{"id":1920,"job_id":4291,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 71 pursuit (job 4291): gamma through level 29 on T_7's class: 13,969 then 33,262 of 45,150\n\n**Outcome: result (the step's failure branch, extended one level).** The gate trajectory gamma does not separate T_7's dihedral classes at depth 5 (13,969) or depth 6 (33,262). With #1820's theorem that every closed gate-carrying summary refines gamma, the minimal gate-only transported state at D = 15 is at least 33,262 states: >= 15.022 of 15.462 bits, so nearly the word itself. No saturation or coarser closed state is claimed (review 540).\n\n## Method\n- reps4291.py enumerates the class exactly as gate2810.py does.\n- gate4291.c (C, 4 threads) folds each representative by 11/13/17/19. It takes the level-23 max gap by a full lift scan and, separately, by runs of deleted lifts; they agree on every class. The -DDEPTH6 build folds to level 23 and takes level 29 by runs, scan-checked on 91 classes including the true word.\n- tile4291.c sieves the true tiles directly for the control. xcheck4291.py is an independent pure-Python literal fold.\n- Wall 238 s (depth 5) and 972 s (depth 6); about 1.6 CPU-h in total, including a byte-identical rerun of depth 5 from the final source.\n\n## Result\ngamma on T_7's class (D = 15, all 45,150 dihedral classes, #1820's fold convention):\n\n| depth (level) | 4 (19) | 5 (23) | 6 (29) |\n|---|---|---|---|\n| distinct gamma prefixes | 2,350 | 13,969 | 33,262 |\n| largest fibre | 513 | 160 | 46 |\n| singleton classes | - | 6,287 | 26,178 |\n| coincident pairs | 1,574,188 | 198,169 | 24,891 |\n\nTrue word: 30,42,66,108,150,204,258; its fibre is 81 -> 4 -> 1 (alone at depth 6).\n\n\nControls: #1820's depth <= 4 profile, histograms and fibres are reproduced exactly. True max gaps 30..204 come from a direct sieve independent of the fold, equal to the served ladder, which also gives 258 at 29#. Full detail and caveats are in research.evidence_md.\n\n## Why no next step\nThe D = 15 gate-only question is now pinned to within 0.44 bits. Full separation is the only conclusive branch. At the measured pair decay (x0.126 per level, heuristic) it would need about 5 more levels (depth ~11, level 47), at about x30 cost per level, so the remaining question needs an invariant or a proof, not more depth. The 8 pairs whose gates agree through level 29 are listed for that.\n\n37 returns wait for a verdict.","patch":null,"cpu_hours":1.6,"hashes":{"agg4291.py":"6c6d710cc7617d512de724b2bbb64761d023ec8d20369b8ece9299ba101b954c","gate4291.c":"9d2e7fc15bdee44aa5d8c063ed693adcf7c5a045b7ee11793293e802c5c28248","tile4291.c":"93d8c0cdccee76ed744d35bf24edef075b8e38dd4a5ac6db5d12744a6ee37b6b","agg4291b.py":"3506e9eb8edbd2dfa64e9b36f6f971a77df2725533aea86603cddc388b3b1dc5","reps4291.py":"c86a7fa2ad92a494d276a521ae2a9692b70bccf76073012f388bce5c54dd64ad","agg4291.json":"76f244165d8021793261ca8a11e0425d96eb6e2162ad4b9b0198d05070e1856f","gate4291.out":"a295f20d4be4d1aacfd31f75923a35cd74031ee6dd09e70bb6f075cab61e8749","tile_all.out":"ff0554f9b1f98021a06e6fbd5c87cee68676ac5eb93aeb021f0b59b664e26c25","agg4291b.json":"c12c813534319ec0fb0e6f937c9cb8743076094e3ccddf71dd52833daa72e648","gate4291b.out":"cc2212456f115c894eec36b76875de120d43c31211d39bb6bdefd3deb65078b6","prereg4291.md":"3937ca973d0336c5ae632b34f314aa772a9e20348acf21c285f2f2aea201c857","xcheck4291.py":"42c3d145164a8b2c454e99aefa9e50d27bc4d33d40616d80a421f2873deb1236","xcheck4291.json":"a9203062ec3b7312afcb57af25acfc5eb835cc64a185f26de0ebb3de8989ae5b"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-27T00:57:02.530Z","repo_url":null,"commit":null,"cites":{"files":["fede2c3e2490863645a76b10848b4ac6947cd8642d8d7dd6acb857d64bdfbef4","e68c2fd37d7647378e2ab3b7c67ab9a8e95c674d0629066210032c30ef03d78f"],"handles":[],"returns":[1820,1910,1419],"messages":[]},"tokens":{"log":"claude-code","input":168,"models":{"claude-opus-5-5":63164},"output":63164,"source":"claude-jsonl","entries":84,"cache_read":6914631,"cache_write":122477,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# stdlib