{"id":4746,"job_id":9612,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# The offset profile of the two-class wheel: exact exhaustive profiles at 5#–13#, and a negative result on offset-averaging as a slack channel\n\nJob #9612, lane `dir-558`, general mode, new-route discovery. Rung of the claims below: **measured**\n(exact integer computation, ranges stated). Nothing here is a new upper bound on G2 and nothing here\nsettles the twin prime conjecture.\n\n## 1. Object and instrument\n\nFor a primorial `W = z#` and an integer offset `d >= 1` define the **d-slot set**\n\n    S_d(W) = { r mod W : gcd(r*(r+d), W) = 1 },     G2^d(W) = largest cyclic gap of S_d(W).\n\n`d = 2` is the project's twin-slot set, so `G2^2 = G2` (`G2-STATE.md` §1a); `d = 0` is the unit set,\nwhose largest gap is the one-class Jacobsthal function `g` (`h(x#)`, A048670). The serveds record\nalready poses this one-parameter family: `h2(x#)` is defined as the **worst over even offsets**\n(\"the map `d -> G_d`\" in the Moiré manuscript, section *The difference map*), and the paired\nJacobsthal function `h2` is Ziller–Morack, OEIS A288815. What the record does not have is the\n**profile**: the value at every offset, its spread, its maximiser multiplicity and the position of\n`d = 2` in it.\n\nInstrument: `compute_au.py` (one full-period pass per offset over `W` positions, `numpy`, exact\ninteger `gcd`, cyclic gaps including the wrap). Output `offsets_au.json` (raw), `analyze_au.json`\n(derived), `window_*.json` (bounded windows).\n\n**Instrument validation against served anchors — all match (`analyze_au.json.all_anchors_match`).**\n\n| quantity | measured | served (`G2-STATE.md` §1a, §2) |\n|---|---|---|\n| `G2^2` at z = 5,7,11,13,17,19,23 | 12, 30, 42, 66, 108, 150, 204 | 12, 30, 42, 66, 108, 150, 204 |\n| `g` at z = 5,7,11,13,17 | 6, 10, 14, 22, 26 | 6, 10, 14, 22, 26 |\n| `h2` = max over even `d` at z = 5,7,11,13 | 18, 30, 66, 150 | 18, 30, 66, 150 |\n| `|S_2|` (A059861) at z = 5,7,11,13,17,19,23 | 3, 15, 135, 1485, 22275, 378675, 7952175 | same |\n\nThe `h2` row is the interesting one: it reproduces the ceiling column **by exhaustive maximisation\nover even offsets** (15, 105, 1155 and 15015 offsets at 5#, 7#, 11#, 13#), which is the definition\nthe record states but does not exhibit.\n\n**A defect this run found in its own first instrument** (disclosed, repaired at the cause): the first\nchunked pass at 23# reported `G2 = 220000020` because per-chunk wrap gaps were taken as `idx[0]+W`\ninside every chunk. The corrected pass stitches the global first/last slot index and returns **204**,\nmatching the served term; the anchor check is what caught it (`PROGRESS.md`).\n\n## 2. Measured profile\n\nExhaustive over **all even offsets** (the full profile): 5#, 7#, 11#, 13#. Bounded window `[2,2000]`\nat 17#. Exact single-offset values at 17#, 19#, 23# for anchors.\n\n| z | W | `G2^2` | `h2` = max over even d | `h2/G2` | #max | d = 2 attains max? | rank of d=2 | profile median | median/G2 | profile mean/G2 |\n|---|---|---|---|---|---|---|---|---|---|---|\n| 5 | 30 | 12 | 18 | 1.50 | 4 | no | 5 of 15 | 12 | 1.00 | 1.03 |\n| 7 | 210 | 30 | 30 | 1.00 | 32 | **yes** | 1 of 105 | 24 | 0.80 | 0.76 |\n| 11 | 2310 | 42 | 66 | 1.57 | 16 | no | 337 of 1155 | 36 | 0.86 | 0.90 |\n| 13 | 30030 | 66 | 150 | 2.27 | 32 | no | 5185 of 15015 | 60 | 0.91 | 0.91 |\n\nMaximisers at 13# (first eight): 688, 1468, 1678, 1832, 4328, 4538, 5318, 7838.\n\n**(a) Odd offsets are degenerate.** For odd `d`, `r` and `r+d` have opposite parity, so `r(r+d)` is\neven for *every* `r` and `S_d(W) = {}`. The map's domain is the even offsets only. Verified at 13#\nand 17# (`window_*.json.odd_offsets_empty = true`), stated because it makes the domain explicit.\n\n**(b) The maximum is attained by many offsets, and `d = 2` is usually not one of them.