{"id":1369,"job_id":2733,"problem_id":1,"lane_id":4,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #2733 (explore, lane measure, triage): route 113 — basin inheritance of the free paired maximum\n\nAttempt `ef4b45ef7598a881dd2741b01f896e6b`, run `run_7c5c6bb20f3bc591c6d9cc6a`, session\n`59a164a787334fa4079848b3`; model `deepseek/deepseek-v4-flash` at thinking level `max` (two\nharness fields agree for this thread). Assignment 4, general direction. Outcome reported:\n`blocked` — a scoped obstruction, with the route's own pre-registered decider applied.\n\n**Verdict in one line.** The route's cheapest experiment was run and its pre-stated sufficiency\ntest fails at both pairs: at 13 → 17 the band that lifts to the exact optimum is 256 offsets\n(8× the attaining set, 4× the top-two cover classes, not the pre-stated \"top two\"), and at\n17 → 19 the band does not lift within the pre-registered cap at all — enlarging it 14× gains\nexactly zero. Seed-chaining from a level's own near-optimum does not transfer the free optimum\nat these bandwidths.\n\n## 1. Object, gates, and what was computed (all MEASURED)\n\n`W = x#`, `tau` even in `[0,W)`, `adm(tau) = { r : gcd(r,W)=1 and gcd((r+tau) mod W, W)=1 }`,\n`cover(tau) = (longest cyclic gap between consecutive elements of adm(tau)) - 1`. Return #675's\nobject, reused; a lift of `tau` from level `x` to the next prime is any even `tau' == tau (mod W)`.\n\nAll three gates were re-derived before any new number was read, by full sweep of every even\noffset at x = 5, 7, 11, 13 **and x = 17** (255,255 offsets, 103 s):\n\n    x     W        cover(0)+1  A048670   cover(2)  A144311   1+max cover  A288815   argmax\n    5     30           6          6         11       11          18          18        4\n    7     210          10         10        29       29          30          30       32\n    11    2310         14         14        41       41          66          66       16\n    13    30030        22         22        65       65         150         150       32\n    17    510510       26         26       107      107         192         192      128\n\nEvery gate passes. Published values from the OEIS pages and b-files fetched this run\n(A048670, A144311, A288815). `cover(2) = A144311(n)` is additionally confirmed by an\nindependent construction from A144311's own OEIS **name** (\"longest sequence of consecutive\nintegers each equal to 1 or -1 modulo at least one of the first n primes\"), which shares no\ncode with the `cover` producer and reproduces 11, 29, 41, 65, 107 at x = 5, 7, 11, 13, 17.\nNote the route's cited level-17 histogram (179: 320, 161: 1088, 1728 offsets at d <= 30) is\nshort of my complete sweep (179: 352, 161: 1120, 1792); mine is a full even-offset sweep and is\nthe one used below.\n\n**New against #1353's recorded scope.** #1353 left the full x = 17 sweep undone (\"the mid\nfacets are unmeasured, so at x = 17 my facet statement is conditional on the published\noptimum\"). It is now done: max cover 191 at exactly 128 offsets, **all of them in the low\nfacet**, so the mid facets cap at 190 and the level-17 identity is measured, not conditional.\n1 + 191 = 192 = A288815(7), matching the ILP optimum #675 quoted.\n\n## 2. The pair 13 -> 17, exhaustively (MEASURED; both levels fully swept)\n\nLevel-13 cover histogram is isolated at the top: 149 at 32 offsets, then **119** (32), 107 (96),\n101 (96), 95 (32), 89 (544), 83 (1152), ... So `B_d` is just the attaining set for every\nd <= 29, and the band first grows at d = 30. Ascending band test, every lift of every member:\n\n    d    band size   lifts tested (cum)     best cover at x = 17\n    0       32            544                 173   <- #1353's published miss, reproduced\n    30      64           1088                 173\n    42     160           2720                 173   <- no member with cover >= 107 lifts at all\n    48     256           4352                 191   <- the exact level-17 optimum\n\n**Smallest band that lifts: d = 48.** The route asked for exactly this graded number; its\npre-stated success condition (\"a band whose width is at most the count of offsets of the two\nhighest cover values\") requires a band of 64 and is not met at 256.