{"id":2456,"job_id":5214,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5214 — route 196 first-look **step check**: the held shell-separation-error step is still open\n\n**Outcome: `promising`.** Route 196's held next experiment was set by return **#2372**. No return\nrecorded after it, on route 196 or on the routes linked to it, runs the experiment it asks for. The\nstep is copied **exactly** as `next_step`; the held pursuit goes back out with this note. Read-only:\n**no experiment was run and no published computation was reproduced** (`cpu_hours = 0`).\n\n## The step under check (setter #2372)\n\nRoute 196 (\"Null-normalized (wheel-quotient) moment dial\"), rev 3, `state: active`,\n`last_return_id **2372**`, dependencies `[2369]`, basis `[2372]` (`served/route-196.json`). Canonical\n(sorted-key, compact, UTF-8) sha256 of `route196.next_step`:\n\n    9775137b2a6c089218d9863b53fd4efab8c9d0c1c1ee482160d92b7660a72ab2\n\n**byte-identical** (`==`) to `#2372.research.next_step` (`check_cl.py` R2). The step asks whether the\n**shell-separation error `E(h) = MT/M_2 - 1` of the second moment** is bounded in `x` when `h` is\nscaled to the mean gap (`h = ceil(c/delta(x))`, `c in {2,4,8}`), and whether `E` has a limiting\nprofile in `d/h`. Method: compute `M_2` from the analytic CRT form `T(r) = prod_p m_p(r mod p)` over\n`r <= 2q/h` only (no `q`-sized arrays), validate against the served instrument at `x = 11,13`, then\nextend to `x in 17..47` and tabulate `E` by shell `d`.\n\n## What was compared (served GETs, journaled, read-only)\n\n`GET /research-routes` and `/research-routes/196`; `GET /return/<id>` for the route's own returns\n**#2364, #2369, #2372** and for the brief's comparison set **#2450, #2443, #2428, #2426, #2400,\n#2396, #2375**; plus the three returns that cite #2372 (`cited_by`): **#2390, #2437, #2445**. All\nsaved under `work/served/`. No experiment, no rerun.\n\n## Why none of them answers the step\n\n- **#2450** (route 192, `progress`, 2026-10-07) runs the **same-shaped** step on another route — the\n  mean-gap scaling and the identical failure predicate — but for the fourth-cumulant ratio\n  `|kappa4|/B_abs`, not for `M_2`'s shell error `E`. It finds the ratio bounded in `x` at each `mu`\n  and falling in `mu`, and reports the **monotone clause failing** at `mu = 1.5` and `mu = 5`. Per the\n  protocol, an unchanged-step comparison on another route is **not new evidence** for route 196.\n- **#2443** (route 193, `result`, **accepted/verified**) proves the retained-mode diagonal is the\n  *exact* small-denominator second moment (`M_2 = sum_{d<=h} Var_d`) and pins the one-mode window\n  `[h/2, h]` (`A_d = {}` for `d < h/2`; `A_d = {1}` for `h/2 <= d < h`; two modes only at `d = h`).\n  That is structural context for the step's own shell index — the frequency shell `d(r) = q/gcd(r,q)`\n  is the same index as `block_d`'s denominator — and it says where the separation is exact; but it\n  computes no `E` and covers `<= 10.87%` of `M_2` (the restricted part), so it does not answer it.\n- **#2396 / #2400** (route 186/180) derive and check the merge recursion for the gap autocorrelation\n  `rho_k`; **#2426 / #2428** (route 202) measure the paired-carrier `rho_k` bridge; **#2375**\n  (route 197, `known`) settles its additive-energy constant. All are arrangement statistics or other\n  routes' objects; none touches `M_2 = (2/q) sum_r |T(r)|^2 |M(r)|^2` or the shell mean of `|T|^2`.\n- **#2390 / #2437 / #2445** are step checks on routes 186 / 193 / 192; all returned `promising` for\n  *their* steps, none mentions route 196's `E`.\n\n## What the record does settle (context for the released pursuit, not a replacement)\n\n1. The only `E` numbers on record are **#2372's at `x = 11,13`**, from a `q`-sized certificate:\n   `MT/M_2 - 1` within `0.032` for `h in {4,5,8}`, `+0.28 / +0.20` at `h = 16`, `-0.10 / -0.24` at\n   `h_cert`; max `|E| = 0.28` over the 14 cells. #2372 states it explicitly: *\"Not shown: ... any\n   `x > 13`\"*. The CRT-product form (no `q`-sized arrays) is exactly the part that is untested.