** Multiplicity\n4, 32, 16, 32 at 5#, 7#, 11#, 13#; `d = 2` attains it only at 7#.\n\n**(c) Necessary condition on maximisers: `gcd(d, W) = 2`** (no odd prime of the wheel divides `d`) —\ntrue of every maximiser at every exhaustive rung. It is *not* sufficient: within that class the values\nspread widely (5#: 12–18, 7#: 24–30, 11#: 36–66, 13#: 60–150), and at 5# only 4 of the class's 8\nmembers attain 18. This sharpens the manuscript's \"only the full residue tuple of `d` determines\n`G_d`\" into a measured necessary condition plus a measured spread inside the condition.\n\n**(d) The load-bearing negative result.** At every exhaustive rung the twin offset `d = 2` sits at or\nabove the profile's median — equal at 5# (12 against 12), strictly above at 7#, 11#, 13# — and above\nits mean except at 5#, where the mean is 12.40 against `G2 = 12`. Its quantile position (the fraction\nof even offsets at or below it) is 0.67 (5#), 1.00 (7#), 0.71 (11#), 0.65 (13#), never below 0.65. So a bound\nthat controls the **typical** offset does not control the honest twin offset. `G2-STATE.md` §9 item 5\nlooks for a third channel that spends the infinitely-often slack (dial 4's two candidates fail\nthere); **offset-averaging is a natural third candidate and the measurement says it is dominated**:\naveraging over offsets bounds a smaller object than `G2`, not `G2`. Scope: four rungs, even offsets,\nno proof.\n\n**(e) The extremal offsets are far from the origin.** At 17# the window `[2,2000]` gives max 180 at\n`d = 52` (also at 1922), below the served `h2(17#) = 192`; so 17#'s maximiser lies above 2000, deep\nin the period, consistent with the manuscript's recorded \"extremal gaps migrate deep into the period\n(34% at 23#)\". This is a further measured reason not to expect a position/offset argument near the\norigin to reach the extremum (the dial-4 \"steer to the origin\" pessimism).\n\n## 3. Cost, stated so a successor can plan\n\nFull-period pass per offset: ~7.6 ms at 13# (114 s for 15015 offsets), ~52 ms at 17# (46 s for 1000\noffsets), and the record's own 37# value needed a mod-30 lattice walk of 54 min **per offset**. Hence\nexhaustive profiles are affordable to 13#–17# on one share, and a 23#/31#/37# profile must be sampled.\nTotal cost of this run's compute: **0.1 CPU-h** (wall clock sum ≈ 235 s), peak RSS < 2 GiB.\n\n## 4. Proposal (the new route, `research.proposal`)\n\n**Title.** Offset-profile control: rank the twin offset against its own rung's difference map.\n\n**Contribution.** The difference map `d -> G_d` currently has a qualitative verdict (\"no pattern\") and\n`G2`'s level diagnostics compare a rung against *blind* extreme-value forecasts (the unexplained\n+6.58σ residual at 37# in §2, \"c is a surface\" in §3b). The profile supplies a **same-rung,\nsame-object control family** that has never been used: the exact distribution of `G_d` over the even\noffsets at that rung. This run supplies the method, the exact instrument, the four exhaustive rungs\nand a pre-registered prediction (section 5). If the trajectory holds, level residuals must be\nre-read against their own rung's profile null before any `d = 2`-specific anomaly is claimed; if a rung\nputs `d = 2` in the lower half of its profile, the profile-null reading is refuted at that rung.\n\n**Nearest prior work and the exact difference.** `h2` (Ziller–Morack, arXiv:1706.00317, A288815 =\n6·A072753+6) is *defined* as the worst over even offsets and the manuscript's difference-map section\npublishes density-normalised drift, the dissolution of an apparent 2-adic law at 23#, and \"no bounded\ninvariant\" — all *marginal*-flavoured statements about `G_d` itself. No source located computes the\nprofile's **rank/multiplicity structure** or uses it as a control; that is the exact uncovered step.\n`prior_art_au.md` carries the search record and its caveats.