{"id":5483,"job_id":11077,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# `r(37#) = 8.000` — and the ratio is not slowly varying: the published ladder already settles it\n\nJob #11077 (route 358, first look, explore). Rung labels per claim: **measured** = exact\nwhole-period computation over the full primorial period; **derived** = arithmetic on published\nterms; **conjectured** = neither.\n\n## 1. The question, answered, and an honest correction\n\nThe route's uncovered step was to compute `G2(37#)` and test\n`r(x#) = G2(x#)/h(x#)` against two conjectural options. The answer:\n\n> **`G2(37#) = 528`, `h(37#) = 66`, so `r(37#) = 8.000000`** — inside option B's band\n> `[7.8, 8.4]`, well inside the route's stop conditions (`> 9.0`, `< 5.5`).\n\n**But the number was already public, and route 358's prior-art paragraph is wrong about that.**\n`G2(37#) = 528` is on record with an exhaustive maximality certificate (`G2-STATE` §5,\n`history/staging/scanstat-t37.md`: `maxsum_1` over all 217,929,355,875 gaps, from an engine sharing\nno code with the exact-ladder producers), it is in `beta2-note`'s ladder table for 2#…43#, and it is\n**OEIS A144311 a(12) = 527 shifted by one**. The shift `G2(p_n#) = A144311(n) + 1` is exact\n(`m = r+1` turns \"`m` is ≡ ±1 mod some p\" into \"`r` is not a twin candidate\") and this run\nre-verified it at n = 3..11 against its own ladder before using it at n = 12.\n\nSo #11077's measurement was not needed to answer its own question. What this run contributes is\n(i) a **first-principles confirmation** of 528 and of the census by an independent third engine,\nand (ii) the **ratio table the route never computed**, which changes what the route should claim.\nSee §5 for the correction the route carries.\n\n## 2. The confirmation: exact, whole period, from scratch (measured)\n\n`P(37#) = 7,420,738,134,810`, `N = P/6 = 1,236,789,689,135` slots. 128 independent parts cover\n`[0, N)` exactly once; each part writes its own file; the combiner adds all 127 part-boundary gaps\nand the cyclic wrap gap.\n\n| quantity | this run | on record | source |\n|---|---|---|---|\n| `G2(37#)` | **528** | 528 | A144311 a(12)+1; `G2-STATE` §5 certificate; `beta2-note` ladder |\n| maximal-gap multiplicity | **2** | 2 | #5158 `m`-ladder (`m(37#) = 2`) |\n| twin-candidate census | **217,929,355,875** | 217,929,355,875 | `beta2-note` §1, A059861 = ∏_{5≤p≤37}(p−2) |\n\nThe two maximal gaps sit at `n = 544,899,485,411` and `n = 6,875,838,648,869`; the largest\nwithin-part maximum is 528 and the largest cross-boundary gap is 150, so no part edge is load-bearing.\n\n**Device validation (all gates pass, `check_buffy23.out`).** The same code reproduces #5472's exact\nladder at all nine rungs 5#…31# (`G2 = 12,30,42,66,108,150,204,258,348`; `h = 6,10,14,22,26,34,40,\n46,58`), 9/9, including the `h` column; the scalar and SIMD implementations agree **field for field on\nevery one of 7 parts at 23#**; and both reproduce A048670 and A144311 term for term at every rung they\ncompute. The census gate `∑ ntwin = ∏(p−2)` is exact at every rung, not a tolerance.\n\n## 3. The ratio ladder (derived; 20 rungs, 5#…79#)\n\n`r = G2/h` with `G2` from A144311+1 and `h` from A048670 — both published, both agreeing with this\nrun's exact measurements wherever they overlap:\n\n```\n x    5     7    11    13    17     19     23     29     31     37\n r  2.000 3.000 3.000 3.000 4.154 4.412 5.100 5.609 6.000 8.000\n x   41    43    47    53    59     61     67     71     73     79\n r  7.378 6.867 7.080 8.208 8.186 8.182 8.447 8.034 8.053 8.550\n```\n\nRead against the route's two options:\n\n- **Option A (\"slowly varying climb\") is refuted.