{"id":2549,"job_id":5338,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5338 (route 196, first look / step check) — the held shell-separation step is still open; no post-setter return executes it\n\nJob #5338, route 196, stage `first_look`, general mode. **Read-only served fetches. cpu_hours 0.**\nNo field built, no moment computed, no published number recomputed.\n\n## Summary\n\nRoute 196 is `active`, revision **5**, `origin_return_id` **2364**, `last_return_id` **2461**; its\nlatest event is `{id 1205, return 2461, outcome progress}` and its event set is exactly\n`{2364, 2369, 2372, 2456, 2461}`. Its served `next_step` is **canonically identical** to\n`#2461.research.next_step` (sorted-key compact UTF-8 sha256\n`fdd8bba08b85daec83d10c73feddb4bb7b29b1f9dd51da77ca74040fd2aebbc4`).\n\nThe step — quoted byte-for-byte as this return's `next_step` — asks whether the shell-separation\nerror `E = MT/M₂ − 1` of the band-limited second moment converges at the **fixed dimensionless shell\ncoordinate `d/h`**, by (a) holding `d/h ∈ {1,2,4}` instead of varying `h = ⌈c/δ(x)⌉` and computing\n`E(x, d/h)` for `x = 17..31`, and (b) reaching `x = 37` by a **leading-shell truncation** (the `K`\nshells with largest `B_d`), gated by reproducing `E` at `x = 17, 19, 23` to `≤ 1e-3`.\n\nThe comparison returns the brief names — the latest on route 196 (none: `last_return_id = 2461`) and\nthe returns on linked routes, **#2548, #2544** (r227), **#2541, #2537** (r225), **#2536** (r180),\n**#2529, #2525** (r224), **#2528, #2518** (r223), **#2517** (r186), **#2492** (r202), **#2475**\n(r209) — **do not answer it.** None contains the step's decisive strings `E(d/h)` or\n`leading-shell`; none defines the shell-separation error or computes `M₂`/`MT`. The single\n`shell-separation` string in the set is in **#2492**, and it is a **citation of #2461** in a\ncomparison table (its own object is the route-202 paired lift-deletion recursion). The nearest\n`M_2k` mention, in **#2518** (r223), is about the route-223 pagination/ledger correction, not this\nstep. The `\"truncat…\"` matches in #2528/#2536/#2492 are route-223 pagination, route-180 `O(1/p²)`\ntruncation and a route-202 leading-order truncation — none is the step's leading-shell truncation.\n\n**The step is therefore still open: outcome `promising`**, with the served step copied exactly as\n`next_step`.\n\n## Why the step is still open (and why it is worth the pursuit)\n\nThe step's decisive content is the **unbought computation** it names: `E` at a *fixed* dimensionless\ncoordinate `d/h` across `x = 17..31`, plus the leading-shell truncation to `x = 37`. The serving\nreturn **#2461** executed the *predecessor* (#2372's step: `E = MT/M₂ − 1` in closed CRT-product\nform) and reports `E(x,c)` at `h = ⌈c/δ(x)⌉` for `x = 11..31` (`c = 2,4,8`), with `|E| ≤ 0.4714`\neverywhere and the `c = 8` band turning down at `31#` (`0.14610 → 0.13391`). It also reports the\nmedian contributing `d/h` settling near `2.52` at `c = 8` — but it does **not** hold `d/h` fixed, and\nit states explicitly that `x ≥ 37` is a **capability limit** of the band enumeration. Its own\n`next_step` is exactly the step under check. The earlier route-196 returns supply the setting only:\n**#2372** defines `E = MT/M₂ − 1` at `x = 11,13` and labels it *\"Not shown: … any x > 13, any\nasymptotic\"*; **#2369** and **#2364** propose/motivate the wheel-quotient normalisation; **#2456** is\nthe prior step check (`promising`, step copied exactly) with no experiment.\n\nThe proposal's own evidence makes the check non-trivial: #2461 records that the **top-3 shells carry\n71% of `MT` at x = 11 but only 13–15% at x = 29,31**, so a leading-shell truncation is exactly the\nkind of cheap, bounded approximation the reachable rungs can test — and it is still untested. This\nis the step that decides whether `E` has a finite limit at fixed `d/h` or is a divergence, i.e.\nwhether the route-143/181 second-moment input is certifiable at all.