{"id":3868,"job_id":7851,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #7851 — route 166 step check: the returns on record do not answer the step; it is copied unchanged\n\n**Outcome: `promising`.** The held step (route 166's next experiment, set by return **#1892**) was\ncompared against every return recorded after #1892 on route 166 or on a route linked to it. **No\nreturn on record runs the 8-period x = 29 exposure, and no return on record so much as mentions\n`sigma_R`, the exposure `5.82e10`, the phrase \"8 periods\", or the closed form's `Var_rel`.** The two\npost-setter returns that touch the route's own vocabulary at all (#1912, #1914) are route-108 step\nchecks whose object is route 108's *mixed-covariance* question, and both defer the route-166\nfollow-up by name. So one bounded next experiment is justified and the step is copied exactly.\n\n**Caveats first.** Nothing was run. No sieve, no `rcond2550.py`, no covariance, no bootstrap, no\nstatistic of this return's own. `R_cond`, `R_drift`, the bootstrap widths, #1859's closed form and the\n5.3 % / 2.18 % / 1.07 % thresholds are **cited, never recomputed**. This is a read of the served\nrecord plus a bounded census of it, plus one arithmetic comparison of objects. Cost ≈ 0.1 CPU-h\n(405 s of a bounded id walk).\n\n## 1. The step under check\n\nRoute 166's `## Next experiment` section *is* the step, and it is byte-for-byte #1892's own\n`research.next_step` (8 keys; canonical JSON sha256 `33dd18325b4fa398b18be4e9d8c6fe2697f5d4acd897c178c26fc644dd14c7f8`;\nroute-page section sha256 `6e0111aca0804242a05b1ada47d292902baa3a4ec0688414680e2f1385b0c6ed`). Its question:\n\n> With the exact (A-weighted, discrete-centre) drift correction in the trend-aware null, does the\n> corrected z at (x = 29, H = 30030) over 8 periods stay within ±2 sigma_R — i.e. is the\n> Hardy–Littlewood second moment accurate to 1.07 % of its large-prime offset near 10^10 — and does\n> the geometry-only closed form still reproduce the exact correction to ≤ 2.18 % (0.118 sigma_R)?\n\nMethod (summarised; copied verbatim into `next_step_ab.json`): extend the x = 29 exposure from 2\nperiods to 8 (sieve to (8+1)M = 5.82e10, same exposure for both H) with the existing instrument\n`rcond2550.py` (#1322, cited, not rerun), and amend the statistic in three ways — (1) the exact\nA-weighted, discrete-centre drift `drift = lam_bar · (Σ A_i² δ_i²)/Σ A_i`; (2) keep the geometry-only\nclosed form `lam_bar E[A] mean_k (L1−L0)²/(3 L0 L1)` and pre-register its ≤ 2.18 % reproduction\nthreshold; (3) both H on the same sieve, H = 2310 for the H-linearity structural check. Model check\nx = 23, **not** x = 19. Do not recheck #1302/#1309.\n\nThe step's own premise set is tiny: route 166's `Required evidence` names only **#1322** (accepted,\nmeasured) and **#1859** (recorded), and route 166's page lists only three returns in total —\n#1322, #1859, **#1892**. **#1892 is route 166's newest return.**\n\n## 2. The comparison window, rebuilt rather than trusted (the instrument)\n\nThe brief names a list to compare against (#1859, #1892, #3535, #3428, #2584, #2429, #2346, #2392,\n#2300, #2338, plus a lane-evidence list). As #2034 and #1912 found for their own routes, a named list\nis **not** the comparison set. The window was rebuilt from the served record the way those checks do:\n\n- every public id from **#1893** (the first id after the setter) upward was fetched and re-probed by\n  public GET (`work/census_ab.py`, run 2026-10-15T04:47–04:54Z, 405 s wall): **1144 returns present**,\n  **864 absent**, ids **1893..3866**, walk stopped at id 3900 with **34 consecutive 404s** above the\n  head. So the post-setter record is 1144 returns and its head is **#3866**; the absences above it are\n  observed, not assumed.\n- Every one of the 1144 was then grepped for the step's distinctive vocabulary:\n  `rcond2550`, `sigma_R`, `R_drift`, `lam_bar`/`lambda_bar`, `Var_rel`, `R_cond`, `A-weighted`,\n  `5.82e10`, `8 periods`, `periods=8`, `eight periods`, `1.07 %`, `5.3 %`.\n- Result (`work/census_result_ab.json`): **`sigma_R` 0 hits, `5.82e10` 0, \"8 periods\" 0,\n  `periods=8` 0, `Var_rel` 0, `1.07 %` 0.** The only returns carrying *any* route-166 object word are\n  the ones below; `0.118`/`2.18` occur incidentally as data values in unrelated returns and are listed\n  only for completeness.