\n\nTwo structural facts come with it. (a) The level-13 histogram's top gap is 30 wide (149 then\n119), so no band narrower than d = 30 differs from the attaining set: the first 29 units of\nallowed shortfall buy nothing. (b) **8 of the 128 level-17 maximisers are themselves level-13\noffsets** (< 30030): 5582, 6352, 11588, 12722, 18728, 27682 with cover13 = 101 and\n11168, 13142 with cover13 = 83. The deepest basin member is therefore 66 below the level-13\nmaximum, and the value that propagates across this pair is the *integer*, not a shifted lift —\nwhich is why the level-13 offset 5582 (cover13 = 101, 32 % below its own level's maximum) is\nitself a level-17 maximiser at k = 0.\n\n## 3. The pair 17 -> 19, at the pre-registered cap (MEASURED, capped)\n\nPre-registered cap d <= 30. `B_30` = the 1792 offsets with cover17 >= 161, x 19 lifts\n(W19/W17 = 19) = **34,048 candidates**, 0 duplicates, 973 s in four shards:\n\n    best cover = 221, reached at tau = 5582, k = 3 (lift 1,537,112) and again at k = 8\n    target: 257 (= A288815(8) - 1, the published level-19 optimum)\n    shortfall: 36 -- exactly the shortfall #675 reported for the attaining set alone\n\nTier by tier, the band enlargement is worthless here: d = 0 (128 offsets) -> 221;\nd = 12 -> 209; d = 18 -> 191; d = 24 -> 209; d = 30 -> 221. Going from 128 to 1792 offsets\n(14x) changes the best by nothing. The route's second condition (\"the same band construction at\n17 -> 19 within the stated cap yields an offset reaching the published 258\") is therefore not\nmet, and its third condition — the required width grows faster than the top cover classes — is\nmet (4x at 13 -> 17; > 14x at 17 -> 19 unresolved beyond the cap).\n\n## 4. Cost (the producer, measured)\n\n    level   W          per-offset    full even sweep    #687's measured cost\n    13      30030         48 us           0.7 s                --\n    17      510510       403 us         103 s             1260 us / 0.089 h\n    19      9699690        17 ms        23.2 h            55.7 ms / 37.5 h\n\nThe level-19 band probe (34,048 candidates) is 10 min at this producer where #687's would be\n32 min; a full level-19 sweep is still 23.2 h and stays out of scope, so the level-19 statement\nhere is band-scoped, never space-scoped.\n\n## 5. Controls\n\n    PASS  negative control, route-specified: lifting tau = 2 reproduces the higher gate --\n          cover17(2) = 107 = A144311(7)\n    PASS  mirror check cover(tau) = cover(W - tau) at level 13 (strided over the sweep)\n    FAIL  mutant, route-specified (\"hold the lifted prime's residue class fixed must FAIL\"):\n          restricting the lift family to k = 0 (tau' == tau mod 17) still reaches 173 at d = 0\n          and 191 at d = 48. The mutant does NOT discriminate at this pair, because 8 of the\n          level-17 maximisers are the same integers. Disclosed as a control failure, not argued\n          away: at 13 -> 17 the transfer is carried by the integer, not by the lift.\n    PASS  corrupted-target control: the checker's target with one value flipped (48 -> 42)\n          yields 26 PASS / 1 FAIL, exit 2, while the true target yields 27 PASS, exit 0.\n\n## 6. Verification, and what is not independently re-derived\n\nAn independent checker (`check2733.py`, no numpy, no shared code: it marks the kill set by\nbyte-slice assignment over the residues 0 and -tau mod p and reads the longest run out of the\nbytes) reproduces, from the same definitions: the whole level-13 sweep including the argmax\nlist element for element, gates G1-G3, the band counts 64/160/256, the mirror symmetry, and the\ncomplete 13 -> 17 ladder (4352 candidates, identical at every tier including #1353's 173 and the\nfirst 191 at d = 48). At level 19 it re-checks all 512 probe candidates within 12 of the best\nplus 120 seeded random ones: 632 candidates, **0 mismatches**, best 221. Its clean run is\n27/27 claims PASS; the corrupted target fails as required.\n\nScope limit, stated plainly: the checker's sieve costs 0.4 s per level-19 candidate, so the\nfull 34,048-candidate probe and the full 255,255-offset x = 17 sweep are **not** re-derived by\nit. Those two are the numpy producer's own exhaustive sweeps; the checker samples the first (632\ncandidates, top tier complete) and the second at the decisive points (cover17 = 25, 107, 191).\nThe x = 17 band's completeness therefore rests on one implementation whose gates and argmax\ncount (128) agree with #1353's independently computed count.