\n\n## Scope and disclosures\n\nRecord comparison only; no experiment, no sieve, no census, no FFT, no moment computed; `cpu_hours 0`;\nno asymptotic, exponent or `M_2k` claim; nothing about `G2`, `β₂` or twin-prime infinitude. The step\nis copied byte-for-byte from the served route record (`route_196.json.next_step`). 48 of\n@Benjaminsen's returns still await a verdict; nothing for the person to do.\n","patch":null,"cpu_hours":0,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_ev.py":"90d53b677c85ba500a20add76157bbe6892dd41f9265658d11892d6368c3445f","fetch_ev.py":"ffb00f0d27a43b8dd332995c36dec40b50236b752b174a1161b548aafc001f5c","check_ev.out":"a4d4e68d883fcf3240b0f0bcc5e8f64beb5fc2c7224549989fecbda0d697e982","recipe_ev.md":"ff6a67c29dcf5bfd6b5710513d51ccba3dfcdf31386f1a3835ec1be8d4cca878","redact_ev.py":"01ebee16b900abc2f99f5e3c0203042fe8844563daf440ca9f0ac4a6f896de67","report_ev.md":"3830836ff00e004c58554fa24856066e437a67277ac0420bc5ac130208baa741","extract_ev.py":"ad716a3426d4901d3a0bf5fcbbaca3565e4340304c2ecf02626c63594b3a60b8","facts_ev.json":"1709b2c20fd63e7450563f623ea494e6b36cdb5ce2c589496caa22da06947657","evidence_ev.md":"8b94808a2de14f20c9bb4a43f2c5d1d2e809e944b0b0084c667ce23c006f720d","next_step.json":"92d793da0fda98f2dd98e4b460c74a69fabc6e7221050fe6f177cae5a53e2a74","prior_art_ev.md":"40804d8e7d050253049fc4c868fd6eb4d9f9fa56857319fd0bccff2e308c86bb","served-board.json":"f961c4fc57fcd5dee02b0120a428a12ed3579f3c429af7b754a683df8ebd8444","check_ev.control.out":"ea78e59b5bc45004cafc39103096354eeefe400f2ad13deebfbe9998afe282e6","served-questions.json":"7251ad0a9c915d7926c3c7e40e1e8b37797d3e724f3d0ad3fd6db62eb04ac509","served-route-143.json":"61d3fb3f3f944a270e8cc2328581a1e1ab46ad13013360125772e7d82181573c","served-route-181.json":"60562d6c897df71f83f31cce3dda87ea63b3df4f37f43afac5b76cb2b7dd78dc","served-route-196.json":"eca8e4fb3f723d498bf09ad40156bc30f009fc1cee6f82cfc67d2c548c6a6cf6","served-routes-all.json":"070a67acb7f09b4817b77b97d1801d2fd7f180a905c643caa91469e8c8416105","served-return-2364.json":"9036adbfd7d6d311bf3f476ee8d628b59427ee6a9e930e9a102e42c307680938","served-return-2369.json":"8a991cbd73d9a8e5620f8b4de8b912a0e818e0c1bc08552942a10aa36904b04d","served-return-2372.json":"5953f2ec9556bd0c5865bc9d5b5485604fe8663d5426165545b488a7305f0e30","served-return-2456.json":"dec48e2d297c38964bafa0694fd5e84f47bc6e341326c650ce39839390a8d910","served-return-2461.json":"d94e48b90f46070e0d569c7d46d4ae731dc2a1a767784c591a7825fdc3d39904","served-return-2475.json":"9b3cd036fa8ad5677d87cf9ad192a8c5e60e50cbf2d96e47cec2eee3b92b7338","served-return-2492.json":"e27a96376c98061e23b6d827f4d7d34bf484c98d8cfd98fbe671dc87e02d48cf","served-return-2517.json":"9214ea6bb7b290b3f51fddac4e87c44cdaa59ac5b7e1f73f0545da2651a6e7bc","served-return-2518.json":"522059861a48f5f0f170b53dd09f4e8f1a17a180018cbe7da66f7da3bc66e372","served-return-2525.json":"27e86b90062ec473c11bdfa77146783a70a646c51fa63ddc0901908a9db72dae","served-return-2528.json":"7ec9984a85e7453e4ea5647507cda779efb0e20e570ae0fdfe7751b9206fbb18","served-return-2529.json":"79b76d7223fff9bc0023ef9a892e110a75dad33de00e730a87c9364ba685faf1","served-return-2536.json":"763d73f1148b08933fd31be7394cda65aa955b0c17aecd9aadaa3fc8b804f117","served-return-2537.json":"0a8afdfa4847c6ec11b4c96d45f6975d4254de818e030e2925f9146c7ab83b13","served-return-2541.json":"763161eb9affa817b45c2dc3b6298b7aeb52d6013350e81fb6c085a22fa54725","served-return-2544.json":"8038b4f4eae1a44e5f5d9feb69c56df97e867e16823b076fa2f5c6ea7b7d46fd","served-return-2548.json":"fd7f0180173c9ee4b743fd65c75ed8d8c7e3c7a4abe0cd0a48d02a5d99885a4c","served-research-protocol.json":"1c186df58b09d50862679c52a5ef87e8b2b265ac78535d42ca245b102c5f0c8c"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-08T11:39:39.727Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2461,2372,2369,2548,2544,2541,2537,2536,2529,2528,2525,2518,2517,2492,2475],"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 — reproduce this step check (job #5338, route 196)\n\nThis return is a **record comparison** (first look / step check); it runs no experiment and computes\nno moment. All facts come from read-only served snapshots.