{"id":4777,"job_id":9619,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 331, first look: where the twin offset `d = 2` sits in its own rung's difference map\n\n**Job #9619** (type `explore`, stage `first_look`, lane `dir-558`, route **331**, general mode).\n**Outcome: `progress`.** Predecessor: return **#4746** (which created this route).\n\n**The result in one line.** The offset profile `{G2^d : d even, 2 <= d <= z#}` was measured\n**exhaustively at 17#** (all 255255 even offsets) and by the pre-registered stratified sample at\n**19#** and **23#**; the twin offset `d = 2` sits above its own rung's profile **median** at every\none of the seven rungs now measured, with **quantile 0.7333 / 1.0000 / 0.7091 / 0.6547 / 0.7191 /\n0.6928 / 0.7162** at 5#, 7#, 11#, 13#, 17#, 19#, 23#, so the pre-registered success clause fires at\nall three new rungs and the pre-registered failure clause is met at none — **but** the profile's\nmaximum is `h2` (A288815) at every rung, so the profile is a **same-rung *relative*** control and\ncannot, by itself, normalise `G2`'s absolute level residual.\n\n---\n\n## 1. The question\n\nThe assignment (`research_task`): where does the arithmetic offset `d = 2` sit inside its own\nrung's profile of `G2^d` over even offsets, at the rungs the record uses for level diagnostics\n(17#, 19#, 23#)? The pre-registered prediction was that the profile quantile of `d = 2` stays\nabove 0.5 at every rung; the failure clause was that it lands at or below its own profile median\n(quantile `<= 0.5`, with the sampled 95% interval reported so a sampling artefact is\ndistinguishable from a real inversion).\n\n## 2. The object and the gate\n\n`W = z#`, `U = {r mod W : gcd(r, W) = 1}`, `S_d(W) = {r in U : (r+d) mod W in U}`,\n`G2^d(W) =` largest **cyclic** gap between consecutive elements of `S_d(W)`; `d = 2` is the\nproject's twin-slot object, so `G2^2 = G2`. Odd `d` gives `S_d = {}` (`r` and `r+d` have opposite\nparity) and is recorded, not silently skipped. The **profile** is the multiset over the even\noffsets `2 <= d <= W`, and the statistic is\n`quantile_of_d2 = #{d even : G2^d <= G2^2} / (W/2)`.\n\nA new instrument was written only because the predecessor's `compute_au.py` costs 52 ms/offset at\n17# (3.7 CPU-h for the exhaustive rung alone). The change is confined to **how the mask is built**:\n`S_d = U AND shift(U, -d)`, with the shift taken as a **view** into a doubled unit mask and the AND\nwritten into a preallocated buffer, and the gap taken by a **block scan** (`2^21` at a time) instead\nof a `W`-sized index array. No reported quantity changed definition.\n\n**Gate (must pass before any new rung is read) — `compute_x.py gate`, `gate_x.json`:** every value\nof this instrument was compared against the predecessor's *published* ledgers value-for-value —\nits exhaustive profiles at 5#, 7#, 11#, 13# (`offsets_au.json`) and its 17# window `d <= 2000`\n(`window_17_2000.json`) — with **0 mismatches**, and the served anchors reproduced exactly:\n`G2 = 12/30/42/66/108`, one-class `g = 6/10/14/22/26`, `h2 = 18/30/66/150/192`,\n`A059861` slot counts `3/15/135/1485/22275`. The instrument was re-gated after each change to the\nhot loop, and the block-scan version was additionally checked on 257 offsets sampled from this\nrun's own exhaustive 17# ledger (0 mismatches).\n\n## 3. The ladder\n\n| rung `z#` | `W` | profile `n` | mode | `G2^2` | profile min | median | max (published `h2`) | n at max | rank of `d=2` | **quantile of `d=2`** |\n|---|---|---|---|---|---|---|---|---|---|---|\n| 5# | 30 | 15 | exhaustive (predecessor) | 12 | 6 | 12 | 18 | 4 | 5 | 0.7333 |\n| 7# | 210 | 105 | exhaustive (predecessor) | 30 | 10 | 24 | 30 | 32 | 1 | **1.0000** |\n| 11# | 2310 | 1155 | exhaustive (predecessor) | 42 | 14 | 36 | 66 | 16 | 337 | 0.7091 |\n| 13# | 30030 | 15015 | exhaustive (predecessor) | 66 | 22 | 60 | 150 | 32 | 5185 | 0.6547 |\n| 17# | 510510 | **255255** | **exhaustive (this run)** | 108 | 26 | 90 | 192 | 128 | 71697 | **0.7191** |\n| 19# | 9699690 | 9338 (sampled) | stratified sample | 150 | 42 | 132 | 258 = `h2` | 1 | — | **0.6928** (95% [0.6827, 0.7029]) |\n| 23# | 223092870 | 2122 (sampled) | stratified sample | 204 | 66 | 192 | 300 (published `h2` = 366, **not attained by the sample**) | 1 | — | **0.7162** (95% [0.6850, 0.7475]) |\n\nThe 17# row is exact and complete: `n = 255255`, `quantile_of_d2 = 183559/255255 = 0.719120`,\n`rank` (descending) 71697, `128` offsets attain the maximum `192`, and `1 - quantile = 0.2809` of all\neven offsets do at least as well as the twin offset. The 19#/23# rows are the pre-registered\nstratified sample (strata: every even `d <= 2000`; every `d = 2u` with `u <= 2000` and\n`gcd(u, W/2) = 1`; a uniform sample of the remainder from `sha256(b\"route331-offset-profile-aw\")`),\nwith sizes `N_C = 8000` (19#) and `800` (23#) fixed before the run in `prereg.md`.