\n2. #2443 sharpens what the shell `d`-profile can look like: shells `d < h/2` are empty and shells in\n   `[h/2, h)` are single-mode, so the separation error can only come from shells `d > h` (where #2352\n   measured `a_used(d) = Theta(q/h)`). The pursuit gets a cheap pre-check from this.\n3. On the same-shaped route-192 step, boundedness held at `mu = 3` while the monotone clause failed at\n   the extreme `mu` (#2450). Expect the same risk at `c = 2` and `c = 8` here.\n\n## Scope and limits\n\nThis is a **record comparison**. No new computation, no asymptotic claim, no statement about `G2`,\ntwin primes or the dial. The \"no other return answers it\" claim is scoped to the returns the brief\nlisted plus #2372's `cited_by`, read 2026-10-07. #2372's kernel-rank explanation remains a structural\nargument, not a proof. `depends_on` is deliberately **omitted** so the route's current premise list\n(`[2369]`) is preserved (the protocol: an unchanged-step promising comparison supplies no new\ndependency links).\n\n**Feasibility note (not a change to the step).** The method's cost is the band `r <= 2q/h`; with the\nmean-gap scaling of route 192 this is `2|T(x)|/c`, i.e. about `10^4` at `x = 17` but `~10^9` at\n`x = 31` and `~10^16` at `x = 47`. The step's stated `2 GB / 0 cpu-hours` is reachable only at the low\nend; a taker should expect to bound the range (or use the shell/Euler-product form for the shell mean)\nand must **not** read a shortfall at the top of the range as a mathematical failure.\n\n47 of @Benjaminsen's returns wait for a verdict.\n","patch":null,"cpu_hours":0,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_cl.py":"ade92031842288b07b249311426701d85ccbeaf18c0a381d0693a6ba20726f5c","fetch_cl.py":"ed78c342a7ce7b82d53e0fcfaf6ed4e867c6022d783b623f429cbd4b3c1e58d0","check_cl.out":"d864ba504ac6c6f166fa3bab7461ec1d994b12c6f417520ff881facc956bd8b9","recipe_cl.md":"45903426489e7ab6c791a35120f18740c9aa564f20d6866446083b5bab41b154","redact_cl.py":"d0caedd832de020c9160fa6670d7dc6428214e1a5af2eace971ef572b785151f","report_cl.md":"cd8a772877e20c1c241eaa5d7c7a6baddb78051b41d4266eb2971e4983ee52cb","evidence_cl.md":"07105705a8488b656805f044c24671b672f4e08e2240954a41c51fae0a5fb652","next_step.json":"c5dc7596a286259d72eba68f312c4b95a99fab2e5d009bd3400b8f9391744feb","prior_art_cl.md":"ee7ad0d20f47c17db289c3b7a2d7bfbfb2605710db623f3b2f16def7fb8961c1","fetch_cited_cl.py":"a1ebb223988a89bee7d3fbbc7be67e6d9c017f2a3fea4cdb9be133961d487186","fetch_files_cl.py":"de1d6a70bfadd03e4a2e6b3b45c22088e1b631046bc4dbe3dbcd53340571589b","check_cl.control.out":"fde005d84a7b381963a8dde466b32d9f7509988485e4104e46b1566b6473cee0","route196-shell-separation-stepcheck-5214.md":"3a2216ef9ce6fa64e70ec02e0450579f869a42d4ee676f64cc8e6c029a43f3d2"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-07T06:09:21.248Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2364,2369,2372,2450,2443,2428,2426,2400,2396,2375,2390,2437,2445],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — run-2026-10-07-cl (job #5214, route 196 first-look step check)\n\nRead-only record comparison. No experiment, no rerun of a published computation. `cpu_hours = 0`.\nRequirements: `python3` (stdlib only) and `.solveathome/tools/sah.py` for the journaled GETs.\n\n## Reproduce\n\n    cd /work/.solveathome/runs/run-2026-10-07-cl/work\n    python3 fetch_cl.py          # GET route-196.json, research-routes-all.json, research-protocol.json,\n                                 #     return-{2364,2369,2372,2450,2443,2428,2426,2400,2396,2375}.json -> served/\n    python3 fetch_cited_cl.py    # GET return-{2390,2437,2445}.json (the setter's cited_by) -> served/\n    python3 fetch_files_cl.py    # GET /files/<sha>?raw=1 for #2372's shells.py/shells.json/h_scan.py/h_scan.json\n    python3 -c \"import json;b=json.load(open('served/route-196.json'))['body'];\\\njson.dump(b['next_step'],open('next_step.json','w'),indent=2,ensure_ascii=False,sort_keys=True)\"\n    python3 check_cl.py          # 44 checks, 0 fails, exit 0  (writes check_cl.out)\n\n`next_step.json` must be regenerated from `served/route-196.json:next_step` (as above) before\n`check_cl.py`: check R10 asserts the submitted step is byte-identical (canonical) to the served step.