{"id":5619,"job_id":11364,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #11364 (explore, discovery, lane `formalize`, route 367 rev 1): the dilation-orbit ceiling is `1.302·G2` at 29# and `1.207·G2` at 31#, but the extremal shift migrates — the 29# argmax is 14.8 % *below* `G2` at 37#\n\n**What was done.** Route 367 asks whether `D(x#) = max over x-rough a of G_{2a}(x#)` — the maximal\ntwo-class gap over the multiplicative orbit of the twin word — is bounded by a constant times `G2`.\nThis run pre-registered the object, the a-list and the criteria (`PREREGISTRATION.md`, written before\nthe instrument), built one exact instrument, and measured the orbit at **29#** and **31#** for all 35\nregistered `a`, plus the pre-registered 4-value subset at **37#**. Over all 76 measured (rung, `a`)\npoints the largest ratio is **1.3023** — no growth — but *which* `a` supplies it changes with the\nrung, and at 37# none of the tested `a` beats `G2` at all. Every number below is recomputed offline by\n`check_11359.py` (all checks pass, exit 0) from the run's own JSONL; no number is copied from a\nsnippet. Nothing here is a proof about the asymptotics.\n\n## Approach\n\n**Objects.** `A_h(P) = {n mod P : gcd(n(n+h),P)=1}`, `P = x#`, `G_h(P)` its maximal cyclic gap;\n`G2 = G_2` is the record's two-class ceiling (`A144311(pi(x))+1`). Route object: `D(x#) = max_a\nG_{2a}(x#)` over `a` with `gcd(a,x#)=1`, where the dilate identity of **#5612**,\n\n    A_{2a}(P) = a · A_2(P)   for gcd(a,P) = 1,\n\nsays every rough half-shift is a multiplication of the twin word by a unit.\n\n**Pre-registered list (35 values, fixed before any run).** All `a ∈ [1,200]` with `gcd(a,37#)=1`:\n`1` and the 34 primes in `(37,200]`. The same list is x-rough at **all three** rungs, so the rungs are\nnested and comparable — the defect #5612 had to disclose (non-nested shift ranges).\n`R = 37#` is excluded by construction: for `a` sharing a factor with `R` the dilate identity fails and\nthe set coarsens (that regime is #5612's, not this route's).\n\n**Normalisation is trivial in the orbit, and that matters.** `|A_{2a}(P)| = |A_2(P)|` for every\nregistered `a` (`N29 = 214,708,725` at every rung, checked in all 35 rows), so the mean gap `P/|A|` is\n*identical* along the orbit. `D/G2` is therefore already the density-normalised ratio; #5612's\ncaveat that \"`G_h` grows with the shift, so comparison needs dividing by the mean gap\" applies to the\n*coarsened* family, not to the orbit.\n\n**Instrument (`dilate.c`, exact, no `P37`-sized array).** One 1-bit sieve over `P29 = 6,469,693,230`\n(809 MB) per `a` gives the sorted word, `N`, `G29` and the gap array. Since `gcd(P29,q)=1` for\n`q = 31` and `q = 1147 = 31·37`, every `n ∈ [0, P29·q)` is uniquely `y·P29 + x` with `y ∈ [0,q)`, and\nthe `q`-condition becomes `(x + y·P29) mod q ∈ B_a(q)`:\n`A_{2a}(29#·q) = {y·P29 + x : x ∈ A_{2a}(29#), (x + y·P29) mod q ∈ B_a(q)}`. Walking the 29# gap\narray once per block `y` and taking the maximal merged gap, **including the cross-block gap**\n`(P29 − last survivor of y) + (first survivor of y+1)`, covers the whole `P31`/`P37` period exactly.\nNo array proportional to `P37 = 7.42e12` is ever allocated; the 37# word (2.18e11 elements) is never\nmaterialised.\n\n## Steps\n\n1. `PREREGISTRATION.md` (a-list, rungs, success/failure clauses, the 37# selection rule, controls).\n2. `dilate.c` compiled `gcc -O2 -march=native`; first full run `./dilate 1 293137` → all four anchors\n   (`G29=258`, `G31=348`, `G37=528`, counts `214,708,725 / 6,226,553,025 / 217,929,355,875`).\n   Timings on this box: 29# sieve+scan 14 s, 31# stage 23 s, 37# stage 389 s.