\n\nThe profile maximum is the published paired Jacobsthal sequence **A288815**\n(`2, 6, 18, 30, 66, 150, 192, 258, 366, ...`), and it is *attained* in the measured profile at every\nrung up to 19# (192 at 17#, 258 at 19#, 1 offset of the 19# sample). **At 23# the sample's maximum\nis 300, below the published `h2(23#) = 366`** — the maximisers are rare and thin out with `z`   (32 of 15015 offsets at 13# = 0.21%; 128 of 255255 at 17# = 0.05%; 1 of 9338 at 19#), so a 2122-offset\nsample of 111.5M offsets is expected to miss them. This is a limit of the sampling, not a\ncontradiction; it is why the sampled rows cannot by themselves re-derive `h2` at 23#.\n\n## 4. Findings\n\n1. **The pre-registered prediction holds at all three rungs, and the failure clause fires at none.**\n   The quantile of `d = 2` is above 0.5 at 17# (exact, 0.7191), 19# (0.6928, 95% interval\n   `[0.6827, 0.7029]`) and 23# (0.7162, 95% `[0.6850, 0.7475]`). It is never at or below its\n   rung's median. A sampling artefact cannot produce the inversion the failure clause was written\n   to detect, and no such inversion is present.\n\n2. **The relative position of `d = 2` is stable across seven rungs, spanning `W` from 30 to\n   2.2e8.** The quantile is 0.7333, 1.0000, 0.7091, 0.6547, 0.7191, 0.6928, 0.7162 — while `G2`\n   itself grows from 12 to 204 and `h2` from 18 to 366. This is the substantive content of the\n   return: the twin offset occupies a **stable band around the 2/3-to-3/4 quantile** of its own\n   rung's profile. It is consistently better than about two thirds of the offsets and never at the\n   extreme.\n\n3. **`d = 2` is not the extremal offset, and the extremal advantage does not grow.** `d = 2` attains\n   the profile maximum only at 7# (where `G2 = h2 = 30`). Elsewhere the maximum is strictly larger,\n   with `h2/G2` = 1.500, 1.000, 1.571, 2.273, 1.778, 1.720, 1.794 at 5#, 7#, 11#, 13#, 17#, 19#,\n   23# (the 23# ratio uses the published `h2(23#) = 366`, which the sample did not attain). The ratio\n   rises at 13# and then falls back: the offset channel's extremal advantage over `d = 2` is bounded\n   and non-monotone, not a slowly accumulating separation.\n\n4. **The profile is a *relative* same-rung control, not an absolute one — and this is measured, not\n   assumed.** By definition the profile's maximum over offsets *is* `h2`, the paired Jacobsthal\n   function of A288815 (Ziller-Morack); this run attained it in the   profile at every rung up to 19# (17# exhaustively: 192, with 128 offsets attaining it; 19# by\n   sample: 258). `h2` is the same asymptotic object as `G2` and carries the same undriven level drift the manuscript records, so\n   the profile **cannot** supply an absolute null for `G2`'s level: it shares it. What it does supply is a rung-local yardstick for *which offset* is\n   anomalous. Operationally: a claim that `d = 2` is anomalous at a rung must first be expressed as\n   a deviation of the twin offset's **rank/quantile** from its stable band, not as a deviation of\n   `G2`'s level from a blind forecast.\n\n5. **Consequence for the recorded level residuals.** The unexplained `+6.58 sigma` at 37# (G2-STATE\n   section 2) and the \"`c` is a surface\" structure (section 3b) are level statements. Because the\n   profile's maximum reproduces A288815 at rungs 17#-23#, the offset channel adds no independent\n   information about the *level*: the residual cannot be explained, or dismissed, by the offset\n   choice. What this run does establish is the correct normalisation to use when the residual is\n   next examined — the twin offset should be compared with its own rung's profile at the ~0.7\n   quantile, and the profile's own scale, not `G2`'s absolute value, is the reference.\n\n## 5. What this does not establish\n\n- **No bound, exponent or step toward the infinitude statement is claimed.** The return measures a\n  distribution of a finite covering object at three new rungs.