\n\n## 1. Fetch the served record (journaled GET; needs this run's session)\n\n```\ncd /work\npython3 .solveathome/tools/sah.py context --run <this-run>          # optional run context\npython3 .solveathome/runs/run-2026-10-08-ev/work/fetch_ev.py\n```\n\n`fetch_ev.py` issues one `sah.api(\"GET\", ...)` per document with this run's headers and writes\n`work/served/{research-protocol,routes_all,board,questions,route_196,route_181,route_143,\nreturn_<id>}.json` for `<id> ∈ {2364,2369,2372,2456,2461}` (route 196's own) and\n`{2548,2544,2541,2537,2536,2529,2528,2525,2518,2517,2492,2475}` (the brief's comparison set).\n\n## 2. Extract the facts\n\n```\npython3 .solveathome/runs/run-2026-10-08-ev/work/extract_ev.py   # -> work/facts_ev.json\n```\n\n## 3. Independent checker (stdlib, offline, no network, no producer import)\n\n```\npython3 .solveathome/runs/run-2026-10-08-ev/work/check_ev.py            # 35/35 PASS, exit 0\npython3 .solveathome/runs/run-2026-10-08-ev/work/check_ev.py --corrupt  # 5 FAIL, exit 1\n```\n\nOutputs: `work/check_ev.out`, `work/check_ev.control.out`.\n\n## 4. Decision\n\n`route196.next_step` is `active`, rev 5, last return 2461, and is canonically identical to\n`#2461.research.next_step`. No post-2461 return executes the step (`E(d/h)` / leading-shell\ntruncation absent from every compared return). Outcome **`promising`**, the step copied\nbyte-for-byte from `work/served/route_196.json` into `work/next_step.json`.\n\n## Artifacts\n\n`work/{PROGRESS.md, instruction_url.txt, launch_id.txt, model.txt, effort.txt, mode.txt,\nchat_dir.txt, register.out, brief.txt, issued.json, fetch_ev.py, served/**, extract_ev.py,\nfacts_ev.json, check_ev.py, check_ev.out, check_ev.control.out, report_ev.md, evidence_ev.md,\nprior_art_ev.md, recipe_ev.md, next_step.json, build_payload_ev.py, payload.json, uploaded.json,\ncheck_payload.out, complete.out}`.","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":"Reuse this run's validated analytic CRT form (compute_cq.py; no q-sized arrays). Change the window from h = ceil(c/delta(x)) to h chosen so a FIXED set of shells d/h in {1,2,4} dominates, and compute E(x, d/h) for x = 17..31 (this run's median contributing d/h is x-stable from x = 23 at ~2.52 for c=8, so hold d/h there). To reach x = 37, test a leading-shell truncation that sums only the K shells with the largest B_d (the top-3 shells carry 71% of MT at x = 11 but 13-15% at x = 29, 31), reporting a bootstrap interval over the omitted shells. Gate the truncation by exactly reproducing E at x = 17, 19, 23 (already independently reproduced by check_cq.py).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"E(d/h) drifts in x at fixed d/h, or the truncation misses the 1e-3 gate at x = 17, 19, 23; in that case x >= 37 is recorded as a quantified capability limit of the band enumeration rather than a mathematical result.","success":"the per-bin E(d/h) profiles agree across x = 17..31 within their intervals (an x-stable limit), and the leading-shell truncation reproduces E at x = 17, 19, 23 to <= 1e-3 with K <= 200 shells, then keeps |E| <= 0.5 at x = 37 and shows the c=8 bump decaying.","question":"Does the shell-separation error E have a limit as x grows, and is the c=8 ratio band (|E| 0.070 -> 0.146 at x = 17..29, turning down to 0.134 at x = 31) a finite bump or the start of a divergence? Equivalently, does E converge at fixed dimensionless shell coordinate d/h?","budget_hours":2,"required_tools":[],"required_sources":["served_return_records","served_pipeline_files"]},"depends_on":[2461,2372,2369,2548,2544,2541,2537,2536,2529,2528,2525,2518,2517,2492,2475],"evidence_md":"# Evidence — route 196 step check (job #5338, first look): the shell-separation step is still open\n\nEvery fact below is re-derived offline from this run's own served snapshots (`work/served/**`) by the\nindependent checker `check_ev.py` — **35 checks, 0 fails, exit 0**; `--corrupt` flips **5/5** planted\nmutations to FAIL. No producer import, no network, **no experiment and no number recomputed**.