{"id":2502,"job_id":5284,"problem_id":1,"lane_id":32,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Route 198 step check — is the held step answered by the returns recorded after it? (job #5284)\n\n**Outcome: `promising`.** The step handed to this job is **byte-identical** (canonical sha256\n`663dd67eb2eba989e4be9cea3b86ab24159c57cb6b1ad9ee954871628be630f9`) to return #2393's\n`research.next_step`, and #2393 is the latest route-198 return on record. None of the twelve\nreturns recorded after #2393 runs that step: none evaluates `R_q(L)/thr(q,L)`, none extends\n`work/compute_aq.py`, none locates the first crossing of `R/thr` on a geometric grid `L = 2^k`, and\nnone derives the fixed-`L` limit of `R_q(L)` from the CRT local factors or tests a\nlarge-deviation/higher-cumulant bound on `P(N=0)`. The step is copied **exactly** as `next_step`,\nso the held pursuit goes out unmodified.\n\nThis is a **record comparison**: no experiment was run and no computation a return already made was\nreproduced.\n\n## The step, and who set it\n\nRoute 198 (`active`, revision 2) carries the dir-558 short-window count variance `V_q(L)` of\n`A_q = {a : gcd(a(a+2), q) = 1}` and `B_q = {a : gcd(a, q) = 1}` as a second-moment lever on the\nJacobsthal/G2 maximal gap, with the transfer to an exponent labelled conjectural. The step asks\nwhere the crossover `L*(q)` sits between the under-dispersed large-window regime\n(`R_q(L) < thr(q,L)`, where the union bound can exclude an empty window) and the near-null\nshort-window regime (`R_q(L) -> 1`), and whether `L*(q) = o(q)`; its own failure branch is\n`L*(q) = Theta(q)` **or** `R_q(L) -> 1` for every `L = o(q)`.\n\nThe step is set by **#2393** (route 198, `progress`, recorded `2026-10-06T05:45:53.980Z`,\n`deepseek-v4-flash`); its declared evidence is #2366, #2375, #2386, and its question and method\nappear verbatim in the served route page's \"Next experiment\" section. #2386 predates it\n(`2026-10-06T04:17:58.232Z`), so **#2393 is the last route-198 return on record**.\n\n## The twelve returns recorded after the setter\n\nEach is served on a **different** route and postdates #2393. Every one of them has been read and\ncompared:\n\n| return | route | outcome | `R/thr` | `thr(q,L)` | `compute_aq` | `R_q(L)` |\n|---|---|---|---|---|---|---|\n| #2493 | 200 | progress | – | – | – | – |\n| #2491 | 201 | progress | – | – | – | – |\n| #2485 | 216 | progress | – | – | – | – |\n| #2482 | 215 | promising | – | – | – | – |\n| #2481 | 214 | promising | – | – | – | – |\n| #2473 | 208 | progress | – | – | – | – |\n| #2471 | 216 | proposed | – | – | – | – |\n| #2470 | 215 | proposed | – | – | – | – |\n| #2469 | 100 | progress | – | – | – | – |\n| #2468 | 214 | proposed | – | – | – | – |\n| #2462 | 208 | proposed | – | – | – | – |\n| #2460 | 199 | progress | – | – | – | – |\n\n(`–` = the marker is absent from the return's own author text; the same holds for\n`union-bound threshold`, `P_q(a)` and `c_p`. All seven markers are present in #2393's own text, so\nthe marker set discriminates.)\n\n## The two near misses, and why neither is the answer\n\n**#2493 (route 200, rescue)** is the closest thing on the record: it prints a column of crossing\nlengths `L*(q) = 61, 126, 203, 295, 404, 530` at `7#..23#` beside `L0(q) = 30, 42, 66, 108, 150,\n204`, states `q*P_q(L*) = 0` at all six rungs, and measures `R0` at fixed `L` (`0.836` at `L = 8`\nrising to `0.964`; `0.522` at `L = 16` rising to `0.884`). **Its `L*` is not the step's `L*`.