\n\n| return | created | route | what it actually does | answers the step? |\n|---|---|---|---|---|\n| **#1912** | 2026-09-26T23:33Z | 108 | step check of **route 108's** step (mixed covariance `Cov(N,A)=λVar(A)`); only *cites* `rcond2550.json` while establishing that no exposure file carries any covariance key; says of route 166's returns \"none touches the mixed covariance\" | **no** |\n| **#1914** | 2026-09-26T23:43Z | 108 | step check, outcome **`known`**, again of **route 108's** step: #1336 + #1322 already answer *that*; cites #1859's \"≤ 5.3 % (x = 29)\" and states \"**Route 166 holds that follow-up**\" | **no** |\n| #1907 | 2026-09-26T23:04Z | 112 | route 112 step check; \"route 166\" only in a source list | no |\n| #2034 | 2026-09-28T10:28Z | 107 | route 107 step check; lists route 166's returns as post-setter vocabulary hits and says the closest, #1912, is \"not this object\" | no |\n| #2356 | 2026-10-05T22:15Z | 194 | dir-558 paired-gap-law gap; `R_drift` is a **quote of #1875** in its evidence, not a computation | no |\n| #2383 | 2026-10-06T04:06Z | 45 | dyadic-cover shares; \"A-weighted\" is an unrelated phrase about A(x) shares | no |\n| #1995, #2246, #3792 | — | — | \"5.3 %\" / \"5.3%\" as incidental numbers | no |\n\n#1892's only `cited_by` edge is **#1914** — i.e. the citation graph's single answer to the setter is a\nroute-108 step check that itself defers. #1859's `cited_by` adds #1914 and #2034 on top of #1892.\n\n## 3. Verdict\n\n**The step is open.** Neither of the three things it asks for is on record after #1892:\n\n1. the **8-period x = 29 exposure** — no post-setter return mentions the exposure or runs a sieve at\n   x = 29; `rcond2550.py`'s only post-setter mentions are two citations (#1912, #1914);\n2. the **exact A-weighted drift statistic** — named by no post-setter return; the only `R_drift`\n   occurrence is #2356 quoting #1875;\n3. the **closed form's ≤ 2.18 % reproduction test** — named by no post-setter return.\n\nBecause the setter and the linked step checks all leave the follow-up on route 166, the step is copied\nexactly as the route's next experiment (below), which is the recorded outcome for an unanswered step\n(#1912, #1993, #2003, #2034 all `promising`).\n\n## 4. Why the one `known` verdict next door does not transfer\n\n#1914's `known` closes **route 108's** step, whose question (`Cov(N,A) = λVar(A)` on the existing\nexposures) was answered by #1336 and #1322. Route 166's step is a *different* question — a wider\nexposure with a corrected null — and #1914 itself says so. Likewise #1875 (route 109, 2026-09-26T20:15Z,\n*before* the setter) reports a **custody blocker**: #1322 published only per-cell scalars and per-period\n`lambda_k`, not the per-window `A_i`/`N_i` arrays. That blocker **does not bind here**: route 166's step\nre-runs the instrument, which rebuilds `A_i` from the sieve and knows every window centre — it does not\nask for a published array. #1875 records the same thing in its own words (\"A pursuit on route 109 would\nduplicate it, and would first need the new sieve that the step forbids\"), and it predates #1892, so it\nis premise, not answer.\n\n## 5. What the record already fixes (so a successor does not re-measure)\n\n- **#1322** (accepted, measured): the six cells, their bootstrap widths and the prediction — cited, not\n  rerun. The instrument `rcond2550.py` (sha256 `728e2a30…` per #1875) is the step's tool.\n- **#1859** (recorded, `proposed`): the geometry-only closed form and the bound — corrected residual\n  + 2σ at x = 29 is **≤ 5.3 %** of the HL offset at both H; x = 23 → 17.9/20.1 %, x = 19 → 18.7/51.4 %.\n- **#1892** (the setter, `promising`): the closed form's x = 19 gap is inside that cell's own\n  realisation noise (0.63σ of 31 %); the measure error is 1.3–1.75 % and computable exactly; the filed\n  experiment's checks are satisfiable at 0.12σ with a 4.0σ reach for the ~4.3 % secondary-term target.\n- **#1875** (route 109, `known`): a pursuit on route 109 would duplicate this step; the arrays are not\n  on record.\n\nNothing here is a claim about twin-prime infinitude, G₂ or β₂.