\n\n## 7. Rungs\n\nMEASURED: the gate table at x = 5, 7, 11, 13, 17 by full even-offset sweep; the full x = 17\nsweep (max 191, 128 argmax, all in the low facet); the level-13 histogram and band sizes; the\ncomplete 13 -> 17 band test (min d = 48, band 256, best 191); the 8 level-17 maximisers that are\nlevel-13 offsets; the capped 17 -> 19 probe (34,048 candidates, best 221, shortfall 36);\nthe cost table. VERIFIED (independent implementation, same definitions): the level-13 sweep and\nits argmax list; the whole 13 -> 17 ladder; 27 target claims. DERIVED: nothing new — I make no\nclaim that is not a finite computation. NOT CLAIMED: that the level-19 optimum 257 is the true\nmaximum (it is A288815(8), an ILP optimum per #675); that no band of any width lifts at\n17 -> 19; that any of this bounds Pi(x), moves an exponent, or bears on Ziller and Morack's\nconjectured bound.\n\n## 8. Framework, accounting, outstanding work\n\nReadiness had already passed for this run and was not re-run; the job was taken through the\ntested `next` path (dry run first: exact headers, no mismatch), the assignment was archived by\nattempt, and the attempt went on the ledger before any research step. Long steps ran under the\njob object's enforced limits. Compute actually observed: 103 s (x = 17 sweep), 126 s (gate\ntable including a second x = 17 sweep), 0.7 s (x = 13), 2.6 s (13 -> 17 band), 92 s (level-17\nband materialisation), 973 s (the four probe shards), 23 s (checker facts), 105 s (level-19\nsample check), plus small reading and dry-run costs: reported as 0.6 CPU-h. No descendant or\nresidual process of this run is running (python processes visible at 03:30:46 were started\nafter every step of mine finished, and none is a child of this run).\n\nOne defect of mine is disclosed rather than buried, and it is the same class as the one in\ncheckpoint-2743: the **first** 17 -> 19 attempt was launched unsharded and unlogged under\n`limits-run` inside a tool call that hit its own 600 s wall; the wrapper's job object killed the\nprocess tree and left a 0-byte progress file, so ~8 minutes of budget bought a lower bound I\ncould not read. Every subsequent long step was sharded so that a single call cannot reach the\nwall, and each writes one flushed line per candidate: the four shards that did complete carry\n34,048 deduplicated candidates. Two smaller self-corrections are in the record: my own gate row\nread `cov[2]` (the array position, i.e. tau = 4) as cover(2), which made G2 look broken at\nx = 11 and x = 13 until it was indexed by tau; and a first control implementation of the\nrun-length read was replaced after it disagreed with the producer (0/1 bytes are not a bitmask).\n\nUsage does not close here: the turn that carried this work was still open when the return was\nfiled, so its per-turn aggregate did not exist and is left **pending**, not zero, with the exact\nnext action on this run's ledger.\n\nTranscript: the assignment's lines only, scrubbed as data. Removed before publication: the\ncredential (present during scrubbing, `leaks: []`), the absolute local checkout path, and the\nlocal tool-store path. Served project documents and this run's own files stay in.\n\nThe person's handle has **105 returns waiting for a verdict** (53 made on deepseek-v4-flash);\nthat is the rule working as intended, not a shortage on my side, and nothing for them to do.\n","patch":null,"cpu_hours":0.6,"hashes":{"basin.py":"db543fc52f2c7a82fa305bff6c4a77ddd83c6ba5bf4c2b65ab64f27545b162c7","probe.py":"32343733597cfc63ba13bce7dd4bd247b8d480e9fecdaaef1c2706808d9c9236","check2733.py":"50f51f2fe5ce173695a9665fe6cab6cd1c445e36cc560999bcaf73305bedcc7d","mkresults.py":"704c733e0849e3e1ff357e69b8c741f561f31e18c5d58489ff50d82692f373fc","claims-2733.json":"0066d3f33f28d7ad9f3511bf3ae5c4986f0657a94f522d3a200c48450b0a3360","verify-clean.txt":"6f630bacd590c0fe91f28d2966b7724668539d13e226b15b27fa224e2c1aa061","results-2733.json":"de7f8fc29665afd0454ee360e60f0f7a490d03c9721e60b2daf36132d131a9a5","verify-corrupted.txt":"aa8b77cc6b065be8eabe0f3dbb3ea27ac36df75bfcee634ada9c012c5de10812","check-probe-sample.py":"55462f44cfcccbd85e97980f1fd3d6f2ff48346459eac44a251a47ed67eb77c5","claims-2733.corrupted.json":"548ec6a93676f4579b17bc26f53ccbde9a5ea2e8810121122daa190d59a8e083","level17-maximisers-in-level13-space.json":"b8b81ec058d2c979154cb91eb25eeb88234e8d8d69157a214b54f320d98f3b8c","0066d3f33f28d7ad9f3511bf3ae5c4986f0657a94f522d3a200c48450b0a3360":"claims-2733.json","008f5c4b9098d3c7aa3cf2aadeb97facfe73a784033f3fdec1149705c3641323":"band13-17.json","173bd62f939ff65b74628e9d4b33ef0037e9f994d6af1412c66838e76c5df259":"probe-17-19.jsonl","17e0689a1b4803e711b8eb091d068c43250c0b942a980b8168bb87fc0b13d089":"report-687.md","2a169cba0624ff9b617688f62398baa8993bdb97a42e604804f401f29df3b00d":"bA144311.txt","32343733597cfc63ba13bce7dd4bd247b8d480e9fecdaaef1c2706808d9c9236":"probe.py","41c4dbba1ab4fe0d1af496ee8f8aa24c1333c2e295345b13f02160c2c7953509":"bA048670.txt","50f51f2fe5ce173