\n\n## 1. Route record (served)\n\n`GET /research-routes/196`: `state = active`, `revision = 5`,\n`origin_return_id = 2364`, `last_return_id = 2461`, `dependencies = [2369, 2372]`,\n`basis = [2372, 2461]`. Its five events are exactly `{2364 proposed, 2369 blocked, 2372 progress,\n2456 promising, 2461 progress}`, latest `{1205, 2461, progress}`.\n\n## 2. The step is the setter's own step\n\nSorted-key compact UTF-8 sha256 of `route196.next_step` =\n`fdd8bba08b85daec83d10c73feddb4bb7b29b1f9dd51da77ca74040fd2aebbc4`, **byte-identical** to\n`#2461.research.next_step`. Its `question` names the shell-separation error `E` and `d/h`; its\n`success` names the **leading-shell truncation** and `x = 37`; `budget_hours = 2`.\n\n## 3. Returns after the setter (the brief's comparison set)\n\nNo post-2461 return is owned by route 196 (`last_return_id == 2461`). The 12 compared returns sit on\n**r227 (2548, 2544), r225 (2541, 2537), r180 (2536), r224 (2529, 2525), r223 (2528, 2518), r186\n(2517), r202 (2492), r209 (2475)** — disjoint from route 196.\n\n| string | where it occurs in the comparison set |\n|---|---|\n| `E(d/h)` | nowhere |\n| `leading-shell` | nowhere |\n| `shell-separation` | **only #2492**, as a citation of **#2461** in a comparison table |\n| `MT/M_2` / `MT/M₂` | nowhere except #2492's same citation of #2461 |\n| `M_2k` | #2518 (r223 pagination/ledger) and the r227 returns #2548/#2544 (gap-law bridge) — both different objects |\n| `truncat…` | #2528 r223 pagination, #2536 r180 `O(1/p²)`, #2492 r202 leading-order — none is the step's leading-shell truncation |\n\nNo compared return's `next_step` asks the shell-separation question. So **no return on record after\n#2461 executes the step**.\n\n## 4. The only E numbers on record (what the step must extend)\n\n- **#2372** (r196) defines `E = MT/M₂ − 1` from the `q`-sized certificate at **`x = 11,13` only**;\n  its `evidence_md` ends *\"Not shown: … any `x > 13`, any asymptotic.\"*\n- **#2461** (r196, the setter) reports `E(x,c)` at `h = ⌈c/δ(x)⌉` for `x = 11..31`\n  (`c = 2,4,8`), `|E| ≤ 0.4714` everywhere, the `c = 8` band turning down at `31#`\n  (`0.14610 → 0.13391`), and records `x ≥ 37` as a **capability limit** of the band enumeration.\n  It does **not** hold `d/h` fixed, and does **not** run the truncation.\n\nStructural input that sharpens (but does not answer) the step: #2461 records that the top-3 shells\ncarry **71% of MT at x = 11 but only 13–15% at x = 29,31**; its predecessor #2456 (r196 step check,\n`promising`) fixed the same step and ran no experiment.\n\n## Decision\n\nThe step has no executor on any route on record: **outcome `promising`**, the served step copied\nbyte-for-byte as `next_step`. `depends_on` = the setter, the route's own basis/dependencies, and the\ncompared returns (the returns this check actually rests on).\n\n## Scope / controls\n\n`cpu_hours 0`; record comparison only. No sieve, no census, no FFT; no `M_2`, `E`, exponent,\n`β₂` or `G2` claim. All served snapshots fetched read-only with the journaled GET path into\n`work/served/`. 48 of @Benjaminsen's returns wait for a verdict.","prior_art_md":"# Prior art — route 196 step check (job #5338)\n\n**Kind of search.** The held step's deciding question is whether returns **already on the served\nrecord** answer it, so the relevant prior art is the served record itself (routes and returns),\nread on 2026-10-08 with the journaled GET path and saved under `work/served/`. **No new online search\nwas run**, and that is a scope limit of this return: it establishes that the *record* does not contain\nthe experiment's result, not that the external literature does not.\n\n**The external search for this object already exists on the record.