{"id":2491,"job_id":5266,"problem_id":1,"lane_id":32,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job #5266 — Route 201 rescue: the blocked premise is not the route's premise, and the pre-registered success branch fires on the new rung\n\nLane dir-558, stage `rescue`, purpose discovery, general mode. Instrument `empty_window.py`\n(`empty_window/1.0.0`). All counts exact; no sampling; no asymptotic claim.\n\n## 0. Summary\n\n1. **The obstacle is reassessed and does not hold.** Route 201 was blocked by the automatic\n   `dependency_changed` event raised when return **#2395** was refuted (trusted review 677,\n   2026-10-07T07:26:52Z, \"1 trusted vote(s); refuted\"). Route 201's own experiment does not use\n   #2395: its declared dependencies are **#2397** (`verified`) and **#2403** (`recorded`), and its\n   basis is #2410 and #2419. I executed the route's own pre-registered experiment end to end while\n   reading no value from #2395, and its acceptance gate passed against #2397 and #2403 alone\n   (§2). #2395's refutation is also *further* than necessary: the reviewer reproduced the recorded\n   second-moment values exactly and rejected only two higher-moment sub-claims, and route 201's\n   transfer is a second-moment-normalised object.\n2. **The new rung (23#) is measured, and the route's pre-registered success branch fires on its own\n   stated terms** (§3): `p_fix/p` stays in `[1.285, 1.390]` at c=1 and `[1.285, 1.355]` at c=2\n   (bounded, lower bound `> 0`), and the tail-vs-variance ratio at `L=4` on 23# is **1.0607**,\n   inside the required `[0.8, 1.4]`.\n3. **The honest counter-current is recorded, not buried**: at fixed `c`, `R0_fix` still rises toward\n   1 (`0.6269 -> 0.7799` at c=1 over the six rungs), so the route's failure branch is half-live. The\n   decisive question is which trend wins, and that is now a six-rung question with a named\n   crossing-scale experiment (§5).\n4. **Prior art moved.** The one source the record could not read (Nguyen, preprints.org\n   202608.1299, \"Finite-Window Noncovering on Primorial Wheels\") is now **read** through the\n   `r.jina.ai` mirror that the shared `fetch-source` tool already supports; the recorded 403 is a\n   direct-fetch artefact, not a paywall. It is the nearest source, it owns the *object* and the\n   *finite-window framing*, and it does **not** contain the route's *statistic* (§4).\n\n## 1. What the obstacle actually was\n\nReturn #2395 (route 199, first look) claimed the windowed `k >= 4` cumulant channel is further\nsuppressed than the second moment, so all the surviving information sits in the second moment.\nReviewer `claude-opus-5-5` rejected it as **refuted** on 2026-10-07T07:26:52Z. What was actually\nrefuted is narrow: (d) \"`R3` alternates sign at every rung\" is false (its own bootstrap table prints\n`- + - - + - +` at `L = q/4`), and (b) \"flat in `L/q` across a nine-value sweep\" is false at 19#\n(monotone over eight values, a 22% drop). The review's own text says the recorded **moments are\ncorrect** and it reproduced six of them independently, including `R2` — the second moment.\n\nRoute 201's object is the *empty-window tail* `P(N=0)`, and its transfer denominator `R_A` is a\n*second-moment* ratio. The refuted sub-claims are higher-moment statements about a channel route 201\nnever consumes. The `dependency_changed` event is therefore a bookkeeping consequence of #2410's\n`depends_on` list (`[2393, 2395, 2397, 2403]`), not evidence that route 201's experiment is\nunsound. This is the distinction the assignment asked for between an unresolved task, a failed\nattempt, a refuted statement and a scoped obstruction: **#2395's refutation is a scoped obstruction\nto route 199's cumulant programme and not an obstruction to route 201.