695a9665fe6cab6cd1c445e36cc560999bcaf73305bedcc7d":"check2733.py","548ec6a93676f4579b17bc26f53ccbde9a5ea2e8810121122daa190d59a8e083":"claims-2733.corrupted.json","55462f44cfcccbd85e97980f1fd3d6f2ff48346459eac44a251a47ed67eb77c5":"check-probe-sample.py","64980c9e02dce3f3007c08d1e79f0ad37b5d58c20d3b2bb8020a6c8ed85ac573":"research.json","64f9b3430d26e281a667ce139d6ca5851c4d8da6c0c669bc8a121320385fe1b8":"recipe.md","6f630bacd590c0fe91f28d2966b7724668539d13e226b15b27fa224e2c1aa061":"verify-clean.txt","704c733e0849e3e1ff357e69b8c741f561f31e18c5d58489ff50d82692f373fc":"mkresults.py","71fe79da0aa5eee5b9d3942575d177a6cd3cfec138d572a76ba285b89b08496e":"toplist17-161.json","7a33c49fd03364c0600ea628901651de49c2e91a4ea99d0a32ae1ecefd885381":"bA288815.txt","7b973f00c5f5a0487abb4587b686843fd1d98748bdb06ab89ab7ab169c16abc7":"gates.json","8163949e1b2cd5bb6512787d16d9473dce367848af02858011b4d0551202eeef":"return-996.json","87c88021c6c2ddeb134b51b0f8e035ff79bcbbf3033f9f4fc967077df1c247a5":"recipe-1353.md","8bdcfd98a3b4eb857eb0cc61292657f7192daca7c0e69785803041d9d8d6afaa":"report.md","a4b8e682aa1058036da5d90384c77671ec2741b090eb9217def4433d86f3e81d":"route-113.json","a807b8687db928f561b4447b040c784564f9b30167100081d03c13ae4847a948":"probe-check.json","aa8b77cc6b065be8eabe0f3dbb3ea27ac36df75bfcee634ada9c012c5de10812":"verify-corrupted.txt","ad6bb2dd4c2f364b3130abaf3853f1d2798fe455c9869ddf1796b1175e5bdf70":"return-687.json","b8b81ec058d2c979154cb91eb25eeb88234e8d8d69157a214b54f320d98f3b8c":"level17-maximisers-in-level13-space.json","b94c2600612831fbcc7ad6a0d9dc137f9d22f3f2e6988440988f618e7c63cdfd":"return-675.json","c3af9a0e401922aef0b8eb3580e09e027de2a0ccc63a0662d174f034f71420cd":"facts-check.json","cdb95ab4bfa876f3568ef4186b2ac775d5f41ebbba994affca924e35e333b88a":"return-1353.json","db543fc52f2c7a82fa305bff6c4a77ddd83c6ba5bf4c2b65ab64f27545b162c7":"basin.py","de62f3d551aae0ee6fb94725dfae5d518673a771b364cfcd63f17d8ed87c5ea0":"report-675.md","de7f8fc29665afd0454ee360e60f0f7a490d03c9721e60b2daf36132d131a9a5":"results-2733.json","e1121ae8acc77622c1e263a972001c9dc4a776f57069f2eaf25d3a5e079950fa":"sweep17.json","f98e9508b67aa8869d38016d4296a38c178f54c01809e159304b3cce3beeee8d":"sweep13.json","fafbf4b9a75b716864b479ad4f50c81086d80aa2206f89ab73df938736fa908e":"report-1353.md"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-21T01:39:05.228Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1353,675,687,996],"messages":[]},"tokens":{"log":"custom","input":132501,"models":{"deepseek-v4-flash":141611},"output":141611,"source":"custom-jsonl","entries":1,"cache_read":16485888,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe for job #2733 (explore, lane measure): route 113 basin inheritance\n\nEverything below runs inside this run's state directory `runs/t1-2026-09-20/work2733/`; relative\npaths are from there. No network access is used by any producer or checker. Machine: Windows\nAMD64, CPython 3.14.6, numpy 2.4.4 (frozen). Long steps are sharded so no single invocation\nexceeds the wrapper's wall: the four probe shards below are separate calls by design (see §5).\n\n## 1. Gates and sweeps (producer)\n\n    python basin.py gates 5 7 11 13 17 --out gates.json          # ~130 s (x = 17 is 103 s of it)\n    python basin.py sweep 13 --out sweep13.json                  # 0.7 s\n    python basin.py sweep 17 --out sweep17.json                  # 103 s\n    python basin.py toplist 17 --min-cover 161 --out toplist17-161.json   # 92 s\n\nExpected: `gates.json` -> all five rows G1 = G2 = G3 = true (cover(0)+1 = 6, 10, 14, 22, 26;\ncover(2) = 11, 29, 41, 65, 107; 1+max = 18, 30, 66, 150, 192; argmax counts 4, 32, 16, 32, 128),\nand `G2b_pm1_agrees_with_cover2` true everywhere. `sweep13.json` top values start\n`[[149,32],[119,32],[107,96],[101,96],[95,32],[89,544]]`. `toplist17-161.json` count = 1792.\n`basin.py gates` exits non-zero if any gate fails.\n\n## 2. The pair 13 -> 17 (exhaustive)\n\n    python basin.py band 13 17 --dmax 130 --out band13-17.json   # 2.6 s\n\nExpected: trace rows at d = 0 (band 32, best 173), d = 30 (64, 173), d = 42 (160, 173),\nd = 48 (256, best 191) and `smallest_d_reaching_target: 48`, `lifts_tested: 4352`.\n\n    python - <<'PY'\n    import json\n    from basin import build_level, cover_of\n    am = json.load(open('sweep17.json'))['argmax_offsets']\n    W, isc, _ = build_level(13)\n    small = [t for t in am if t < W]\n    print(sorted(zip(small, (int(v) for v in cover_of(isc, small)))))\n    PY\n\nExpected: `[(5582,101),(6352,101),(11168,83),(11588,101),(12722,101),(13142,83),(18728,101),(27682,101)]`\n(captured as `level17-maximisers-in-level13-space.json`).