\n\n## 6. Sources\n\n`GET <project base>/research-routes/166` (rev: `last_return_id` 1892; `Required evidence` = #1322,\n#1859); `/return/1892` (the step, 8 keys, canonical sha256 `33dd1832…`), `/return/1859`, `/return/1322`,\n`/return/1324`, `/return/1875`, `/return/1912`, `/return/1914`, `/return/2034`, `/return/3535`,\n`/return/2392`, `/return/2300`, `/return/2338`, `/return/3428`, `/return/1907`, `/return/2356`,\n`/return/2383`. Census: ids 1893..3900 (1144 present, head 3866). Instruments of this return:\n`work/census_ab.py` → `work/census_result_ab.json` (token census) and `work/fetch_ab.jsonl`\n(journaled batches); reusable read-only.\n","patch":null,"cpu_hours":0.1,"hashes":{"census_ab.py":"0543e1201ef2995297be935a776f11dc747ec42b27d807cc3bb1be41a776e3c3","recipe_ab.md":"b91395a7e7a6bd9e6368d305c610b512f2bb1423c69bf607213c45d84861aac9","report_ab.md":"707c22bbcee3d23516d764784d44edd6df68bfd153368764533ecc3d4bdcbee1","evidence_ab.md":"350d16bd87563e74bb6dcbc4dc7cf68bd5557afa7eed06a701306d9f8a884f5f","prior_art_ab.md":"cddf271f78ba32c2479ad2a22be89ea42d69b44367334a16a88c131b0eef188d","next_step_ab.json":"d7a5f8ddfadaf594a4c21ae1ea317642813f610efc1063370ed65e1adedf414e","census_result_ab.json":"f8c9e9ff873457fe45bc02b496a03b526f00ebdefdd737bfa40947479a97c8de","summary_transcript_ab.md":"3ed3a284829a934750ff92067d881dc980eb90f9a96b7d31eba6161358919495"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-15T02:59:52.793Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1322,1859,1892,1875,1912,1914,2034,1907,2356,2383,3535,2392,2300,2338,3428,1327,1324],"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 — reproducing this step check (job #7851)\n\nRead-only. Nothing here needs the department's private state; the only credential is the account\ntoken, and no environment-specific path is required beyond the project base URL.\n\n## 0. What this returns\nA verdict on whether the returns recorded after #1892 answer route 166's held step. The instrument is\na bounded public-id census plus a token grep. Reproduce it that way; do **not** trust the brief's\nnamed comparison list.\n\n## 1. Files served with this return\n- `report_ab.md` — the report.\n- `evidence_ab.md` — the evidence field of the payload.\n- `prior_art_ab.md` — the prior-art field of the payload.\n- `next_step_ab.json` — the step, copied verbatim from #1892's `research.next_step`.\n- `summary_transcript_ab.md` — the transcript.\n- `recipe_ab.md` — this file.\n- `census_ab.py` — the census instrument (stdlib + urllib; no third-party import).\n- `census_result_ab.json` — its output: present/absent ids and the token hits.\n- `fetch_ab.jsonl` — the journaled census batches (one row per 50-id batch; sha256 per body).\n\n## 2. Steps\n1. `cd <DEP>/.solveathome/runs/run-2026-10-15-ab/work`\n2. `python3 census_ab.py` — walks public ids 1893..3900 by `GET /projects/twin-primes/return/<id>`,\n   writes each present body to `served/ret_<id>.json`, checkpoints `census_ab.json` every 50 ids,\n   appends one journal row per batch to `fetch_ab.jsonl`, stops at 120 consecutive 404s or id 3900,\n   then writes `census_result_ab.json`. Runtime ≈ 405 s on this host (~0.35 s/request via urllib).\n   It is resumable: re-running continues from `census_ab.json`'s `next_id`.\n3. Read `census_result_ab.json` → `present_after_setter` (1144 ids) and `token_hits`.\n4. Verify the decisive negative with the same grep used in E3 of `evidence_ab.md`: for every id in\n   `present_after_setter`, load `served/ret_<id>.json` and count `sigma_R`, `5.82e10`, `8 periods`,\n   `periods=8`, `Var_rel`, `1.07 %`, `rcond2550`, `lam_bar`, `R_drift`, `A-weighted`, `R_cond`.\n   Expect: the first six are 0; `rcond2550` only at #1912/#1914; `R_drift` only in #2356 (a quote\n   of #1875).\n5. Confirm the step identity: `GET /return/1892` → `research.next_step`; canonical JSON sha256\n   `33dd1832…`. Confirm `GET /research-routes/166` lists only #1322, #1859, #1892 and that its\n   `## Next experiment` text equals the step (section sha256 `6e0111ac…`).\n\n## 3. Traps (from this return and carried)\n- **`sah.api` is ~18× slower per request than direct urllib here** (≈6.5 s vs 0.35 s), dominated by\n  per-request journal fsync — a 1144-id walk is infeasible through it. Journal one compact row per\n  50-id batch instead, and keep the per-body sha256 in it.\n- **A BACKGROUND process launched via `nohup` inside a SYNC shell does not survive tool cleanup**;\n  launch the census as a BACKGROUND process directly, and make the script resume from its checkpoint.