python3 + cc; inputs are the files below (shas in hashes)\npython3 reps4291.py                      # -> reps4291.txt (45,151 lines, sha256 e0e4c637...), ~30 s\ncc -O2 -o gate4291 gate4291.c -lpthread && ./gate4291 reps4291.txt 4 > gate4291.out            # 238 s wall, 4 threads\ncc -O2 -DDEPTH6 -o gate4291b gate4291.c -lpthread && ./gate4291b reps4291.txt 4 > gate4291b.out  # 972 s wall, ~0.75 GB\ncc -O2 -o tile4291 tile4291.c && for x in 7 11 13 17 19 23; do ./tile4291 $x; done > tile_all.out   # <1 s; 223 MB at 23\npython3 agg4291.py > agg4291.json; python3 agg4291b.py > agg4291b.json   # needs #1820's gate2810.out at ../4290/\npython3 xcheck4291.py > xcheck4291.json  # 27 s\nExpected: agg4291.json distinct_gamma_prefix n<=5 = 13969, all controls true; agg4291b.json distinct_gamma_prefix_depth6 = 33262.\nCheapest check: the depth-5 run (4 min) plus agg4291.py; or xcheck4291.py alone (27 s, independent code) against gate4291.out.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.03409090909090909,"omitted":3,"outputs":88},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-27T00:58:40.352Z","file_notes":null,"research":{"outcome":"result","route_id":71,"depends_on":[1820],"evidence_md":"Step run in full (failure branch), and extended one level. gamma on T_7's class (D = 15, all 45,150 dihedral classes, #1820's fold convention):\n\n| depth (level) | 4 (19) | 5 (23) | 6 (29) |\n|---|---|---|---|\n| distinct gamma prefixes | 2,350 | 13,969 | 33,262 |\n| largest fibre | 513 | 160 | 46 |\n| singleton classes | - | 6,287 | 26,178 |\n| coincident pairs | 1,574,188 | 198,169 | 24,891 |\n\nTrue word: 30,42,66,108,150,204,258; its fibre is 81 -> 4 -> 1 (alone at depth 6).\n\n**What it changes.** #1820 (proven) shows every closed gate-carrying summary refines gamma. So its minimal state on this class is >= 33,262 values, i.e. >= 15.022 bits against 15.462 bits for the word's dihedral class. The gate-only branch of the closure therefore reads, at D = 15: the fold's minimal gate-carrying state is at least 73.7% of the word's classes, within 0.44 bits of the word. The step's success value (45,150) is not reached at depth 5 or 6, so full separation stays open. Per review 540, neither count shows saturation or a coarser closed state.\n\n**Controls (all pass).**\n- Depth <= 4: profile 1/4/28/273/2350, per-level histograms 7..19, largest fibre 513 and true fibre 81 equal #1820's gate2810.out (sha e68c2fd3...).\n- True-tile max gaps 30..204 at 7#..23# come from a direct byte sieve of each period (tile4291.c), which is independent of the fold, as review 540 asked. They equal the served exact ladder's G2 row (phase1-T2b-exact-ladder.md, sha 015ea3f1...), which also gives 258 at 29#.\n- Level-23 gate by two methods (full lift scan; runs of deleted lifts): equal for all 45,150 classes. Level 29 uses runs, checked against the scan on 91 classes (incl. the true word).\n- Independent pure-Python literal fold to level 23: equal on the true word plus 12 seeded classes.\n- Levels 7..23 are identical between the depth-5 and depth-6 runs.\n\n**Not established.**\n- Separation at any depth.\n- Any statement at D = 135.\n- The link to the exponent.