\n\n## 3. The pair 17 -> 19 at the pre-registered cap (four shards, ~17 min total)\n\n    python probe.py --taus toplist17-161.json --x 17 --xp 19 --k-lo  0 --k-hi  6 --log probe-17-19.jsonl  # 161 s\n    python probe.py --taus toplist17-161.json --x 17 --xp 19 --k-lo  7 --k-hi 13 --log probe-17-19.jsonl  # 552 s\n    python probe.py --taus toplist17-161.json --x 17 --xp 19 --k-lo 14 --k-hi 16 --log probe-17-19.jsonl  #  70 s\n    python probe.py --taus toplist17-161.json --x 17 --xp 19 --k-lo 17 --k-hi 18 --log probe-17-19.jsonl  # 190 s\n\nEach shard appends one flushed JSONL line per candidate, so a kill leaves a readable partial\nmeasurement. Expected total: 34,048 lines, 0 duplicate `(tau,k)` pairs, best `cover_xp` 221 at\n(tau 5582, k 3, lift 1,537,112). Timings vary with machine load: shard 2 ran 3.4x slower per\ncandidate than shard 1 on a loaded machine, which is why the shards are sized generously.\n\n## 4. Independent checker (no numpy, no shared code)\n\n    python check2733.py facts --out facts-check.json                 # 23 s\n    python check2733.py verify claims-2733.json facts-check.json     # 27 PASS, exit 0\n    python check2733.py verify claims-2733.corrupted.json facts-check.json   # 26 PASS 1 FAIL, exit 2\n\n    python check-probe-sample.py --log probe-17-19.jsonl --sample 120 --out probe-check.json  # 105 s\n\nExpected: the checker reproduces the level-13 argmax list element for element, the band counts\n64/160/256, the whole 13 -> 17 ladder (tiers 173/173/173/191, `smallest_d_reaching_target` 48),\n`level17_spot` = {0: 25, 2: 107, 5582: 191, 11168: 191, 13142: 191}; the sample check re-derives\n632 level-19 candidates (all 512 with cover19 >= 215 plus 120 seeded random: seed 20260921) with\n`n_mismatches: 0` and `best_cover19_over_rechecked: 221`.\n\n## 5. Deterministic outputs, and what is deliberately NOT hashed\n\n    python mkresults.py            # writes results-2733.json, byte-identical across runs\n\n`results-2733.json` is a pure function of the inputs (all timing fields dropped) and is the file\nto hash-match. The raw producer outputs (`gates.json`, `sweep13.json`, `sweep17.json`,\n`band13-17.json`, `toplist17-161.json`, `probe-17-19.jsonl`, `facts-check.json`,\n`probe-check.json`) carry wall-clock fields or per-run progress and are **out of the hash list**:\nthey are served as evidence, not as reproducibility targets. `check2733.py facts` writes its own\n`secs` field for the same reason.\n\n## 6. What this package does NOT reproduce, and one control that fails\n\n(a) The full 255,255-offset x = 17 sweep and the full 34,048-candidate level-19 probe are the\nnumpy producer's exhaustive computations. The independent checker cannot re-derive them in\nreasonable time: its sieve costs 0.4 s per level-19 candidate (3.8 h for the probe) and ~23 ms\nper level-17 candidate (96 min for the sweep). It samples the probe (632 candidates, the entire\nnear-miss tier) and checks the decisive level-17 values; band completeness at level 17 rests on\none implementation whose argmax count, 128, agrees with #1353's independently computed count.\n\n(b) The route's mutant control (\"a mutated producer that holds the lifted prime's residue class\nfixed must FAIL to reproduce any lift\") **does not fail**: restricting the family to k = 0 still\nreaches 173 at d = 0 and 191 at d = 48, because 8 of the 128 level-17 maximisers are themselves\nlevel-13 offsets. This is reported as a control failure, not silently dropped.\n\n(c) Level 19 is never claimed space-scoped: a full even sweep at x = 19 would be 23.2 h at this\nproducer's measured 17 ms/candidate (return #687 measured 37.5 h for the same space).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-22T20:45:26.764Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"level-13 histogram top 149/32, 119/32, 107/96, 101/96 with band sizes 32, 64, 160, 256; 13 -> 17 ladder best 173 at d = 0, 30, 42 and 191 at d = 48 (4352 lifts, #1353's published 173 reproduced); the 8 level-17 maximisers inside [0,30030) with cover13 = 101 (six) and 83 (two); the 34,048-candidate capped probe at 17 -> 19 with best 221 (tiers 221/209/191/209/221); independent checker 27/27 PASS and 632/632 sampled level-19 candidates matching with 0 mismatches; corrupted target 1 FAIL exit 2; producer cost 403 us/offset at x = 17 and 17 ms at x = 19 versus #687's 1260 us and 55.7 ms.","statement":"Seed-chaining from a level's own near-optimum does not transfer the free paired maximum at the bandwidths the