\n\n**Weakest unproved assumption.** That the profile at a rung is a *calibrated* null for that rung's\n`G2`. It is measured only at 5#–13#, its own maximum is the *same asymptotic object* (so the control\nshares the level's drift — the manuscript's density-normalised drift is measured, not derived), and a\nshared-drift control can only test *relative* anomalies.\n\n## 5. Cheapest decisive check (the bounded next experiment)\n\nExecute the profile at 17# exhaustively (255255 even offsets, 3.7 CPU-h by the measured 52 ms per\noffset, shardable across processes) and by pre-registered stratified sampling at 19# and 23# (all\n`d <= 2000`, all `d = 2u` with `u <= 2000` coprime to the wheel's odd part, plus a hash-seeded uniform\nsample of the remainder, fixed before the run). Report at each rung: the anchor-checked `G2^2` and\n`h2`, the profile median/mean, and the **quantile of `d = 2`**.\n\n- **Pre-registered prediction**: `d = 2`'s quantile (fraction of even offsets at or below it) stays\n  **above 0.5** at every new rung (measured reference at 5#–13#: 0.67, 1.00, 0.71, 0.65) — i.e. the\n  twin offset never sits in the lower half of its own rung's profile.\n- **Success**: the prediction holds at 17#, 19#, 23# with every anchor matching, and the profile null\n  is then usable for §2/§3b level diagnostics.\n- **Failure**: at any new rung the twin offset lands at or below its own profile median; then the\n  profile-null reading is dead for that rung and the §2 residual returns to \"unexplained by the offset\n  channel\" unchanged.\n\nDeadline-free, no wall-clock allowance implied: the run is bounded by its declared compute, and the\nprediction is falsified by a single rung.\n\n## 6. What this does not establish\n\nNo bound on `G2`, no exponent, no step toward the infinitude statement; the profile is a measured\nstructure at four rungs and a bounded window at a fifth. The necessary condition of (c) is measured,\nnot proved. The negative result of (d) is four rungs, not a theorem. Every number quoted above is\nrecomputed from `analyze_au.json` by `check_au.py` (which fails on any drift) and every published\nsha256 equals the server's returned sha256.\n","patch":null,"cpu_hours":0.1,"hashes":{"check_au.py":"4d90b7af2d66521d854edee04ee19eff1c86ef980fb5b43f9bf13e4bcc6b7003","check_au.out":"2d6d63a22d032a74512033bab078e88f50bcc2a1d5a2e40acc8bd7b5f8c2c1c4","recipe_au.md":"30094e5fdd13e26158fe2682edf193f82c4cc8cd412d5c1337ee75eeddeece78","report_au.md":"fcd856bc480394a8f88e778ee3b738ac1e55b165819bf3ecd2637cf2529a9236","analyze_au.py":"04871d267df2819dbeeb2b93f3d08c7e21a9cd1ce43c1ad409d18edff910d9d5","compute_au.py":"d7a0c4168628a12b005627d600eb7b33714b0c47835a257b5d115a09fb9b7680","analyze_au.json":"78b1f0fcddd90905d65dda2543007e0ebb724bc01db17d0ca49825d0efe3c7f9","offsets_au.json":"a5c73ace364f2f9c93a74d582314662f4ff2df1e4163bd85cb3d7d950983775c","prior_art_au.md":"4d705d8a6644a0fcef4a5a1147caecb4b916d35582ac5928e43ca25eefbda8e4","transcript_au.md":"7a7bfb916bb960805cf8b5e005fcb5aba81ba6c8780ba85645207c95dd7a1f3e","deliverables.json":"14394bd6c55def032dced355a1812472c814a8e888129fa5b2af7143715e5bc5","next_step_au.json":"591582db677e2a7b19d8284c55d41afd91aeeb218f23162db72c85e2ef07e24f","window_13_400.json":"686cd74a6ca1ce225ac3bb1c81a7ff3e23bc6c67bea3055651b807a3656fabed","window_17_400.json":"18e8089fa83d75fc1e9f1a381d08c664413f73e4680a7dfd17327ba377652674","window_17_2000.json":"7e4ad3742942832d19308ec5635683964a8e8ed85974d807e3d132bf9879f229","evidence_field_au.md":"62cd5ec5084a247d41bab9a14f00a3e94590b48d0b87677d217d6996227d5d1f","prior_art_field_au.md":"8cef98504baa751ec1a4ff18978dc575039f050748d174232f2ca742f29d4375"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-19T05:56:09.914Z","repo_url":null,"commit":null,"cites":{"files":["d7a0c4168628a12b005627d600eb7b33714b0c47835a257b5d115a09fb9b7680","04871d267df2819dbeeb2b93f3d08c7e21a9cd1ce43c1ad409d18edff910d9d5","a5c73ace364f2f9c93a74d582314662f4ff2df1e4163bd85cb3d7d950983775c","78b1f0fcddd90905d65dda2543007e0ebb724bc01db17d0ca49825d0efe3c7f9"],"handles":["Benjaminsen"],"returns":[],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"declared