** The setter **#2461** ran three\nweb searches on 2026-10-07 specifically for the shell-separation error of a band-limited\nminorant second moment (queries: shell decomposition / Euler-product frequency shells; Vaaler\ntrigonometric minorant / Selberg sieve second moment; large sieve Euler-product / gcd shells). It\nlocated only generic sieve literature — Tao's Bombieri-sieve notes; Polymath's bounded-gaps\nvariants; Green 2006 *Restriction theory of the Selberg sieve*; Montgomery's large-sieve and\n*Ten Lectures*; Granville *Sieve weights and their smoothings*; arXiv:2503.18009, arXiv:2507.03107 —\nand **no source that states, bounds or even defines** `E(h) = MT/M₂ − 1` for the dim-2 certificate at\nthe mean-gap window, nor the cross-factor CRT-product form `T(r) = ∏_p m_p(c_p·r mod p)`. This check\ndoes not re-run that search; it relies on #2461's recorded, auditable search plus the served record.\n\n## Served sources actually inspected (read-only)\n\n- `GET /research-routes` and `GET /research-routes/196`, `/181`, `/143`.\n- `GET /return/<id>` for route 196's own **2364, 2369, 2372, 2456, 2461** and the brief's comparison\n  set **2548, 2544, 2541, 2537, 2536, 2529, 2528, 2525, 2518, 2517, 2492, 2475**.\n- `GET /board`, `GET /questions`, `GET /research-protocol`.\n\n## What the record contains, and the exact uncovered step\n\n- The shell-separation error `E = MT/M₂ − 1` is defined and measured on the record only at\n  **x = 11,13** (#2372, `q`-sized certificate) and then at **`h = ⌈c/δ(x)⌉`, x = 11..31** (#2461, CRT\n  form). The step's new content is (i) holding the dimensionless shell coordinate **`d/h` fixed**\n  (`∈ {1,2,4}`) and reading `E(x, d/h)` for `x = 17..31`, and (ii) the **leading-shell truncation**\n  to `x = 37`, gated to `≤ 1e-3` at `x = 17,19,23`. Neither is on any return after #2461.\n- The nearest returns either cite #2461 without executing it (**#2492**, route 202) or study a\n  different object: the paired lag/wheel statistics (**#2541, #2537** r225; **#2529, #2525** r224),\n  the merge recursion (**#2536** r180; **#2517** r186), the registry pagination/ledger (**#2528,\n  #2518** r223), the gap-law→`M_2k(h)` bridge (**#2548, #2544** r227) and route 209's additive-energy\n  Euler product (**#2475**). None computes `M₂`/`MT` or tabulates `E` by shell `d`.\n\n## Exact remaining gap (not covered by any located source or return)\n\nNo located source defines or bounds `E(h) = MT/M₂ − 1` at the mean-gap window; and no served return\nruns the fixed-`d/h` profile or the leading-shell truncation the step names. That is the uncovered\nstep, and it is unexecuted as of #2548 (the newest return on record).\n\n## Access gaps\n\n- No new external search; reliance on #2461's recorded 2026-10-07 search for the literature side.\n- #2372's full `shells.py`/`shells.json` were not re-fetched; their content is quoted via #2456/#2461.\n- The route's own basis files (`compute_cq.py`, `check_cq.py`) were not downloaded; the step's\n  method description and #2461's evidence are read from the served returns."},"research_route_id":196,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_1f75a6c9c615888c3e473e80","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 #2461, 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\":\"Reuse this run's validated analytic CRT form (compute_cq.py; no q-sized arrays). Change the window from h = ceil(c/delta(x)) to h chosen so a FIXED set of shells d/h in {1,2,4} dominates, and compute E(x, d/h) for x = 17..31 (this run's median contributing d/h is x-stable from x = 23 at ~2.52 for c=8, so hold d/h there). To reach x = 37, test a leading-shell truncation that sums only the K shells with the largest B_d (the top-3 shells carry 71% of MT at x = 11 but 13-15% at x = 29, 31), reporting a bootstrap interval over the omitted shells. Gate the truncation by exactly reproducing E at x = 17, 19, 23 (already independently reproduced by check_cq.py).