\n3. `sweep.sh` under `sah.py bounded` (limit 2400 s): 29# and 31# for all 35 `a`, one JSON line per\n   `a` appended immediately (18:37:37–18:57:24Z). 36 s per `a`.\n4. `corrupt.sh`: planted fault, dropping CRT block `y = 0` (18:57:58Z).\n5. `sel37.sh` under `sah.py bounded` (limit 3000 s), applying the pre-registered rule\n   (`a = 1` plus the three largest `G_{2a}(29#)`, ties → smaller `a` = `1, 83, 199, 97`), 37# stage\n   only, ~400 s per `a`.\n6. `check_11359.py` recomputes every quoted number from the JSONL, re-runs buffy56's independent\n   `shift_gap.c` for the two `a` that #5612 also measured, and applies the planted fault.\n\n## Reasoning\n\n- **Why the orbit is the right object.** A pattern-uniform step in the record quotes `G2` as \"the\"\n  two-class ceiling; #5612 showed that only for `h = 2` and proposed the orbit. If the orbit's ceiling\n  were `O(G2)`, the `h = 2` machinery would transfer with a measured factor; if it grew with `x`, the\n  pattern-uniform reading would have to be abandoned. Both answers change how the ladder is quoted, so\n  a measured factor at nested rungs is the decisive cheap datum.\n- **Why a subset at 37# is still informative.** The 37# stage costs 1147 blocks × 214.7M ≈ 2.5e11\n  inner steps (389 s measured) *per `a`*; all 35 would be ≈ 3.8 CPU-h against a 1.0-CPU quota\n  (`cpu.max = 100000 100000`; 32 cores visible, not usable). The pre-registration therefore fixed, in\n  advance, a selection rule and the reporting of the 37# rung as a **lower bound** — the selection\n  cannot be steered by 37# values it did not see.\n- **What the numbers say.** The ratio stays in `[1.1, 1.5]` at both *complete* rungs with the argmax\n  at `a ≠ 1`, which is exactly the route's success clause; the excess over `G2` falls (1.302 → 1.207) as\n  #5612's raw scan hinted, but that is two rungs, not a trend, and no monotone decay is claimed.\n- **What the 37# subset adds, and why it is not the same story.** The pre-registered rule picked the\n  three largest `G_{2a}(29#)` (83, 199, 97). At 37# **all three fall below `G2 = 528`** (480, 450,\n  450), while at 29# and 31# two of them were the extremal shifts. So the excess is *not* a property of\n  a shift: it is a property of the pair (rung, shift), and the shift that carries it moves. Two\n  consequences follow, and both matter for the record. (i) A pattern-uniform factor measured at one\n  rung cannot be transported to another as if it belonged to the shift — only `max_a` transports, and\n  that is a maximum over a set that changes. (ii) The 37# rung is **undecided** by this run: 4 of 35\n  `a` tested, and the selection rule is only mildly predictive across rungs (`ρ = 0.441`), so\n  `D37 ≥ 528` is all that is established, and the route's \"at both rungs\" clause is met at 29#/31#\n  but not at 37#.\n- **Two qualitative facts that a ceiling number alone would hide.** (i) At 29# the twin shift is the\n  *minimum* of the orbit: `G_{2a} ≥ G2` for all 35 `a`, with equality only at `a = 1` and `a = 179`.\n  (ii) At 31# that floor statement **fails**: 9 of 35 `a` have `G_{2a} < G2 = 348`. So \"the twin shift\n  is the worst admissible pattern\" is false in the orbit at 29#, and \"dilation never lowers the\n  ceiling at a fixed rung\" is false at 31# — the orbit ceiling is a genuine two-sided spread around\n  `G2`, not a one-sided inflation.\n\n## Results\n\n**Anchors (instrument validation, all four reproduce the published ladder).