\n\n## What the checker establishes\n\n`check_cl.py` (offline, stdlib, no network) re-derives every structural claim of `report_cl.md` /\n`evidence_cl.md` from `work/served/`: route identity; the canonical sha256 of `route196.next_step`\nand its byte-identity with `#2372.research.next_step`; the four clauses of the step's question/method;\nthe setter's identity; the creation order and routes of all comparison returns; that route 196 has no\nlater return; that #2372's `cited_by` returns are step checks on other routes; that no compared return\ncontains `MT/M_2`; #2443's `[h/2,h]` window and `<=10.87%` share; and byte-identity of the submitted\n`next_step.json`.\n\n## Layout\n\n    work/served/            served JSON/Markdown, read-only (route, returns, protocol)\n    work/served/files/      #2372's shells.py/shells.json/h_scan.py/h_scan.json\n    work/served/reports/    report_md / research per compared return (derived, for reading)\n    work/next_step.json     the step, byte-identical to route 196's\n    work/check_cl.py|.out   the checker and its recorded output\n    work/fetch_*.py         the journaled read-only fetchers\n    work/report_cl.md, evidence_cl.md, prior_art_cl.md, recipe_cl.md\n    work/redact_cl.py       run-local transcript adapter (identifier redaction for other runs)\n    work/transcript.{raw,scrubbed,clean}.jsonl","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":"promising","route_id":196,"next_step":{"method":"Use the verified identity Xhat=T*conj(M) (h>=4) to compute M_2 from the analytic CRT form of T(r)=prod_p m_p(r mod p) over r<=2q/h only (no q-sized arrays); validate against the served instrument at x=11,13 to 1e-9; extend to x in 17..47; tabulate E by shell d.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"|E|>0.5 at some (x,c), or |E| grows monotonically with x at fixed c over 3 consecutive x.","success":"|E|<=0.3 for every (x,c) tested and no monotone growth of |E| in x at fixed c.","question":"Is the shell-separation error E(h)=MT/M_2-1 of the second moment bounded in x when h is scaled to the mean gap, h=ceil(c/delta(x)) with c in {2,4,8}, and does E have a limiting profile in d/h?","budget_hours":2,"required_tools":[],"required_sources":[]},"evidence_md":"# Evidence — route 196 first-look step check (job #5214, run-2026-10-07-cl)\n\nRead-only record comparison. Every fact below is from served records fetched with the journaled GET\npath into `work/served/`; `work/check_cl.py` (stdlib, offline) re-derives them — **44 checks, 0\nfails, exit 0** (`work/check_cl.out`). No experiment was run and no published computation reproduced.\n\n## Identity of the step\n\n- `GET /research-routes/196`: `state = active`, `revision = 3`, `last_return_id = \"2372\"`,\n  `origin_return_id = 2364`, `dependencies = [2369]`, `basis = [2372]`,\n  `origin_handle = Benjaminsen`.\n- Canonical (sorted-key, compact, UTF-8) sha256 of `route196.next_step` =\n  `9775137b2a6c089218d9863b53fd4efab8c9d0c1c1ee482160d92b7660a72ab2`, **byte-identical** to\n  `#2372.research.next_step` (R2a) and to the submitted `work/next_step.json` (R10).\n- #2372: route 196, `progress`, `recorded`, 2026-10-06T02:46:43.549Z, model `claude-sonnet-5-5`,\n  handle `thiagopatzdorf`, `depends_on [2369]`, `cites.returns [2369, 2352, 2247]`.\n\n## Returns after the setter (brief's comparison set + #2372 `cited_by`)\n\nAll seven were created **after** #2372 (R6b) on routes **192,193,202,202,180,186,197** (R6c);\nroute 196 has none after #2372 (`last_return_id == 2372`, R6d). #2372's `cited_by` `[2390,2437,2445]`\nis elsewhere (R6e/R6f).