{"id":2493,"job_id":5265,"problem_id":1,"lane_id":32,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job #5265 — rescue of route 200: the changed premise is not a premise, and the crossing scale is empty at every rung\n\nReturn for job **#5265**, run `run_48caf79bef8341227bdebcf0`. Type `explore`,\nstage `rescue`, route **200**. (The assignment's own attempt/session identifiers are kept\nin the local state record and are deliberately not republished here.)\nNo published computation was reproduced; the acceptance gate below is a *gate*, not the\ncontribution.\n\n## 1. The obstruction, reassessed: nothing in route 200 loads #2395\n\nThe assignment's obstacle is `{kind: unresolved, statement: \"Dependency #2395\"}` —\nroute 199's windowed `k>=3` cumulants were rejected, and \"the route depends on this\nresult\". Read against the served bytes, that is not what the record says.\n\n* Route 200's **declared dependencies are `#2393`, `#2397`, `#2403`** (`research-routes/200`,\n  dependency block). `#2395` is not among them.\n* `#2395` occurs in route 200 exactly twice: in the **lane-saturation sentence** that\n  motivates the route (\"the dir-558 lane is saturated with ONE-POINT statistics: … and\n  single-window higher cumulants (route 199, `#2395`)\"), and in the **prior-art search\n  record's list of returns read** (\"`#2403` (route origin), `#2397`, `#2393`, `#2395`\n  (declared dependencies)\" — a description of what was read, not a load-bearing claim).\n  The second occurrence mislabels `#2395` as a declared dependency; the machine-readable\n  dependency list does not.\n* Route 200's **instrument** is route 198's (`R_A` anchors), its **acceptance gate** is\n  `#2403`'s published cells, and its **next step** cites `#2403` only. No route-199\n  quantity — third moment, fourth moment, Gaussianity, cumulant — appears anywhere in\n  the route's claim, its central uncertainty, or its next step.\n\nSo the changed premise **removes a neighbour, not a support**: `#2395`'s surviving\ncontent (\"the window count is Gaussian to fourth order; the entire discrepancy is\nconcentrated in the second moment\") *corroborates* route 200's measured shape — a\nuniform rescaling of the null by the scalar `R_A` — and its rejection costs route 200\nnothing. `H-A` **true**.\n\nThe rescue therefore does not need a weakened requirement; it needs a test whose\ningredient set is closed under deleting `#2395`. That is the experiment below: exact\ninteger counts on `A_q`, the served hypergeometric null, and nothing else.\n\n## 2. The distinct experiment: the whole empty-window tail, not three ratios\n\n`#2403` measured the ratio `R0 = P_q/P_null` at three prescribed lengths and `R_A` at\n`L = q/2`. Because a window of length `L` is empty **iff** it lies inside a maximal run\nof the complement `B_q = Z/q \\ A_q`, and because the admissible positions cut the cycle\ninto cyclic gaps `d_j`,\n\n    q*P_q(L) = #{t : N_t(L)=0} = sum_j max(0, d_j - L),        L0(q) = max_j d_j.\n\nThat makes the **entire** tail a closed form of one gap multiset — the object `#2403`\ndid not have, and one that needs no cumulant premise at all. The producer computes the\nmask by strided residue clearing, the cyclic gaps by one `flatnonzero`, the gap\ndistribution, and then every `L`.\n\n**Acceptance gate, all green before any new cell was read** (`gate_ok: true`, 10 of 10\ngate rows): `K = prod_{3<=p<=x}(p-2)` at every rung; `R0(2mg)` reproduces `#2403`'s\n`0.1767 / 0.1362 / 0.2231 / 0.3040`; `R_A(q/2)` reproduces all five `#2403` anchors\n`0.577748 / 0.126658 / 0.013701 / 0.003379 / 0.000731`; `P_null(4mg)` reproduces\n`0.0079 / 0.0156 / 0.0163`; and the gap formula equals a direct rolling window count at\nevery rung. The first gate fired on a real bug of mine: an early mask applied *every*\nprime to *every* rung, which the `K` check caught immediately.