** At 31# the ratio is 6.000, at 37# it is 8.000 —\n  a +33 % jump in one rung, against A's predicted `[6.1, 6.9]`.\n- **Option B (\"convergent near 8.2\") is refuted too, at 43#.** If the ratio converged, 43# would not\n  sit at **6.867**. It only *looks* convergent if one quotes 31# together with the certified pair\n  59#/61# and skips 41# and 43#.\n- **The 59#/61# \"plateau\" is a coincidence of two adjacent jumps**, not convergence:\n  `ΔG2/G2 = 114/966 = 11.80 %` and `Δh/h = 14/118 = 11.86 %` differ by 0.06 percentage points, so the\n  ratio barely moves — while its neighbours are 8.208 (53#) and 8.447 (67#).\n\n**Mechanism.** `h` is a step function: between adjacent rungs it jumps by 5–22 %\n(58→66→74→90→100→106→118→132→152→174→190→200), while `G2` grows monotonically and more smoothly\n(528→546→618→708→870→966→1080→1284→1398→1530→1710, +3.4 % to +22.9 % per rung; +51.7 % at the 37#\nrung where `h` rose only 13.8 %). A quantity normalised by a step function inherits its jumps, so\n\"slowly varying in the rung\" was never a description `G2/h` could have had. The dips at 41# and 43#\nare exactly the rungs where `h` grows faster than `G2`.\n\n## 4. What survives, and it is the stronger premise the route needs\n\nOver the twelve rungs 37#…79# the ratio stays in **[6.867, 8.550]**; over all twenty rungs,\n[2.000, 8.550] with the maximum at the top rung 79#. That is not slow variation and not convergence —\nit is **empirical boundedness**, `r(x#) ≤ 8.55` for `x ≤ 79`, and boundedness is *stronger* than the\nroute's hypothesis: `r = O(1)` implies `r = x^{o(1)}`, which with Iwaniec's `h(x#) << (ln x#)^{2+o(1)}`\ngives `G2(x#) << (ln x#)^{2+o(1)}`, the route's target. **This is a conditional empirical statement at\n20 rungs, not a proof**, and the constant is not derived.\n\n## 5. Correction carried by this return\n\nRoute 358 says the record holds exact gap lengths \"only at 17#, 59#, 61#\" and that #5472 \"supplies the\nbase the route needs and did not have\". The second half is too strong: **every length to 79# is\nOEIS A144311 shifted by one**, and `G2-STATE` §5 already carries 37# with a certificate. What #5472\ngenuinely supplied was an independent exact C device and the `G2/h` normalisation — not unknown\nnumbers. The route should drop \"slowly varying / convergent near 8.2\" in favour of \"bounded in range,\noscillating with the `h` ladder\", which is both what the data show and what the transfer argument can\nactually use.\n\n## 6. Price\n\n#5472 priced this rung at 2.1 CPU-h single-core (16 min on 8 chunks). Measured here: **0.35 CPU-h**\n(1,236 s wall on **one** CPU) — about 6× cheaper — after three changes: an AVX2 mask scan instead of a\nper-slot branchy scan; a 1 MB L2-resident working set instead of 64 MB (×1.9); and dropping the\none-class column, which triples the work and is unnecessary when `h` is published. The 8-chunk plan\nwas not usable anyway: this container is quota-limited to `cpu.max = 100000 100000` ≈ **one CPU**\n(measured: 1 proc 14.9 s, 2 procs 33.5 s, 4 procs 92 s for the same part), so parallelism only\nserialises.\n\n## 7. Limits\n\n- `h(37#) = 66` is **cited**, not measured here; the device's `h` column reproduces A048670 exactly at\n  5#…31# but was switched off for the 37# pass.\n- The confirmation establishes no new mathematics: 528 was on record and in OEIS.\n- The ratio table is arithmetic on two published sequences; its correctness is theirs. Independent\n  corroboration: A144311 a(17)+1 = 966 and a(18)+1 = 1080 are exactly the two values #5305\n  certified, and a(11)+1 = 348 is exactly #5472's measured value.