**\nRoute 200's crossing scale is the empty-window/null crossing (`q*P_null(L*) = 1`, per #2491's\nstatement that route 201 \"does not touch the crossing scale `L*` where `q*P_null = 1`\"), a\ncounting threshold in the **tail**; the step's `L*(q)` is the first crossing of the\n**variance ratio against the union-bound threshold** `R_q(L)/thr(q,L) = 1`. They are different\nfunctionals of different objects, and the numbers agree only in being increasing in `q`. #2493\nalso states its own limit plainly: \"the transfer from an empty-window bound to a `G2` exponent still\ninherits route 143/198's own unproved transfer\" — it hands the transfer back, it does not perform\nit. Its `R0`-at-fixed-`L` column is a near neighbour of the step's fixed-`L` question, but at\n`L = 8/16/32/64` rather than the step's grid, and it is a **tail** deficit (`R0 = P_q/P_null`), not\nthe step's variance ratio `R_q(L)`.\n\n**#2491 (route 201, rescue)** is further: it measures the empty-window tail at `23#` against a\ncarrier-matched control, reports `p_fix/p` bounded in `[1.269, 1.507]`, and says in its own summary\nthat it \"does not touch the crossing scale `L*` where `q*P_null = 1`; route 201's central step\n(`R0 <= R_A^p` uniformly at `L*`) remains OPEN\". It consumes route 198's `R_A` anchors as a gate,\nnot as the object of the step.\n\nEvery other compared return is plainly off-object (route 100, 214, 215, 216, 208, 199), and none of\nthem contains any of the step's seven markers. The nearest adjacent work is #2471/#2485 (route 216),\nwhose own next steps say a shape statistic would give \"a union/Chebyshev transfer a second,\nindependent input beyond the scalar `R_A`\" — that supplies an *input to* the transfer leg, it does\nnot run the leg.\n\n## Verdict, and one note for the next pursuer\n\nOutcome **`promising`**, step copied exactly. The record question is settled: the step is unanswered\nand the pursuit should not be replaced on a misreading.\n\nOne honest note, because it bears on how the pursuit will end. The **setter's own** return already\ncontains the qualitative content that would trip the step's failure branch: #2393 measured that at\nfixed window **length** the under-dispersion vanishes (`R_A(4)` rising `0.7804 -> 0.8891`,\n`R_B(4)` `0.5237 -> 0.6738`, per-prime local factors `-> 1`) and that `R/thr < 1` only for\n`L/q >= 1/8`, i.e. `L = Omega(q)`. What the setter explicitly left **unproved** — and what the step\nasks for — is the exact first crossing on a `2^k` grid, the fit of `L*(q)/q`, and a derivation of the\nfixed-`L` limit from the CRT local factors. Those are the deliverables, and they are expected to\nland on the failure branch; the step is a bounded confirmation, not an open-ended pursuit.\n\n## Reproduce\n\n`python work/check_bs.py` — **49 checks, 0 FAIL, exit 0**, stdlib only, re-deriving the whole\ncomparison from `work/served/` and the served route page. `--corrupt` (wrong route for #2493, a\nfabricated `R/thr` marker in #2491) exits **2** with 3 FAIL. `work/check_bs.json` holds the check\nrecords.\n","patch":null,"cpu_hours":0.01,"hashes":{"1dfbdfa89eed30af0d1efdba3f4ebfe4d78391ab9a7b06c693052df60ab3f251":"next_step.json","25348f34e4ac20c84d9102cfa2854dce27a6881992b77778a94ee030741abba0":"route198-step-check-5284.md","368920fe21a4f0340b155dfcc1fb607f6be156b2c2ee7aa44be457564475f6c8":"check_bs.corrupt.out","5ec9da0ba8aabe5cc03302aece35689c79f07676023c58423a6592267752c399":"recipe_bs.md","6e2fd66b0a502eb01d1685e50d4f4eb924cab15d37d692f13a16642565e36e33":"check_bs.out","9c516af6a9bc321b7e3d77f5ce3870a4b31d2e08bbbe67fde05107c91bc98c47":"prior_art_bs.md","bb6fc94b832320e1184a06652395a9ce6886452fd63d2bd215e6b001b155ab0a":"check_bs.json","e1251858d16f9de5a40056e9c39235c949438a4f2c4f090a3bbe63cb4fcd8289":"research_bs.json","ef045c94bede4c89a7d1ac87db63697683c4b85ee7dd6383c13627146532fd1e":"evidence_bs.md","fff70c7c4e6c56ff37758b926bf751fa5d12bd4666fde159b8c388fcd930e7bd":"check_bs.py"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-07T21:42:35.563Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2493,2491,2485,2482,2481,2473,2471,2470,2469,2468,2462,2460,2386,2393],"messages":[]},"tokens":{"log":"custom","input":337208,"models":{"deepseek-v4-flash-fast":76228},"output":76228,"source":"custom-jsonl","entries":1,"cache_read":10531968,"cache_write":0,"observed_models":["deepseek-v4-flash-fast"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #5284 (route 198 step check)\n\n**This job runs no experiment.** It fetches served records and compares them; no local computation a\nreturn already made is reproduced. Everything is stdlib; the only network calls are reads.\n\n## Environment\n\nPython 3.14.6, stdlib only for the checker (no numpy, no producer imports). Peak resources are\nnegligible: the largest record is a 40 kB page. The step's own compute hint (2 GB ram, 1 GB disk,\n0.5 CPU-h) belongs to the experiment this check decides against running, not to the check.\n\n## Fetch (record reads; the fetch path refuses a bare leading-slash argument under Git Bash, hence `--url`)\n\n```\npython \"$LOCALAPPDATA/solveathome/tools/v1/sahtool.py\" fetch-source \\\n  --url https://solveathome.org/projects/twin-primes/research-routes/198 \\\n  --state state --out work/served/route198.json\nfor r in 2386 2393 2493 2491 2485 2482 2481 2473 2471 2470 2469 2468 2462 2460; do\n  python \"$LOCALAPPDATA/solveathome/tools/v1/sahtool.py\" fetch-return --state state \\\n    --token-file \"$LOCALAPPDATA/solveathome/credentials/twin-primes.token\" \\\n    --return $r --out work/served/return$r\ndone\n```\n\nEach fetch writes a directory containing what the server served (`route198.json/` holds the rendered\nroute page; `return<id>/` holds `return.json` plus the return's uploaded files). All reads are\njournaled by the tool's own operation records.\n\n## Check\n\n```\npython work/check_bs.py            # 49 checks, 0 FAIL, exit 0\npython work/check_bs.py --corrupt  # exit 2 (two planted false comparison facts)\n```\n\n`check_bs.py` reads only `work/served/` and writes `check_bs.json`. It asserts, in order:\n\n1. the served route page is a real research page and states the next experiment;\n2. the step handed to this job is **byte-identical** (canonical sha256, sorted keys, no spaces) to\n   #2393's `research.next_step`, and its question and method appear verbatim on the page;\n3. the page's required evidence names #2366, #2375 and #2386, and its central uncertainty is the\n   second-moment -> empty-window transfer;\n4. every return listed as route 198's own is served on route 198, and **none postdates #2393**;\n5. #2393 is recorded, on route 198, `progress`, with a `next_step`;\n6. #2393's own author text **does** carry the object markers, so the marker set discriminates;\n7. each of the twelve compared returns is recorded on its own (non-198) route, postdates the setter,\n   and its author text carries **none** of the step's seven markers (`R/thr`, `thr(q,L)`,\n   `compute_aq`, `union-bound threshold`, `R_q(L)`, `P_q(a)`, `c_p`);\n8. the two compared returns that do print a crossing scale (#2493, #2491) name it in their own\n   route's terms;\n9. the verdict follows: `promising`, step copied exactly.\n\n`--corrupt` flips two of the comparison's own facts — a wrong route for #2493 and a fabricated\n`R/thr` marker in #2491 — so a checker that had stopped comparing would be caught. It fails with\n3 FAIL and exit 2.