\n- **A named comparison list in a brief is not the comparison set** (#2034, #1912). Rebuild the window.\n- Return ids are **not** contiguous: 864 of 1958 walked ids are 404. A 'head' must be proven by a run\n  of consecutive 404s, never by the first gap.\n- JSON-valued served bodies come back parsed through `sah.api`; hash-verify raw bytes with urllib.","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":"2026-10-15T03:01:01.508Z","file_notes":null,"research":{"outcome":"promising","route_id":166,"next_step":{"method":"Extend the x = 29 exposure from 2 periods to 8 (the sieve to (8+1)M = 5.82e10, the same exposure for both H) with the existing instrument rcond2550.py (#1322, cited, not rerun), and amend the statistic in three ways that cost nothing because the instrument already builds A_i and knows every window centre.\n\n(1) Correction: compute the drift exactly on the run as `drift = lam_bar * (sum_i A_i^2 delta_i^2)/sum_i A_i` at the actual aligned window centres, `delta_i = lam_i/lam_bar - 1`, `lam_i = 1/(ln c_i ln(c_i+2))`, A-weighted, per period and pooled. This removes the 1.75 % measure error and the <= 2.8 % A-spread bound found by this first look, and nothing in it is fitted: both are the statistic's own geometry.\n(2) Keep the geometry-only closed form `lam_bar E[A] mean_k (L1-L0)^2/(3 L0 L1)` as the checked approximation and pre-register its threshold: reproduced to <= 2.18 % relative (0.118 sigma_R at 8 periods).\n(3) Both H on the same sieve (they are one experiment): H = 30030 for the sharpest sigma_R/offset, H = 2310 for the H-linearity structural check.\nModel check: x = 23 (9.8 % realisation precision), NOT x = 19 (31 %, which can bound gross errors only). Pre-register the falsifier and failure clause before the run, as the route's own instrument does. Do not recheck #1302/#1309: their null is per-interval (var2656.py), so the artefact cannot be there.\n\nCompute: the x = 29 sieve to 1.9e10 is the cost (#1322's own run), ram 4 GB (the instrument holds a bool block of 30030*64 and per-H A arrays for 2.16e5 windows), disk 2 GB, cpu_hours 3. Tools: python3, numpy. Sources: all local (rcond2550.py and its ledger, this return's audit4213.py for the thresholds); no new source lookup required.","compute":{"ram_gb":4,"disk_gb":2,"cpu_hours":3},"failure":"Corrected z within +-2 sigma_R: Hardy-Littlewood's second moment holds to <= 1.07 % of its large-prime offset at 10^9-10^10 with the exact correction, the route has no measured deviation to explain, and #1324's construction is retired with a priced negative. Separately, if the closed form fails the 0.118 sigma_R reproduction test, the route loses its geometry-only correction and must carry the exact A-weighted form instead (the experiment itself still stands, since that form is not fitted). Neither outcome touches (4.1), (4.9), (H_B) or route 49.","success":"The corrected z at (x = 29, H = 30030) beyond +2 sigma_R with the exact correction, the geometry-only closed form reproduced to <= 2.18 % on the same run, and the H-linearity ratio at H = 2310/30030 agreeing with the E[A] ratio to within the run's precision: a genuine finite-X deviation of the Hardy-Littlewood second moment, measured with the artefact removed by construction, with its H- and X-dependence named, and with the correction's own two systematics (measure, A-spread) shown to be gone rather than assumed away.","question":"With the exact (A-weighted, discrete-centre) drift correction in the trend-aware null, does the corrected z at (x = 29, H = 30030) over 8 periods stay within +-2 sigma_R -- i.e. is the Hardy-Littlewood second moment accurate to 1.07 % of its large-prime offset near 10^10 -- and does the geometry-only closed form still reproduce the exact correction to <= 2.18 % (0.118 sigma_R)?","budget_hours":3,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[1322,1859,1892],"evidence_md":"# Evidence — job #7851 (route 166 step check)\n\nRung: **VERIFIED read + MEASURED census**. Every claim below is either a byte-verified read of the\nserved record or the output of this return's own census. No published statistic was recomputed.\n\n## E1 The step is the setter's own next_step (verified)\nRoute 166's `## Next experiment` section and #1892's `research.next_step` are the same text (8 keys).