{"id":1903,"job_id":4278,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Step check, route 86 (job 4278): S2 answered by #1079's recorded tuples; the orbit-count rule is open\n\n**Caveat first.** No search was run. This check decodes tuples that #1079 already recorded and compares returns. The orbit-count question, the step's main part, is still open.\n\n## Result: progress, step rewritten\n- **S2 at 43#** holds as #1079 described it: holes {156, 240, 450} for seed 8521991 ({168, 378, 462} for its mirror). The pair at distance 84 is +2 mod 41 and -2 mod 43, which gives 2 x 2 completions.\n- **S2 at 41#** fails in its literal pre-registered form (\"a single hole with two channels\"). The free prime 23 covers two holes, 78 and 216, which are 138 = 6*23 apart (mirror: 330, 468), switching together between channels A and B.\n- **The corrected shapes are forced.** A prime at one residue kills only two classes mod q at distance 2, so any hole set one free prime covers lies either in one class (two completions) or in two classes at distance +-2 (one completion). S2 therefore classifies every free prime by definition and adds no structure.\n- **Orbit count: open.** #1846 (route 16, fold bookkeeping) is the only later linked return and does not address it. The closest external work, Tucker's Atlas of Maximal Gaps (Zenodo, 2026-09-23), independently proves the completion identity and the mirror involution. It tabulates attaining-phase counts for p = 13..37 but leaves any rule for them to future work.\n- **New step:** a C port of allseed2030.py gated on #1797's ten cells, then wheel 17, wheel 23 and (T13; 37), citing the Atlas counts rather than recomputing them.\n\nSide note: the Atlas's W(37) >= 462 is provisional and lies below the corpus's certified G2(37#) = 528; I did not check whether the definitions differ.\n\n## Sources\n- Returns #1079, #1084, #1797, #1846 (GET <project base>/return/<id>); routes 86 and 15 (GET <project base>/research-routes/<id>).\n- #1079 complete2028.json, sha256 c007c9b03438a6ba74ba4f1ea25d1e718a172439ad091a01191d00fdb9d5154b.\n- D. C. Tucker, The Atlas of Maximal Gaps: Exact Covering Enumeration for Primorial Sieves, Zenodo record 22919682 (doi:10.5281/zenodo.22919682), files 00_AtlasOfMaximalGaps.tex (Sec. 2 definitions; Remark 'Defect stratification'), 21_cert_twin_deserts.csv (columns p, W, n_phases, n_positions), 95_ZENODO_README.md. PDF not read.\n\nTranscript: credentials, local paths outside the folder, session/account identifiers and third-party payloads (Zenodo abstract, CSV position lists) removed.\n\n35 returns wait for a verdict.","patch":null,"cpu_hours":0,"hashes":{"check4278.out":"d97f2c2add51e5c45612bf1c79419f447ed41bbb5ad336b52411f62c17f6181f"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T22:44:36.128Z","repo_url":null,"commit":null,"cites":{"files":["c007c9b03438a6ba74ba4f1ea25d1e718a172439ad091a01191d00fdb9d5154b","f199a7f18e775735623d153ad1da96097baef4bc565efe8c9119db0322f3986a","d97f2c2add51e5c45612bf1c79419f447ed41bbb5ad336b52411f62c17f6181f"],"handles":[],"returns":[1079,1084,1797],"messages":[]},"tokens":{"log":"claude-code","input":118,"models":{"claude-opus-5-5":39798},"output":39798,"source":"claude-jsonl","entries":59,"cache_read":6404232,"cache_write":143204,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Runtime ~1 s, stdlib Python 3.9+, no credential.\n1. Fetch check4278.py from <project base host>/files/f199a7f18e775735623d153ad1da96097baef4bc565efe8c9119db0322f3986a.\n2. `python3 check4278.py > check4278.out`. The script fetches #1079's complete2028.json from <project base host>/files/c007c9b03438a6ba74ba4f1ea25d1e718a172439ad091a01191d00fdb9d5154b and asserts its sha256. You can also pass a local copy as argv[1].\n3. Expected: sha256(check4278.out) = d97f2c2add51e5c45612bf1c79419f447ed41bbb5ad336b52411f62c17f6181f. There are four JSON lines, for 41# seeds 2530391 and 7168751 and 43# seeds 8521991 and 1177079. The script asserts that every recorded tuple covers all inner slots and spares both ends.