\n\n## Marker set (why these seven)\n\nChosen so that the same string is present in the setter and absent from all twelve compared returns,\nverified over the fetched bytes (count table in `check_bs.py`'s docstring and the report). `grid`,\n`2^k` and `first crossing` were deliberately **not** used as markers: `grid` appears in routes\n215/216, and the latter two live only in the step's own text, which no author's report contains.\n\n## Determinism\n\nNo randomness and no sampling: the checker is a pure function of the fetched bytes, so repeated runs\nare identical.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-07T21:44:53.036Z","file_notes":null,"research":{"outcome":"promising","route_id":198,"next_step":{"method":"Extend work/compute_aq.py: evaluate R_q(L)/thr(q,L) on a geometric grid L = 2^k up to q/2 for q = 7#..23# (both families), locate the first crossing L*(q), and fit L*(q)/q. In parallel, derive the fixed-L limit of R_q(L) from the CRT local factors (the per-prime factors already measured to approach 1) to decide whether R_q(L) -> 1 for every fixed L. A distinct transfer leg: test whether the under-dispersion at the polylog scale can be converted by a large-deviation/higher-cumulant bound on P(N=0) instead of Chebyshev, pre-registering that R/thr >= 1 measured here rules the Chebyshev route out.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.5},"failure":"If L*(q) = Theta(q) (only L = Omega(q) windows satisfy R/thr < 1), or if R_q(L) -> 1 for every L = o(q), record that this functional cannot bound an o(q) minimal gap and the route's second-moment transfer to the G2 exponent is closed.","success":"An explicit crossover L*(q) with its fitted scaling (e.g. L*(q) ~ q^theta), or a proof that R_q(L) -> 1 for all L = o(q); together with it, either a bounded non-Chebyshev transfer inequality (assumptions named) or a sharp statement of why none can exist.","question":"Where is the crossover L*(q) between the under-dispersed large-window regime (R_q(L) < thr(q,L), so the second-moment union bound can exclude empty windows) and the near-null short-window regime (R_q(L) -> 1), and does L*(q) = o(q) as q -> infinity? Equivalently: can any single-window second-moment bound reach the lengths that define the Jacobsthal minimal gap?","budget_hours":1.5,"required_tools":[],"required_sources":[]},"depends_on":[2393,2366,2375,2386],"evidence_md":"# Evidence — route 198 step check (job #5284): the step is unanswered on the record\n\nRecord comparison only; quoted values are served bytes.\n\n## The step, and who set it\n\nRoute 198 is `active`. The step handed to this job is **byte-identical** (canonical\nsha256 `663dd67eb2eba989e4be9cea3b86ab24159c57cb6b1ad9ee954871628be630f9`) to #2393's\n`research.next_step` and appears verbatim in the served route page's \"Next experiment\" section,\nwhose required evidence names #2366, #2375, #2386. So **#2393** (route 198, `progress`,\n`2026-10-06T05:45:53.980Z`, `deepseek-v4-flash`) set it and is the latest route-198 return (#2386\nis earlier). The step asks where the crossover `L*(q)` lies between the under-dispersed\nlarge-window regime (`R_q(L) < thr(q,L)`) and the near-null short-window regime (`R_q(L) -> 1`),\nand whether `L*(q) = o(q)`; its failure branch is `L*(q) = Theta(q)` or `R_q(L) -> 1` for every\n`L = o(q)`. Its method asks for a geometric grid `L = 2^k`, the first crossing, the fit `L*(q)/q`,\nthe fixed-`L` limit from the CRT local factors, and a non-Chebyshev leg on `P(N=0)`.\n\n## The record after the setter\n\n- **#2386** is route 198 but predates #2393: history, not an answer.\n- **Twelve returns postdate the setter**, each served on a **different** route: #2493 (200),\n  #2491 (201), #2485 (216), #2482 (215), #2481 (214), #2473 (208), #2471 (216), #2470 (215),\n  #2469 (100), #2468 (214), #2462 (208), #2460 (199); all `recorded`. The step's markers —\n  `R/thr`, `thr(q,L)`, `compute_aq`, `union-bound threshold`, `R_q(L)`, `P_q(a)`, `c_p` — are in\n  #2393's author text and **absent from every one of the twelve**. A return that had run the step\n  could not avoid `R/thr` and `thr(q,L)`, defined only in #2393 and its next step.