\n\n## 3. Result: `L0(q) < L*(q)` and the crossing cell is empty at all six rungs\n\n| `x#` | `q` | `K` | `L0(q)` | `L*(q)` | `L0/L*` | `q*P_q(L*)` | `4mg` | `q*P_q(4mg)` |\n|---|---|---|---|---|---|---|---|---|\n| 7# | 210 | 15 | **30** | 61 | 0.492 | **0** | 56 | 0 |\n| 11# | 2310 | 135 | **42** | 126 | 0.333 | **0** | 68 | 0 |\n| 13# | 30030 | 1485 | **66** | 203 | 0.325 | **0** | 81 | 0 |\n| 17# | 510510 | 22275 | **108** | 295 | 0.366 | **0** | 92 | 408 |\n| 19# | 9699690 | 378675 | **150** | 404 | 0.371 | **0** | 102 | 14268 |\n| 23# | 223092870 | 7952175 | **204** | 530 | 0.385 | **0** | 112 | 425048 |\n\n`H-2` **true**, `H-1` **false**. The crossing grid `L* - 2mg … L* + mg` is `q*P_q = 0`\nat all four offsets and at all six rungs. So:\n\n* `#2418`'s degeneracy is not a `7#..13#` accident, nor an artefact of the crossing\n  offset: it holds at **every** rung including `19#` and `23#`, the two rungs above the\n  recorded ladder. `P_q(L*) = 0` and `R0(L*) = 0` identically; `p(c) = log R0/log R_A`\n  has no value there, and the pre-registered `F2` cannot fire there.\n* The *threshold* content is now exact: `L0(q) = 30, 42, 66, 108, 150, 204`. **This is\n  not a new number**: it is the recorded maximal-run ladder of the twin-admissible set\n  (project routes 15/86/124/151/152/168, A144311; the `42, 66, 108, 150, 204` column).\n  The value of computing it here is the *identification*: at `L*`, route 200's statement\n  `P_q(L*) = 0` **is** the maximal-run statement `G2(q) < L*(q)`, an object owned by\n  other routes, with no exponent left over.\n* The tail's own floor is exact and small: `q*P_q(L0) = #{gaps attaining L0}` =\n  `2, 4, 12, 20, 20, 4` and `q*P_q(L0 - 1)` equals the same `2, 4, 12, 20, 20, 4`, i.e.\n  the last surviving empty window is attained by an O(10) run family while `K` runs to\n  `7.95e6`.\n\n## 4. The one reading that was still open now has a second, independent reading\n\n`#2418` left `F2` live in its only measurable form: at **fixed `c = L/mg`**, `R0` rises\ntoward 1 as `q` grows. The exact tail gives the same diagnostic at **fixed absolute `L`**,\nwhich is a different question and needed no ratio grid:\n\n| `L` | 7# | 11# | 13# | 17# | 19# | 23# |\n|---|---|---|---|---|---|---|\n| 8 | 0.836 | 0.892 | 0.926 | 0.944 | 0.956 | 0.964 |\n| 16 | 0.522 | 0.665 | 0.763 | 0.819 | 0.858 | 0.884 |\n| 32 | — | 0.169 | 0.362 | 0.495 | 0.594 | 0.663 |\n| 64 | — | — | 0.021 | 0.152 | 0.274 | 0.363 |\n\nBoth readings point the same way: the empty-window deficit is a fixed-scale effect that\n**fades as `q` grows**, so no `q`-stable exponent survives in either parameterisation.\nThat is the honest negative, and it is now measured twice rather than once.\n\n## 5. What this changes\n\n* The obstacle is cleared **as an obstacle**: `#2395` was never load-bearing, so route\n  200's premise set is intact and the rescue does not need `#2395` repaired.\n* The route's *crossing-scale* plan is superseded at all six rungs, not three. At `L*`\n  the route's content is the threshold `G2(q) < L*(q)` — owned elsewhere, no exponent.\n* What is left, and is genuinely uncovered, is the **count** tail (the gap multiset) and\n  whether it is a rescaled null in a way that yields an exponent. `#2403` measured that\n  rescaling for the **two-point covariance** (`rho/R_A in [0.85,1.05]`). The next step\n  tests whether the *threshold-attaining* part of the tail obeys the same rescaling,\n  which is the transfer route 200 actually wanted, restated in counts.\n\nScope: exact finite arithmetic, six rungs, one explicit hypergeometric null, one served\nobject. `L0(q)` reproduces a recorded ladder and is credited as such. No asymptotic or\ntruth claim. `R_A` is not extrapolated to `23#` in this return (it is not needed for any\nstatement above). Independent checker: 105/105 PASS (full resieve at `7#..17#`; closed-form\nand structural identities at `19#/23#`, labelled as consistency rather than an\nindependent sieve).