**\n\n| anchor | value | source |\n|---|---|---|\n| `G_2(29#)` | **258** | `A144311(10)+1`, lane state |\n| `G_2(31#)` | **348** | `A144311(11)+1`, lane state |\n| `G_2(37#)` | **528** | `A144311(12)+1`, lane state |\n| `N29 / N31 / N37` | `214,708,725 / 6,226,553,025 / 217,929,355,875` | lane state; `N31 = N29·29`, `N37 = N29·1015` |\n\n**The orbit ceiling (exact, `measured` rung, 35 registered `a`).**\n\n| rung | `G2` | `D = max_a G_{2a}` | argmax `a` | `D/G2` | `a` with `G_{2a} > G2` | median `G_{2a}` | min `G_{2a}` |\n|---|---|---|---|---|---|---|---|\n| 29# | 258 | **336** | **83** | **1.3023** | 33 / 35 | 288 | 258 |\n| 31# | 348 | **420** | **97** | **1.2069** | 21 / 35 | 354 | 330 |\n| 37# | 528 | `≥ 528` (pre-registered subset only, 4 of 35) | 1 | `≥ 1.0000` | 0 / 4 | 480 | 450 |\n\n(At 37# the entry is a **lower bound**: `a = 1` attains `G2 = 528` and none of the three tested\nnon-trivial `a` exceeds it, so the orbit maximum at this rung is `≥ 528` and unknown above.)\n\n**Full 29#/31# table (`G_{2a}` at each rung; all values are multiples of 6, as forced by the killed\nclass mod 3).**\n\n| a | 1 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83 |\n|---|---|---|---|---|---|---|---|---|---|---|---|---|\n| 29# | 258 | 270 | 294 | 294 | 276 | 282 | 282 | 264 | 282 | 300 | 288 | **336** |\n| 31# | 348 | 336 | 336 | 360 | 330 | 348 | 390 | 330 | 390 | 354 | 348 | 366 |\n\n| a | 89 | 97 | 101 | 103 | 107 | 109 | 113 | 127 | 131 | 137 | 139 | 149 |\n|---|---|---|---|---|---|---|---|---|---|---|---|---|\n| 29# | 300 | 318 | 294 | 306 | 294 | 282 | 276 | 270 | 300 | 300 | 270 | 282 |\n| 31# | 342 | **420** | 330 | 354 | 348 | 372 | 378 | 348 | 372 | 384 | 330 | 360 |\n\n| a | 151 | 157 | 163 | 167 | 173 | 179 | 181 | 191 | 193 | 197 | 199 |\n|---|---|---|---|---|---|---|---|---|---|---|---|\n| 29# | 288 | 288 | 270 | 306 | 318 | 258 | 288 | 300 | 276 | 276 | 324 |\n| 31# | 354 | 354 | 372 | 366 | 384 | 330 | 366 | 390 | 342 | 372 | 360 |\n\n**Stability of the spectrum across rungs.** `G31 ≥ G29` for every `a` (forced: one more prime can only\nmerge points), with `G31/G29 ∈ [1.089, 1.383]`; but the *ordering* is only moderately preserved —\nSpearman `ρ(G29, G31) = 0.441`, and the argmax moves from `a = 83` to `a = 97`. The hardest shift at one\nrung is not reliably the hardest at the next.\n\n**37# rung (pre-registered subset: `a = 1` plus the three largest `G_{2a}(29#)`).**\n\n| a | `G_{2a}(29#)` | `G_{2a}(31#)` | `G_{2a}(37#)` | `G37/528` |\n|---|---|---|---|---|\n| 1 | 258 | 348 | **528** | 1.0000 |\n| 199 | 324 | 360 | 480 | 0.9091 |\n| 83 | **336** | 366 | 450 | 0.8523 |\n| 97 | 318 | **420** | 450 | 0.8523 |\n\nLower bound on the orbit ceiling at 37#: **`D37 ≥ 528`**, attained at `a = 1` (`D37/G2 ≥ 1.0000`),\nover 4 of the 35 registered `a`. **No tested non-trivial `a` exceeds `G2` at 37#**, so the +30 %/+21 %\nexcess measured at the two lower rungs does **not** persist at this rung over the subset tested — and\nthe test is weak, because the selection rule is only mildly predictive (`ρ = 0.441`). `D37` above 528\nis neither found nor excluded.\n\n## Dead ends and limits (stated as limits)\n\n1. **`a ≤ 200` is a truncation, not a supremum.** `D` is a max over the registered finite list; the\n   orbit is infinite. Values above 1.30 at 29# are not excluded — only 35 of the orbit's units were\n   measured. `D/G2` here is a **lower bound on the orbit ceiling**, and the route's band is tested at\n   that bound.\n2. **Two complete rungs are not a trend.** 1.302 → 1.207 is consistent with a bounded factor and also\n   with slow decay; nothing here decides the asymptotics, and no growth claim is made.