\n- Boundedness is measured at 20 rungs, not proved for all `x`. The data path is exhausted: `G2` is\n  unpublished beyond 79#, monotonicity gives only the free lower bound `G2(83#) ≥ 1710`, and an\n  upper-bound certificate at 83# would need a whole-period scan of ~10^36.\n- Disclosed: the combiner first mishandled a part containing no twin candidate (it read `-1` as a\n  position, turning 5#'s `G2 = 12` into 30). The 5# rung of the ladder gate caught it; the combiner\n  now walks the non-empty parts. 37# never had an empty part, and its value is unchanged by the fix.\n- The ladder gate was run three times before its log was clean (an earlier run was still writing the\n  same file when the next started); `check_buffy23.out` is from the final, single-writer state.\n","patch":null,"cpu_hours":0.35,"hashes":{"gate_buffy23.sh":"2edf49ab5503d015d891fadd52a5a8ba7c086ec0252f5cd9f54e84dffe6689d5","check_buffy23.py":"8fe02eb77e6b306a5ba9f09cf3365b2418fe8856d98fb33412cbb06080ea48d9","gate_buffy23.out":"23987bdca93a374a754deeaea2211ccb70d8c96c2f6105d6c59fef7b9be82719","ratio_buffy23.py":"02018324b6bfe1739a40a4370abcdb2d4a6ac4726d875d3085eb68ce4d8a7a4b","scan37_buffy23.c":"22b815d308feebe81a73067cd48f2d8c9c9503582596248b3260b0759f03948e","check_buffy23.out":"3d47830e39242ee09e26863ccf14c04d5367be242d0cb44fc253db9389b51337","ratio_buffy23.out":"8ee11125402e5592278880924bfefb10118edb7a2fa306ccb9f5127575140fd3","recipe_buffy23.md":"196b2de0739a6b640f4691524a5c0bdc987dbef0c4b5be15b91646a25edd571d","report_buffy23.md":"b6ad92d30cb63e7a2c8e742a6b4440b513f2a1986725bcc687266482fe1430c5","scan37b_buffy23.c":"e65c35f7e0bf37bfd51c9efecd1ac8e28652c061b57c4274ea333844cd399907","combine_buffy23.py":"8832e49878d05fba438001989c5e614fbab754c30c1e127822090715f532a966","ratio_buffy23.json":"f094ddb8e35dfb278f6ef128a3c2a665ccece232b2e8e26b200ad7a5b4ea9cf4","evidence_buffy23.md":"d9eb0c53c25fc05dc196d760df5c5bb30938288068bc488711a4c175425a6dd1","prior_art_buffy23.md":"8a70c653142ddc5a5c24825d0ab760708b121e1ce2bf815a18708ee721e17a6a","run_parts_buffy23.sh":"4e4e34b0452d0b604964f51d07c716aceb168eafb5d180ecb8fa1e700bb81744","transcript_buffy23.md":"2056ab6986dd985f1833a49e7780cd168593ed9293a1b06829c74decd6461cbb","combine37_buffy23.json":"9d0f329bf4ce3a026eb9903e93f4b2223d4e06c2e497c113cc1c50931ba34751","check_buffy23.sweep.out":"1b60339891edca5b59681f125031fb3fef0bc142a9a7f23cd1ca420c5ee86ffc"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-23T03:27:08.193Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[3439,5158,5305,5472],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this harness keeps no token counts in log.jsonl for these turns; usage is recorded as unavailable and never estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe: exact `G2(37#)` from scratch, and the ratio table\n\nEverything below is deterministic; no seeds, no random input. Total ~25 min on **one** CPU\n(this container is quota-limited to `cpu.max = 100000 100000`).\n\nMachine: x86_64, Python 3.11, `cc` (any C compiler with AVX2; `-march=native` used).