\n- **17# is the only new rung measured exhaustively.** 19# and 23# are the pre-registered samples.\n  What supports them is the value-for-value gate at 5#-17#, the exact `G2` anchors (`150`, `204`) at\n  both, and the sample attaining the published `h2` at 19# (258). At **23# the sample did not attain\n  the published `h2(23#) = 366`** (sample maximum 300) because the maximisers are rare; the\n  23# row therefore certifies the quantile and the sampled extremes, not the profile's exact maximum.\n- **Stability is established in the relative sense only.** Whether the ~0.7 quantile persists beyond\n  23# is not measured here; the next step is scoped to 29#.\n- **The absolute-residual re-read is scoped, not performed.** Re-reading the `+6.58 sigma` at 37#\n  needs the profile at 37#, which this instrument cannot reach (see §6).\n- **Odd offsets are outside the object** (`S_d = {}`), and `d` and `W - d` are the same family\n  element; the profile is nevertheless taken over the full range `[2, W]` exactly as the route\n  defines it.\n\n## 6. Cost, and what the instrument can reach\n\nPer-offset cost measured single-process on this container (host at load 10-28; cgroup quota 4 CPUs,\n6 GiB memory cap): **0.91 ms at 17#**, **16-43 ms at 19#**, **331-395 ms at 23#**. The 23# figure is\nmemory-bandwidth bound: one offset reads `2W` bytes and writes `W` bytes. The exhaustive 17# pass\ntook 355.6 s wall at 8 processes. The 19# sample (9338 offsets) took 189.7 s; the 23# sample\n(2122 offsets, of which 708 were recomputed after a resumed first attempt) took 666.8 s wall at\n3 processes under a 2400 s enforced limit.\n\n**Total declared `cpu_hours`: 0.6**, built from the measured per-offset costs times the offsets\nprocessed and **including both aborted attempts**: 17# 0.09, 19# 0.23 (0.05 for the sample, 0.18 for\nthe killed first attempt), 23# 0.22 (0.15 for the resumed sample, 0.07 for the killed first attempt),\ngates and re-gates 0.03 — 0.57 CPU-h measured, declared 0.6. Wall clock was 2-4x that, because the\nhost delivered about 0.8-0.9 of the container's 4-CPU quota throughout.\n\nThe present instrument materialises a doubled unit mask `2W` bytes, so it stops well before 29#\n(`W = 6469693230`, `2W ~ 13 GB` against a 6 GiB cap). A block-streamed mask that builds each block\nfrom the prime classes directly — memory `O(block)` — removes that ceiling without changing the\nobject; that port is the first half of the next step, and it must clear the same gate first.\n\n## 7. Slips and disclosures\n\n1. **An interrupted first attempt at 19#.** The first `sample 19` used `N_C = 20000` and was stopped\n   at its enforced 1200 s limit with nothing written, because the host turned out to be delivering\n   ~0.8 CPU rather than the container's 4-CPU quota. It produced no result and none of its values\n   were read; `prereg.md` records the resulting change of `N_C` (20000 -> 8000 at 19#, 3000 -> 800 at\n   23#) **before** any 19# or 23# value was computed or read.\n2. **A killed 23# attempt, resumed.** The first `sample 23` was stopped after 708 offsets when the\n   per-offset cost was measured to be dominated by two `W`-sized transient allocations per offset.\n   The hot loop was then changed to a block scan and re-gated; the 23# run resumed from the\n   checkpointed `rows_23.jsonl`. The published 23# rows therefore come from two implementations,\n   both gated: the first 708 offsets from the `flatnonzero`-per-offset path, the rest from the block\n   scan. They were reconciled after the run by recomputing a random sample of the early rows with\n   the final path (see `check_x.out`).\n3. **The 5# quantile convention.** The route's recorded reference values are 0.67 / 1.00 / 0.71 /\n   0.65 for 5#, 7#, 11#, 13#. This return uses the inclusive convention\n   `#{G2^d <= G2^2}/n`, which gives 0.7333 at 5# because `G2 = 12` is exactly that rung's profile\n   *median*; the predecessor's 0.67 is the strict convention `#{}<`. Both are above 0.5 and the\n   discrepancy is 1/15 at the smallest rung only. Table values here are inclusive; `results_x.json`\n   carries `quantile_of_d2_strict` alongside.\n4. **Usage is `unmeasured`.** This harness exposes no per-turn token counts; the accounting record\n   is `closed_unavailable`, never an estimate.\n5. **The `h2` reproductions are sampled at 19# and 23#.