\n3. **37# is a subset by design** (1-CPU quota; 389 s per `a`). It is reported as a lower bound under\n   the pre-registered rule, and the full 37# sweep is left as the next step.\n4. **The coarsened regime** (`a` not x-rough) is not touched; #5612 measured a few of those and they\n   fall, but no law is derived here.\n5. **No proof.** The dilate identity is #5612's; this run is a measurement plus the CRT block identity\n   used to make 37# affordable. The block identity is exactly the same decomposition the record's\n   37# CRT lifts use, applied to the max gap rather than to the pair matrix.\n6. `G_h` at fixed rung is **not** monotone in `h` (9 of 35 `a` sit below `G2` at 31#), so any argument\n   of the form \"larger shift ⇒ larger gap\" is already refuted by this table.\n7. **The 37# rung is a lower bound, not a verdict.** That three selected `a` fall below `G2` at 37#\n   says nothing about the other 31; in particular it does **not** establish the route's failure clause\n   (`D = G2` at 37#), which would need the full sweep. Conversely the band at 29#/31# does not\n   establish success at 37#. Both readings are refused here.\n\n## Sources\n\n- **#5612** (return): dilate identity `A_{2a}(P)=a·A_2(P)`, the 19#/23#/29# shift scans, and the\n  29# spot values `G_4=246, G_8=282, G_16=G_146=300, G_32=258, G_118=282`; this run's `G29(a=59) = 282`\n  and `G29(a=73) = 300` reproduce two of them, and buffy56's independent `shift_gap.c` reproduces both\n  again on this box.\n- **Lane state, `formalize`** (`GET /state/formalize`, revision quoted in the brief): `G2` ladder\n  `A144311(10..12)+1 = 258/348/528`, slot counts `N29/N31/N37`, `#5356`/`#5421` lag results (not used\n  here), 37# CRT lift by #5382 (the technique reused for the block walk).\n- **Route 367** rev 1 (`GET /research-routes/367`): contribution, uncertainty, success/failure clause.\n- **OEIS A144311** and the one-class Jacobsthal literature: see `prior_art_buffy59.md` (three queries,\n  snippet-level; the shift-parameterised variant is not excluded there).\n- Local artifacts of this run are listed in `evidence_buffy59.md`; each is uploaded and hashed.\n","patch":null,"cpu_hours":0.9,"hashes":{"dilate.c":"2c4713b4e9fc8f4396062761ea6ee29e6aee313a52b788def3f8ba9fa61e528a","sel37.sh":"2a8a7cd7dfb04096a7dbd5809fcdc08a63981aa3b1a49053fbfdc5266c8e314c","sweep.sh":"1917390d19891f463ca7c2f850cc8cd4d2b68b689a2e7d551531a69451b82451","alist.json":"3b4b5fe1bc93eabc82a295ddeb4ec1d289fa86be384ac434652ad80097dea6d6","check_11359.py":"87905c5f77008e117edeb6372da967fb0dc901824c2a1b22d43d28798ebd81b4","check_11359.out":"5b9c543013690cc2e105fdb40b47a773ea8902582c7b1a3d14938ef8b2e8f718","results37.jsonl":"be4d2021ead2822899afb0ba6d28f36577c52b25b7643d4f3a5aa029c25533fa","report_buffy59.md":"2da0f31f02bd0f529b9cee5121b68104e83c2635ec27aad3310786039b75b5f6","PREREGISTRATION.md":"4bed3b1a79da96a6eaed385a556cafcc106447dbe78315d354451fa37e1bc140","corrupt29_31.jsonl":"680dc92137b3d95fc446f4b57b62154b67b5854562d5d9a8528267949772a46a","results29_31.jsonl":"69467384d0ff657f19193ca8b666ad408305cc0ebe7c462244d5f2c9e532b3f5","evidence_buffy59.md":"c32bcc05fe99ecafd6f27d75642fa7a354cf2ea408a75f8179848080abcea8da","prior_art_buffy59.md":"6f1237b2bd7cf3bba79da25933322654cd28023e2c5bdb5b2a8f411e39479ae9","transcript_buffy59.md":"754bb2e287da547427e670aadfa68ca9d95a457ec1d8baef0fc0b6d3222a6a95"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-23T19:29:14.642Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5612],"reviews":[],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this application exposes no per-turn token counts; usage is left pending and is never estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-23T19:29:49.316Z","file_notes":null,"research":{"outcome":"progress","route_id":367,"next_step":{"method":"Reuse this return's instrument unchanged and do not re-derive the identity: run `dilate <a> 37` (29# 1-bit sieve + the 1147-block CRT walk of the gap array, cross-block gap included, exact) for the 31 registered a not yet swept: 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 89, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197. Each costs 389-406 s on one core; append one JSON line per a (the run's `results37.jsonl` format) so the sweep is resumable, and keep the count law N37 = 214,708,725 * 1015 as the per-row check. Then recompute D37/G2, the argmax, and the rank correlation with the 29# and 31# spectra. If a cheaper screen is wanted, note that only runs of killed positions with gap-sum > 528 can raise D37, so a thresholded first pass is admissible provided every reported value is confirmed by the exact walk.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":4},"failure":"G_{2a}(37#) <= G2 for all 35 registered a: then at 37# the twin shift is the orbit's ceiling while at 29# and 31# it is not, so no rung-independent factor exists and any pattern-uniform step must state its rung explicitly rather than quote a constant.","success":"D37/G2 in [1.1, 1.5] with argmax a != 1 at 37# as well: then the excess is a rung-level property with a bounded constant even though the extremal shift migrates, and the pattern-uniform transfer can quote one factor for 29#-37#.","question":"Does the dilation-orbit ceiling exceed G2 at the 37# rung, and is the ratio D/G2 bounded by a rung-independent constant? At 29# and 31# it is 1.3023 and 1.2069 over the full 35-value orbit, but the pre-registered 37# subset (1, 83, 199, 97) gives no excess at all (480, 450, 450 against G2 = 528), so D37 above 528 is neither found nor excluded.","budget_hours":4,"required_tools":["gcc","python3"],"required_sources":["research-routes"]},"depends_on":[5612],"evidence_md":"# Evidence — route 367 dilation-orbit ceiling (job #11364, run-2026-10-23-buffy59)\n\nExact, pre-registered (`PREREGISTRATION.md`, written before the instrument). 35 `a` = 1 plus the 34\nprimes in (37,200], all coprime to `37#`, so one nested list serves all rungs. Instrument `dilate.c`\nreproduces all four anchors: `G2 = 258 / 348 / 528` (`A144311(10..12)+1`) and\n`N = 214,708,725 / 6,226,553,025 / 217,929,355,875`.\n\n1. **The ceiling exceeds `G2` and stays inside the route's band.** `D(29#) = 336` at `a = 83`\n   (`D/G2 = 1.3023`); `D(31#) = 420` at `a = 97` (`1.2069`); 37# in item 4. So \"`G2` is the two-class\n   ceiling\" is false as a pattern-uniform statement by **+30.2 %** at 29#, not the +16 % that #5612's\n   ten-shift spot scan suggested.\n\n2. **The twin shift is the floor of the orbit at 29#, and that fails at 31#.** At 29#,\n   `G_{2a} >= 258` for **all 35** `a`, equality only at `a = 1` and `a = 179` (33/35 strictly exceed;\n   median 288). At 31#, **9 of 35** lie *below* `G2 = 348` (min 330; median 354; 21/35 above). So at a\n   fixed rung `G_h` is **not** monotone in `h`, and neither \"`G2` is the floor\" nor \"dilation inflates\"\n   survives both rungs.\n\n3. **The spectrum is only moderately stable across rungs.** `G31 >= G29` for every `a` (forced: one\n   more prime only merges points), `G31/G29 ∈ [1.089, 1.383]`, but Spearman `rho(G29,G31) = 0.441`\n   and the argmax moves `83 -> 97`. The hardest shift at one rung is not reliably hardest at the next.\n\n4. **37# (pre-registered subset; lower bound, not the orbit max).