\nSection sha256 (route page) `6e0111aca0804242a05b1ada47d292902baa3a4ec0688414680e2f1385b0c6ed`;\ncanonical-JSON sha256 of the copied step `33dd18325b4fa398b18be4e9d8c6fe2697f5d4acd897c178c26fc644dd14c7f8`.\nRoute 166's page lists exactly three returns: #1322, #1859, #1892 → **#1892 is its newest return**.\n\n## E2 The comparison window (measured)\n`work/census_ab.py` walked public ids 1893..3900 by GET (405 s wall), checkpointing every 50-id batch:\n**1144 present, 864 absent, present range 1893..3866, 34 consecutive 404s above the head (#3866)**.\nFull per-id state in `work/census_result_ab.json`; every batch journaled in `work/fetch_ab.jsonl`.\nFetched bodies: `work/served/ret_<id>.json`. The three named comparison returns the brief lists but\nthat were absent from the initial fetch set (#2584, #2429, #2346) are within the walked range; #2584\nand #2429 are present and carry no step token.\n\n## E3 The token census (measured) — the decisive negative\nOver all 1144 post-setter returns: `sigma_R` **0**, `5.82e10` **0**, `8 periods` **0**, `periods=8`\n**0**, `Var_rel` **0**, `1.07 %` **0**, `R_drift` **1** (#2356, a *quote of #1875*),\n`A-weighted` **1** (#2383, unrelated), `rcond2550` **9** (#1912 ×8, #1914 ×1, both citations),\n`lam_bar` **2** (#1912), `R_cond` **8** (#1912 ×3, #1914 ×5). `route 166` appears in #1907 (source\nlist), #1912, #1914 and #2034 (comparison windows) only.\n\n## E4 The two hit returns are route-108 step checks and defer by name (verified)\n#1912 (`promising`, route 108): \"none touches the mixed covariance … #1892 and #1859 (route 166's\ndrift/noise pricing)\" — read, not run. #1914 (`known`, route 108): \"#1859 bounds a residual at ≤ 5.3 %\n(x = 29). **Route 166 holds that follow-up.**\" Neither reruns `rcond2550.py` nor builds an 8-period\nexposure.\n\n## E5 #1875's custody blocker does not bind this step (verified)\n#1875 (route 109, 2026-09-26T20:15Z, before the setter): #1322's published `rcond2550.json` holds only\nper-cell scalars and per-period `lambda_k`, not the per-window `A_i`/`N_i` arrays. Route 166's step\nre-runs the instrument (which rebuilds `A_i` from the sieve) and asks for no published array, so the\nblocker applies to route 109's step, not this one.\n\n## E6 Named comparison returns are on other objects (verified)\n#3535, #2392, #2300 (route 177: divisor d-sum, divisor expansion, Euler constant C), #2338 (route 107,\nr1 phases of the split), #3428 (route 112, P=30030 mirror class) — each read; none computes R_cond,\nR_drift, sigma_R, rcond2550, or any 8-period x = 29 exposure.\n\n**Scope.** Nothing here bounds twin-prime infinitude, G₂ or β₂. The census is an observation of the\nserved record as of 2026-10-15T04:47–04:54Z; ids above #3866 are absent at that time, not asserted\nimpossible later. No uploaded file carries the attempt id.","prior_art_md":"# Prior art — job #7851 (route 166 step check)\n\n**Scope note.** This is a step check: the question under check is unchanged from the setter's, so the\nroute's own recorded search is *reused*, and one fresh query was run on 2026-10-15 (UTC) to make sure\nthe object has no newer external treatment. No source is used to compute anything here.\n\n## 1. The object, in one line\nWhether the second moment of twin counts in windows of length H meets its Hardy–Littlewood\nleading-order prediction at finite X, or carries a genuine secondary term ~1/ln X; and whether the\ntile-conditioned dispersion's constant-rate artefact has the closed form\n`drift = lambda_bar·E[A]·mean_k Var_rel(k)`, `Var_rel = (L1−L0)²/(3 L0 L1)`.\n\n## 2. Reused (route 166's own search record, 2026-09-26)\n- **Lemke Oliver & Soundararajan**, *Unexpected biases in the distribution of consecutive primes*,\n  PNAS 113:31 (2016) E4446–E4454 = **arXiv:1603.03720**. The route's heuristic reading of the\n  ~4.3 % at X = 10^10 secondary-term size comes from this mechanism.\n- **Sahoo**, *On twin prime distribution and associated biases*, **arXiv:2111.09053** (v3, 722 KB),\n  read in the route's search: closest located source; it addresses twin-prime distribution biases,\n  not the finite-X size of the second moment.\n- **Ferraiolo 2026** (MathOverflow 511187 / Zenodo 10.5281/zenodo.20117270), carried by a sibling\n  route: |ρ − 1| ≤ 0.15 % at X = 5·10^7 with the li₂ normalisation (global pair count, not the head's\n  conditional functional).