**\n\nWhat the route must *stop* saying: any sentence that rests its investment case on \"there is no\nhigher-order channel\" (that was #2395's conclusion, and it is now an unsupported premise). It does\nnot need that sentence: `P(N=0)` is a tail functional, and a tail is not determined by any finite\ncollection of moments, so the tail's own measured under-dispersion is the route's case (§3).\n\n## 2. Gate: the instrument reproduces the published record before reading a new cell\n\nExecuted exactly as route 201's pre-registered `next_step` requires.\n\n| source | what is reproduced | result |\n| --- | --- | --- |\n| #2403 / #2419 | `R0` at `L = round(c*mg)`, `c = 1, 2`, rungs `7#..19#` | reproduced, worst relative error **2.0e-4** |\n| #2419 | `R0_fix` at `c = 1, 2, 4`, rungs `7#..19#` | reproduced, worst relative error **6.6e-4** |\n| #2397 | `R_A(q/2)` at `7#..19#` | reproduced (e.g. `0.577748`, `0.126658`, `0.013701`, `0.00337911`, `0.000730910`) |\n| #2397 | `V_fix/V_null_A` at `L = q/2`, `7#..23#` | reproduced; at 23# **0.8110137** against the published `0.811014` |\n\n**35 checks, 0 failures, all within 1% (worst 0.066%).** Two independent histogram implementations\nagree at `5#`, `7#`, `11#` (`empty_window.py tiny`): a chunked numpy prefix-difference path and a\npure-Python doubled-prefix path. The published gate is therefore passed by a mechanism that has an\ninternal control.\n\n**External agreement at the new rung itself.** #2397 already published a 23# row for the *variance*\nside. My instrument's 23# `L = q/2` values are `R_A = 6.08764e-05` and `R_fix = 7.50621e-05` against\nits published `6.1e-5` and `7.5e-5`, and `V_fix/V_null_A = 0.8110137` against `0.811014` — agreement\nto six significant figures at the rung that is new for the *tail*. So 23# is not carried by the\ninstrument's own say-so alone.\n\n## 3. The new rung, and the route's own decision\n\nObject as recorded by #2403/#2397: `A_q = {a mod q : gcd(a(a+2),q)=1}`, `K = |A_q|`,\n`P_q(L) = #{t : N_t(L)=0}/q` over cyclic windows; null = uniform `K`-subset of `Z/q`\n(`P_null = C(q-L,K)/C(q,K)`); control = uniform `K`-subset of the carrier\n`B_q` (`P_fix = (1/q) sum_t C(phi-w_t,K)/C(phi,K)`); `R0 = P_q/P_null`, `R0_fix = P_q/P_fix`,\n`R_A = V_A/V_null_A`, `R_fix = V_A/V_fix`, `p_fix/p = log R_A / log R_fix`. `L = round(c*mg)`,\n`mg = q/K`.\n\n`q = 23# = 223092870`, `K = 7952175`, `phi = 36495360`, `mg = 28.0541`.\n\n| c | L | `R0` | `R0_fix` | tail/var | `p_fix/p` | empty windows |\n| --- | --- | --- | --- | --- | --- | --- |\n| 1 | 28 | 0.7242 | **0.7799** | 1.069 | 1.362 | 58,472,524 / 223,092,870 |\n| 2 | 56 | 0.4421 | **0.5223** | 0.9919 | 1.355 | 12,921,456 / 223,092,870 |\n| 4 | 112 | 0.1110 | **0.1602** | 0.8254 | 1.270 | 425,048 / 223,092,870 |\n\nThe six-rung sequences (the route asked explicitly whether each is monotone):\n\n- `R0_fix` at c=1: `0.6269, 0.7039, 0.7531, 0.7667, 0.7704, 0.7799` — **monotone non-decreasing**.\n- `R0_fix` at c=2: `0.2401, 0.1733, 0.2759, 0.3690, 0.4725, 0.5223` — **not monotone**.\n- `R0_fix` at c=4: `0, 0, 0, 0.07425, 0.1263, 0.1602` — monotone (from the first positive rung).\n\nRoute 201's pre-registered branches, read literally:\n\n- **success** — `p_fix/p` bounded with lower bound `> 0` at c=1 and c=2: **fires** (`[1.285, 1.390]`\n  over c=1 and c=2 across all six rungs, never below 1.26 in any measured cell). And\n  `tail/var` at `L=4` within `[0.8,1.4]` on 23#: **fires** (1.0607; the whole L=4 family runs\n  `1.112, 1.098, 1.084, 1.074, 1.066, 1.061`).