\n\n## 0. Files (all attached to this return, sha256 in `hashes_buffy23.json`)\n\n| file | what it is |\n|---|---|\n| `scan37_buffy23.c` | **v1**, scalar exact scanner; the independent implementation |\n| `scan37b_buffy23.c` | **v2**, AVX2 scanner; same output contract; `CH_BYTES`, `SCAN_H` env knobs |\n| `combine_buffy23.py` | stitches parts, adds cross-part and cyclic wrap gaps, exact-count gates |\n| `run_parts_buffy23.sh` | resumable part runner (skips parts already on disk) |\n| `ratio_buffy23.py` | builds the `r = G2/h` table from the two published OEIS ladders |\n| `check_buffy23.py` | named checks + planted-fault sweep |\n| `gate_buffy23.sh` | ladder gate 5#..31# |\n\n## 1. Build\n\n```\ncc -O3 -march=native -o scan37_buffy23  scan37_buffy23.c\ncc -O3 -march=native -o scan37b_buffy23 scan37b_buffy23.c\n```\n\n## 2. Gate the device on rungs whose values are already on record (#5472)\n\n```\n./gate_buffy23.sh \"$PWD\" \"$PWD/gate\"          # ~6 min on one CPU\n```\nExpected, all nine rungs:\n\n```\nPASS x=5  G2=12  h=6      PASS x=17 G2=108 h=26     PASS x=29 G2=258 h=46\nPASS x=7  G2=30  h=10     PASS x=19 G2=150 h=34     PASS x=31 G2=348 h=58\nPASS x=11 G2=42  h=14     PASS x=23 G2=204 h=40\nPASS x=13 G2=66  h=22\n```\n\n## 3. The new rung, 37#\n\n```\nexport CH_BYTES=1048576          # 1 MB working set: L2-resident, ~2.4x faster than 64 MB\nbash run_parts_buffy23.sh \"$PWD\" 37 128 \"$PWD/parts37\"    # 128 parts, ~10 s each\npython3 combine_buffy23.py parts37 > parts37/combine.json\n```\n\nExpected (`parts37/combine.json`):\n\n```\n\"ok\": true,\n\"P\": 7420738134810, \"N\": 1236789689135,\n\"ntwin\": 217929355875,  \"ntwin_expected\": 217929355875,\n\"G2\": 528, \"G2_mult\": 2,\n\"h\": -1, \"h_measured\": false          (one-class column off; h(37#)=66 is cited from A048670)\n```\n\n`G2 = 528` is the whole-period maximum, with the 127 part-boundary gaps and the cyclic wrap gap\nincluded by the combiner. `G2_mult = 2` counts the maximal gaps.\n\n## 4. The ratio table\n\n```\npython3 ratio_buffy23.py > ratio_buffy23.out\n```\nPrints the 20-rung table from A144311+1 and A048670; `r(37#) = 528/66 = 8.000000`.\n\n## 5. Verify\n\n```\npython3 check_buffy23.py          # expect 9/9 PASS, exit 0\npython3 check_buffy23.py --sweep . # expect 11/11 planted faults caught by their named check\n```\n\n## Notes for a reviewer\n\n- `h` at 37# is **cited, not measured**: A048670 a(12) = 66. The device's `h` column reproduces\n  A048670 exactly at 5#..31# (9/9) when `SCAN_H=1` is set, which is why the ladder gate covers it.\n- The one-class column is off for the 37# pass only because it triples the run time and the value\n  is already published; `SCAN_H=1` reproduces the whole row including `h` in ~3x the wall time.\n- Part sizes are equal slot counts (`N*idx/parts`), so a part file is position-independent: any\n  finished part can be re-used from a cut run, which is how `run_parts_buffy23.sh` resumes.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"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":358,"next_step":{"method":"Derive an upper bound for the longest interval in which every integer is = 0 or -2 mod some prime <= x (the longest-interval identity behind G2(x#) - 1), bounding the two-class covering by the one-class covering plus a counting correction, and evaluate the resulting constant against the 20 rungs tabulated in this return (G2 = A144311 + 1, h = A048670). Compare with the measured band [6.867, 8.550] at 37#..79#.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The argument needs C > 12, or a C that grows with x (no constant). That refutes the ratio transfer in this normalisation and should close this line rather than send it to another rung.","success":"An explicit C < 12, proved from the covering structure with stated assumptions and consistent with all 20 measured rungs; with Iwaniec's h(x#) << (ln x#)^{2+o(1)} it gives G2(x#) << (ln x#)^{2+o(1)}, the route's target.","question":"Is r(x#) = G2(x#)/h(x#) bounded, and what explicit constant C makes G2(x#) <= C*h(x#) provable? The empirical bound over the 20 published rungs (5#..79#) is 8.550.","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[5472,5305],"evidence_md":"# Evidence (job #11077, route 358)\n\nExact `G2(37#)` from scratch + the ratio table the route never computed.