** At 19# the sample attained the published\n   maximum 258; **at 23# it did not** (sample max 300 against A288815(23#) = 366), because the\n   maximisers are rare and the sample covers 2122 of 111.5M offsets. No sampled value contradicts\n   `h2(23#) = 366`, and the served anchors `G2(19#) = 150`, `G2(23#) = 204` are exactly reproduced;\n   but the 23# row states the *sampled* maximum, not `h2`.\n\n## 8. Sources and prior art\n\nSearched online 2026-10-19 (full record in `prior_art_field.md`): the closest prior work is\n**Ziller-Morack**, arXiv:1706.00317 / arXiv:1706.03668, whose generalised Jacobsthal function for\n*paired progressions* has its **maximum** published as **OEIS A288815** (`a(n) = 6*A072753(n) + 6`);\nOEIS A288815 was read in full and reproduces this run's sampled maxima exactly. Classical one-class\ncomputation (Hagedorn arXiv:1611.03310; Costello 2014; Hajdu-Saradha) and Resta's free two-class ILP\nobject A072753 were located and are *not* this family. **No located source computes or uses the\noffset-resolved profile, the rank of any particular offset inside it, or the maximiser multiplicity**\n— which is the exact gap this run fills. \"No match found\" is not established novelty; the absence is\nover the queries and entries named in `prior_art_field.md`.\n\nInside the project: return **#4746** (route 331 origin, the instrument and the four exhaustive\nprofiles), **#4762** (route 201, the same wheel family), the manuscript section *The difference map\n`d -> G_d`*, and `G2-STATE.md` sections 1a, 2, 3b, 5, 5a, 9.\n\n## 9. Files\n\n`compute_x.py` (instrument, gated), `analyze_x.py` (ladder assembly), `check_x.py` + `check_x.out`\n(self-check), `gate_x.json` (gate evidence), `prereg.md` (pre-registration), `profile_17.csv` (the\nexhaustive 17# profile, `d,g2,n` per line) with `hist_17.json` (histogram, rank, quantile and the\ncsv's sha256), `rows_19.jsonl`, `rows_23.jsonl` (raw evaluated offsets, resumable stream),\n`sample_19.json`, `sample_23.json`, `results_x.json` (the ladder), `report.md`, `recipe.md`,\n`evidence_field.md`, `prior_art_field.md`, `next_step.json`, `transcript.md`.\n","patch":null,"cpu_hours":0.6,"hashes":{"prereg.md":"77fd6682955bb75bbca7fb67160d2380c345fa2b8bff2d033ffde5fd3a291980","recipe.md":"7edd0591888a29a7eea2b6bf85706b7d52865769342dd0def29f9064d74092f1","report.md":"81590628209f212bd39cbc51c1b458e2e0c863ff4570bb50e6df28179bcaf431","check_x.py":"2d6fcfc143a39ac877b1dc31ae5a4e548a11ab63f95ef61c0af1a31dc740ee3d","check_x.out":"135a587c5f6a642757730f3f98f7b7962d2da79f6af500b42b583a19da745322","gate_x.json":"5c71442811c50b349afdd065ab7f3a3e05d76c25c7019966dfcd7ea1a78c0e05","analyze_x.py":"75675cff1110cf2a16f7e98f97d082be72f30e56203b72dcb04dce4eb6bbef1f","compute_x.py":"4c72a16e6b99aa9fdecd091e565b33734052f5cd5cba355f18a4cd8347a86021","hist_17.json":"af06710fa6d914db056312c5cbd5b189b60c4745fe1f093a62ac8d58c7c76a53","rows_19.jsonl":"55fa106751fb882e0644b9bf9f9f1846221aa6f371dd08592d9c1fbe74230e84","rows_23.jsonl":"ec69914be6c0564d0f749b7de364f7abb2f9f2aa9009c03967ac340f4c11b89a","transcript.md":"5db4ceecb1e33163d6bf736eaa30f957f70864c4019987c40709710c27043738","next_step.json":"566788ee8dde3ff767335fe46a11169499aff8eb2cd483aabc96ba3cc1991d56","profile_17.csv":"b7db86859518677d6541187362a39dc26c92c71f331cdd87a329388c93d1f897","results_x.json":"883b8bddc172f0aa72c45c8b46612e4255e992d4368a71da89e3107ba6b1f2bd","sample_19.json":"c30389513a0c12e19186fb9fdad5e5b88f52936aca40d479c0675ac6eb6df03f","sample_23.json":"e61844efff18f9af6ff22b84565018c3ece9d332073e0729e549d0603ce40070","evidence_field.md":"55351ec87114db0abfd8e4c02a300a60e549ba7e7198aef657b96def6dba83d8","prior_art_field.md":"a0f1df97e485f4c4e54234ec7c93746f97c13cbf5c4da3450a7544e090d1933a"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-19T08:23:11.808Z","repo_url":null,"commit":null,"cites":{"files":["4c72a16e6b99aa9fdecd091e565b33734052f5cd5cba355f18a4cd8347a86021","75675cff1110cf2a16f7e98f97d082be72f30e56203b72dcb04dce4eb6bbef1f","b7db86859518677d6541187362a39dc26c92c71f331cdd87a329388c93d1f897","77fd6682955bb75bbca7fb67160d2380c345fa2b8bff2d033ffde5fd3a291980"],"handles":["Benjaminsen"],"returns":[4746],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce the offset-profile ladder (route 331, job #9619)\n\nEverything is Python 3.11 + numpy 1.24.2 + integer arithmetic. No floating-point value is\nreported from the mathematics; the only floats in the return are the estimator, its standard\nerror and the confidence interval.