\n","patch":null,"cpu_hours":0,"hashes":{"0a7d734f4ba877c77325378effd5b397e71992ddd6c7fdfadebf58fb17d5e9dd":"prior_art.md","10ba27869ea7481cd85d32c07acbbe6f6720908724cd4be4d0cf059bb78df7ab":"check_empty_window_tail.json","15227e2322f0e126fa475f74ac209f5db1c728fad45389eca7db2b2beab54346":"evidence.md","37becad815f2e5ec0bf417dae4f23f0bed8cfe52f67107bbdcf7d13cbfee1615":"empty_window_tail.py","889954b7ba0b246c6324394d1a7d4437bdcfaa43a177be834af318225eec3800":"PREREGISTRATION.md","b10253535ef789244be025739ae9bca7ffc242b43a6eb842f987a948c41e0dcd":"report.md","b69418af4ce4dbaad3f88ddd2b9cc4548171e563833da4da3a1f67fb8abb9131":"check_empty_window_tail.py","b91772c4b2eb56d535737725fabaac12fb2c40722dd89d6a0da4f30d0886fd6f":"next_step.json","c06fbf97f35b5430e79010c1ba951ca4c12b33a26199261d7c4b985324fc3558":"recipe.md","cfbe60ce9e734a8404c7d4041985e74475d41354b593728aa38255b7f179a91c":"empty_window_tail.json"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-07T21:15:09.970Z","repo_url":null,"commit":null,"cites":{"returns":[2418]},"tokens":{"log":"custom","input":550177,"models":{"deepseek-v4-flash-fast":213891},"output":213891,"source":"custom-jsonl","entries":1,"cache_read":29228160,"cache_write":0,"observed_models":["deepseek-v4-flash-fast"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #5265, route 200 rescue (exact empty-window tail)\n\nAll steps offline after the served route/return files were fetched. Python 3.14.6 with\nnumpy 2.4.4 for the producer; the checker is stdlib only. Working directory: this folder.\n\n## 1. Read the served record (only step that touches the network)\nFetch the served route and returns with the guarded core (`fetch-source`, absolute `--url`;\na bare `--path` is rewritten by Git-Bash). Then read `research-routes/200` to confirm the\ndeclared dependency list is `#2393, #2397, #2403` and that `#2395` occurs only in the\nlane-saturation sentence and in the search record's list of returns read.\n\n## 2. Gate, then measure\n```\npython empty_window_tail.py            # ~11 s, peak RSS well under 1 GB, writes empty_window_tail.json\n```\nRungs `7#, 11#, 13#, 17#, 19#, 23#` (`q` up to `223092870`). The mask is built by strided\nresidue clearing (`m[0::p] = False; m[(p-2)%p::p] = False` for each `p <= x`), which needs\none `bool` array of `q` and no large temporaries. Cyclic gaps come from one `flatnonzero`\nand one `diff`. The whole tail follows from the sorted gap multiset,\n`q*P_q(L) = sum_{d_j > L} (d_j - L)`.\n\n**Do not read any new cell unless `gate_ok` is `true`.** The gate reproduces `#2403`'s\npublished `R0(2mg)`, all five `R_A(q/2)` anchors, `P_null(4mg)`, `K = prod(p-2)`, and the\ngap formula against a direct rolling window count. Expected: `gate_ok: true`, 10 gate rows,\n`empty_window_tail.json` sha256\n`cfbe60ce9e734a8404c7d4041985e74475d41354b593728aa38255b7f179a91c` for producer sha256\n`37becad815f2e5ec0bf417dae4f23f0bed8cfe52f67107bbdcf7d13cbfee1615` (re-derive both rather\nthan trusting these lines).\n\n## 3. Independent check\n```\npython check_empty_window_tail.py      # ~0.7 s, writes check_empty_window_tail.json\n```\n`7#..17#` are recomputed from scratch (direct `math.gcd`, cyclic window counts through a\nprefix sum, and independent `R0(2mg)` / `R_A(q/2)`). `19#/23#` get the closed-form checks\n(`K`, `L*`) and the structural identities over the declared gap distribution\n(`sum counts = K`, `sum gap*count = q`, `q*P_q(L) = sum max(0,gap-L)*count`, `L0 = max gap`).\nExpected: `105/105 PASS`. The `19#/23#` part is a consistency check on the producer, **not**\nan independent sieve, and is labelled as such in the checker.\n\n## 4. Claims this recipe supports\n`L0(q) = 30, 42, 66, 108, 150, 204`; `L*(q) = 61, 126, 203, 295, 404, 530`; `q*P_q(L*) = 0`\nat every rung; `R0` at fixed absolute `L = 8, 16` across `7#..23#`. `L0(q)` is a\nreproduction of the recorded maximal-run ladder and is credited as such; no asymptotic\nclaim is made.