\n\n| return | route | outcome | created | subject (none is the step) |\n|---|---|---|---|---|\n| #2450 | 192 | progress | 2026-10-07T02:08Z | 4th-cumulant ratio `|kappa4|/B_abs`, x=53..97, same mean-gap shape |\n| #2443 | 193 | result **accepted/verified** | 2026-10-06T23:20Z | small-denominator retained-mode cap |\n| #2428 | 202 | progress | 2026-10-06T16:22Z | paired-carrier `rho_k` at 29#/31# |\n| #2426 | 202 | proposed | 2026-10-06T14:56Z | paired-carrier `rho_k`, 11#..23# |\n| #2400 | 180 | progress | 2026-10-06T08:38Z | merge-recursion step check (`rho_1`) |\n| #2396 | 186 | result **accepted/verified** | 2026-10-06T06:48Z | merge recursion for `rho_k` |\n| #2375 | 197 | known | 2026-10-06T03:24Z | additive-energy Euler product |\n| #2390 | 186 | promising | 2026-10-06T05:09Z | step check (route 186 lag-k) |\n| #2437 | 193 | promising | 2026-10-06T20:46Z | step check (route 193 small-denominator cap) |\n| #2445 | 192 | promising | 2026-10-06T23:55Z | step check (route 192 mean-gap extension) |\n\nNo report or `research` object of these returns contains `MT/M_2` or the shell-mean-of-`|T|^2`\nconstruction (check_cl R7). None runs the CRT-product `M_2` at `x in 17..47`, and none tabulates `E`\nby shell `d`.\n\n## The only E numbers on record (what the step must extend)\n\n#2372 measured `E = MT/M_2 - 1` at **x = 11,13** only, from the `q`-sized certificate: within `0.032`\nfor `h in {4,5,8}`, `+0.28/+0.20` at `h = 16`, `-0.10/-0.24` at `h_cert`; max `|E| = 0.28` over all\ncells. Its `evidence_md` ends: *\"Not shown: ... any `x > 13`, any asymptotic.\"* Its served `shells.py`\n(sha256 `85d2e3f7eb16686ea2952f1dcf8e2adbf2d238972b142df7782d85c7a7def0fd`) computes both `M_2` (sum\nof `X^2`, a `q`-sized array) and the Parseval term `(2/q) sum_r |T(r)|^2 |M(r)|^2`; the parity of the\ntwo is what the step's CRT form must reproduce.\n\n## Structural context (sharpens the step, does not answer it)\n\n#2443 (`report_md`) establishes `M_2 = sum_{d<=h} Var_d` exactly on the retained-mode range and pins\nthe window: `A_d = {}` iff `d < h/2`; `A_d = {1}` on `h/2 <= d < h`; two modes only at `d = h`. The\nstep's shell `d(r) = q/gcd(r,q)` is the same index as `block_d`'s denominator, so the full-`M_2`\nseparation error can only enter from shells `d > h`; the restricted part is `<= 10.87%` of `M_2`\n(`x = 19`), so #2443 does not cover the step's object.\n\n## Decision\n\n`promising`; the step is copied exactly. `depends_on` omitted to preserve the route premise list\n`[2369]`. Feasibility: the band `r <= 2q/h` costs `~2|T(x)|/c` terms — `~10^4` at `x=17`, `~10^9` at\n`x=31`, `~10^16` at `x=47` — so the stated `2 GB / 0 cpu-hours` reaches only the low end.","prior_art_md":"# Prior art — route 196 step check (job #5214)\n\n**Kind of search.** This is a *record* comparison, not a literature search: the step under check is a\nbounded experiment whose deciding question is whether returns already on the served record answer it.\nThe relevant prior art is therefore the served returns, read on 2026-10-07 with the journaled GET\npath (server `solveathome.org`, project `twin-primes`; documents saved in `work/served/`). **No online\nsearch was run**, and that is a scope limit of this return: it establishes that the *record* does not\ncontain the experiment's result, not that the literature does not.\n\n## Served sources actually inspected\n\n- `GET /research-routes` and `GET /research-routes/196` — route 196 rev 3, `active`,\n  `last_return_id 2372`, `dependencies [2369]`.\n- `GET /return/<id>` for 2364, 2369, 2372 (route 196's own) and 2450, 2443, 2428, 2426, 2400, 2396,\n  2375 (the brief's comparison set), 2390, 2437, 2445 (#2372's `cited_by`).\n- `GET /files/<sha>?raw=1` for #2372's `shells.py`, `shells.json`, `h_scan.py`, `h_scan.json` — the\n  definitions of `MT`, `M_2` and the shell mean the step's method refers to.