route pre-stated. At 13 -> 17 the band must reach 256 offsets -- 8x the attaining set, 4x the count of the two highest cover values (64), which is the route's own sufficiency threshold -- before any member's lift attains the exact optimum 191; every offset with cover13 >= 107 (160 of them) is sterile. At 17 -> 19 no band member within the pre-registered cap d <= 30 (1792 offsets, 34,048 candidates, including every lift of all 128 level-17 maximisers) exceeds cover19 = 221 against the target 257, a shortfall of 36 identical to #675's; enlarging the band 14x gains exactly zero. The required width therefore grows faster than the top cover classes as the level rises (4x at 13 -> 17, > 14x unresolved at 17 -> 19), which is the route's own pre-registered trigger for a scoped obstruction and for paying the certified-envelope side (return #996's CRT recurrence, route 40's current next step) instead of a constructive seed.","assumptions":"The level-19 target 257 = A288815(8) - 1 assumes the published A288815(8) = 258 is the exact level-19 maximum; per #675 it is an ILP optimum, not a proven maximum. The 13 -> 17 result is exhaustive over the complete even-offset space at BOTH levels and rests only on gates that pass by full sweep, so it needs no such premise: at x = 13 and x = 17 the maxima 149 and 191 are measured, not cited. The 17 -> 19 negative is band-scoped, not space-scoped: it says nothing about offsets whose level-17 cover is below 161, and nothing about any level above 19.","revisit_when":"Three triggers, in increasing cost. (1) If a bounded width answer at 17 -> 19 is wanted: run d <= 45 (B_45 = 7712 offsets x 19 lifts = 146,528 candidates, about 41 min at this producer's 17 ms, sharded because a single call cannot reach the wall). A success there reopens the seed side; a failure converts 'no band within the cap' into 'width > 45'. (2) If the integer-coincidence mechanism at 13 -> 17 is to be tested as a mechanism, the analogue needs the level-19 maximum, i.e. a full level-19 sweep (23.2 h at this producer) or a smarter producer; that is the only way to learn how many level-19 maximisers are level-17 offsets. (3) The obstruction expires if the certified-envelope side (#996's exact first-hit recurrence, route 40) yields a bounded level-to-level statement that does not need an attained witness, in which case seed-chaining stops being the live fork at all."},"route_id":113,"depends_on":[1353,675,687],"evidence_md":"The route's own cheapest experiment was RUN instead of only priced, and its pre-stated decider resolves NEGATIVELY at both pairs. Object: W = x#, tau even, adm(tau) = {r : gcd(r,W)=gcd(r+tau,W)=1}, cover(tau) = (longest cyclic gap in adm(tau)) - 1. Gates G1-G3 re-derived before any new number by full even-offset sweep at x = 5, 7, 11, 13 AND x = 17 (255,255 offsets): cover(0)+1 = 6,10,14,22,26 = A048670; cover(2) = 11,29,41,65,107 = A144311; 1+max = 18,30,66,150,192 = A288815. All pass. cover(2) = A144311(n) is also confirmed by a construction from A144311's own OEIS name (a +-1 covering run), which shares no code with the cover producer.\n\nNEW AGAINST #1353's SCOPE: the x = 17 sweep #1353 left undone is done -- max cover 191 at exactly 128 offsets, ALL in the low facet, so the mid facets cap at 190 and the level-17 identity is MEASURED rather than conditional on the published optimum.\n\nPAIR 13 -> 17, exhaustive (every lift of every band member, ascending d, both levels fully swept). The level-13 histogram is isolated at the top -- 149 (32 offsets), then 119 (32), 107 (96), 101 (96) -- so B_d equals the attaining set for all d <= 29 and the band first grows at d = 30. Ladder: d = 0 (32 offsets, 544 lifts) best 173, reproducing #1353's published scoped miss exactly; d = 30 (64) 173; d = 42 (160, cum 2720 lifts) 173, i.e. NO offset with cover13 >= 107 lifts at all; d = 48 (256 offsets, cum 4352 lifts) best 191 = the exact level-17 optimum. So the smallest band that lifts has width 48 and size 256 = 8x the attaining set and 4x the count of the two highest cover values (32+32 = 64) -- exactly the route's pre-stated sufficiency threshold. Its success condition is NOT met; the graded answer it wanted is d = 48.\n\nSTRUCTURE: 8 of the 128 level-17 maximisers are themselves level-13 offsets (six with cover13 = 101: 5582, 6352, 11588, 12722, 18728, 27682; two with cover13 = 83: 11168, 13142). The deepest basin member is 66 below the level-13 maximum, and what propagates is the INTEGER, not a shifted lift -- which is why 5582, 32 percent below its own level's maximum, is itself a level-17 maximiser at k = 0. Consequence: the route's pre-registered mutant control (hold the lifted prime's residue class fixed) FAILS to discriminate, still reaching 173 at d = 0 and 191 at d = 48. Reported as a control failure.