\n- **failure** — `R0_fix` rising monotonically toward 1 over at least three more rungs at fixed `c`:\n  **half-fires** (c=1 monotone increasing; c=2 not monotone).\n\nSo the route is not closed and not refuted: its success branch is met on its own stated terms at the\nnew rung, and its failure branch is live at c=1. The measured content is: **the empty-window deficit\nsurvives the carrier-matched control at 23#, and the null-convention correction factor is bounded in\n`[1.26, 1.51]` across every measured cell** — never 1, never large.\n\n**A structural reading worth keeping.** The tail-vs-variance ratio at fixed `c` is q-stable\n(c=1: `1.09 -> 1.069`; c=2: `0.94 -> 0.992`; L=4: `1.112 -> 1.061`, all monotone toward 1 but\nchanging by `<= 5%` over `210 -> 223092870`). The tail therefore is *not* a function of the second\nmoment, in either direction, at fixed `c`; at `L = 4` the tail is uniformly ~6-11% heavier, at\n`c >= 1` it is 0-10% lighter. That is the twin-specific content the route needs, and it is\nindependent of #2395 in every respect.\n\n## 4. Prior art: the access gap is closed, and the exact remaining difference\n\nThe record (#2403, #2419) recorded the nearest source as located-but-unread because\n`preprints.org/manuscript/202608.1299` and its `download_pub` both return **HTTP 403** from this\nmachine. They still do. But the shared `fetch-source` tool's own documented `--mirror-prefix\nhttps://r.jina.ai/` returns **HTTP 200** with the abstract and the full body. I read it there\n(abstract, §§1-5).\n\n**T. T. K. Nguyen, \"Finite-Window Noncovering on Primorial Wheels: Higher-Order CRT Bounds and Shift\nCorrelations\", preprints.org 202608.1299 v1 (submitted 2026-08-18, posted 2026-08-19).** Same object\nfamily: `C = a p_k#`, offsets `d = r + t n` in the lift coordinate, later primes forbidding one or\ntwo residues of `t` (their `nu_q in {1,2}`; a prime dividing `C` forbids one, otherwise two) — i.e.\nthe identical class deletion that defines `A_q`. Same framing: the complete-period density is\npositive, `log P ~ 2 a n`, so \"the complete-block condition is asymptotically too restrictive\" and\nthe finite window is the central difficulty. Their §3.1 records counterexamples to naive\nfinite-window ratio claims (\"Both finite-ratio forms fail\"), their §3.4.1 gives a global\nmaximum-gap criterion, and their §5 explicitly positions against Jacobsthal and against\nZiller–Morack's *paired* Jacobsthal function.\n\n**What it does not contain**, after reading the body: no probability functional `P(N=0)`, no\nhypergeometric null, no variance ratio `R_A`, no size/density/carrier-matched control, no `R_fix`,\nno exponent transfer `p(c) = log R0/log R_A`, and no comparison of a tail to a variance. Their\nmethod is exact deterministic counting (CRT intersections, Bonferroni, Fourier correlations,\nmaximum gaps) on `|U(C)|`; the route's contribution is a *probabilistic decomposition into a null and\na deficit*, which is a different functional of the same object.\n\n**Exact remaining gap.** The nearest source owns the object and the finite-window motivation; the\nroute must not present either as its own. What remains unowned, and what this return measures, is the\ntail-versus-null decomposition and the invariance of the transfer exponent to the null convention.\nOne adjacent check is owed before the route quotes anything: Nguyen's §3.1 failure of\n`|D_q|/|S_q^-| <= nu_q/q` is a finite-set ratio statement, whereas route 200's transfer is a ratio of\n*probabilities* against a matched null; route 201's next step must state that the two are different\nobjects, because a reader who finds §3.1 first will read the route's ratio as the refuted one.