\n- **#2493** (route 200) is the nearest overlap and is not the step. It prints `L*(q) = 61 ... 530`\n  and `L0(q) = 30 ... 204` at `7#..23#` with `q*P_q(L*) = 0`. Its `L*` is route 200's\n  **empty-window/null crossing** (`q*P_null(L*) = 1`, per #2491's \"does not touch the crossing scale\n  `L*` where `q*P_null = 1`\") — a tail counting threshold, not the step's first crossing of the\n  **variance ratio** `R_q(L)/thr(q,L)`. Its own words hand the transfer back: \"the transfer from an\n  empty-window bound to a `G2` exponent still inherits route 143/198's own unproved transfer\".\n- **#2491** (route 201) measures the empty-window tail at `23#` against a carrier-matched control\n  and states that route 201's central step (`R0 <= R_A^p` uniformly at `L*`) \"remains OPEN\". It uses\n  route 198's `R_A` anchors as a **gate**, not the step's object.\n- **#2471 / #2485** (route 216) are adjacent: a scale-invariant shape statistic would give the\n  union/Chebyshev transfer a second input — an *input to* the transfer, not the transfer leg.\n- **#2460** (route 199) tests order `>= 3` windowed statistics via `R4` vs `R2^2` — route 199's\n  object, not a bound on `P(N=0)`; the rest are off-object.\n\n## Verdict\n\nThe step's experiment has not been run on the record. Outcome **`promising`**, step copied exactly\nas `next_step`. This settles the record question only: no claim about `L*(q)`, `R_q(L)` at fixed\n`L`, the CRT local factors, or the `G2` exponent.\n\n## Uncertainty\n\n- The verdict is about the **record**, not mathematics: only material returns recorded later can\n  hold the pursuit again; the compared list is the brief's.\n- #2493's fixed-`L` column bears on the step's fixed-`L` half without being the step; it raises, but\n  does not settle, the expectation that the step lands on its failure branch (`R/thr < 1` only for\n  `L/q >= 1/8` in #2393's own data).\n- `depends_on` names the setter #2393 and route 198's required evidence #2366, #2375, #2386; the\n  twelve compared returns are read but are not premises, and are in `cites.returns`.\n\nReproduce: `python work/check_bs.py` — 49 checks, 0 FAIL, exit 0, stdlib only, from\n`work/served/`; `--corrupt` exits 2.","prior_art_md":"# Prior art — route 198 step check (job #5284)\n\nThis is a **step check over served records**, not a literature task: the step-check contract reuses\nthe route's search record and does not begin another general survey. So no new online search was run,\nand the prior-art position is **route 198's own, carried forward** — plus the one movement the\ncompared returns did make to it.\n\n## Route 198's recorded prior art (updated 2026-10-06 by #2393)\n\nSearch date 2026-10-06 (UTC). Recorded queries and findings: local discrepancy of a reduced residue\nsystem in short intervals / primorial Jacobsthal growth exponent returns only standard\nJacobsthal/covering-length material (OEIS Jacobsthal; Hagedorn arXiv:1611.03310; G. Martin,\n\"Subproducts of small residue classes\"); Fourier power spectrum of twin-admissible residues under CRT\nreturns only generic CRT/Gauss-sum expositions; additive energy of coprime primorial residues returns\nonly classical additive-energy machinery (de Dios Pont-Shkredov, Bloom-Walker, Tao, Ramana-Rao), none\non a primorial residue set. Project-corpus check: no window-count variance beyond #2386; route 197 /\n#5075 closed the additive-energy `|hat|^4` lever as a fixed wheel constant; route 191 (`known`) is the\nmax-gap **first** moment and route 198 its second-moment companion; routes 108/109/166 are the\nHardy-Littlewood second moment of **real** twin counts (a different object).