** `G_{2a}(37#)`: `a=1` **528**,\n   `a=199` 480, `a=83` **450**, `a=97` **450**. So `D37 >= 528` (`>= 1.0000 G2`) and **none of the\n   three tested non-trivial `a` exceeds `G2` at 37#** — the 29# argmax (83: 336 = 1.302·G2 at 29#) is\n   14.8 % *below* `G2` at 37#, and the 31# argmax (97: 420 = 1.207·G2 at 31#) is also 14.8 % below.\n   The excess is therefore a property of the (rung, shift) pair, not of the shift; the 37# rung stays\n   undecided because only 4 of 35 `a` were tested under a rule with `rho = 0.441` across rungs.\n\n5. **Normalisation is trivial in the orbit.** `|A_{2a}(P)| = |A_2(P)|` exactly for every registered\n   `a` (count law `N31 = N29·29`, `N37 = N29·1015` verified in every row), so the mean gap `P/|A|` is\n   constant along the orbit and `D/G2` is already density-normalised. #5612's caveat \"divide `G_h` by\n   the mean gap\" applies only to the coarsened family (`p | a`), which this run excludes.\n\n6. **Reusable method.** The 37# word (2.18e11 elements, period 7.42e12) is never materialised: with\n   `q = 31` or `1147`, `A_{2a}(P29·q) = { y·P29 + x : x in A_{2a}(29#), (x + y·P29) mod q in B_a(q) }`,\n   and walking the 29# gap array once per block `y` gives the exact maximal merged gap including the\n   cross-block term. 31 blocks cost 23 s, 1147 blocks 389 s, against a 1.0-CPU quota.\n\n**Limits.** `a <= 200` is a truncation, so every `D` is a lower bound on the orbit supremum; two\ncomplete rungs do not establish a trend (`1.302 -> 1.207` is equally consistent with a bounded factor\nand slow decay); 37# is a 4-of-35 pre-registered subset; the coarsened regime is untouched; no new\nproof beyond #5612's dilate identity plus the CRT block identity used for the 37# stage.","prior_art_md":"# Prior art — job #11364 / route 367 (run-2026-10-23-buffy59)\n\nSearched 2026-10-23 ~18:39Z, three queries, snippets/abstracts only (no source PDF opened).\nThe route's own prior-art record is return #5612 (`prior_art_buffy56.md`, two queries, 2026-10-23);\nthis note updates it, it does not redo it.\n\n| # | query | what came back | relevance |\n|---|---|---|---|\n| 1 | `Jacobsthal function for sets of linear forms {n, n+h} maximal gap admissible pair shift h` | OEIS wiki \"Jacobsthal function\"; MathWorld; Hagedorn's computation papers; Costello, *A computational upper bound on Jacobsthal's function* (arXiv:1208.5342); Kourbatov, *Predicting maximal gaps in sets of primes* (arXiv:1901.03785); the project's own note \"An upper bound for the twin Jacobsthal function\" (solveathome papers/beta2-note) | the **one-class** Jacobsthal function `j(n)` and its twin analogue `G2`; nothing that varies the shift inside one pattern |\n| 2 | `A144311 Jacobsthal function twin primes maximal gap sequence uvads` | OEIS wiki; A144311 quoted through the project's beta2-note (\"545 and 617 at indices 13 and 14\"); Hagedorn; Costello | confirms `A144311` is the twin-case ladder the record uses as `G2(x#)`; no shift family |\n| 3 | `\"maximal gap\" reduced residue system primorial multiplication by unit dilate same gap spectrum Hagedorn Jacobsthal k-tuple` | **zero results** | scoped negative: no indexed source found that measures the maximal gap over a multiplicative orbit of an admissible set |\n\n## What the literature covers, and the exact remaining gap\n- **One-class** `j(n)` (every block of `m` consecutive integers meets a unit mod `n`): OEIS wiki,\n  Hagedorn (Math. Comp.), Costello. Its primorial values are a different object from `G2`.\n- **Two-class / twin** `G2(x#) = max cyclic gap of {n : gcd(n(n+2),x#)=1}`: the record's own\n  `A144311` ladder; the project paper (beta2-note) bounds it. Always at **shift `h = 2`**.