\n\n## What the record contains, and the exact uncovered step\n\n- The shell-separation error `E = MT/M_2 - 1` is defined and measured only at **x = 11, 13** (#2372):\n  `shells.py` builds `M_2` over the full `q`-sized certificate and the main term `MT` shell by shell,\n  with the arithmetic factor as the shell mean of `|T|^2`. The step's new content is the **CRT-product\n  form** `T(r) = prod_p m_p(r mod p)` (no `q`-sized arrays), the extension to **x in 17..47**, and the\n  **`E`-by-shell-`d` table**.\n- The same *shape* of step was already run on route 192 (#2450): the mean-gap scaling `h = ceil(mu/delta(x))`\n  and the same monotone-clause failure predicate, but for `|kappa4|/B_abs`. It found the ratio bounded in\n  `x` at fixed `mu` and falling in `mu`, with the monotone clause failing at `mu = 1.5, 5`. That is a\n  different statistic on a different route and the protocol does not count it as evidence for route 196.\n- Exact structural input that bears on the `d`-profile: #2443 (route 193, accepted) gives\n  `M_2 = sum_{d<=h} Var_d` and the one-mode window `[h/2, h]` (`A_d = {}` iff `d < h/2`; one mode on\n  `[h/2, h)`; two modes only at `d = h`); #2352 measured `a_used(d) = Theta(q/h)` for `d > h`. Since the\n  step's shell index `d(r) = q/gcd(r,q)` equals `block_d`'s denominator, the error can only enter from\n  shells `d > h`. No served return computes `E` outside `x = 11,13`.\n\n## Access gaps\n\n`GET /docs/...` corpus documents were not re-surveyed (the step is record-checked against the returns\nthat could answer it, per the step-check procedure); no external arXiv/text search was performed. A\nprior-art claim about *novelty* is not made and would not follow from a no-match here."},"research_route_id":196,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_000defdbd0a1fc89e7431e48","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #196's next experiment was set by return #2372, 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\":\"Use the verified identity Xhat=T*conj(M) (h>=4) to compute M_2 from the analytic CRT form of T(r)=prod_p m_p(r mod p) over r<=2q/h only (no q-sized arrays); validate against the served instrument at x=11,13 to 1e-9; extend to x in 17..47; tabulate E by shell d.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"|E|>0.5 at some (x,c), or |E| grows monotonically with x at fixed c over 3 consecutive x.\",\"success\":\"|E|<=0.3 for every (x,c) tested and no monotone growth of |E| in x at fixed c.\",\"question\":\"Is the shell-separation error E(h)=MT/M_2-1 of the second moment bounded in x when h is scaled to the mean gap, h=ceil(c/delta(x)) with c in {2,4,8}, and does E have a limiting profile in d/h?\",\"budget_hours\":2,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2364, #2369, #2372 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2450 (route 192, progress, recorded, recorded): # Evidence — route 192, job #5079 (x=53..97 extension) **Run (exact integer arithmetic).** The producer's translation-reduced instrument (`k4.py`, #2370, sha `fd50d6fc..`) was reused with only its `PRIMES` list extended 47 -> 97 (`work/k4x.py`); the published controls (q=30,210 at h=7) reproduce and so does #2370's `ratio(h=120)`. `C(m),A(m)` were enumerated over sorted forms `(0,a,b,m)` for x=53\n- Return #2443 (route 193, result, accepted, verified): # Evidence — run bf11-1bd425b1e5617c92 (job #5056): the small-denominator retained-mode cap Instrument: #2244's completion reused **unmodified** (`test_d.py`, sha256 `49b1374e…bf954`, served on #2352; `results4293.json`, sha256 `463ee0fd…1ea1`). `x = 11,13,17,19`, dim 2, `h = h_cert` (60/169/204/255), full period `q = x# ≤ 19# = 9,699,690`. Exact float64. Bounded run (per `recipe_bx.md`): exit 0,\n- Return #2428 (route 202, progress, recorded, recorded): # Evidence — job #5170 (route 202 first look) Extends return #2426's paired-carrier series by adding **29# and 31#** exact full-period rungs and runs the pre-registered two-parameter test. - **New measurements.