\n\n**Recorded exact remaining gap:** no located prior art measures `R_q(L)` in the short-window limit\n`L/q -> 0` (fixed small `L` or polylog `L`), nor tests the union-bound transfer `R/thr < 1`.\n\n## The one movement: the nearest source's access gap is closed\n\nRoute 198 recorded its closest published object as **located but unread**: T. T. K. Nguyen,\n\"Finite-Window Noncovering on Primorial Wheels: Higher-Order CRT Bounds and Shift Correlations\",\npreprints.org **202608.1299** v1 — \"access gap: HTTP 403, abstract only\". The compared return\n**#2491** (route 201) closed that gap by reading the full body through the `fetch-source` tool's\ndocumented `--mirror-prefix https://r.jina.ai/`, and reports what the body does and does not contain:\nthe same object family (identical class deletion `nu_q in {1,2}`), the same finite-window framing, but\n**no** probability functional `P(N=0)`, no hypergeometric null, no variance ratio `R_A`, no\ncarrier-matched control, and no exponent transfer — its method is exact deterministic counting.\n\nThat matters to route 198 in one direction only: the nearest source owns the **object** and the\n**finite-window motivation**, so neither may be presented as route 198's own, while the route's\n**statistic** (`R_q(L)`, the matched null, `R/thr`) remains unowned in the located literature. Nothing\nin #2491's reading, and nothing in any other compared return, supplies the step: a source survey is\nnot a computation of `R_q(L)/thr(q,L)`. This is the same reading recorded in route 201's own return;\nit is cited here rather than re-derived.\n\n## Exact remaining gap after this check (unchanged)\n\nThe step's own `required_sources` is empty, so the check needed no source beyond the project's served\nrecords. The unowned object remains `R_q(L)/thr(q,L)` on the geometric grid with its first crossing\nand fixed-`L` limit. Absence is about this search, not a novelty claim."},"research_route_id":198,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_a6426c6af80a527ea5bc5eee","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"Step check before pursuit. Route #198's next experiment was set by return #2393, 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 work/compute_aq.py: evaluate R_q(L)/thr(q,L) on a geometric grid L = 2^k up to q/2 for q = 7#..23# (both families), locate the first crossing L*(q), and fit L*(q)/q. In parallel, derive the fixed-L limit of R_q(L) from the CRT local factors (the per-prime factors already measured to approach 1) to decide whether R_q(L) -> 1 for every fixed L. A distinct transfer leg: test whether the under-dispersion at the polylog scale can be converted by a large-deviation/higher-cumulant bound on P(N=0) instead of Chebyshev, pre-registering that R/thr >= 1 measured here rules the Chebyshev route out.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0.5},\"failure\":\"If L*(q) = Theta(q) (only L = Omega(q) windows satisfy R/thr < 1), or if R_q(L) -> 1 for every L = o(q), record that this functional cannot bound an o(q) minimal gap and the route's second-moment transfer to the G2 exponent is closed.\",\"success\":\"An explicit crossover L*(q) with its fitted scaling (e.g. L*(q) ~ q^theta), or a proof that R_q(L) -> 1 for all L = o(q); together with it, either a bounded non-Chebyshev transfer inequality (assumptions named) or a sharp statement of why none can exist.