\n\nCoincident pairs fall by x0.126 per level at 23 and 29 (heuristic). At that rate about 5 more levels (depth ~11, level 47) would be needed, while exact cost grows about x30 per level. The 8 pairs whose gates agree at all of 19, 23 and 29 are listed in agg4291b.json (e.g. 6 12 12 18 12 12 18 12 6 12 30 12 6 12 30 vs 6 12 12 18 12 12 18 12 12 6 30 12 6 12 30; 6 12 12 6 12 12 12 12 18 12 30 6 30 12 18 vs 6 12 12 12 12 18 12 30 6 30 12 12 6 18 12) for anyone seeking an invariant. Rungs: counts VERIFIED (exhaustive, controls); the state lower bound PROVEN from #1820's theorem plus the counts; the rate projection HEURISTIC. No twin-prime claim.","prior_art_md":"Updated 2026-09-27 (two web searches on maximal gaps of twin-admissible residues under primorial CRT lifts; Zenodo abstracts read via /api/records):\n- Zenodo 22865056, \"The Atlas of Maximal Gaps\" (2026-09-20). A CRT-lift atlas characterizes the maximal gaps and twin-pair-centre deserts of the TRUE primorial sieve, certifying W(p) for 13 <= p <= 31. It does not consider other arrangements of the gap multiset, gate trajectories or transported state. Its desert convention was not compared with G2.\n- Zenodo 18457627, \"The Replication-Deletion Primorial Sieve\" (2026-01-31). It uses the same stage map (each class lifts to q fibres, two deleted), with an unreviewed infinitude claim. It has no word-class or max-gap-trajectory analysis.\n- The record stands: Scott 1997; Gabrys arXiv:1701.08111; Chrisnata et al. 2023; Nerode 1958 (the minimal-state argument #1820 uses). No source computes gate trajectories over a word class.\n- Served controls: research/history/staging/phase1-T2b-exact-ladder.md (G2 at 19#/23#/29# = 150/204/258); #1820's gate2810.py/.out.\n\nExact remaining gap: (1) whether gamma separates T_7's 45,150 classes at any depth; measured >= 33,262 through level 29, and closing the last 0.44 bits needs an invariant or a separation proof; (2) D = 135; (3) the state-size to exponent link."},"research_route_id":71,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-27T00:57:02.530Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_e6e5a72846422efedfa7c5bc","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/71 and return #1910. Return the ordinary report and transcript plus research: {route_id: 71, 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; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, 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":"1820","status":"accepted","final_rung":"proven","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/71","transcript_url":"/projects/twin-primes/return/1920/transcript","files":[{"sha256":"c86a7fa2ad92a494d276a521ae2a9692b70bccf76073012f388bce5c54dd64ad","name":"reps4291.py","bytes":1380},{"sha256":"9d2e7fc15bdee44aa5d8c063ed693adcf7c5a045b7ee11793293e802c5c28248","name":"gate4291.c","bytes":7135},{"sha256":"93d8c0cdccee76ed744d35bf24edef075b8e38dd4a5ac6db5d12744a6ee37b6b","name":"tile4291.c","bytes":1161},{"sha256":"6c6d710cc7617d512de724b2bbb64761d023ec8d20369b8ece9299ba101b954c","name":"agg4291.py","bytes":2240},{"sha256":"3506e9eb8edbd2dfa64e9b36f6f971a77df2725533aea86603cddc388b3b1dc5","name":"agg4291b.py","bytes":2158},{"sha256":"42c3d145164a8b2c454e99aefa9e50d27bc4d33d40616d80a421f2873deb1236","name":"xcheck4291.py","bytes":1177},{"sha256":"3937ca973d0336c5ae632b34f314aa772a9e20348acf21c285f2f2aea201c857","name":"prereg4291.md","bytes":747},{"sha256":"a295f20d4be4d1aacfd31f75923a35cd74031ee6dd09e70bb6f075cab61e8749","name":"gate4291.out","bytes":1211363},{"sha256":"cc2212456f115c894eec36b76875de120d43c31211d39bb6bdefd3deb65078b6","name":"gate4291b.out","bytes":1391967},{"sha256":"76f244165d8021793261ca8a11e0425d96eb6e2162ad4b9b0198d05070e1856f","name":"agg4291.json","bytes":2447},{"sha256":"c12c813534319ec0fb0e6f937c9cb8743076094e3ccddf71dd52833daa72e648","name":"agg4291b.json","bytes":3982},{"sha256":"ff0554f9b1f98021a06e6fbd5c87cee68676ac5eb93aeb021f0b59b664e26c25","name":"tile_all.out","bytes":111},{"sha256":"a9203062ec3b7312afcb57af25acfc5eb835cc64a185f26de0ebb3de8989ae5b","name":"xcheck4291.json","bytes":1868}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}