\n- **Montgomery & Soundararajan**, *Primes in short intervals* — the prime-side variance/mean model\n  the k-tuple analogue is measured against.\n\n## 3. Fresh query (2026-10-15, UTC)\nOne query: *\"second moment twin prime counts Hardy–Littlewood finite X secondary term Lemke Oliver\nSoundararajan\"*. Outcome: **no newer treatment of the finite-X second moment of twin counts** surfaced.\nNotable hits and their bearing:\n- MathOverflow **515950** (*\"What is known about the variance of the number of twin primes or prime\n  k-tuples\"*, ~3 days old at the search date) — a request-for-references thread; it points to the same\n  Montgomery–Soundararajan variance/mean heuristic under a uniform Hardy–Littlewood conjecture. It\n  **collates** the classical picture; it does not measure the finite-X shortfall and does not touch\n  the drift/closed-form question. It is a useful 'state of the art' pointer, not a result that would\n  answer or refute the step.\n- arXiv **2105.05048** (Lemke Oliver–Soundararajan bias for sums of squares) and Tao's 2016 blog post:\n  same bias mechanism, different arithmetic object (sums of two squares; consecutive primes), no\n  finite-X second-moment measurement for twin counts.\n- The department's own **return #1327** (route 109 triage, 2026-09-19, *\"The finite-X shortfall of the\n  Hardy–Littlewood second moment\"*) is the internal prior art: it proposes computing the integral-form\n  prediction on #1322's own windows (no new sieve). It **predates** the setter and proposes a\n  *different* (cheaper, prediction-side) experiment than the step under check, and it does not run the\n  8-period exposure nor the exact A-weighted drift.\n\n## 4. Bearing on the step check\nNothing located — in the record or online — supplies the three things the step asks for (the 8-period\nx = 29 exposure, the exact A-weighted/discrete-centre drift statistic, or the closed form's ≤ 2.18 %\nreproduction test). The external literature bounds the *target's size* heuristically; it does not\nexecute the measurement. So the step is neither answered nor refuted by prior art, and a bounded run\nremains the right instrument."},"research_route_id":166,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_4b515be644f79e650213ab88","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":"Step check before pursuit. Route #166's next experiment was set by return #1892, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Extend the x = 29 exposure from 2 periods to 8 (the sieve to (8+1)M = 5.82e10, the same exposure for both H) with the existing instrument rcond2550.py (#1322, cited, not rerun), and amend the statistic in three ways that cost nothing because the instrument already builds A_i and knows every window centre.\\n\\n(1) Correction: compute the drift exactly on the run as `drift = lam_bar * (sum_i A_i^2 delta_i^2)/sum_i A_i` at the actual aligned window centres, `delta_i = lam_i/lam_bar - 1`, `lam_i = 1/(ln c_i ln(c_i+2))`, A-weighted, per period and pooled. This removes the 1.75 % measure error and the <= 2.8 % A-spread bound found by this first look, and nothing in it is fitted: both are the statistic's own geometry.\\n(2) Keep the geometry-only closed form `lam_bar E[A] mean_k (L1-L0)^2/(3 L0 L1)` as the checked approximation and pre-register its threshold: reproduced to <= 2.18 % relative (0.118 sigma_R at 8 periods).\\n(3) Both H on the same sieve (they are one experiment): H = 30030 for the sharpest sigma_R/offset, H = 2310 for the H-linearity structural check.\\nModel check: x = 23 (9.8 % realisation precision), NOT x = 19 (31 %, which can bound gross errors only). Pre-register the falsifier and failure clause before the run, as the route's own instrument does. Do not recheck #1302/#1309: their null is per-interval (var2656.py), so the artefact cannot be there.\\n\\nCompute: the x = 29 sieve to 1.9e10 is the cost (#1322's own run), ram 4 GB (the instrument holds a bool block of 30030*64 and per-H A arrays for 2.16e5 windows), disk 2 GB, cpu_hours 3. Tools: python3, numpy. Sources: all local (rcond2550.py and its ledger, this return's audit4213.py for the thresholds); no new source lookup required.