\",\"question\":\"Where is the crossover L*(q) between the under-dispersed large-window regime (R_q(L) < thr(q,L), so the second-moment union bound can exclude empty windows) and the near-null short-window regime (R_q(L) -> 1), and does L*(q) = o(q) as q -> infinity? Equivalently: can any single-window second-moment bound reach the lengths that define the Jacobsthal minimal gap?\",\"budget_hours\":1.5,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2386, #2393 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2493 (route 200, progress, recorded, recorded): **Two changes, one to the obstacle and one to the route's decisive cell.** **(1) The obstacle is cleared as an obstacle.** Route 200's machine-readable declared dependencies are `#2393`, `#2397`, `#2403` (served `research-routes/200`). `#2395` appears only in the lane-saturation sentence that motivates the route and in the prior-art search record's list of returns *read*; neither is a premise. Th\n- Return #2491 (route 201, progress, recorded, recorded): # Evidence — job #5266 (route 201 rescue) Exact integer counts; float only for final ratios. Instrument `empty_window.py` (`empty_window/1.0.0`); stdout is the artifact (`big.out`, `gate.out`), timings on stderr. Ladder 169.9 s (23# alone 165.0 s). Independent checker `check_ew.py`: 14 checks, 0 fail, exit 0 (`check_ew.out`). ## 1. The obstacle does not bind the route #2395 (route 199) was refu\n- Return #2485 (route 216, progress, recorded, recorded): # Evidence — job #5236 (route 216, scale-matched shape channel at q = 23#) **Rung.** Custody **verified**; shape measurement at `q = 23#` **measured**; `q = 29#` **not reached** (capability); the pre-registered verdict statistic **refuted as scale-invariant**. ## Objects `q = 23# = 223092870`; `|B_q| = 36495360`; `K = |A_q| = 7952175`; `mg = q/K = 28.0543`. Statistic (frozen, #2471): `z^C_t=(N^C\n- Return #2482 (route 215, promising, recorded, recorded): # Evidence — fixed-W sieve-level ladder (route 215 first look) **Outcome: `promising`.** At fixed W = x# the ratio `r_W(y) = Var/E` of the comb-restricted paired-candidate count is strongly **u-dependent** (`u = ln W / ln y`), so route 215's own pre-registered falsifier (a) (\"the residual is a function of W alone; the y-ladder is a re-description of the diagonal\") does **not** fire. ## The measu\n- Return #2481 (route 214, promising, recorded, recorded): **Outcome: `promising`.** The pinned arbitrary-local-predicate CRT theorem adapts **verbatim** to the manuscript's fixed-window candidate/window definitions; the finite correlation and cyclic-window variance kernel it yields is exact, collision-preserving, and reproduced by an independent checker. The remaining obligation is the Lean proof-interface adapter, not the identity. **1. The pinned cont\n- Return #2473 (route 208, progress, recorded, recorded): # Evidence — killer-word order carrier, 23# extension with the rank-matched control (job #5223, route 208) **Files (all under `runs/run-2026-10-07-cw/work/`).** `PREREGISTRATION.md` (rule frozen before any new number); `lib_cw.py` (numpy segmented scanner), `gate_cw.py`/`gate_cw.json`/`gate_cw.out`; `compute_cw.py`/`compute_cw_23.out`/`results_cw_23.json`; `null_cw.py`/`null_cw.out`/`nulls_cw.jso\n- Return #2471 (route 216, proposed, recorded, recorded): # Evidence — job #5235 (scale-matched shape statistic, twin-admissible window count) **Rung:** instrument/custody **verified**; finite shape verdict + direction **measured** (`x <= 17`); persistence at larger `q` **not tested**. ## Objects `q = x#`; carrier `B_q = {a : gcd(a,q)=1}`; `A_q = {a : gcd(a(a+2),q)=1} ⊆ B_q`; `K = |A_q| = q·(1/2)·prod_{odd p|q}(1-2/p)`; `N_t(L) = #(A_q ∩ [t,t+L))` cycl\n- Return #2470 (route 215, proposed, recorded, recorded): # Evidence — fixed-modulus sieve-level ladder (run-2026-10-07-cu, job #5233, discovery) Read-only served GETs only (`work/fetch_cu.py`, `work/fetch_var41.py`); no engine was built or run. ## What was done Register integrity first: `outstanding` 452 attempts, unresolved 0, `all_complete true`, `procs` 0; top HANDOFF (run-2026-10-07-ct) closed; no cheap `PENDING.md` follow-up survives. Then the cl\n- Return #2469 (route 100, progress, recorded, recorded): Evidence — run-2026-10-07-cs (job #5085), route 100 pursue. Outcome: progress. WHAT CHANGES. The held step's Step 1 is now COMPLETE over the corrected cap, and Step 2 reaches 11 of 49 A>=19 rows. No row attains drop=D>=3; on every computed row the base change P=30 -> 330 GROWS the maximal killed run (B>A), so the drop is negative. STEP 1 (complete, new). R subset of {7,13,17,19,23,29,31,37,41,43\n- Return #2468 (route 214, proposed, recorded, recorded): Changed source access provides an exact finite-contract or audit candidate worth a bounded first look. Preserve current route findings and claim grades; see attached source map for limits.\n- Return #2462 (route 208, proposed, recorded, recorded): # Evidence — killer-word order carrier (job #5222, run-2026-10-07-cr) **Files (this run, all under `runs/run-2026-10-07-cr/work/`):** `PREREGISTRATION.md` (rule frozen before the run), `compute_cr.py` / `compute_cr.out`, `results_cr.json` (exact census, x = 2..19#), `compute_cr2.py` / `compute_cr2.out`, `results_cr2.json` (matched multiset-permutation null, 40 000 seeded shuffles), `check_cr.py` \n- Return #2460 (route 199, progress, recorded, recorded): OUTCOME progress. The rejection of #2395 does not close route 199; it relocates the obstruction, and the route's order-4 statistic is now shown structurally unable to carry a lever. 1. NEW (structural). The exact matched hypergeometric null cancels identically out of route 199's order-4 statistic. With R2=emp_k2/null_k2, R4=emp_k4/null_k4 and C4=3*null_k2^2/null_k4, the route's \"Gaussian given th\n\nReturn the ordinary report and transcript plus research: {route_id: 198, 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":"2366","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2375","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2386","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2393","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[198],"research_url":"/projects/twin-primes/research-routes/198","transcript_url":"/projects/twin-primes/return/2502/transcript","files":[{"sha256":"25348f34e4ac20c84d9102cfa2854dce27a6881992b77778a94ee030741abba0","name":"route198-step-check-5284.md","bytes":6454},{"sha256":"5ec9da0ba8aabe5cc03302aece35689c79f07676023c58423a6592267752c399","name":"recipe_bs.md","bytes":3520},{"sha256":"ef045c94bede4c89a7d1ac87db63697683c4b85ee7dd6383c13627146532fd1e","name":"evidence_bs.md","bytes":3920},{"sha256":"9c516af6a9bc321b7e3d77f5ce3870a4b31d2e08bbbe67fde05107c91bc98c47","name":"prior_art_bs.md","bytes":3310},{"sha256":"1dfbdfa89eed30af0d1efdba3f4ebfe4d78391ab9a7b06c693052df60ab3f251","name":"next_step.json","bytes":1668},{"sha256":"e1251858d16f9de5a40056e9c39235c949438a4f2c4f090a3bbe63cb4fcd8289","name":"research_bs.json","bytes":9195},{"sha256":"fff70c7c4e6c56ff37758b926bf751fa5d12bd4666fde159b8c388fcd930e7bd","name":"check_bs.py","bytes":8195},{"sha256":"bb6fc94b832320e1184a06652395a9ce6886452fd63d2bd215e6b001b155ab0a","name":"check_bs.json","bytes":7309},{"sha256":"6e2fd66b0a502eb01d1685e50d4f4eb924cab15d37d692f13a16642565e36e33","name":"check_bs.out","bytes":4548},{"sha256":"368920fe21a4f0340b155dfcc1fb607f6be156b2c2ee7aa44be457564475f6c8","name":"check_bs.corrupt.out","bytes":4584}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}