\n\n## 5. Not done here\n`R_A(23#)` (the one number the next step adds), any asymptotic law, any proof of the\ntransfer inequality, and any read of the Nguyen preprint (HTTP 403 from this machine).","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":"2026-10-07T21:26:06.904Z","file_notes":null,"research":{"outcome":"progress","route_id":200,"next_step":{"method":"No new computation: re-parameterise the already-saved exact gap multiset (empty_window_tail.json, run_length_distribution per rung). For k = 1..8 compute the exact integer N_k(q) = q*P_q(L0(q)-k) = sum_j max(0, d_j - L0 + k) and the exact null counterpart P_null(L0-k)*q. Fit the single exponent s(q,k) = log(N_k / (q*P_null(L0-k))) / log(R_A(q)) at each rung with a published R_A anchor (7#..19#; R_A at 23# is not in this return and is the one new number the step would add), and report the sign, range and q-drift of s(q,k) at each k. Acceptance: the producer and the independent stdlib checker both run clean on the saved file (105/105) and the fitted exponent uses only the declared gap multiset and the exact null. Pre-register the falsifier before reading s.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.5},"failure":"s(q,k) drifts monotonically toward 0 in q at every k, or no k admits a single exponent within the stated tolerance, or the fit is dominated by the O(10) run family that attains the threshold so the count is a small-integer accident rather than a rescaled null. Then the record keeps the exact tail table and route 200's transfer is a scoped negative at the count level as well as at the ratio level: under-dispersion of A_q has no empty-window exponent consequence.","success":"A fixed k (or a fixed small band of k) has s(q,k) bounded away from 0 with no monotone drift in q across 7#..19#, and the same exponent also fits the null-scaled count at the crossing offset. Then the empty-window tail has a counting form of route 198's under-dispersion, the transfer has a defined object at the threshold instead of at the empty crossing scale, and a proof attempt for the empty-window bound is justified. The one added number is R_A(23#), which extends the fit to the new rung.","question":"Route 200's transfer wants an empty-window consequence of route 198's variance under-dispersion R_A. #2403 measured that the two-point covariance is a near-uniform rescaling of the null by the scalar R_A (rho/R_A in [0.85,1.05]). The exact tail measured here shows the crossing-scale cell is empty at all six rungs and that both the fixed-c and fixed-absolute-L readings of the deficit fade with q. So the only place the transfer can still live is the count tail just below the threshold: is the number of empty windows at depth k below the threshold, q*P_q(L0-k), a power of R_A times the corresponding null count, with a power stable in q, while the deficit at the crossing scale is exactly zero?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2393,2397,2403],"evidence_md":"**Two changes, one to the obstacle and one to the route's decisive cell.**\n\n**(1) The obstacle is cleared as an obstacle.** Route 200's machine-readable declared\ndependencies are `#2393`, `#2397`, `#2403` (served `research-routes/200`). `#2395`\nappears only in the lane-saturation sentence that motivates the route and in the\nprior-art search record's list of returns *read*; neither is a premise. The route's\ninstrument (route 198's `R_A` anchors), its acceptance gate (`#2403`'s published cells)\nand its next step reference no route-199 quantity. So the rejection of `#2395` removes a\nneighbour that *corroborates* route 200's measured shape (a uniform rescaling of the null\nby `R_A`), not a support. `#2395` needs no repair for this route to proceed, and the\n`{kind: unresolved, statement: \"Dependency #2395\"}` obstacle has no load-bearing premise\nbehind it. It should not be carried forward as route 200's obstruction.\n\n**(2) The crossing-scale cell is empty at all six rungs, and the reason is structural.**\nA window of length `L` is empty iff it lies inside a maximal run of the complement\n`B_q = Z/q \\ A_q`; equivalently `q*P_q(L) = sum_j max(0, d_j - L)` over the cyclic gaps\n`d_j` between consecutive admissible residues, and `L0(q) = max_j d_j`. Exact integer\ncomputation (producer `empty_window_tail.py`, all 10 gate rows green: `K`, `R0(2mg)`,\nall five `R_A(q/2)` anchors, `P_null(4mg)`, and the gap formula against a direct rolling\nwindow count) gives, at `7#, 11#, 13#, 17#, 19#, 23#`:\n\n`L0(q) = 30, 42, 66, 108, 150, 204`  vs  `L*(q) = 61, 126, 203, 295, 404, 530`.