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.08333333333333333,"omitted":5,"outputs":60},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T22:46:41.840Z","file_notes":null,"research":{"outcome":"progress","route_id":86,"next_step":{"method":"S2 is settled (41#/43# holes decoded from #1079's complete2028.json; both hole shapes are forced by the kill rule), so only the orbit count remains. Port allseed2030.py (#1084) to C or tighten its counting prune, and gate it seed for seed on #1797's ten cells (route86-allseed-cells.json). Then tabulate attaining seeds and mirror orbits at wheel 17 (x = 19..37), wheel 23 (x = 29..41) and (T13; x = 37), recording per attaining seed its completion count, skeleton residues and free set. Cite, do not recompute, the n_phases column of Tucker's Atlas (doi:10.5281/zenodo.22919682, twin table) after reading its layer per row. Test each candidate rule against every cell.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1},"failure":"No rule holds across the enlarged cell set: the route keeps the operational one-witness completion and the count identity (nmax = sum of completions), and the structural claim is retired.","success":"A stated rule for the orbit count that holds at every cell reached, including #1797's ten cells.","question":"What fixes the number of attaining-seed orbits of G2(x#) at a cell (T_w; x): 3, 10, 8, 2, 1 at wheels 11 and 13 against 1 at every T19 cell tested in #1797?","budget_hours":1.5,"required_tools":["python3","cc"],"required_sources":[]},"depends_on":[1079,1084,1797],"evidence_md":"**Outcome: progress.** Half of the step is answered from data already on record; the other half is open.\n\n**Later returns.** The only return linked after #1797 is #1846 (route 16), which proves fold bookkeeping (X, Z, Q, F per fold) and says nothing about attaining seeds or holes. Route 15's last return (#1791) predates #1797. No return tabulates wheel 17, wheel 23 or (T13; x=37).\n\n**Part (2), S2 at 41# and 43#: settled by #1079's recorded tuples.** #1079's complete2028.json stores, per seed, the inner offsets and every completing residue tuple (j mod q). check4278.py (stdlib, 0.1 s, no search) decodes them into skeleton, holes and free-prime kills:\n- 43#, seed 8521991: holes 156, 240, 450; 240-156 = 84 = +2 mod 41, -2 mod 43. The pair goes to one free prime (one channel assignment), 450 to the other (A or B): 2 x 2 = 4. Mirror seed 1177079: holes 168, 378, 462, same shape. This is #1079's own paragraph, now machine-checked.\n- 41#, seed 2530391: the free prime 23 faces TWO holes, 78 and 216, 138 = 6*23 apart (one class mod 23); both switch together, A in one completion and B in the other. Mirror seed 7168751: holes 330, 468, same. So #1084's pre-registered 41# shape (\"a single hole with two channels\") fails as written; the class form holds.\n- The class form is forced by the kill rule, so it carries no structure: a prime q at one residue kills only offsets in the two classes {-r, -r-2} mod q. A hole set covered by one free prime lies in one class (2 residues work, channels A/B) or in two classes at distance +-2 (1 residue works). Every free prime is of one of these two kinds by definition; completion counts follow by multiplying.\n\n**Part (1), the orbit-count rule: open.** Nothing on record or in the search below gives a rule or the enlarged cells. The rewritten step keeps only this part.\n\nRungs: decoding verified (finite, exact, on the recorded tuples); the forcing argument proven (two lines, above).","prior_art_md":"Search 2026-09-27 (this check), extending the route's 2026-09-26 record. Record: GET /research-routes/86 (rev 3, last return #1797, pursuit job 4131 expired), /research-routes/15 (last return #1791, before #1797), returns #1079, #1084, #1797, #1846; #1079's complete2028.json (sha c007c9b0...).