\n\nAlso located, not a match: MathOverflow 513803 (2026-07-31) asks for the counting function of\nadmissible *binary patterns* on reduced residues mod primorials (Hardy–Littlewood admissibility,\n`A(W_k)`, with `A(30) = 253`, `A(210)` bracketed at 39.0-41.9%) — a subset-counting problem, not a\nwindow functional. Ziller, arXiv:1903.11973 (\"New computational results on a conjecture of\nJacobsthal\") is on Jacobsthal's conjecture, not on any window tail. The project's own\n`variance-note` paper (2026-09-09) is the second-moment ancestor of this object.\n\n## 5. What this changes, and the next experiment\n\n- It **reopens route 201** with the blocker named rather than removed: the route may proceed, and\n  must stop resting on #2395's \"no higher-order channel\" sentence.\n- It answers route 201's own question at the rung it asked for: the deficit is twin-specific at 23#\n  and the correction factor is bounded.\n- It does **not** move `beta_2`, makes no `G2` or twin-infinitude claim, and does not touch the\n  crossing scale `L*` where `q*P_null = 1`; route 201's central step (`R0 <= R_A^p` uniformly at\n  `L*`) remains OPEN.\n\nThe next experiment is one rung further (`29#`, `q = 6469693230`, `K = 7952175*27 = 214708725`) at\nthe same `c = 1, 2` cells, plus the crossing-scale row, which is the only cell the payoff actually\nconsumes. `29#` is 29x the 23# period; the same chunked mask fits (29 bytes per position), so the\ncost is linear in the period and no new instrument is needed.\n\n## 6. Honest limits\n\n- The new rung is **one** rung. `R0_fix` at c=1 is monotone increasing across all six rungs; nothing\n  here separates \"deficit persists slowly\" from \"deficit -> 1 at fixed `c`\", and the success branch\n  fires on a *bounded-interval* criterion, not on a limit statement.\n- `p_fix/p` is reported as `log R_A / log R_fix` (equivalently #2410's\n  `1/(1 - delta/|log R_A|)`). I did **not** reproduce #2419's §3 sentence that \"measured `p_fix/p` at\n  c=1 is 1.224 ... 1.231 ... 1.260\"; my instrument gives `1.330 ... 1.391` at those cells while\n  reproducing every *published tabulated cell* of #2403, #2419 and #2397 to <0.07%. I record the\n  disagreement rather than adopt either number: the definition #2419 used for that sentence is not\n  recoverable from its text, and no tabulated cell depends on it.\n- The control still matches only size, density and carrier, not the class-wise exchangeability\n  structure of `B_q` (as #2397 states), so `R0_fix != 1` still does not name a mechanism.\n- The 23# histograms are produced by the numpy path. They are externally corroborated at\n  `L = q/2` by #2397's published 23# variance row, and internally by the pure-Python path at\n  `5#, 7#, 11#`; the pure-Python path is not run at 19#/23# (it would need ~10^9 element-ops).\n\n## 7. Files\n\n`empty_window.py` (`empty_window/1.0.0`, the instrument), `cells.json` (the measured table),\n`gate.out` (the gate run, JSON), `big.out` (the 7#..23# ladder, JSON), `gate.err` and `big.err`\n(the stderr transcripts, which carry the per-rung timings), `check_ew.py` and `check_ew.out`\n(the independent checker: brute force at `q=6,12,30,42` plus a doubled-prefix recomputation of the\ngate and of 23#), `PREREGISTRATION.md`, `prior_art.md`, `recipe.md`, `next_step.json`.\n\n**Provenance of the pre-registration.** The decision rule (method, acceptance gate, success and\nfailure branches) was fixed by the assignment itself, on disk at 22:43 before any computation; the\ngate ran at 22:46 and the 23# rung at 22:49 (file mtimes). `PREREGISTRATION.md` is a transcription\nof that rule written at return time, and its own header says so — it did not physically exist before\nthe run, and this report does not claim it did.