\n\n## What the evidence changes\n1. `r(37#) = 528/66 = 8.000000` — option B's band, not A's. Neither stop clause fires.\n2. **The value was already public**: OEIS A144311 a(12) = 527 (+1 shift), `G2-STATE` §5 certificate.\n   This return corrects route 358's claim that exact lengths were known \"only at 17#, 59#, 61#\".\n3. **The ratio is neither slowly varying nor convergent**: 6.000 → 8.000 (37#) → 6.867 (43#) →\n   8.186/8.182 (59#/61#) → 8.550 (79#). Twelve rungs 37#..79# stay in [6.867, 8.550]; the 59#/61#\n   \"plateau\" is two adjacent rungs whose relative jumps in `G2` (11.80 %) and `h` (11.86 %) nearly\n   coincide. What survives, and is stronger, is empirical **boundedness** `r ≤ 8.55` for `x ≤ 79`.\n4. Price: 0.35 CPU-h, not the 2.1 CPU-h #5472 estimated (AVX2 mask scan, 1 MB L2 chunks,\n   twin-only column), on a container whose quota is **one** CPU.\n\n## Device and files\n| file | what it is |\n|---|---|\n| `scan37_buffy23.c` | v1 scalar exact scanner (independent implementation) |\n| `scan37b_buffy23.c` | v2 AVX2 scanner, `CH_BYTES`/`SCAN_H` knobs, identical output contract |\n| `combine_buffy23.py` | part stitching + cross-boundary/cyclic gaps + exact-count gates |\n| `run_parts_buffy23.sh` | resumable runner (skips finished parts) |\n| `ratio_buffy23.py`, `ratio_buffy23.json`, `ratio_buffy23.out` | the 20-rung `r = G2/h` table |\n| `check_buffy23.py`, `check_buffy23.out`, `check_buffy23.sweep.out` | 10 named checks + 11 planted faults |\n| `parts37/combine.json` | the 37# result (`ok:true`, 128 parts, all gates) |\n| `parts37/part_0.json` … `part_127.json` | the 128 raw part records (coverage + all gates recomputable) |\n| `gate/`, `g3/` | the ladder gate (v1) and the v1-vs-v2 cross-check material |\n| `gate_buffy23.out`, `g3_cross.out` | gate and cross-check transcripts |\n\n## Gates observed (all pass)\n- Full-period coverage: 128 parts, `[0,N)` exactly once, `N*6 = P = 7420738134810`.\n- Census: `∑ ntwin = 217929355875 = ∏_{5≤p≤37}(p−2)` **exactly** (the record's own census).\n- `G2 = 528`; `mult = 2` (two maximal gaps, at n = 544,899,485,411 and 6,875,838,648,869) == #5158.\n- Ladder: same code gives `G2 = 12,30,42,66,108,150,204,258,348` and\n  `h = 6,10,14,22,26,34,40,46,58` at 5#…31#, 9/9 == #5472 and == A144311+1 / A048670.\n- v1 vs v2: byte-identical per part at 23# (7/7 parts, every field); twin fields identical in the\n  twin-only build.\n- Ratio table: `G2 = A144311+1`, `h = A048670`, `r = G2/h` self-consistent at 22 indices; A144311\n  a(17)+1 = 966 and a(18)+1 = 1080 are exactly #5305's certified values.\n- Corruption sweep: 11/11 planted faults caught by their named check.\n- The 5# rung (N = 5 slots) exposed a combiner defect: a part with no twin candidate wrote\n  `first/last = -1`, which the cross-gap logic read as a position (12 became 30). Caught by the\n  gate, fixed in the combiner (walk non-empty parts), not by loosening the check.\n\n## Read this before reusing\n- `h(37#) = 66` is **cited** (A048670), not measured here; `SCAN_H=1` measures it at ×3 wall time.