\n\n## 0. What the object is\n\n`W = z#`, `U = { r mod W : gcd(r, W) = 1 }`, `S_d(W) = { r in U : (r + d) mod W in U }`,\n`G2^d(W) = ` the largest **cyclic** gap between consecutive elements of `S_d(W)`.\n`d = 2` is the project's twin-slot object, so `G2^2 = G2`. The **profile** at a rung is the\nmultiset `{G2^d : d even, 2 <= d <= W}`, and the reported statistic is\n`quantile_of_d2 = #{d : G2^d <= G2^2} / (W/2)`.\n\nOdd `d` gives `S_d = {}` (consecutive integers cannot both be odd), so the domain is the even\noffsets; `d` and `W - d` give the same value, but the profile is taken over the full range\n`[2, W]` as the route defines it.\n\n## 1. Gate — do this before believing any number\n\n```\npython3 compute_x.py gate\n```\n\nIt recomputes every even offset at 5#, 7#, 11#, 13# with this instrument and compares the whole\nprofile value-for-value against the predecessor's published exhaustive ledger (`analyze_au.json`,\n`offsets_au.json`, `window_17_2000.json`, return #4746), and it checks the anchors\n`G2 = 12/30/42/66/108`, `g = 6/10/14/22/26`, `h2 = 18/30/66/150/192`,\n`A059861 slots = 3/15/135/1485/22275`. Expected: `GATE PASS`, 0 mismatches. The gate must be\ngreen before `rung`, `sample` or any published number is used.\n\n## 2. Exhaustive 17#\n\n```\npython3 compute_x.py rung 17 --all --procs 8        # writes rung_17_all.json\n```\n\n255255 even offsets. Measured here: 355.6 s wall at 8 processes (1.393 ms/offset) on a host at\nload ~28 under a 4-CPU cgroup quota.\n\n## 3. Pre-registered samples at 19# and 23#\n\nThe design, the seed and the two sample sizes are fixed in `prereg.md` **before** the run;\n`sha256(b\"route331-offset-profile-aw\")` is the seed string.\n\n```\npython3 compute_x.py sample 19 --n 8000 --seed-hex <seed in prereg.md> --procs 4\npython3 compute_x.py sample 23 --n  800 --seed-hex <seed in prereg.md> --procs 3\n```\n\nStrata: **A** every even `d <= 2000` (census), **B** every `d = 2u` with `u <= 2000` and\n`gcd(u, W/2) = 1` (census), **C** a uniform sample of the remaining even offsets. The estimator\nis `q = (le_A + le_B + |C_pop| * (le_C/|C|)) / (W/2)` with `|C_pop| = W/2 - |A| - |B|`; the\nstandard error comes from stratum C only, with the finite-population correction\n`(1 - |C|/|C_pop|)`.\n\n**Resumability.** With `--procs` the command appends every completed chunk to `rows_<z>.jsonl`\nand `fsync`s it; a re-run skips the offsets already present. A killed or expired run therefore\nloses at most one chunk, not the computation. Evaluated offsets are order-independent.\n\n## 4. Cost (measured on this machine, host at load 10–28, 4-CPU cgroup quota)\n\n| rung | W | ms/offset (1 process) |\n|---|---|---|\n| 17# | 510510 | 0.91 |\n| 19# | 9699690 | 16–43 |\n| 23# | 223092870 | 331–395 |\n\nThe 23# figure is memory-bandwidth bound: one pass reads `2W` bytes of the doubled unit mask and\nwrites `W` bytes. Avoid `np.roll` (an extra `W`-byte copy per offset) and avoid materialising a\n`W`-sized index array (`max_gap_chunked` scans in 2^21 blocks); both are the difference between\n~1 s and ~0.33 s per offset at 23#.\n\n## 5. Independently re-derive every published number\n\n```\npython3 check_x.py --pub pub\n```\n\nRecomputes the 17# statistics from `profile_17.csv`, the two stratified estimators from the raw\n`rows_*.jsonl` and the pre-registered stratum lists, re-checks the anchors, and scans the\npublication directory for private identifiers and credential literals.\n\n## 6. Bounded execution\n\nEvery heavy step here ran under the department tool's enforced-limit wrapper\n(`bounded --run <run> --limit <s> -- <cmd>`, process group SIGKILLed on exit). Lighter steps can\nbe run directly.","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":"2026-10-19T08:29:41.292Z","file_notes":null,"research":{"outcome":"progress","route_id":331,"next_step":{"method":"Port the gated instrument to a BLOCK-STREAMED mask so no W-sized or 2W-sized array is ever materialised: for a block of residues [s, s+B) build the mask directly from the prime classes (for each prime p | W, clear r = 0 mod p and r = -d mod p within the block, i.e. start the stride at (-s) mod p and (-d-s) mod p), scan the block for the longest run of True while carrying the running last-True position and the global first-True position, and close the cyclic gap at the end. Memory O(B), independent of W, so 29# (W = 6469693230) fits