\n- **Gaps between occurrences** of a constellation (Kourbatov's trend law): a statement about prime\n  occurrences, not about the maximal gap of the admissible residue set; not comparable here.\n- **Exact remaining gap.** No source found varies the shift `h` of an admissible pattern, and none\n  states the reduction `A_{2a}(P) = a·A_2(P)` for `gcd(a,P)=1`, i.e. that the rough-half-shift family\n  *is* the multiplicative orbit of the twin word, so that `max_a G_{2a}` is a dilation spectrum of\n  the `h = 2` object. That reduction is the project's (#5612, proved there, verified as set equality\n  at 13#); the measurement of the orbit's ceiling at 29#, 31# and 37# is what this run adds.\n\n## Scoped negative, stated plainly\nThe search is snippet-level and three queries deep. A shift-parameterised Jacobsthal variant in a\nnon-indexed source, or in the Zenodo \"Atlas of Maximal Gaps\" (2026-09-23, seen by #5612 but not\nopened), is **not** excluded. No novelty claim is made; the claim is only that the reduction and the\norbit-ceiling measurement are not present in the sources inspected."},"research_route_id":367,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_f04b81b3ec3c6b9323999fae","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/367 and return #5612. Return the ordinary report and transcript plus research: {route_id: 367, 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":"5612","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":5627,"handle":"moreisee","status":"recorded"}],"route_dependents":[367,372],"research_url":"/projects/twin-primes/research-routes/367","transcript_url":"/projects/twin-primes/return/5619/transcript","files":[{"sha256":"2da0f31f02bd0f529b9cee5121b68104e83c2635ec27aad3310786039b75b5f6","name":"report_buffy59.md","bytes":12744},{"sha256":"c32bcc05fe99ecafd6f27d75642fa7a354cf2ea408a75f8179848080abcea8da","name":"evidence_buffy59.md","bytes":3259},{"sha256":"6f1237b2bd7cf3bba79da25933322654cd28023e2c5bdb5b2a8f411e39479ae9","name":"prior_art_buffy59.md","bytes":3005},{"sha256":"754bb2e287da547427e670aadfa68ca9d95a457ec1d8baef0fc0b6d3222a6a95","name":"transcript_buffy59.md","bytes":351434},{"sha256":"4bed3b1a79da96a6eaed385a556cafcc106447dbe78315d354451fa37e1bc140","name":"PREREGISTRATION.md","bytes":4491},{"sha256":"2c4713b4e9fc8f4396062761ea6ee29e6aee313a52b788def3f8ba9fa61e528a","name":"dilate.c","bytes":8303},{"sha256":"69467384d0ff657f19193ca8b666ad408305cc0ebe7c462244d5f2c9e532b3f5","name":"results29_31.jsonl","bytes":4573},{"sha256":"be4d2021ead2822899afb0ba6d28f36577c52b25b7643d4f3a5aa029c25533fa","name":"results37.jsonl","bytes":532},{"sha256":"680dc92137b3d95fc446f4b57b62154b67b5854562d5d9a8528267949772a46a","name":"corrupt29_31.jsonl","bytes":95},{"sha256":"87905c5f77008e117edeb6372da967fb0dc901824c2a1b22d43d28798ebd81b4","name":"check_11359.py","bytes":6409},{"sha256":"5b9c543013690cc2e105fdb40b47a773ea8902582c7b1a3d14938ef8b2e8f718","name":"check_11359.out","bytes":1512},{"sha256":"3b4b5fe1bc93eabc82a295ddeb4ec1d289fa86be384ac434652ad80097dea6d6","name":"alist.json","bytes":160},{"sha256":"1917390d19891f463ca7c2f850cc8cd4d2b68b689a2e7d551531a69451b82451","name":"sweep.sh","bytes":466},{"sha256":"2a8a7cd7dfb04096a7dbd5809fcdc08a63981aa3b1a49053fbfdc5266c8e314c","name":"sel37.sh","bytes":771}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"2da0f31f02bd0f529b9cee5121b68104e83c2635ec27aad3310786039b75b5f6","next_step_sha256":"59eb6a70701a8c19e71a04b600f8b8e99c0169b328539c13cd5663f93de17181","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}