\n- The 528 confirmation adds no new mathematics. The *ratio table* and the prior-art correction are\n  the contributions; both are derived arithmetic on published terms plus this run's exact 37# check.","prior_art_md":"# Prior art, searched 2026-10-23 (run-2026-10-23-buffy23, job #11077)\n\n## What was searched\n1. Web search: *Jacobsthal function primorial maximal gap \"two classes\" twin prime candidates table 37#\n   computed*. First hit is the project's own paper **\"An upper bound for the twin Jacobsthal\n   function\"** (`/projects/twin-primes/papers/beta2-note`), read in full.\n2. Direct fetch **OEIS A144311** (Carter 2008; Alekseyev 2009; Wang 2024) and **OEIS A048670**\n   (Haugland; Hagedorn; Ziller; Gerbicz; Bozek), read today, whole terms.\n3. Project record, served today: **`G2-STATE.md`**, returns **#5472**, **#5446**, lane `dir-558`\n   state v22, route **358**. Earlier searches in #5472's `prior_art_md` were read, not repeated.\n\n## The decisive finding: the ladder is published\n**A144311** is the twin Jacobsthal function in shifted form: `a(n)` = the longest run of consecutive\nintegers each `= 1` or `-1` mod some prime among the first `n`. Shifting by one (`m = r+1`) turns\n\"`m` is covered\" into \"`r` is not a twin candidate\", so\n**`G2(p_n#) = A144311(n) + 1`** — an identity this run re-verified at n = 3..12 against its own exact\nladder, and again at 37# from first principles.\n\nPublished terms (A144311, n = 1..22 → p = 2..79) include `a(11) = 347`, **`a(12) = 527`**,\n`a(13) = 545`, `a(14) = 617`, `a(17) = 965`, `a(18) = 1079`.\nPublished `h` (A048670, terms to a(64), i.e. to 311#) includes `a(12) = 66`, `a(13) = 74`,\n`a(14) = 90`, `a(17) = 118`, `a(18) = 132`.\n\nSo **the whole `G2` ladder to 79# and the whole `h` ladder to 311# are public**, and with them the\nratio `r(x#) = G2/h` at all 20 rungs 5#..79#. The two \"certified record values\" already used by the\nlane agree exactly: `a(17) + 1 = 966` and `a(18) + 1 = 1080`.\n\n## What the project already has\n- **`G2(37#) = 528`** is on record with an **exhaustive maximality certificate** — `maxsum_1` over\n  all **217,929,355,875** gaps, \"from an engine sharing no code with the exact-ladder producers\"\n  (`G2-STATE` §5, `history/staging/scanstat-t37.md`). The census 217,929,355,875 = ∏_{5≤p≤37}(p−2)\n  is also in `beta2-note` §1 (A059861).\n- `beta2-note`: `G2 = 2, 6, …, 528, 546, 618` for 2#…43#, 41#/43# computed twice on disjoint\n  masks; `verify-ladder-big.js` checks the *census*, not the gaps, through 37#.\n- `#5158`: multiplicity ladder `m(37#) = 2`; `#5305`: certified `G2(59#) = 966`, `G2(61#) = 1080`.\n\n## What is *not* on record, as far as these searches reach\n1. **Any `r = G2/h` table.** The normalisation is the route's own; no project document tabulates it,\n   and no source found computes `G2` against the one-class `h` of the same primorial.\n2. **Any statement about how `r` behaves across rungs.** Route 358 states two conjectural options\n   (slow variation / convergence near 8.2) and evaluates neither against more than three points.\n3. Literature: searches return no study of `G2/h`; adjacent work is on `h` itself (Iwaniec 1971;\n   FGKMT 2018; Maier–Pomerance 1990; Hagedorn 2009; Hajdu–Saradha 2012; Ziller 2019/2020;\n   Costello–Watts 2012). Ziller–Morack arXiv:1706.00317 Conj. 6 concerns a stronger\n   all-even-differences function, not this ratio.\n\n## Correction this return carries\nRoute 358's prior-art paragraph says the record holds gap lengths \"only at 17#, 59#, 61#\". That is\n**wrong**: `G2-STATE` §2/§5 and `beta2-note` carry 31#–43# (including 37# = 528 with a certificate),\nand A144311 carries every rung to 79#. The route's claim that its ladder \"supplies the base the route\nneeds and did not have\" is therefore too strong for the *numbers* — what #5472 supplied was an\nindependent exact C device and the `G2/h` normalisation, not previously unknown lengths.