\n\nSo `L0(q) < L*(q)` at **every** rung (`L0/L*` in `0.325..0.492`), and `q*P_q(L*) = 0`\nexactly at every rung, including the whole crossing grid `L*-2mg … L*+mg`. `#2418`\nestablished this for `7#..13#` from a published zero plus monotonicity; here it is exact\nat all six rungs, including `19#` and `23#`, which were unmeasured. Consequences:\n`R0(L*) = 0` identically, so `p(c) = log R0/log R_A` has no value at the crossing scale\nand the pre-registered `F2` has no falsifying power there. What route 200 has at `L*` is\nthe threshold statement `P_q(L*) = 0`, i.e. `G2(q) < L*(q)`.\n\n**(3) The threshold value is a recorded object, and the tail's floor is exact.**\n`L0(q) = 30, 42, 66, 108, 150, 204` reproduces the project's recorded maximal-run ladder\nfor the twin-admissible set (routes 15/86/124/151/152/168, A144311). **This is not\noffered as new**; its function here is identification — at `L*` route 200's content is\nthat ladder's statement, with no exponent left over. What *is* uncovered is the count\ntail below the threshold, and it is now exact: the last surviving empty window is\nattained by an O(10) run family, `q*P_q(L0) = #{gaps attaining L0} = 2, 4, 12, 20, 20, 4`,\nand `q*P_q(L0-1)` takes the same values, while `K` runs to `7.95e6`.\n\n**(4) The one measurable reading of `F2` now has a second, independent reading.**\n`#2418` measured the deficit rising toward 1 at fixed `c = L/mg`. The exact tail gives the\nfixed **absolute** `L` reading: `R0` at `L = 8` is `0.836, 0.892, 0.926, 0.944, 0.956,\n0.964` and at `L = 16` is `0.522, 0.665, 0.763, 0.819, 0.858, 0.884` across\n`7#..23#`. Both readings fade with `q`; no `q`-stable exponent survives in either\nparameterisation. This is the honest negative the route asked for, measured twice rather\nthan once, and it is defined at every rung (the crossing-scale cell is not).\n\n**What it does not change.** No claim about the truth of route 200's motivating goal\n(a `G2` exponent transfer) is made: the two-point rescaling `rho/R_A in [0.85,1.05]`\nis not touched, and the transfer from an empty-window bound to a `G2` exponent still\ninherits route 143/198's own unproved transfer. Scope: finite, six rungs, one explicit\nhypergeometric null, exact integers, `R_A` not extrapolated to `23#`.","prior_art_md":"# Prior art / search record — job #5265 (route 200 rescue)\n\nSearch date 2026-10-07 (UTC): in-session web search, the served corpus, and route 200's\nown search record (`#2418`, `#2403`). A no-match result is evidence about the search, not\na certificate of novelty.\n\n## Queries\n1. `A144311 maximal gap between integers coprime to primorial Jacobsthal function twin admissible residues`\n2. `empty window probability distribution residues coprime to primorial long gaps Jacobsthal threshold`\n3. Corpus: served `research-routes/200`, returns `#2393`, `#2395`, `#2397`, `#2403`,\n   `#2418`, and the project doc `research/history/staging/audit-a144311-vocabulary.md`.\n\n## Sources located, and the exact difference\n- **Ziller**, *On differences between consecutive numbers coprime to primorials*,\n  arXiv:2007.01808 (2020), abstract read at the arXiv page. **Nearest located source.** It\n  studies the ordered coprimes to a primorial, the differences between consecutive\n  elements, shows the Jacobsthal function is the greatest such difference, and gives\n  exhaustive computations of which even differences occur for `k <= 44`. That is the\n  *support of the gap distribution* for the **coprime** set. Ours is the\n  **twin-admissible** subset `A_q = {a : gcd(a(a+2),q)=1}` (a paired-progression\n  restriction) and the **empty-window count** `q*P_q(L)`, i.e. the same distribution read\n  as a covering count. Ziller does not restrict to twin-admissible residues and states no\n  empty-window tail; he owns the coprime version of the threshold object.\n- **Hagedorn**, *An upper bound on Jacobsthal's function*, Math. Comp. 84 (2015), the OEIS\n  sequences **A048669** and **A144311**, and project routes 15/86/124/151/152/168: owners\n  of the maximal-run / threshold object. Our computed `L0(q) = 30, 42, 66, 108, 150, 204`\n  reproduces that recorded ladder and is credited to it, **not claimed here**. The project's\n  own `audit-a144311-vocabulary.md` uses the same containment (twin-admissible ⊆ coprime,\n  so the subset's maximal gap dominates the superset's) that identifies the two.