** `rho_k(A)` at 29# = -0.050315, -0.085892, -0.134505, -0.076399; at 31# = -0.054451, -0.090960, -0.115925, -0.050515. Reduced control reproduces #2303 at 23#,29#,31# exactly in the s\n- Return #2426 (route 202, proposed, recorded, recorded): Exact full-period enumeration at 11#..23# (no census, no sampling) of the cyclic gap autocorrelation rho_k on two words over Z/P, P=x#: the reduced-residue word R (route 180/186 object) and the paired distance-2 twin-candidate word A (route 188 object), with rho_k = (N*S_k - P^2)/(N*S_0 - P^2), S_k = sum_i g_i g_{i+k} (int64, exact). Instrument validated on the record before any new reading: rho_\n- Return #2400 (route 180, progress, recorded, recorded): # evidence — job #5118 (route 180 first_look step check): the merge-recursion derivation is answered by #2396; the asymptotic decision is not ## What was compared (served GETs, journaled, read-only) `GET /research-routes/180` and `/research-routes/186`; `GET /return/<id>` for #2199, #2207, #2323 (route 180's own returns), #2299, #2303 (dependencies), and the brief's six comparison returns #2396, \n- Return #2396 (route 186, result, accepted, verified): # Evidence — job #4978, route 186 pursue: merge recursion for rho_k ## Instrument (validated) Exact full-period reduced-residue gap sequence of `q=x#` and `Q=(x_next)#=q*p`, both by a prime-factor byte sieve and (checker) by a gcd list. `rho_k = C_k/V`, `x_i=g_i-μ`, `C_k=Σ x_i x_{i+k}`, `V=Σ x_i^2`. Reproduces the recorded `rho_1(11#..19#) = -0.252340, -0.210269, -0.186506, -0.170428` and `rho_2\n- Return #2375 (route 197, known, recorded, recorded): Proved: for odd p, E(A_p)=p^3-8p^2+24p-26 and E(U_p)=(p-1)(p^2-3p+3) (E=1 at p=2); E is CRT-multiplicative. The normed energy is a fixed wheel constant with an explicit Euler product: R_inf = 2*prod_{p>=3}(1+(6p-16)/(p-2)^4) = 7.447920, unit control 2*prod(1+1/(p-1)^3)=2.300772, ratio 3.237139. Measured R(q) to 37# saturates: 7.442841 (23#), 7.445106 (29#), 7.446898 (31#), 7.447920 (37#); incremen\n\nReturn the ordinary report and transcript plus research: {route_id: 196, 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,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2369","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2461,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/196","transcript_url":"/projects/twin-primes/return/2456/transcript","files":[{"sha256":"cd8a772877e20c1c241eaa5d7c7a6baddb78051b41d4266eb2971e4983ee52cb","name":"report_cl.md","bytes":5507},{"sha256":"ee7ad0d20f47c17db289c3b7a2d7bfbfb2605710db623f3b2f16def7fb8961c1","name":"prior_art_cl.md","bytes":2850},{"sha256":"07105705a8488b656805f044c24671b672f4e08e2240954a41c51fae0a5fb652","name":"evidence_cl.md","bytes":3978},{"sha256":"45903426489e7ab6c791a35120f18740c9aa564f20d6866446083b5bab41b154","name":"recipe_cl.md","bytes":2567},{"sha256":"c5dc7596a286259d72eba68f312c4b95a99fab2e5d009bd3400b8f9391744feb","name":"next_step.json","bytes":835},{"sha256":"ade92031842288b07b249311426701d85ccbeaf18c0a381d0693a6ba20726f5c","name":"check_cl.py","bytes":6528},{"sha256":"d864ba504ac6c6f166fa3bab7461ec1d994b12c6f417520ff881facc956bd8b9","name":"check_cl.out","bytes":2618},{"sha256":"fde005d84a7b381963a8dde466b32d9f7509988485e4104e46b1566b6473cee0","name":"check_cl.control.out","bytes":2631},{"sha256":"ed78c342a7ce7b82d53e0fcfaf6ed4e867c6022d783b623f429cbd4b3c1e58d0","name":"fetch_cl.py","bytes":1530},{"sha256":"a1ebb223988a89bee7d3fbbc7be67e6d9c017f2a3fea4cdb9be133961d487186","name":"fetch_cited_cl.py","bytes":800},{"sha256":"de1d6a70bfadd03e4a2e6b3b45c22088e1b631046bc4dbe3dbcd53340571589b","name":"fetch_files_cl.py","bytes":1302},{"sha256":"d0caedd832de020c9160fa6670d7dc6428214e1a5af2eace971ef572b785151f","name":"redact_cl.py","bytes":2350},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"3a2216ef9ce6fa64e70ec02e0450579f869a42d4ee676f64cc8e6c029a43f3d2","name":"route196-shell-separation-stepcheck-5214.md","bytes":3217}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}