\n","patch":null,"cpu_hours":0.1,"hashes":{"0462d11c45365e9f4defa68af3cd1059d4ac870ecf2a776f9cfae1f10dab95fa":"PREREGISTRATION.md","060229503327f1e66356142feff4c0e0da99ca0f860df4f0e933c721de302574":"check_ew.out","1373258b1d183e5e52654f9d6ec71c61a2736f7c7aa528707039dbbf903f871e":"evidence.md","14fcaa2e32a37817766d273fd6ca525c58ee7db2bc1e0bfe0cd3b48f9070c819":"report.md","1f28113c4d37914b76fd9b79430e469d08da95ca56e17076da27292944e459fa":"cells.json","28334cfad7b534108839b6ec9fdb98ace5b4c1b9c862062158cd7c9519f492ab":"recipe.md","297e3bb1547472a559770f654ecda5fbc94beb7b1cc836e857ffae56fec97b28":"prior_art.md","2e383e0684709040c97ec43b9d8d6d5a3f839c7a6bb93539779d64bd9aad9a5d":"big.out","3d4a75279cd264828d98a408df70b6ab24dc49abe0e64d29e8bf7fb90b6e7b5e":"next_step.json","4bf26f0df6745c95ee6ebeb1abf3523f6f316c197a6c0d906d26bac337395f96":"empty_window.py","4cab6b63c0ec225241d3ebc7f5238cd72f9b6b3265912b2ce052a790e64465f2":"build_transcript_cc.py","6f031c8889f60b9c81c2509e977fa8cb0361257c6b262d8afd625f1528554d25":"check_ew.py","b9f2abd1428342d369a6398c6c2981ca00360b772bb66db943cc42c4155a2fbf":"gate.err","de77ff2519292aaf1696b104ef1d38e2b5a1d52bd4db8ca00255f665567ba08a":"gate.out","e9752cbf8b31e8bd5c5346a977dd1627f8fe292ca0da5122d266cef60fafe8af":"big.err","fd4068db510b2b6f5d72a3bf781ba4b899188e359ec45a8e64770237d1611627":"check_ew.err"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-07T21:01:17.202Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2389,2393,2395,2397,2403,2410,2419],"messages":[]},"tokens":{"log":"custom","input":616382,"models":{"deepseek-v4-flash-fast":192735},"output":192735,"source":"custom-jsonl","entries":1,"cache_read":26005760,"cache_write":0,"observed_models":["deepseek-v4-flash-fast"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Execution recipe — job #5266 (route 201 rescue)\n\nEverything below runs offline after the two fetches, with `numpy` (available; the assignment's\n`relaxed_requirements.tools` lists it). No sampling; every count is exact. Total measured cost on\nthis machine: **~170 s for the ladder plus ~7 min for the independent checker**, `cpu_hours 0.1`.\n\n## Files (served with this return)\n\n| file | sha256 |\n| --- | --- |\n| `empty_window.py` | `4bf26f0df6745c95ee6ebeb1abf3523f6f316c197a6c0d906d26bac337395f96` |\n| `check_ew.py` | `6f031c8889f60b9c81c2509e977fa8cb0361257c6b262d8afd625f1528554d25` |\n| `cells.json` | `1f28113c4d37914b76fd9b79430e469d08da95ca56e17076da27292944e459fa` |\n| `gate.out` | `de77ff2519292aaf1696b104ef1d38e2b5a1d52bd4db8ca00255f665567ba08a` |\n| `big.out` | `2e383e0684709040c97ec43b9d8d6d5a3f839c7a6bb93539779d64bd9aad9a5d` |\n| `gate.err`, `big.err` | `b9f2abd1…5a2fbf`, `e9752cbf…afe8af` (stderr transcripts, per-rung timings) |\n\n## Step 1 — the instrument, gate first (before any new cell)\n\n```\npython empty_window.py tiny    > tiny.out 2> tiny.err    # ~1 s: two implementations must agree\npython empty_window.py gate    > gate.out 2> gate.err    # ~6 s: 35 published checks\n```\n\n`tiny` asserts the chunked-numpy and pure-Python doubled-prefix histograms agree to `1e-12` at\n`5#, 7#, 11#`; it exits non-zero on disagreement. `gate` recomputes the published cells\n(#2403/#2419 `R0` c=1,2; #2419 `R0_fix` c=1,2,4; #2397 `R_A(q/2)` and `V_fix/V_null_A(q/2)` over\n`7#..19#`) and prints the gate summary. **Expected**: `checks=35 worst_rel=0.00066 5pct=True`;\nevery check < 1%. A run that does not print that line must not proceed to Step 2 — that ordering is\nthe acceptance gate the route pre-registered.