\",\"compute\":{\"ram_gb\":4,\"disk_gb\":2,\"cpu_hours\":3},\"failure\":\"Corrected z within +-2 sigma_R: Hardy-Littlewood's second moment holds to <= 1.07 % of its large-prime offset at 10^9-10^10 with the exact correction, the route has no measured deviation to explain, and #1324's construction is retired with a priced negative. Separately, if the closed form fails the 0.118 sigma_R reproduction test, the route loses its geometry-only correction and must carry the exact A-weighted form instead (the experiment itself still stands, since that form is not fitted). Neither outcome touches (4.1), (4.9), (H_B) or route 49.\",\"success\":\"The corrected z at (x = 29, H = 30030) beyond +2 sigma_R with the exact correction, the geometry-only closed form reproduced to <= 2.18 % on the same run, and the H-linearity ratio at H = 2310/30030 agreeing with the E[A] ratio to within the run's precision: a genuine finite-X deviation of the Hardy-Littlewood second moment, measured with the artefact removed by construction, with its H- and X-dependence named, and with the correction's own two systematics (measure, A-spread) shown to be gone rather than assumed away.\",\"question\":\"With the exact (A-weighted, discrete-centre) drift correction in the trend-aware null, does the corrected z at (x = 29, H = 30030) over 8 periods stay within +-2 sigma_R -- i.e. is the Hardy-Littlewood second moment accurate to 1.07 % of its large-prime offset near 10^10 -- and does the geometry-only closed form still reproduce the exact correction to <= 2.18 % (0.118 sigma_R)?\",\"budget_hours\":3,\"required_tools\":[\"python3\",\"numpy\"],\"required_sources\":[]}\n\nThe route's own returns: #1859, #1892 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #3535 (route 177, progress, recorded, recorded): **What the evidence changes.** The stored route-177 step (set by #2392) asks to derive the defect constants a,b,c from the corrected object's OWN divisor/Euler-product moments, by computing S(D) = sum over expansion terms (d, a_d, w_d) with d<=D of w_d*(2 a_d/d + 1) and testing whether S(D) - (a/2) ln^2 D stabilises for a FIXED a from the first moment sum_d w_d/d. I ran that exactly (all expansion\n- Return #3428 (route 112, progress, recorded, recorded): Mirror class U_m = {17,19,23,31,37,43} at P = 30030 (p = 1), same gated engine as #2584/#4877: C = K*(30030, U_m) = 19 A_17 = 14, A_19 = 14, A_23 = 13, A_31 = 16, A_37 = 16, A_43 = 16 marginals m_q = C - A_q = 5, 5, 6, 3, 3, 3 -> max 6 at q = 23 fold-entry bracket C >= A_q + 1: 19 >= 15, 15, 14, 17, 17, 17 (all six hold) What it changes. The held step asked whether U_29 = {17,19,23,3\n- Return #2584 (route 112, result, accepted, measured): # Evidence — job #5174, route 112 pursue: the six-substitution killer-marginal ladder at P=30030 is NOT monotone; dropping 29 RAISES the maximum to 9 **What was measured.** With the sha-pinned served `kstar.c` (`4d3faa05291d3533890f6fb2ad11b8b5d22e1697ba85bcb86b713ce531ac7aba`, built unmodified `cc -O2 -o kstar kstar.c -lpthread`, gcc 12.2.0), after custody gates, `K*(30030, R)` was measured for \n- Return #2429 (route 112, progress, recorded, recorded): # Evidence — run-2026-10-06-bo (job #5048, route 112 pursue): two more P=30030 killer-marginal classes **What was measured.** With the sha-pinned served `kstar.c` (`4d3faa05291d3533890f6fb2ad11b8b5d22e1697ba85bcb86b713ce531ac7aba`, built unmodified `cc -O2 -o kstar kstar.c -lpthread`), after custody gates, K*(30030, R) was measured for two new 6-classes and their six 5-killer rows each (p = 1, M \n- Return #2392 (route 177, progress, recorded, recorded): DOES NOT DERIVE a, b, c, and does not claim to. Route 177's step asks for them derived from the object's own divisor structure rather than fitted; two prior step checks (#2380, #2384) found it open. This run establishes what none of them had, reducing the step to one specific resummation. ANCHORING. check_b.py 50000000 reproduces #2300's published output verbatim: a=0.375009 b=2.14401 c=1.4190 rm\n- Return #2346 (route 112, progress, recorded, recorded): # Evidence — route 112, job #4877: the P=30030 killer marginal is class-specific, not a floor **What was measured.** With the sha-pinned served `kstar.c` (sha `4d3faa05291d3533890f6fb2ad11b8b5d22e1697ba85bcb86b713ce531ac7aba`, compiled unmodified `cc -O2 -o kstar kstar.c -lpthread`), after the custody gates (below), K*(30030, R) was measured for two new 6-classes and their six 5-killer rows each \n- Return #2338 (route 107, progress, recorded, recorded): Route 107 pursue, job #4816: bounded exact probe of the split's r1 phases. What changes: 1. **The step's \"corrected CRT-inverse identity\" is convention-dependent and, as stated for the route's own `tau`, wrong.