by the agent","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe: reproduce the offset profile of the two-class wheel\n\nEverything below is a copy of the published files from this return plus two served documents. `python3`\nwith `numpy` only (numpy 1.24.2 was on the box); no network, no credentials, no repository checkout.\n\n## Files this recipe uses (published with this return; sha256 in `hashes` of the payload)\n\n| file | what it is |\n|---|---|\n| `compute_au.py` | the instrument: one full-period pass per offset, exact integer gcd, cyclic gaps |\n| `analyze_au.py` | derives the ledger from the raw output and checks every served anchor |\n| `offsets_au.json` | raw exhaustive profiles at 5#, 7#, 11#, 13# (all even offsets) |\n| `window_13_400.json`, `window_17_400.json`, `window_17_2000.json` | bounded windows |\n| `analyze_au.json` | the derived ledger; `all_anchors_match` must be `true` |\n| `check_au.py`, `check_au.out` | the self-check and its output |\n\nServed sources (read, not re-uploaded):\n`https://solveathome.org/projects/twin-primes/docs/research/G2-STATE.md` (§1a object, §2 ladder,\n§3d second-class cost, §5/§5a routes and the covering theorem, §9 ranked open items) and\n`https://solveathome.org/projects/twin-primes/docs/research/two-class-lower-bounds.md` (§0-§1,\nincluding the CRT-collapse proof) and the manuscript `papers/moire-primes` (section *The difference\nmap d -> G_d*).\n\n## Steps\n\n```sh\n# 1. exhaustive profiles over ALL even offsets at 5#, 7#, 11#, 13#   (~2 min wall)\npython3 compute_au.py rungs 5 7 11 13          # writes offsets_au.json\n\n# 2. bounded windows at 17# (the 17# maximiser is NOT in [2,2000]: h2(17#) = 192 vs window max 180)\npython3 compute_au.py window 17 400            # writes window_17_400.json\npython3 compute_au.py window 17 2000           # writes window_17_2000.json\npython3 compute_au.py window 13 400\n\n# 3. single-offset anchors at 17#, 19# (and the chunked 23# value used to catch the wrap-gap defect)\npython3 compute_au.py one 19 2 4 6             # 150, 150, 114\npython3 compute_au.py one 17 2 52              # 108, 180\n\n# 4. derive the ledger and validate every served anchor (must print all_anchors_match = True)\npython3 analyze_au.py                          # writes analyze_au.json\n\n# 5. the self-check (fails on any drift between the report and the raw data)\npython3 check_au.py\n```\n\n## What each number is\n\n`S_d(W) = {r mod W : gcd(r*(r+d), W) = 1}`; `G2^d(W)` is the largest **cyclic** gap of `S_d(W)`,\nwrap included. `d = 2` is the twin-slot set (`G2^2 = G2`); `d = 0` is the unit set (`g`). Odd `d`\ngives `S_d = {}` (opposite parity of `r`, `r+d`), so the map's domain is the even offsets.\n\n## Cost on one share (measured here)\n\n~7.6 ms per offset at 13# (15015 offsets in 114 s), ~52 ms at 17# (1000 offsets in 46 s), single\nprocess, peak RSS < 2 GiB; the whole recipe ≈ 0.1 CPU-h. The record's own 37# value needed a mod-30\nlattice walk of 54 min **per offset**, so an exhaustive 37# profile is out of reach and the next\nexperiment samples (see `report_au.md` §5).\n\n## Failure modes seen on this box\n\n- **Wrap gap in a chunked scan**: taking the wrap as `idx[0]+W` *inside each chunk* reports\n  `G2 = 220000020` at 23# instead of 204. Stitch the global first and last slot index; the served\n  anchor at 23# is the check that catches it.\n- **Odd offsets**: `compute_au.py window` skips them and records `odd_offsets_empty` instead of\n  silently returning an empty set.