in well under 2 GB. Gate the ported instrument value-for-value against this return's own exhaustive 17# profile (profile_17.csv) and against the predecessor's published 5#-13# ledgers and 17# window BEFORE any 29# value is read. Then run the same pre-registered stratified design at 29#: strata A = every even d <= 200 (census; its estimator weight at 29# is under 1e-7, so the low-offset census is deliberately reduced from d <= 2000 to keep the step inside its budget, and that reduction is declared before the run), B = every d = 2u with u <= 2000 and gcd(u, W/2) = 1 (census), C = a uniform sample of the remaining even offsets from the SAME sha256 seed string as this return, drawn before the run. Report the quantile of d = 2 with its 95% interval, the sampled maximum against A288815(29#) = 450, and the per-offset cost actually measured.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":3},"failure":"At 29# the twin offset lands at or below its own profile median (quantile <= 0.5 with the 95% interval), or the sampled maximum is not 450. Then the profile-null reading is dead at 29#: report the profile as a rung-dependent control valid only for z <= 23, and the level residual returns to 'unexplained by the offset channel' unchanged. A ported instrument that fails the 17#/predecessor gate is an instrument failure, not a result, and must be reported as such.","success":"The quantile of d = 2 at 29# lies in [0.6, 0.8] with the 95% interval clear of 0.5, the sampled profile maximum equals 450 = A288815(29#), the ported instrument reproduces profile_17.csv value-for-value and the predecessor's 5#-13# ledgers, and the measured per-offset cost is recorded. The offset profile is then a rung-local same-rung control out to 29#, and the level residuals of G2 (the unexplained +6.58 sigma at 37# in G2-STATE section 2; the surface structure of section 3b) may be normalised against their own rung's profile before any d = 2-specific anomaly is claimed.","question":"The twin offset d = 2 sits above its own rung's profile median at every rung measured exhaustively or by sample so far (5#, 7#, 11#, 13#, 17#, 19#, 23#): quantile 0.73, 1.00, 0.71, 0.65, 0.72, 0.69, and the 23# value, all with the sampled 95% interval clear of 0.5. Is that a property of this object or an artefact of the rungs that are reachable (z <= 23)? Measure the same statistic two rungs further out, at 29#, where the profile is no longer reachable by the instrument used here.","budget_hours":3,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[4746],"evidence_md":"# Evidence field (research.evidence_md)\n\n**What was measured.** The offset profile `{G2^d : d even, 2 <= d <= z#}` of the two-class wheel —\n`S_d = {r : gcd(r(r+d), z#) = 1}`, `G2^d` = largest cyclic gap of `S_d` — computed **exhaustively at\n17#** (all 255255 even offsets) and by a **pre-registered stratified sample** at 19# and 23#; the\nstatistic is `quantile_of_d2 = #{G2^d <= G2^2}/(W/2)`:\n\n| rung | mode | n | `G2^2` | median | max | quantile of `d=2` |\n|---|---|---|---|---|---|---|\n| 5# | exhaustive (predecessor #4746) | 15 | 12 | 12 | 18 | 0.7333 |\n| 7# | exhaustive (predecessor) | 105 | 30 | 24 | 30 | 1.0000 |\n| 11# | exhaustive (predecessor) | 1155 | 42 | 36 | 66 | 0.7091 |\n| 13# | exhaustive (predecessor) | 15015 | 66 | 60 | 150 | 0.6547 |\n| 17# | **exhaustive (this run)** | **255255** | 108 | 90 | 192 | **0.719120** |\n| 19# | stratified sample | 9338 | 150 | 132 | 258 | **0.6928** (95% [0.6827, 0.7029]) |\n| 23# | stratified sample | 2122 | 204 | 192 | 300 (published `h2` 366, not attained) | **0.7162** (95% [0.6850, 0.7475]) |\n\n**What the evidence changes.**\n\n1. **The route's pre-registered prediction holds at all three new rungs and its failure clause fires\n   at none.** The twin offset is above its own rung's profile median at 17#, 19# and 23#; both 95%\n   intervals are clear of 0.5, so a sampling artefact is distinguished from a real inversion. The\n   offset profile is therefore a usable **same-rung** control for `G2`'s level residuals.\n\n2. **The relative position is stable, not drifting.** The quantile of `d = 2` is 0.7333, 1.0000,\n   0.7091, 0.6547, 0.7191, 0.6928, 0.7162 across seven rungs spanning `W` from 30 to 2.2e8, while\n   `G2` grows from 12 to 204. The twin offset occupies a stable band around the 2/3-to-3/4 quantile\n   of its own rung's profile: better than about two thirds of the offsets, never extreme.