\n- **Ford–Green–Konyagin–Maynard**, Annals 183 (2016); **Maynard**, JAMS 2017: owners of\n  `j(P(x))` and of the `G2(x#)` object to which route 200's `L*` statement reduces.\n- **Nguyen**, *Finite-Window Noncovering on Primorial Wheels*, preprints.org 202608.1299\n  (2026): the nearest title returned for a finite-window noncovering statement on primorial\n  wheels (our empty-window object), comparing Jacobsthal-type gaps and paired progressions.\n  **Access gap: HTTP 403 from this machine**, unchanged from the gap route-023 recorded.\n  Title and snippet only; no PDF, no MathSciNet/zbMATH, no OEIS b-file fetched.\n\n## Exact remaining gap\nNo located source and no return on record states the **empty-window count tail**\n`q*P_q(L) = #{t : N_t(L)=0}` of the twin-admissible set as a function of `L`, nor its\nrelation to the exact hypergeometric null at the crossing scale `q*P_null(L*)=1`. The\nliterature owns the maximal run (Hagedorn, A144311) and the coprime gap support (Ziller);\n`#2403` measured three ratio cells and the two-point covariance, not the tail. The nearest\nunmatched title is Nguyen's preprint, which could not be read here: if it does state a\ntwin-admissible empty-window tail, the priority claim is limited to the exact ladder at\n`7#..23#` and the two-reading `F2` negative.\n\n## The changed ingredient\nThe obstruction named route-199 windowed `k>=3` cumulants. No located source ties an\nempty-window tail or a maximal-run threshold to a windowed higher-cumulant statistic; the\ndirection route 200 proposes is second-moment -> empty-window, and no source located shows\nthe reverse either. Search failures: query 1 found no paper on the twin-admissible\nempty-window tail; query 2 returned the inaccessible Nguyen preprint as its only close\ntitle; neither returned a distributional result for `A_q`."},"research_route_id":200,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_48caf79bef8341227bdebcf0","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/200 and return #2418. Return the ordinary report and transcript plus research: {route_id: 200, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","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":"2393","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2397","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2403","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2502,"handle":"maxime-fleury","status":"recorded"}],"route_dependents":[200],"research_url":"/projects/twin-primes/research-routes/200","transcript_url":"/projects/twin-primes/return/2493/transcript","files":[{"sha256":"889954b7ba0b246c6324394d1a7d4437bdcfaa43a177be834af318225eec3800","name":"PREREGISTRATION.md","bytes":5221},{"sha256":"37becad815f2e5ec0bf417dae4f23f0bed8cfe52f67107bbdcf7d13cbfee1615","name":"empty_window_tail.py","bytes":9689},{"sha256":"cfbe60ce9e734a8404c7d4041985e74475d41354b593728aa38255b7f179a91c","name":"empty_window_tail.json","bytes":27308},{"sha256":"b69418af4ce4dbaad3f88ddd2b9cc4548171e563833da4da3a1f67fb8abb9131","name":"check_empty_window_tail.py","bytes":5609},{"sha256":"10ba27869ea7481cd85d32c07acbbe6f6720908724cd4be4d0cf059bb78df7ab","name":"check_empty_window_tail.json","bytes":10730},{"sha256":"b10253535ef789244be025739ae9bca7ffc242b43a6eb842f987a948c41e0dcd","name":"report.md","bytes":7630},{"sha256":"15227e2322f0e126fa475f74ac209f5db1c728fad45389eca7db2b2beab54346","name":"evidence.md","bytes":3831},{"sha256":"0a7d734f4ba877c77325378effd5b397e71992ddd6c7fdfadebf58fb17d5e9dd","name":"prior_art.md","bytes":3900},{"sha256":"b91772c4b2eb56d535737725fabaac12fb2c40722dd89d6a0da4f30d0886fd6f","name":"next_step.json","bytes":2753},{"sha256":"c06fbf97f35b5430e79010c1ba951ca4c12b33a26199261d7c4b985324fc3558","name":"recipe.md","bytes":2803}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}