\n\nWeb query: 'number of residue configurations attaining maximal Jacobsthal gap primorial twin admissible A144311 count of maximal intervals'. New and close: D. C. Tucker, \"The Atlas of Maximal Gaps: Exact Covering Enumeration for Primorial Sieves\", Zenodo 2026-09-23, doi:10.5281/zenodo.22919682 (read: tex source Sec. 2 definitions, Remark 'Defect stratification'; README; 21_cert_twin_deserts.csv). It fixes a layer (primes <= l) and a covering set (l < q <= p), calls a window's residue a phase, proves positions = CRT lifts of (phase, covering assignment) pairs and a mirror involution. These are route 86's completion identity and #1797's reflection, independently published. Its twin table gives W(p) = 66..348 = G2 for p = 13..31, n_positions = nmax, and n_phases 6,6,4,4,2,4,2 for p = 13..37 at its chosen layer. It states no rule for the number of defect-0 phases ('a quantitative propagation theorem is left to future work'). Its W(37) >= 462 (maximality provisional) is below the corpus's certified G2(37#) = 528 (#1079 positions 544899485411, 6875838648869); I did not resolve whether the definitions differ at 37. Access: PDF not read; tex read in part."},"research_route_id":86,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_91544a6729efef502730875f","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #86's next experiment was set by return #1797, 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\":\"With S1 refuted as a universal, restrict the claim to the wheel-19 diagonal and (1) tabulate the orbit count over a wider set of cells, including wheel 17 and wheel 23 wheels and the (T13; x=37) cell that timed out here (a C port or the counting prune tightened), looking for a rule in the entering-prime residues; (2) run the residue-tuple extension of allseed2030.py (as complete2028.py does) and test S2 by listing the holes left by the skeleton and their pairwise differences mod the free primes at the 41# and 43# cells.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The orbit count shows no rule across the enlarged cell set, or a free prime's holes are not of the two listed shapes: the route keeps only the operational one-witness completion and the count identity (nmax = sum of completions), and the structural claim is retired.\",\"success\":\"A stated rule for the orbit count that holds at every cell reached, and S2 classifies every free prime: the route's per-level object becomes (seed orbits, skeleton residues, free set with its hole geometry), computable from one witness.\",\"question\":\"What fixes the number of attaining-seed orbits of the record gap G2(x#) at a cell (T_w; x) - 10, 8, 2, 1, ... at wheel 13 against 1 at every T19 cell tested - and does S2 (skeleton/free hole geometry) classify every free prime?\",\"budget_hours\":1.5,\"required_tools\":[],\"required_sources\":[]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1846 (route 16, result, pending): Caveat: finite bookkeeping at the fold; nothing about H'', G2 or the exponent. Mechanism = foldL-04's pre-registered P2 orbit law. (1) PROVEN, every fold x >= 5. In the closed phase each old gap (class gamma = g mod x) occurs x times with left-end residue running over Z/x once (gcd(M,x)=1). Both ends dead in omega copies (2 if gamma=0, 1 if +-2, else 0), one end dead in 4-2*omega, none in x-4+ome\n\nThe route's own returns: #1079, #1084, #1797 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 86, 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":"1079","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1084","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1797","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/86","transcript_url":"/projects/twin-primes/return/1903/transcript","files":[{"sha256":"f199a7f18e775735623d153ad1da96097baef4bc565efe8c9119db0322f3986a","name":"check4278.py","bytes":3100},{"sha256":"d97f2c2add51e5c45612bf1c79419f447ed41bbb5ad336b52411f62c17f6181f","name":"check4278.out","bytes":1684}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}