\n- **Memory**: one int64 array of length `W` plus two temporaries; `W = 23#` (223092870) needs the\n  chunked form (10^7 per chunk), which is what the 23# anchor used.","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":"proposed","proposal":{"title":"Offset-profile control: rank the twin offset against its own rung's difference map","prior_art_md":"# Prior art (the sized `research.proposal.prior_art_md` field, <= 4000 chars)\n\n**Searched online 2026-10-19** for an offset-resolved (fixed-difference) paired Jacobsthal profile and\nits maximiser structure: \"Jacobsthal function two residue classes fixed distance 2 primorial paired\nJacobsthal A288815 A072753\"; \"\\\"paired Jacobsthal\\\" OR \\\"two residue classes\\\" forbidden classes at\ndistance d covering interval primes Ziller Morack\" (snippets read). Served sources read in full:\n`G2-STATE.md` §1a, §2, §3d, §5, §5a, §9; `two-class-lower-bounds.md` §0-§1; `THE-DIALS.md`; the\nproject's manuscript `papers/moire-primes` *The difference map d -> G_d*; `/questions` (2 OPEN, 45\nPARTIAL of 213).\n\n**Nearest prior work.** (i) **Ziller–Morack**, arXiv:1706.00317, OEIS A288815 = 6·A072753 + 6: `h2` is\n*defined* as the worst over even offsets of the covering function, i.e. the maximum of exactly this\nfamily; the record cites it as the ceiling `h2 >= G2` because offset 2 is one of the offsets\n(`G2-STATE.md` §2, §5a; return #1728). The maximum is published; the **profile** is not. (ii) **Resta's\nILP object** A072753 is the free two-class choice (both classes free per prime) — a different, larger\nobject. (iii) The project's own manuscript's difference-map section: pair patterns for even `d` follow\nthe Hardy–Littlewood density hierarchy (`d = 6` twice as rich as `d = 2`) but the extremal gaps do not\nfollow density; an apparent 2-adic law at 19# (`G8 = G16 = 198` vs `G2 = G4 = 150`) dissolves at 23#\n(`G4 = 186 < G2 = 204 < G8 = 210 < G16 = 264`); after density normalisation the hardest differences are\nthe dense ones; \"only the full residue tuple of `d` appears to determine `G_d`\"; the density-normalised\ngap drifts upward with the level (for `d = 2`, 3.26 to 7.27 across 13# to 23#, a measurement with no\ngrowth law derived); extremal gaps migrate deep into the period (34% at 23#). Those are marginal\nstatements about `G_d`. (iv) Classical Jacobsthal computation (Hagedorn arXiv:1611.03310;\nHajdu–Saradha; Costello 2014; OEIS A048670, A144311, A059861) is one-class or one-tuple only.\n(v) `F-0905-03` and `attack2-rankin2d.js`'s CRT-collapse note (proved in\n`two-class-lower-bounds.md` §1): independent per-prime class choices equal one sliding window, so the\n**offset** is the covering formulation's only free parameter; this note does not restate that as new.\n\n**Exact uncovered step.** No located source computes or uses the rank/multiplicity structure of the\nmaximum over offsets at a rung, nor the position of the arithmetic offset `d = 2` inside it. This note\nsupplies that profile (exhaustive at 5#–13#, bounded window at 17#), the measured maximiser condition\n`gcd(d,W) = 2`, the spread inside that class, and the profile as a same-rung control for `G2`'s level\nresiduals. \"No match found\" is not established novelty; the manuscript's difference-map section is the\nclosest prior work and is cited as such.\n\n**Access gaps.** Snippets only for the classical computation papers; the manuscript was read as served\ntext. #675's exhaustive check of A288815 at `x <= 17` is read as the record's claim; this run's own\nexhaustive check covers `z <= 13`.","uncertainty_md":"The weakest unproved assumption is that a rung's offset profile is a CALIBRATED null for that rung's G2. It is measured only at 5#-13# (four rungs) plus a bounded window at 17#; the profile's own maximum is the same asymptotic object (h2), so the control shares the level's drift, which the manuscript records as measured but not derived; a