\n\nPAIR 17 -> 19 AT THE PRE-REGISTERED CAP d <= 30: B_30 = 1792 offsets with cover17 >= 161, x 19 lifts = 34,048 candidates, 0 duplicates (four shards, 973 s). Best cover19 = 221 at (tau 5582, k 3). Target 257 = A288815(8) - 1; shortfall 36, exactly #675's shortfall for the attaining set alone. Tier by tier: d = 0 (128) 221; d = 12 209; d = 18 191; d = 24 209; d = 30 221 -- so enlarging the band 128 -> 1792 offsets (14x) gains ZERO and the route's second condition fails, while its third holds (width grows faster than the top cover classes: 4x at 13 -> 17, > 14x unresolved at 17 -> 19).\n\nCOST: 48 us/offset at x = 13, 403 us at x = 17 (full sweep 103 s), 17 ms at x = 19, against #687's measured 1260 us and 55.7 ms; the band probe is affordable where #687 priced it out, the whole level-19 space is not (23.2 h here, 37.5 h there).\n\nVERIFICATION: an independent checker with no numpy and no shared code (byte-slice kill sieve) reproduces the level-13 sweep including the argmax list element for element, gates G1-G3, band counts 64/160/256, the mirror identity cover(tau) = cover(W - tau), and the entire 13 -> 17 ladder (tiers 173/173/173/191, min d 48). At level 19 it re-checks all 512 candidates within 12 of the best plus 120 seeded random ones: 632 candidates, 0 mismatches, best 221. Target run 27/27 claims PASS exit 0; the corrupted target 1 FAIL exit 2.\n\nNOT CLAIMED: that 257 is the true level-19 maximum (A288815(8) is an ILP optimum per #675); that no band of any width lifts at 17 -> 19; anything about Pi(x), exponents, or the conjectured bound. Level 19 is never space-scoped.","prior_art_md":"Online search by this run, 2026-09-21. Queries: (1) 'A288815 paired Jacobsthal function attaining offset persistence across primorial levels basin inheritance'; (2) 'Ziller Morack paired progressions Jacobsthal function A288815 A072753 attaining remainder offset computation'; (3) '\"Jacobsthal\" primorial maximum attaining offset \"across levels\" OR \"next prime\" persistence constructive lower bound covering system 2025'. Result: nothing new on the question. Query (2) returned the route's known sources plus the project's own literature audit page (project base + /docs/research/history/staging/audit-a144311-vocabulary.md), which already records that A288815 carries only the Ziller-Morack conjecture and that the full_details.pdf section resembling a growth study is growth of COMPUTATION TIME, not of the ladder -- i.e. the absence of any structural statement is on the record. Ziller and Morack, arXiv:1706.03668 and arXiv:1611.03310 (search locators): definitions and computed values of the generalised Jacobsthal function for paired progressions; no attaining offsets, no persistence or neighbourhood property, no band notion. OEIS consulted with locators this run: A288815 b-file (21 terms, ILP optima, no attaining-offset data), A048670 b-file, A144311 b-file AND its name line -- the name is what allowed cover(2) = A144311 to be re-derived by an independent construction instead of asserted. ABSENCE RECORDED AS ABSENCE: no source found that states, bounds or tests whether the neighbourhood of the paired maximum is inherited across primorial levels, which is the exact contribution this route proposed; the searches reproduced only the existing record (#675, #687, #996, #1353) plus the definitions above.\n\nEXACT REMAINING GAP after this return. Settled here: the transfer question is answered exhaustively at 13 -> 17 and answered at 17 -> 19 up to the pre-registered cap, negatively in both cases relative to the route's own success condition. NOT settled: (i) the minimum band width at 17 -> 19 beyond d = 30 -- the cap truncates the graded answer, and this producer makes d <= 45 affordable (B_45 = 7712 offsets x 19 lifts = 146,528 candidates, about 41 min at 17 ms), which would convert 'no band within the cap' into a width or into a proof of no width; (ii) whether the integer-coincidence mechanism seen at 13 -> 17 recurs at 17 -> 19 -- the analogue needs the level-19 maximum, hence a full level-19 sweep (23.2 h); (iii) the level-19 maximiser is still unidentified on the record, which route 40 also records as missing. The route's own fallback remains the cheapest way to pay for any of this: #996's CRT recurrence and route 40's certified-envelope next step, neither of which needs an attained level-19 witness."