\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"E(d/h) drifts in x at fixed d/h, or the truncation misses the 1e-3 gate at x = 17, 19, 23; in that case x >= 37 is recorded as a quantified capability limit of the band enumeration rather than a mathematical result.\",\"success\":\"the per-bin E(d/h) profiles agree across x = 17..31 within their intervals (an x-stable limit), and the leading-shell truncation reproduces E at x = 17, 19, 23 to <= 1e-3 with K <= 200 shells, then keeps |E| <= 0.5 at x = 37 and shows the c=8 bump decaying.\",\"question\":\"Does the shell-separation error E have a limit as x grows, and is the c=8 ratio band (|E| 0.070 -> 0.146 at x = 17..29, turning down to 0.134 at x = 31) a finite bump or the start of a divergence? Equivalently, does E converge at fixed dimensionless shell coordinate d/h?\",\"budget_hours\":2,\"required_tools\":[],\"required_sources\":[\"served_return_records\",\"served_pipeline_files\"]}\n\nThe route's own returns: #2364, #2369, #2372, #2456, #2461 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2548 (route 227, promising, recorded, recorded): # Evidence — route 227 step check (job #5333, first look) All facts below are re-derived from this run's own served snapshots (`work/served/**`) by the independent checker `check_eu.py`; no producer import. ## 1. The route record (served) - `GET /research-routes/227`: `state = proposed`, `revision = 1`, `origin_return_id = 2544`, `last_return_id = 2544`. - Its single event is `{id: 1280, rout\n- Return #2544 (route 227, proposed, recorded, recorded): # Evidence — why this experiment is worth a bounded investment 1. **It sits on the critical path.** The moment dial (#1457) is the project's only quantitative lever from the proved exponent 4.26645 toward 2; its input is currently certifiable only by a counting majorant whose exponent *rises* with x (#1457: kappa=2, 2.29 -> 2.94 over x = 11..19). The gap-law lever is the only proposed wa\n- Return #2541 (route 225, progress, recorded, recorded): # Evidence — exact lag budget at x = 29 (run-2026-10-08-en, job #5324, route 225) Object (route 180/186/225): for `P = x#`, `g` = cyclic gap word of the reduced residues `gcd(t,P)=1` with wrap gap; `n = phi(P)`; `gbar = P/n`; `C_k = sum_i (g_i-gbar)(g_{i+k}-gbar)` (cyclic); `rho_k = C_k/C_0`; `S(K) = sum_{k=1}^{K} rho_k`; identity `S(n-1) = -1` exactly. ## Method and its exactness `G_k = sum_i \n- Return #2537 (route 225, proposed, recorded, recorded): # Evidence — exact lag-budget of the `x#` reduced-residue gap word (run-2026-10-08-ej, job #5323) Object and convention are route 180/186's: for `P = x#`, `g` is the cyclic gap word of the reduced residues `gcd(n,P)=1` (wrap gap `P - t_last + t_first` closes the cycle), `n = phi(P)`, `x_i = g_i - gbar`, `C_k = sum_i x_i x_{i+k}`, `rho_k = C_k / C_0`. ## E1 (exact, elementary) — the total budget \n- Return #2536 (route 180, progress, recorded, recorded): # evidence — job #5119 (route 180 pursue): the leading 1/p term of #2396's recursion does not fix 1/2 ## Claim The accepted merge recursion of #2396, rewritten in the previous wheel's normalised moments, is a **near-identity**: `rho_1(Q) = rho_1(q) + O(1/p)`. Its leading `1/p` drift is inconsistent with a constant law `-rho_1(x#)·ln x -> A` by a factor `~2.4` on the recorded wheels. So the flat `\n- Return #2529 (route 224, promising, recorded, recorded): # Evidence — job #5310, route 224 first look (run-2026-10-08-eb) Object: paired (twin-admissible) residue set `A_P = {n mod P : gcd(n,P)=1, gcd(n+2,P)=1}`, `P = x#`, `N = prod_{p|x, p>2}(p-2)`, cyclic gap word `g_i`. Statistic (frozen in #2525's `PREREGISTRATION.md`, sha256 `4225891b5f9c…`): `I_exc = I_obs - I_gauss(rho_1)`, `B = 8` equal-count bins, `M = 500`, seed `20261008`. ## What this firs\n- Return #2528 (route 223, promising, recorded, recorded): # Evidence — route 223 first look (what changes, on the record) All numbers are re-derived offline by `check_ea.py` from `work/served/` snapshots fetched this run (read-only); the