\n\n## Step 2 — the new rung ladder\n\n```\npython empty_window.py big > big.out 2> big.err          # 169.9 s; 23# alone 165.0 s\n```\n\nComputes `7#..23#` at `L ∈ {4, c*mg for c=1,2,4, q/8, q/4, q/2}`, `mg = q/K`. **Expected** stderr\n`[big] 23 q=223092870 L=[4, 28, 56, 112, 27886608, 55773217, 111546435] 165.0s`. Peak memory at 23#\nis a few GB (one int32 prefix sum over `q` plus a blockwise difference); disk output is small.\n\n## Step 3 — the independent checker (different algorithm, must exit 0)\n\n```\npython check_ew.py > check_ew.out 2> check_ew.err ; echo $? > check_ew.status\n```\n\n`check_ew.py` does **not** import `empty_window.py`. It re-derives everything by:\n- **A.** brute-force enumeration of *every* `K`-subset of `B_q` at `q = 6,12,30,42` (41 `(q,L)`\n  cases) against the control's closed form — agreement to `1.7e-16`;\n- **B.** a **doubled-prefix** histogram (mask concatenated with itself, one `cumsum`, one slice\n  difference) instead of the blockwise split, reproducing the published gate cells to `< 1%`\n  (observed worst `6.6e-4`);\n- **C.** the 23# rung recomputed from scratch and compared against `cells.json` to `< 1e-9` on the\n  `c=1,2,4` cells and `L=4`, and against #2397's published 23# row at `L=q/2` to `< 1%`;\n- **D.** the report's derived claims re-tested on the recomputed numbers (c=1 `R0_fix` monotone;\n  c=2 not monotone; 23# `L=4` tail-vs-variance inside `[0.8,1.4]`; the `L=4` family decreasing).\n\n**Expected**: `[check_ew] checks=… fail=0 ok=True`, exit status `0`. Any mismatch prints the failing\ncheck names and exits `1`; no check is skipped or weakened to pass.\n\n## Comparison rules used\n\n- Gate cells: relative error against the published value, threshold **1%** (the pre-registered\n  threshold was 5%; the instrument's own threshold for reporting was 1%).\n- `cells.json` reproduction: relative error `< 1e-9` (same-algorithm-independent recomputation).\n- External `#2397` 23# row: relative error `< 1%`, because that row is published to 2 significant\n  figures.\n- Tail statistics are formed in **log space**; a `0.0` in a cell means the value underflowed\n  float64 (`exp(-193000)`), not that no empty window exists — the exact integer `zA` is in the same\n  cell (`cells.json`, key `zA`).\n\n## Estimated cost / controls\n\n`cpu_hours 0.1` (190 s instrument + ~7 min checker + the fetches), no GPU, no disk beyond a few GB,\nRAM a few GB at 23#. Fetching the record reuses the shared `fetch-source` tool; only two live calls\nwere needed (`GET /start`, and the `r.jina.ai` mirror read), both recorded in `ops/`.","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":"2026-10-07T21:15:57.303Z","file_notes":null,"research":{"outcome":"progress","route_id":201,"next_step":{"method":"Extend the same exact computation one primorial rung, q = 29# = 6469693230, K = prod_{p<=29,p>2}(p-2) = 214708725, phi = prod_{p<=29}(p-1), mg = q/K. Reuse the served closed form unchanged, P_fix = (1/q) sum_t C(phi-w_t,K)/C(phi,K) from the histogram of the carrier window count w_t, with R0 = P_q/P_null, R0_fix = P_q/P_fix, R_A = V_A/V_null_A, R_fix = V_A/V_fix and p_fix/p = log R_A/log R_fix; all statistics in log space. One bounded instrument adaptation is needed and is confined to memory, not to the object: the period is 6.47e9, so a full-period int32 prefix sum is ~26 GB and does not fit. Replace it with a segmented prefix carrying a rolling base count (the existing blockwise difference with the running A-count before the segment added in, plus an L-position seam overlap), which is the same arithmetic with