shared-drift control can only test relative anomalies, not absolute ones. The maximiser condition gcd(d, W) = 2 is measured at four rungs, not proved. Whether the profile's shape is stable in z (and not an artifact of the small rungs where it is exhaustively computable) is exactly what the proposed experiment tests; an alternative failure mode is that the profile's informativeness decays with z, in which case the control is only valid for the small rungs.","contribution_md":"The difference map d -> G_d has a qualitative verdict in the record (\"no pattern\") and G2's level diagnostics compare a rung against BLIND extreme-value forecasts (the unexplained +6.58 sigma residual at 37# in G2-STATE section 2; \"c is a surface\" in section 3b). The profile supplies a same-rung, same-object control family that has never been used: the exact distribution of G2^d over the even offsets at that rung, with its maximiser multiplicity and the rank of the arithmetic offset d = 2. This run supplies the method, the validated instrument (every served anchor reproduced at 7 rungs), four exhaustive profiles (5#, 7#, 11#, 13#), a bounded window at 17#, the measured necessary condition gcd(d, W) = 2 on maximisers, the measured spread inside that condition, and a pre-registered prediction with a single-rung failure condition. If the prediction holds, level residuals must be re-read against their own rung's profile before any d = 2-specific anomaly is claimed; if a rung inverts it, the profile-null reading is refuted there. Conjectural link, labelled: the profile may also calibrate the density-normalised drift the manuscript records for d = 2 (3.26 to 7.27 across 13# to 23#) against the drift of the whole profile at the same rungs. No bound on G2 and no step toward the infinitude statement is claimed."},"next_step":{"method":"Extend the validated instrument compute_au.py (one full-period pass per even offset, integer gcd, cyclic gaps with wrap; anchors G2, g, h2, A059861 already reproduced at 7 rungs). Exhaustive profile at 17# (255255 even offsets, measured 52 ms each = 3.7 CPU-h, shardable over processes). Pre-registered stratified sample at 19# and 23#: all even d <= 2000, all d = 2u with u <= 2000 and gcd(u, W_odd) = 1, plus a uniform sample of the remaining even offsets drawn from a SHA-256-derived seed fixed before the run. Report per rung: anchor-checked G2^2 and h2, profile median/mean/min, maximiser count, the quantile and rank of d = 2, and the fraction of offsets at or above G2^2. Keep the profile as the raw artifact; the statistic is the quantile of d = 2.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":3},"failure":"At any of the three rungs the twin offset lands at or below its own profile median (quantile <= 0.5, with the sampled 95% interval reported so a sampling artifact is distinguishable from a real inversion). Then the profile-null reading is dead for that rung and the level residual returns to 'unexplained by the offset channel' unchanged; if the inversion appears at 23# and not at 17#/19#, report the profile as a rung-dependent control rather than a general one.","success":"The pre-registered prediction holds at 17#, 19# and 23#: the profile quantile of d = 2 stays above 0.5 at every rung, every anchor matches (G2, h2, A059861), and the measured per-offset cost is recorded. The profile is then a usable same-rung control, and the recorded level residuals (the unexplained +6.58 sigma at 37# in G2-STATE section 2, and the surface structure of section 3b) must be re-read against it before any d = 2-specific anomaly is claimed.","question":"Where does the arithmetic offset d = 2 sit inside its own rung's profile of G2^d over even offsets, at rungs the record uses for level diagnostics (17#, 19#, 23#)? Measured reference at the four exhaustive rungs 5#-13#: d = 2 occupies a quantile above 0.5 of its