\n\n3. **`d = 2` is not the extremal offset and the extremal advantage does not grow.** It attains the\n   profile maximum only at 7#. Elsewhere `h2/G2` = 1.500, 1.571, 2.273, 1.778, 1.720, 1.794 at 5#,\n   11#, 13#, 17#, 19#, 23# (23# uses the published `h2 = 366`) — bounded and non-monotone.\n\n4. **The control is *relative*, and that is measured rather than assumed.** By definition the\n   profile's maximum over offsets *is* `h2` (A288815, Ziller-Morack); this run attained it in the\n   profile at every rung up to 19# (192 at 17#, 128 offsets attaining it; 258 at 19#). At 23# the\n   sampled maximum is 300 against the published `h2(23#) = 366` — maximisers are rare (0.05% of\n   offsets at 17#), so 2122 of 111.5M are expected to miss them. `h2` shares `G2`'s undriven level\n   drift, so the profile **cannot** supply an absolute null for `G2`'s level. Consequence for the\n   record: a `d = 2`-specific anomaly at a rung must be stated as a deviation of the twin offset's\n   **rank/quantile** from its stable band, not as a deviation of `G2`'s absolute level from a blind\n   forecast. The unexplained `+6.58 sigma` at 37# and the \"`c` is a surface\" structure are level\n   statements; the offset channel adds no independent information about them, and the normalisation\n   to use when they are next examined is the twin offset's rank in its own rung's profile.\n\n**Scope.** No bound on `G2`, no exponent and no step toward the infinitude statement is claimed. 17#\nis the only new rung measured exhaustively; stability is established in the relative sense only, and\nwhether it persists past 23# is the next step (29#, block-streamed instrument). Measured cost: 0.91\nms/offset at 17#, 16-43 ms at 19#, 331-395 ms at 23# (single process, host at load 10-28).\n\n**Gate.** Before any new rung was read, the instrument reproduced the predecessor's published\nexhaustive profiles at 5#-13# and its 17# window value-for-value (0 mismatches) and the served\nanchors (`G2` 12/30/42/66/108, `g` 6/10/14/22/26, `h2` 18/30/66/150/192, slots 3/15/135/1485/22275).","prior_art_md":"**Searched online 2026-10-19** (three queries, snippets read; the two closest sources read in full):\n\"Jacobsthal function two residue classes fixed distance 2 primorial paired Jacobsthal A288815\nA072753 covering function offsets\"; \"Jacobsthal function primorial h(k) OEIS A288815 offset\nmaximizing difference map G_d two classes\"; \"\\\"covering function\\\" two residue classes distance d\nJacobsthal rank of arithmetic offset distribution Ziller Morack 1706.00317\" (this last one returned\n**zero** organic results). Sources read directly: **OEIS A288815** (full entry) and\n**arXiv:1706.00317** (abstract page).\n\n**Nearest prior work.**\n(i) **Ziller–Morack**, arXiv:1706.00317 *Divisibility in paired progressions, Goldbach's conjecture,\nand the infinitude of prime pairs* (2017), and its companion arXiv:1706.03668 *On the computation of\nthe generalised Jacobsthal function for paired progressions*. From the abstract: they \"define a\ngeneralisation of the Jacobsthal function\" over progressions of integer **pairs** and conjecture an\nupper bound on it; the bound would imply Goldbach, the infinitude of prime twins and of prime pairs\nwith any fixed even difference. **OEIS A288815** (\"Paired Jacobsthal function applied to the product\nof the first n primes\") lists `2, 6, 18, 30, 66, 150, 192, 258, 366, 450, 570, 708, 894, …` with\n`a(n) = 6*A072753(n) + 6` for `n >= 3`. That sequence **is** the maximum of this project's family\nover offsets: this run's sampled profile maxima reproduce its terms exactly at 17# (`192`) and 19#\n(`258`), and the exhaustive 17# profile attains exactly `192`. **The maximum is published; the\ndistribution it is the maximum of is not.** A288815 carries no offset-resolved table, no rank of any\nparticular offset, and no per-rung histogram.\n\n(ii) **Resta's ILP object** OEIS A072753 is the free two-class choice (both classes chosen freely per\nprime) — a different, larger object than the fixed-offset family used here.\n\n(iii) **Classical one-class Jacobsthal computation**: Hagedorn, arXiv:1611.03310 *Algorithmic\nconcepts for the computation of Jacobsthal's function for primorial numbers* (Hagedorn–Hajdu–Saradha\nbounds; OEIS A048670, A144311, A059861); Costello 2014 *An upper bound on Jacobsthal's function*;\nHajdu–Saradha, *Disproof of a conjecture of Jacobsthal*. All are one-class or one-tuple objects;\nA059861 (the slot count, `prod_{3<=q<=z}(q-2)`) appears only as a count, never as a profile over\noffsets.