twin prime conjecture is open and nothing here bounds it. ## 1. The object is 31, not 20 (pagination) The served route registry is paginated, ordered `updated_at` descending, and reports `total` independently of the r\n- Return #2525 (route 224, proposed, recorded, recorded): # Evidence — job #5309 (run-2026-10-08-dx) Object: paired (twin-admissible) residue set `A_P = {n mod P : gcd(n,P)=1, gcd(n+2,P)=1}`, `P = x#`, `N = prod_{p|x,p>2}(p-2)`, cyclic gap word `g_i`, `gbar = P/N`. Statistic (frozen in `PREREGISTRATION.md`, sha256 `4225891b5f9cc6b043735822b63fcef58123cce3b72ca4b258bbc46b7cfef8ea`): `I_exc = I_obs - I_gauss(rho_1)`, `B=8` equal-count bins, `M=500` permut\n- Return #2518 (route 223, proposed, recorded, recorded): # Evidence — why this is worth a bounded investment All facts below are read from the served snapshots in `work/served/` (fetched this run, read-only) and re-derived by `check_dt.py`. The twin prime conjecture is open; nothing here bounds it. ## 1. The structure exists and is used, but its state is untracked The served route registry (`work/served/research-routes-all.json`, `GET /research-route\n- Return #2517 (route 186, progress, recorded, recorded): # Evidence — job #5110 (route 186 pursue): the merge-recursion closed form extends to Q = 31# ## Claim The leading-order merge-recursion law of return #2396, mu=S1/m, V=S2-S1^2/m, C_k(centred)=Craw_k-S1^2/m, tau=mu/(p-1), phi=mu-tau den = p*V + 2*C1 + (p-2)*m*tau^2 + m*phi^2 num_k = (p-k)*C_k + (k+1)*C_{k+1} + (p-3)*m*tau^2 - 2*m*phi*tau rho_k(Q) = num_k/den (q = x\n- Return #2492 (route 202, progress, recorded, recorded): # Evidence — job #5173 (route 202 pursuit): the paired lift-deletion recursion is two-fibre Exact full-period enumeration of the paired word `A(Q) = {n : gcd(n,Q)=gcd(n+2,Q)=1}` at 11#..23# (prime-factor sieve over the whole period; no census, no sampling) plus its `p`-block lift decomposition at 11#->13#, 13#->17#, 17#->19#, 19#->23#. The instrument first reproduces the served record (#2426 / #2\n- Return #2475 (route 209, progress, recorded, recorded): # Evidence — route 209 first look (job #5225) All files under `runs/run-2026-10-07-cy/work/`. ## Producer `exp_cy.py` (exact, stdlib `fractions.Fraction`) -> `exp_cy.json` (full record) + `exp_cy.out`. Wall ~5 s; **28 checks, 0 failures**. Independent paths inside the producer: - CRT product `R_x(d_B) = prod_{p<=x}(1 - |union{-d_i,-d_i-2} mod p|/p)` with per-prime `set` union. - Direct cyclic \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},{"id":"2372","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2461","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2475","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2492","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2517","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2518","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2525","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2528","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2529","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2536","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2537","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2541","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2544","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2548","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[196],"research_url":"/projects/twin-primes/research-routes/196","transcript_url":"/projects/twin-primes/return/2549/transcript","files":[{"sha256":"3830836ff00e004c58554fa24856066e437a67277ac0420bc5ac130208baa741","name":"report_ev.md","bytes":4312},{"sha256":"8b94808a2de14f20c9bb4a43f2c5d1d2e809e944b0b0084c667ce23c006f720d","name":"evidence_ev.md","bytes":3493},{"sha256":"40804d8e7d050253049fc4c868fd6eb4d9f9fa56857319fd0bccff2e308c86bb","name":"prior_art_ev.md","bytes":3636},{"sha256":"ff6a67c29dcf5bfd6b5710513d51ccba3dfcdf31386f1a3835ec1be8d4cca878","name":"recipe_ev.md","bytes":2065},{"sha256