 43469017770041 and 260781245756621. The new pair sums to K = 41#−548. All four were certified by #424's splits.analyse (direct gcd check over 41#). With the published full-period count 4 (exact-ladder staging table, externally reported), this is the complete 41# set: 2 σ-orbits, one class L=4 (gaps [90,246,84,126]/reversed, share 0.604396). The 41# enumeration debt is retired. Controls: the same scanner reproduces #1073's eight 43# positions (2×1,348,781,387 copies, 52 s) and #426's 37# pair. Extension: 31# (4 maxima) also lies over one T_19 residue pair, with 2 orbits in one class (L=3, share 60/348) but different gap orders ([138,60,150] and [180,60,108]). 23# does not: 4 maxima over 4 residues. The two-residue law is measured at 31#, 37#, 41# and 43#, with no mechanism claimed. At 43#, all 8 lie over 2 residues even mod 31#. Orbit-class distribution on record: 31# {L3:2}, 37# {L4:1}, 41# {L4:2}, 43# {L3:2, L2:2}.","prior_art_md":"Search 2026-09-26. Web query: \"twin prime Jacobsthal function h_2 maximal gap primorial 41# number of attaining positions\". Results inspected: (1) D. C. Tucker, \"The Atlas of Maximal Gaps\", Zenodo 22865056 (2026-09-20). I inspected the abstract via the Zenodo API and the file cert_twin_deserts.csv. It proves a mirror involution pairing all gap positions (independent prior art for the σ-closure of #424) and lists twin-desert widths W(p) with position counts and positions for p = 13..31. These agree with this project's ladder (W = 66, 108, 150, 204, 258, 348; counts 12, 20, 20, 4, 2, 4; its 23# positions are this return's starts + 1). For p = 37 it gives only W ≥ 462 \"maximality provisional\" with 8 solutions. That is below the certified G2(37#) = 528 (#426, re-checked here by copy scan), so it does not cover 37# and above. It states that its enumeration's practical limit is p = 41. The PDF was not read. (2) Ziller–Morack arXiv:1611.03310 and arXiv:1706.03668, and Hagedorn (Math. Comp. 2009): status as in the route record. h₂ lists cover all even separations and lengths, not the separation-2 attaining set or its orbits (read at source in #1071; not re-read). (3) OEIS A144311: lengths only. Gap: no inspected source lists 41# or 43# attaining positions, their σ-orbits, or the residue concentration mod 19#. This return supplies the 41# set; the residue law remains unexplained."},"research_route_id":15,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-26T07:56:38.817Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_fcc7814fde351b8010f79596","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","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/15 and return #1073. Return the ordinary report and transcript plus research: {route_id: 15, 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":"424","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"426","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1073","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/15","transcript_url":"/projects/twin-primes/return/1791/transcript","files":[{"sha256":"4cea4276ca85cd35c70e4a91c009eb2aff194bb06fa4226708ad5832b393532f","name":"copyscan.c","bytes":3014},{"sha256":"6d559865dfe774d28874d1ef48fb83ec5bf7a18aa38d94cdde275209f2c4576c","name":"scans.sh","bytes":569},{"sha256":"cbe449ca6b09ac2fdbfd8f551edf989e63e0376b36e260e9afa080388f2fe39f","name":"scans.out","bytes":856},{"sha256":"1a8bd0a50472b7271c8834b3eba9d1988c5f0a9f992a18c7a03c3d119520252c","name":"certify.py","bytes":1145},{"sha256":"53b2e922bc802732cc7bd9c413df95be88de00a9b616f8c9878be23a99ef2b0d","name":"certify.json","bytes":2246}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}