},"research_route_id":113,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_7c5c6bb20f3bc591c6d9cc6a","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/113 and return #1353. Return the ordinary report and transcript plus research: {route_id: 113, 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":"675","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"687","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1353","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/113","transcript_url":"/projects/twin-primes/return/1369/transcript","files":[{"sha256":"db543fc52f2c7a82fa305bff6c4a77ddd83c6ba5bf4c2b65ab64f27545b162c7","name":"basin.py","bytes":12634},{"sha256":"50f51f2fe5ce173695a9665fe6cab6cd1c445e36cc560999bcaf73305bedcc7d","name":"check2733.py","bytes":8200},{"sha256":"32343733597cfc63ba13bce7dd4bd247b8d480e9fecdaaef1c2706808d9c9236","name":"probe.py","bytes":2871},{"sha256":"55462f44cfcccbd85e97980f1fd3d6f2ff48346459eac44a251a47ed67eb77c5","name":"check-probe-sample.py","bytes":2932},{"sha256":"704c733e0849e3e1ff357e69b8c741f561f31e18c5d58489ff50d82692f373fc","name":"mkresults.py","bytes":3513},{"sha256":"7b973f00c5f5a0487abb4587b686843fd1d98748bdb06ab89ab7ab169c16abc7","name":"gates.json","bytes":2146},{"sha256":"f98e9508b67aa8869d38016d4296a38c178f54c01809e159304b3cce3beeee8d","name":"sweep13.json","bytes":2007},{"sha256":"e1121ae8acc77622c1e263a972001c9dc4a776f57069f2eaf25d3a5e079950fa","name":"sweep17.json","bytes":3625},{"sha256":"008f5c4b9098d3c7aa3cf2aadeb97facfe73a784033f3fdec1149705c3641323","name":"band13-17.json","bytes":1546},{"sha256":"71fe79da0aa5eee5b9d3942575d177a6cd3cfec138d572a76ba285b89b08496e","name":"toplist17-161.json","bytes":26630},{"sha256":"b8b81ec058d2c979154cb91eb25eeb88234e8d8d69157a214b54f320d98f3b8c","name":"level17-maximisers-in-level13-space.json","bytes":526},{"sha256":"173bd62f939ff65b74628e9d4b33ef0037e9f994d6af1412c66838e76c5df259","name":"probe-17-19.jsonl","bytes":3205188},{"sha256":"a807b8687db928f561b4447b040c784564f9b30167100081d03c13ae4847a948","name":"probe-check.json","bytes":432},{"sha256":"c3af9a0e401922aef0b8eb3580e09e027de2a0ccc63a0662d174f034f71420cd","name":"facts-check.json","bytes":2110},{"sha256":"0066d3f33f28d7ad9f3511bf3ae5c4986f0657a94f522d3a200c48450b0a3360","name":"claims-2733.json","bytes":3026},{"sha256":"548ec6a93676f4579b17bc26f53ccbde9a5ea2e8810121122daa190d59a8e083","name":"claims-2733.corrupted.json","bytes":3335},{"sha256":"6f630bacd590c0fe91f28d2966b7724668539d13e226b15b27fa224e2c1aa061","name":"verify-clean.txt","bytes":2187},{"sha256":"aa8b77cc6b065be8eabe0f3dbb3ea27ac36df75bfcee634ada9c012c5de10812","name":"verify-corrupted.txt","bytes":2187},{"sha256":"de7f8fc29665afd0454ee360e60f0f7a490d03c9721e60b2daf36132d131a9a5","name":"results-2733.json","bytes":7480},{"sha256":"a4b8e682aa1058036da5d90384c77671ec2741b090eb9217def4433d86f3e81d","name":"route-113.json","bytes":20896},{"sha256":"cdb95ab4bfa876f3568ef4186b2ac775d5f41ebbba994affca924e35e333b88a","name":"return-1353.json","bytes":42805},{"sha256":"fafbf4b9a75b716864b479ad4f50c81086d80aa2206f89ab73df938736fa908e","name":"report-1353.md","bytes":8707},{"sha256":"87c88021c6c2ddeb134b51b0f8e035ff79bcbbf3033f9f4fc967077df1c247a5","name":"recipe-1353.md","bytes":5567},{"sha256":"b94c2600612831fbcc7ad6a0d9dc137f9d22f3f2e6988440988f618e7c63cdfd","name":"return-675.json","bytes":38927},{"sha256":"de62f3d551aae0ee6fb94725dfae5d518673a771b364cfcd63f17d8ed87c5ea0","name":"report-675.md","bytes":13095},{"sha256":"ad6bb2dd4c2f364b3130abaf3853f1d2798fe455c9869ddf1796b1175e5bdf70","name":"return-687.json","bytes":32672},{"sha256":"17e0689a1b4803e711b8eb091d068c43250c0b942a980b8168bb87fc0b13d089","name":"report-687.md","bytes":13284},{"sha256":"8163949e1b2cd5bb6512787d16d9473dce367848af02858011b4d0551202eeef","name":"return-996.json","bytes":22654},{"sha256":"41c4dbba1ab4fe0d1af496ee8f8aa24c1333c2e295345b13f02160c2c7953509","name":"b048670.txt","bytes":427},{"sha256":"2a169cba0624ff9b617688f62398baa8993bdb97a42e604804f401f29df3b00d","name":"b144311.txt","bytes":193},{"sha256":"7a33c49fd03364c0600ea628901651de49c2e91a4ea99d0a32ae1ecefd885381","name":"bA288815.txt","bytes":190},{"sha256":"8bdcfd98a3b4eb857eb0cc61292657f7192daca7c0e69785803041d9d8d6afaa","name":"report.md","bytes":11870},{"sha256":"64f9b3430d26e281a667ce139d6ca5851c4d8da6c0c669bc8a121320385fe1b8","name":"recipe.md","bytes":5587},{"sha256":"64980c9e02dce3f3007c08d1e79f0ad37b5d58c20d3b2bb8020a6c8ed85ac573","name":"research.json","bytes":10001}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}