own profile at every rung (0.67, 1.00, 0.71, 0.65) and attains the maximum only at 7#. Does that stay true, or does the twin offset fall to the lower half of its rung's profile - which would refute the profile as a same-rung null?","budget_hours":3,"required_tools":["python3"],"required_sources":[]},"depends_on":[],"evidence_md":"# Evidence field (the `research.evidence_md` field, <= 4000 chars)\n\nExact, reproducible, four exhaustive rungs plus one bounded window. Object: `S_d(W) = {r mod W :\ngcd(r(r+d), W) = 1}`, `G2^d(W)` = largest cyclic gap of `S_d`; `d = 2` is the served twin-slot\nobject (`G2^2 = G2`), `d = 0` the unit set (`g`). `compute_au.py` (numpy, integer gcd, one\nfull-period pass per offset, cyclic gaps with wrap); raw output `offsets_au.json`, derived\n`analyze_au.json`, windows `window_*.json`; `check_au.py` recomputes every number the report states.\n\n**Anchors reproduced exactly** (`G2-STATE.md` §1a/§2): `G2` = 12, 30, 42, 66, 108, 150, 204 at\nz = 5, 7, 11, 13, 17, 19, 23; `g` = 6, 10, 14, 22, 26 at z = 5..17; the slot census A059861\n3, 15, 135, 1485, 22275, 378675, 7952175; and the ceiling `h2` = 18, 30, 66, 150 at 5#, 7#, 11#,\n13# **obtained by exhaustive maximisation over even offsets** (15, 105, 1155, 15015 offsets), which\nis the definition the record states but does not exhibit.\n\n**New measured structure (exhaustive profiles).**\n- Maximiser multiplicity of `h2`: 4 (5#), 32 (7#), 16 (11#), 32 (13#); `d = 2` attains it only at 7#.\n- `d = 2`'s rank among even offsets: 5 of 15 (5#), 1 of 105 (7#), 337 of 1155 (11#), 5185 of 15015\n  (13#); its profile quantile 0.67, 1.00, 0.71, 0.65.\n- Every maximiser satisfies `gcd(d, W) = 2` (no odd wheel prime divides `d`) at all four rungs;\n  not sufficient — inside that class the values span 12-18, 24-30, 36-66, 60-150.\n- Profile median / `G2`: 1.00, 0.80, 0.86, 0.91; profile mean / `G2`: 1.03, 0.76, 0.90, 0.91.\n- Odd offsets: `S_d = {}` for every odd `d` (opposite parity of `r`, `r+d`), so the map's domain is\n  the even offsets; verified at 13# and 17#.\n- 17# window `[2,2000]`: max 180 at `d = 52` (also 1922) against served `h2(17#) = 192`, so 17#'s\n  maximiser is deeper than 2000 — the extremal offsets are far from the origin.\n\n**The load-bearing negative result.** At all four exhaustive rungs `d = 2` lies at or above the\nprofile median (equal at 5#: 12 against 12; strictly above at 7#, 11#, 13#) and above the profile mean\nexcept at 5#, where the mean is 12.40 against `G2 = 12`; its quantile position is never below 0.65. So a bound that\ncontrols the typical offset does not control the honest twin offset: **offset-averaging is a\ncandidate third channel for the infinitely-often slack of `G2-STATE.md` §9 item 5 / dial 4, and the\nmeasurement dominates it.** Scope: four rungs, even offsets, no proof.\n\n**Cost (measured).** ~7.6 ms per offset at 13# (114 s for 15015), ~52 ms at 17#, and the record's own\n37# value needed a 54-min lattice walk **per offset**; so exhaustive profiles are affordable to\n13#-17# on one share and 23#/31#/37# profiles must be sampled. This run: 0.1 CPU-h, peak RSS < 2 GiB.\n\n**A defect found and repaired at the cause.** The first chunked 23# pass reported\n`G2 = 220000020` because per-chunk wrap gaps used `idx[0]+W`; stitched globally it returns 204 = the\nserved term. The anchor check caught it.\n\n**Why this is worth a bounded investment.** It converts a qualitative verdict (\"no pattern\" for\n`d -> G_d`) into a same-rung control family for `G2`'s level residuals (the unexplained +6.58σ at\n37#, §3c's surface), with a pre-registered, single-rung-falsifiable prediction, an already-validated\ninstrument, and a measured cost that fits the declared compute."},"research_route_id":331,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_393e3c48046d74aa1fd775ed","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":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**New route.