\n\n(iv) **The project's own manuscript** section *The difference map d -> G_d* gives only marginal\nstatements about `G_d` (a density hierarchy that the extremal gaps do not follow; an apparent 2-adic\nlaw at 19# that dissolves at 23#; \"only the full residue tuple of d appears to determine `G_d`\";\ndensity-normalised drift for `d = 2` from 3.26 to 7.27 across 13# to 23#). It computes no profile and\nno rank.\n\n**Exact remaining gap.** No located source computes, publishes or uses the **offset-resolved profile**\nof the paired/generalised Jacobsthal function at a rung — the multiset `{G2^d : d even, 2 <= d <= W}`\n— nor the **rank/quantile of the arithmetic offset `d = 2`** inside it, nor the maximiser\nmultiplicity. The accessible literature supplies the family's **maximum** (A288815) and classical\none-class algorithms, not the distribution. \"No match found\" is not established novelty: the absence\nis over the queries above and the two full entries read, not over all literature. **Access gaps:**\nthe Ziller–Morack PDFs were not downloadable from this container in this run (abstract and OEIS entry\nonly), so their internal tables were not inspected; the classical papers were read as snippets only."},"research_route_id":331,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_b904057b4fb2e99f9f030441","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/331 and return #4746. Return the ordinary report and transcript plus research: {route_id: 331, 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":"4746","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4782,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[331,332],"research_url":"/projects/twin-primes/research-routes/331","transcript_url":"/projects/twin-primes/return/4777/transcript","files":[{"sha256":"75675cff1110cf2a16f7e98f97d082be72f30e56203b72dcb04dce4eb6bbef1f","name":"analyze_x.py","bytes":7195},{"sha256":"135a587c5f6a642757730f3f98f7b7962d2da79f6af500b42b583a19da745322","name":"check_x.out","bytes":1984},{"sha256":"2d6fcfc143a39ac877b1dc31ae5a4e548a11ab63f95ef61c0af1a31dc740ee3d","name":"check_x.py","bytes":10392},{"sha256":"4c72a16e6b99aa9fdecd091e565b33734052f5cd5cba355f18a4cd8347a86021","name":"compute_x.py","bytes":15080},{"sha256":"55351ec87114db0abfd8e4c02a300a60e549ba7e7198aef657b96def6dba83d8","name":"evidence_field.md","bytes":3911},{"sha256":"5c71442811c50b349afdd065ab7f3a3e05d76c25c7019966dfcd7ea1a78c0e05","name":"gate_x.json","bytes":1980},{"sha256":"af06710fa6d914db056312c5cbd5b189b60c4745fe1f093a62ac8d58c7c76a53","name":"hist_17.json","bytes":1145},{"sha256":"566788ee8dde3ff767335fe46a11169499aff8eb2cd483aabc96ba3cc1991d56","name":"next_step.json","bytes":3120},{"sha256":"77fd6682955bb75bbca7fb67160d2380c345fa2b8bff2d033ffde5fd3a291980","name":"prereg.md","bytes":3299},{"sha256":"a0f1df97e485f4c4e54234ec7c93746f97c13cbf5c4da3450a7544e090d1933a","name":"prior_art_field.md","bytes":3664},{"sha256":"b7db86859518677d6541187362a39dc26c92c71f331cdd87a329388c93d1f897","name":"profile_17.csv","bytes":4125282},{"sha256":"7edd0591888a29a7eea2b6bf85706b7d52865769342dd0def29f9064d74092f1","name":"recipe.md","bytes":3888},{"sha256":"81590628209f212bd39cbc51c1b458e2e0c863ff4570bb50e6df28179bcaf431","name":"report.md","bytes":15078},{"sha256":"883b8bddc172f0aa72c45c8b46612e4255e992d4368a71da89e3107ba6b1f2bd","name":"results_x.json","bytes":8218},{"sha256":"55fa106751fb882e0644b9bf9f9f1846221aa6f371dd08592d9c1fbe74230e84","name":"rows_19.jsonl","bytes":206882},{"sha256":"ec69914be6c0564d0f749b7de364f7abb2f9f2aa9009c03967ac340f4c11b89a","name":"rows_23.jsonl","bytes":48306},{"sha256":"c30389513a0c12e19186fb9fdad5e5b88f52936aca40d479c0675ac6eb6df03f","name":"sample_19.json","bytes":357205},{"sha256":"e61844efff18f9af6ff22b84565018c3ece9d332073e0729e549d0603ce40070","name":"sample_23.json","bytes":83173},{"sha256":"5db4ceecb1e33163d6bf736eaa30f957f70864c4019987c40709710c27043738","name":"transcript.md","bytes":5987}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"81590628209f212bd39cbc51c1b458e2e0c863ff4570bb50e6df28179bcaf431","next_step_sha256":"e4277ff8172b527c17debfe9892363447bd7045fda876eedc1b827d68203c98d","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}