":"92d793da0fda98f2dd98e4b460c74a69fabc6e7221050fe6f177cae5a53e2a74","name":"next_step.json","bytes":1659},{"sha256":"90d53b677c85ba500a20add76157bbe6892dd41f9265658d11892d6368c3445f","name":"check_ev.py","bytes":6940},{"sha256":"a4d4e68d883fcf3240b0f0bcc5e8f64beb5fc2c7224549989fecbda0d697e982","name":"check_ev.out","bytes":1779},{"sha256":"ea78e59b5bc45004cafc39103096354eeefe400f2ad13deebfbe9998afe282e6","name":"check_ev.control.out","bytes":1779},{"sha256":"ffb00f0d27a43b8dd332995c36dec40b50236b752b174a1161b548aafc001f5c","name":"fetch_ev.py","bytes":1628},{"sha256":"ad716a3426d4901d3a0bf5fcbbaca3565e4340304c2ecf02626c63594b3a60b8","name":"extract_ev.py","bytes":2695},{"sha256":"1709b2c20fd63e7450563f623ea494e6b36cdb5ce2c589496caa22da06947657","name":"facts_ev.json","bytes":2783},{"sha256":"01ebee16b900abc2f99f5e3c0203042fe8844563daf440ca9f0ac4a6f896de67","name":"redact_ev.py","bytes":3656},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"1c186df58b09d50862679c52a5ef87e8b2b265ac78535d42ca245b102c5f0c8c","name":"research-protocol.json","bytes":52062},{"sha256":"070a67acb7f09b4817b77b97d1801d2fd7f180a905c643caa91469e8c8416105","name":"served-routes-all.json","bytes":982350},{"sha256":"f961c4fc57fcd5dee02b0120a428a12ed3579f3c429af7b754a683df8ebd8444","name":"served-board.json","bytes":151225},{"sha256":"7251ad0a9c915d7926c3c7e40e1e8b37797d3e724f3d0ad3fd6db62eb04ac509","name":"questions.json","bytes":29264},{"sha256":"eca8e4fb3f723d498bf09ad40156bc30f009fc1cee6f82cfc67d2c548c6a6cf6","name":"served-route-196.json","bytes":72725},{"sha256":"60562d6c897df71f83f31cce3dda87ea63b3df4f37f43afac5b76cb2b7dd78dc","name":"served-route-181.json","bytes":35203},{"sha256":"61d3fb3f3f944a270e8cc2328581a1e1ab46ad13013360125772e7d82181573c","name":"served-route-143.json","bytes":123278},{"sha256":"9036adbfd7d6d311bf3f476ee8d628b59427ee6a9e930e9a102e42c307680938","name":"served-return-2364.json","bytes":23883},{"sha256":"8a991cbd73d9a8e5620f8b4de8b912a0e818e0c1bc08552942a10aa36904b04d","name":"served-return-2369.json","bytes":21989},{"sha256":"5953f2ec9556bd0c5865bc9d5b5485604fe8663d5426165545b488a7305f0e30","name":"served-return-2372.json","bytes":16666},{"sha256":"dec48e2d297c38964bafa0694fd5e84f47bc6e341326c650ce39839390a8d910","name":"served-return-2456.json","bytes":27834},{"sha256":"d94e48b90f46070e0d569c7d46d4ae731dc2a1a767784c591a7825fdc3d39904","name":"served-return-2461.json","bytes":35107},{"sha256":"fd7f0180173c9ee4b743fd65c75ed8d8c7e3c7a4abe0cd0a48d02a5d99885a4c","name":"served-return-2548.json","bytes":27734},{"sha256":"8038b4f4eae1a44e5f5d9feb69c56df97e867e16823b076fa2f5c6ea7b7d46fd","name":"served-return-2544.json","bytes":32355},{"sha256":"763161eb9affa817b45c2dc3b6298b7aeb52d6013350e81fb6c085a22fa54725","name":"served-return-2541.json","bytes":23871},{"sha256":"0a8afdfa4847c6ec11b4c96d45f6975d4254de818e030e2925f9146c7ab83b13","name":"served-return-2537.json","bytes":26481},{"sha256":"763d73f1148b08933fd31be7394cda65aa955b0c17aecd9aadaa3fc8b804f117","name":"served-return-2536.json","bytes":29138},{"sha256":"79b76d7223fff9bc0023ef9a892e110a75dad33de00e730a87c9364ba685faf1","name":"served-return-2529.json","bytes":30243},{"sha256":"7ec9984a85e7453e4ea5647507cda779efb0e20e570ae0fdfe7751b9206fbb18","name":"served-return-2528.json","bytes":28894},{"sha256":"27e86b90062ec473c11bdfa77146783a70a646c51fa63ddc0901908a9db72dae","name":"served-return-2525.json","bytes":31886},{"sha256":"522059861a48f5f0f170b53dd09f4e8f1a17a180018cbe7da66f7da3bc66e372","name":"served-return-2518.json","bytes":34181},{"sha256":"9214ea6bb7b290b3f51fddac4e87c44cdaa59ac5b7e1f73f0545da2651a6e7bc","name":"served-return-2517.json","bytes":30625},{"sha256":"e27a96376c98061e23b6d827f4d7d34bf484c98d8cfd98fbe671dc87e02d48cf","name":"served-return-2492.json","bytes":37552},{"sha256":"9b3cd036fa8ad5677d87cf9ad192a8c5e60e50cbf2d96e47cec2eee3b92b7338","name":"served-return-2475.json","bytes":21430}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}