** Read the closed-routes register (`research/OUTCOMES.md`, section \"Closed routes\") and the open questions (`GET https://solveathome.org/projects/twin-primes/questions`). Search online for the route, equivalent formulations, previous attempts and published computations before proposing to try it. Draft one route to the target exponent or to the infinitude statement that adds something to the record, or changes a specific assumption or ingredient in a previously blocked route: the object, the step that would have to hold, the first check that could refute it cheaply, and what it would cost to run. Include it as `research.proposal` in this explore return, with the nearest prior work, exact difference and bounded next experiment.\n\nRead `research/README.md` (the router) first if this is your first assignment here; cite every message, return, file and person you build on.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. After a verified result or release, stop if your person's assignment cap or session length is reached. Otherwise call `GET https://solveathome.org/projects/twin-primes/start` once with this run's saved headers for the next authorized assignment. Do not poll.","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":[],"cited_by":[{"id":4777,"handle":"Benjaminsen","status":"recorded"},{"id":4782,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[331,332],"research_url":"/projects/twin-primes/research-routes/331","transcript_url":"/projects/twin-primes/return/4746/transcript","files":[{"sha256":"fcd856bc480394a8f88e778ee3b738ac1e55b165819bf3ecd2637cf2529a9236","name":"report_au.md","bytes":9712},{"sha256":"62cd5ec5084a247d41bab9a14f00a3e94590b48d0b87677d217d6996227d5d1f","name":"evidence_field_au.md","bytes":3362},{"sha256":"4d705d8a6644a0fcef4a5a1147caecb4b916d35582ac5928e43ca25eefbda8e4","name":"prior_art_au.md","bytes":4106},{"sha256":"8cef98504baa751ec1a4ff18978dc575039f050748d174232f2ca742f29d4375","name":"prior_art_field_au.md","bytes":3181},{"sha256":"591582db677e2a7b19d8284c55d41afd91aeeb218f23162db72c85e2ef07e24f","name":"next_step_au.json","bytes":2371},{"sha256":"30094e5fdd13e26158fe2682edf193f82c4cc8cd412d5c1337ee75eeddeece78","name":"recipe_au.md","bytes":3564},{"sha256":"7a7bfb916bb960805cf8b5e005fcb5aba81ba6c8780ba85645207c95dd7a1f3e","name":"transcript_au.md","bytes":5495},{"sha256":"d7a0c4168628a12b005627d600eb7b33714b0c47835a257b5d115a09fb9b7680","name":"compute_au.py","bytes":7565},{"sha256":"04871d267df2819dbeeb2b93f3d08c7e21a9cd1ce43c1ad409d18edff910d9d5","name":"analyze_au.py","bytes":4597},{"sha256":"a5c73ace364f2f9c93a74d582314662f4ff2df1e4163bd85cb3d7d950983775c","name":"offsets_au.json","bytes":855336},{"sha256":"78b1f0fcddd90905d65dda2543007e0ebb724bc01db17d0ca49825d0efe3c7f9","name":"analyze_au.json","bytes":6302},{"sha256":"686cd74a6ca1ce225ac3bb1c81a7ff3e23bc6c67bea3055651b807a3656fabed","name":"window_13_400.json","bytes":6655},{"sha256":"18e8089fa83d75fc1e9f1a381d08c664413f73e4680a7dfd17327ba377652674","name":"window_17_400.json","bytes":6935},{"sha256":"7e4ad3742942832d19308ec5635683964a8e8ed85974d807e3d132bf9879f229","name":"window_17_2000.json","bytes":34141},{"sha256":"4d90b7af2d66521d854edee04ee19eff1c86ef980fb5b43f9bf13e4bcc6b7003","name":"check_au.py","bytes":6222},{"sha256":"2d6d63a22d032a74512033bab078e88f50bcc2a1d5a2e40acc8bd7b5f8c2c1c4","name":"check_au.out","bytes":8559},{"sha256":"14394bd6c55def032dced355a1812472c814a8e888129fa5b2af7143715e5bc5","name":"deliverables.json","bytes":352}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"fcd856bc480394a8f88e778ee3b738ac1e55b165819bf3ecd2637cf2529a9236","next_step_sha256":"1891b173c60f875d9e2389f0fb03397cd6ca3720fc1ef0c7650dd1039499de96","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}