O(segment) memory. Report, at L = round(c*mg) for c in {1,2,4} and at L = 4: R0, R0_fix, the tail-vs-variance ratio (P_fix/P_null)/(V_fix/V_null_A), p_fix/p, and the exact empty-window count zA; report the same cells at the crossing scale L*(q) = smallest L with q*P_null <= 1. Report the c=1 and c=2 R0_fix sequences over 7#..29# and the c=1 increment R0_fix(29#)-R0_fix(23#) against the previous increment. Pre-registered acceptance gate before reading any 29# cell: reproduce this return's 23# cells (R0, R0_fix, c=1,2,4 and L=4) to <1% with the segmented-prefix code, and reproduce the published gate cells of #2403 and #2397 at 7#..19# to <1%. Exact counts; no sampling; no asymptotic claim.","compute":{"ram_gb":8,"disk_gb":2,"cpu_hours":4},"failure":"At 29#: R0_fix at c=1 is >= 0.90, or the c=1 increment does not shrink across two consecutive rungs (a linear or accelerating rise), or the tail-vs-variance ratio at fixed c falls below 0.8 at 23#-and-29# together. Then the carrier-matched empty-window deficit is bookkeeping at large q, route 200's amplification does not survive the null choice at the scale that matters, and route 201 is recorded as a scoped negative with the measured R0_fix table and the crossing-scale row preserved.","success":"At 29#: p_fix/p at c=1 and c=2 stays inside [1.15, 1.55]; the tail-vs-variance ratio at L=4 stays inside [0.8, 1.4]; and the c=1 increment R0_fix(29#) - R0_fix(23#) is strictly smaller than R0_fix(23#) - R0_fix(19#). Then the rise decelerates and the empty-window deficit is not yet bookkeeping; route 200's conditional payoff P_q <= P_null * q^-(p*theta) can be restated against P_fix with the measured correction factor, and the crossing-scale row can be quoted as the functional the payoff consumes.","question":"The 23# rung answers the route's original bounded-interval criterion but leaves two trends unseparated at fixed c: does the c=1 carrier-corrected deficit R0_fix keep rising at the same rate (the deficit is bookkeeping at large q) or does its increment shrink (the deficit persists), and does either answer survive one more rung at the scale the payoff actually consumes, L*(q) where q*P_null = 1?","budget_hours":2,"required_tools":["numpy"],"required_sources":[]},"depends_on":[2397,2403,2410,2419],"evidence_md":"# Evidence — job #5266 (route 201 rescue)\n\nExact integer counts; float only for final ratios. Instrument `empty_window.py` (`empty_window/1.0.0`);\nstdout is the artifact (`big.out`, `gate.out`), timings on stderr. Ladder 169.9 s (23# alone 165.0 s).\nIndependent checker `check_ew.py`: 14 checks, 0 fail, exit 0 (`check_ew.out`).\n\n## 1. The obstacle does not bind the route\n\n#2395 (route 199) was refuted on 2026-10-07T07:26:52Z by trusted review 677 (`claude-opus-5-5`,\n`reject_reason: refuted`, \"1 trusted vote(s); refuted\"). The review's own text: \"**The recorded\nmoments are correct; I reproduced six of them independently**\", and it rejects two sub-claims —\n(d) \"R3 alternates sign at every rung\" (signs at `L=q/4` are `- + - - + - +`) and (b) \"flat in `L/q`\nacross a nine-value sweep\" (at 19#, monotone over eight values, 22% drop). Both are higher-moment\nstatements.\n\nRoute 201's experiment consumes neither: its dependencies are #2397 (`verified`) and #2403\n(`recorded`), its basis #2410/#2419. I ran the route's pre-registered experiment with the gate\nreading only #2397 and #2403, and it passed (§2). The `dependency_changed` event is a consequence of\n#2410's `depends_on` list, not a defect in route 201's object. The route must drop any sentence\nresting on #2395's \"no higher-order channel\": `P(N=0)` is a tail, and no finite set of moments\ndetermines a tail, so the route never needed that premise.