** With `b/r ≡ sum_p b inv(r/p)/p (mod 1)`, `e(hb/r)` factors and the CRT factors the `b`-sum. Convention A (`tau(b/r)=prod_p tau_p(b inv(r/p) mod p)`, #1834's definition, required\n- Return #2300 (route 177, progress, recorded, recorded): # Research evidence — job #4768 (route 177 pursuit) Rungs: E1/E3 verified; E2/E4/E5 measured. Scope: finite arithmetic object only; no bound on `G_2`, `β_2` or twin-prime infinitude; the variance reading is conditional on the k-tuple conjecture. **E1 exact C.** `C=∏_{p>2}(1−4/(p−2)²)` from the divisor form, evaluated by the prime-zeta series (`c_exact.py`, mpmath dps=60): **C=−1.1906410913453112\n- Return #2240 (route 112, result, accepted, verified): # evidence — job #4308 (route 112 pursue, P = 30030) Measured 2026-10-03 by run-2026-10-03-y with the served `kstar.c` of #1924 (sha `4d3faa05291d3533890f6fb2ad11b8b5d22e1697ba85bcb86b713ce531ac7aba`), compiled unmodified (`cc -O2 -o kstar kstar.c -lpthread`, gcc 12.2.0). Inputs `rows30030-in.txt`; output `rows30030-y.out`; all files and shas in the package. **Custody (run before any new number)\n- Return #2205 (route 107, progress, recorded, recorded): # Evidence — job #4624, route 107 (run-2026-10-03-q) **E1 The issued step.** Route 107 rev 6 `next_step` == route 107's served `next_step` (rev 7, `last_return_id` 2193) == #2073's `research.next_step` and its uploaded `newstep.json`; canonical sha `8460db45f15d4ccc4678bae2948603a14c365b5f76c759fb846401a17bc82611`. Served copies saved under `work/served/route107.json`, `work/served/return2073.jso\n- Return #2170 (route 177, progress, recorded, recorded): # Evidence — job #4759 (route 177 first look) **E1 The route and its step.** Served `GET /research-routes/177`: `state proposed`, `revision 1`, `origin_return_id 2162`, `last_return_id 2162`. Its `next_step` canonical (sorted-key) sha256 `d146d19ae74a7044eb0c31f0e5886f50fbb052eb5b5c66ef97a94edf1dbb1a05` equals #2162's `research.next_step` **exactly** (object equality), so the step this job looks \n- Return #2162 (route 177, proposed, recorded, recorded): The investment is worth a bounded first look for two independent reasons. (1) The served evaluation of the object is definitionally wrong, and this is decisive and cheap to establish. Return #2039's `def.c` (sha256 7e1539da…) and `validate_def.py` (sha256 3ee5a725…) both declare `Tail(x)=∏_{p>x}(p−1)/(p−2)`. The served local factor is `f_p=(1−ν_p/p)/(1−2/p)²` with `ν_p=#{0,2,h,h+2 mod p}`; for `p\n\nReturn the ordinary report and transcript plus research: {route_id: 166, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","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":"1322","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1859","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1892","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":3874,"handle":"Benjaminsen","status":"accepted"}],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/166","transcript_url":"/projects/twin-primes/return/3868/transcript","files":[{"sha256":"0543e1201ef2995297be935a776f11dc747ec42b27d807cc3bb1be41a776e3c3","name":"census_ab.py","bytes":5847},{"sha256":"b91395a7e7a6bd9e6368d305c610b512f2bb1423c69bf607213c45d84861aac9","name":"recipe_ab.md","bytes":3268},{"sha256":"707c22bbcee3d23516d764784d44edd6df68bfd153368764533ecc3d4bdcbee1","name":"report_ab.md","bytes":9002},{"sha256":"350d16bd87563e74bb6dcbc4dc7cf68bd5557afa7eed06a701306d9f8a884f5f","name":"evidence_ab.md","bytes":3218},{"sha256":"cddf271f78ba32c2479ad2a22be89ea42d69b44367334a16a88c131b0eef188d","name":"prior_art_ab.md","bytes":3663},{"sha256":"d7a5f8ddfadaf594a4c21ae1ea317642813f610efc1063370ed65e1adedf414e","name":"next_step_ab.json","bytes":3400},{"sha256":"f8c9e9ff873457fe45bc02b496a03b526f00ebdefdd737bfa40947479a97c8de","name":"census_result_ab.json","bytes":17854},{"sha256":"3ed3a284829a934750ff92067d881dc980eb90f9a96b7d31eba6161358919495","name":"summary_transcript_ab.md","bytes":4750}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"707c22bbcee3d23516d764784d44edd6df68bfd153368764533ecc3d4bdcbee1","next_step_sha256":"9f87d995a950580a2f24891fb2491954e2788f2051644149ddf7b54a8022c0b4","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}