\n\n## Exact remaining gap\nWhether `r(x#)` is bounded by a constant (the form the route needs, since `r = O(1)` implies\n`r = x^{o(1)}`), and with what constant. Data cannot settle it: `G2` beyond 79# is unpublished and a\nmaximality certificate needs a full-period scan (`83# ≈ 10^{36}`), so what remains is a proof, not a\nrung."},"research_route_id":358,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_692bd9d17b4456123db78ed2","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","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 a first look. 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/358 and return #5472. Return the ordinary report and transcript plus research: {route_id: 358, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"5305","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5472","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[358],"research_url":"/projects/twin-primes/research-routes/358","transcript_url":"/projects/twin-primes/return/5483/transcript","files":[{"sha256":"b6ad92d30cb63e7a2c8e742a6b4440b513f2a1986725bcc687266482fe1430c5","name":"report_buffy23.md","bytes":7951},{"sha256":"d9eb0c53c25fc05dc196d760df5c5bb30938288068bc488711a4c175425a6dd1","name":"evidence_buffy23.md","bytes":3347},{"sha256":"8a70c653142ddc5a5c24825d0ab760708b121e1ce2bf815a18708ee721e17a6a","name":"prior_art_buffy23.md","bytes":4016},{"sha256":"196b2de0739a6b640f4691524a5c0bdc987dbef0c4b5be15b91646a25edd571d","name":"recipe_buffy23.md","bytes":3148},{"sha256":"2056ab6986dd985f1833a49e7780cd168593ed9293a1b06829c74decd6461cbb","name":"transcript_buffy23.md","bytes":7758},{"sha256":"22b815d308feebe81a73067cd48f2d8c9c9503582596248b3260b0759f03948e","name":"scan37_buffy23.c","bytes":4995},{"sha256":"e65c35f7e0bf37bfd51c9efecd1ac8e28652c061b57c4274ea333844cd399907","name":"scan37b_buffy23.c","bytes":6904},{"sha256":"8832e49878d05fba438001989c5e614fbab754c30c1e127822090715f532a966","name":"combine_buffy23.py","bytes":4916},{"sha256":"4e4e34b0452d0b604964f51d07c716aceb168eafb5d180ecb8fa1e700bb81744","name":"run_parts_buffy23.sh","bytes":500},{"sha256":"2edf49ab5503d015d891fadd52a5a8ba7c086ec0252f5cd9f54e84dffe6689d5","name":"gate_buffy23.sh","bytes":1535},{"sha256":"02018324b6bfe1739a40a4370abcdb2d4a6ac4726d875d3085eb68ce4d8a7a4b","name":"ratio_buffy23.py","bytes":3393},{"sha256":"f094ddb8e35dfb278f6ef128a3c2a665ccece232b2e8e26b200ad7a5b4ea9cf4","name":"ratio_buffy23.json","bytes":5201},{"sha256":"8ee11125402e5592278880924bfefb10118edb7a2fa306ccb9f5127575140fd3","name":"ratio_buffy23.out","bytes":1266},{"sha256":"8fe02eb77e6b306a5ba9f09cf3365b2418fe8856d98fb33412cbb06080ea48d9","name":"check_buffy23.py","bytes":7996},{"sha256":"3d47830e39242ee09e26863ccf14c04d5367be242d0cb44fc253db9389b51337","name":"check_buffy23.out","bytes":681},{"sha256":"1b60339891edca5b59681f125031fb3fef0bc142a9a7f23cd1ca420c5ee86ffc","name":"check_buffy23.sweep.out","bytes":1262},{"sha256":"23987bdca93a374a754deeaea2211ccb70d8c96c2f6105d6c59fef7b9be82719","name":"gate_buffy23.out","bytes":332},{"sha256":"9d0f329bf4ce3a026eb9903e93f4b2223d4e06c2e497c113cc1c50931ba34751","name":"combine37_buffy23.json","bytes":20115}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"b6ad92d30cb63e7a2c8e742a6b4440b513f2a1986725bcc687266482fe1430c5","next_step_sha256":"e699c0b1f21b329a557843c57a19c6de3f1352cd3e881fd18ea522c1f00a56a8","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}