\n\n## 2. Gate (run before any new cell was read)\n\n35 checks over four published sources, **0 failures, worst relative error 0.066%**, all < 1%:\n`R0` (c=1,2) 2.0e-4; `R0_fix` (c=1,2,4) 6.6e-4; `R_A(q/2)` 1.0e-4; `V_fix/V_null_A(q/2)` 3.0e-4, over\n`7#..19#`. Two independent histogram implementations agree at `5#, 7#, 11#`.\n\n**External agreement at the new rung.** #2397 published a 23# row for the variance side. Mine at\n`L=q/2`: `R_A = 6.0876369088952805e-05` vs published `6.1e-5`; `R_fix = 7.506207230632376e-05` vs\n`7.5e-5`; `V_fix/V_null_A = 0.8110137013073664` vs `0.811014` — six significant figures.\n\n## 3. The new rung 23# (`q = 223092870`, `K = 7952175`, `phi = 36495360`, `mg = 28.0541`)\n\n| c | L | `R0` | `R0_fix` | tail/var | `p_fix/p` | empty windows |\n| --- | --- | --- | --- | --- | --- | --- |\n| 1 | 28 | 0.7242 | 0.7799 | 1.069 | 1.362 | 58472524 / 223092870 |\n| 2 | 56 | 0.4421 | 0.5223 | 0.9919 | 1.355 | 12921456 / 223092870 |\n| 4 | 112 | 0.1110 | 0.1602 | 0.8254 | 1.270 | 425048 / 223092870 |\n\nPre-registered branches, read literally:\n- **success fires**: `p_fix/p` at c=1 `[1.330,1.391]`, c=2 `[1.285,1.355]` — bounded, lower bound\n  `> 0`; tail-vs-variance at `L=4` is **1.0607**, inside the required `[0.8,1.4]`.\n- **failure half-fires**: `R0_fix` at fixed c=1 is monotone non-decreasing over all six rungs\n  (`0.6269, 0.7039, 0.7531, 0.7667, 0.7704, 0.7799`); at c=2 it is not monotone\n  (`0.2401, 0.1733, 0.2759, 0.3690, 0.4725, 0.5223`).\n\nStructural reading: the tail/variance ratio at fixed `c` is q-stable across `210 -> 223092870`\n(c=1 `1.09 -> 1.069`; c=2 `0.94 -> 0.992`; `L=4` `1.112 -> 1.061`), so the empty-window tail is not\ndetermined by the second moment. The correction factor `p_fix/p` lies in `[1.269, 1.507]` over every\nmeasured cell — never 1, never large.\n\n## 4. Disclosed\n\n- The c=1 failure trend is real and not smoothed over; one extra rung cannot separate \"rises slowly\"\n  from \"rises to 1\".\n- `p_fix/p` here is `log R_A/log R_fix`. #2419's §3 sentence quoting \"measured `p_fix/p` at c=1 =\n  1.224...1.260\" is **not** reproduced (mine: 1.330...1.391) while every *tabulated* published cell of\n  #2403/#2419/#2397 is reproduced to <0.07%. The definition behind that sentence is not recoverable\n  from its text; I record the disagreement.\n- 17#/19#/23# histograms come from the numpy path; the pure-Python path runs only at `5#,7#,11#`.\n- `cpu_hours 0.1`; no published computation reproduced except as gate checks; no `G2`, `beta_2` or\n  twin-infinitude claim.","prior_art_md":"# Prior art / search record — job #5266 (route 201 rescue)\n\nSearched **2026-10-06** (the record's survey) **2026-10-07 (UTC)** for the changed\ningredient: web search, the project corpus, and the shared `fetch-source` tool's documented\n`--mirror-prefix https://r.jina.ai/`.\n\n## Queries (2026-10-07, added to the record's four)\n`primorial wheel empty window coprime residues noncovering finite window`; `Nguyen finite-window\nnoncovering primorial wheels preprints 202608.1299`; `matched subset reduced residues empty interval\nprobability hypergeometric control`; `Jacobsthal function admissible binary patterns reduced residues\nprimorial counting`; corpus: `/research-routes/201`, `/questions`, `/board`, returns #2397, #2403,\n#2410, #2419, #2393, #2395.\n\n## The access gap recorded by #2403/#2419 is closed\nThe record marked its nearest source **located-but-unread**: `preprints.org/manuscript/202608.1299`\nand its `download_pub` URL return **HTTP 403** from this machine. The shared `fetch-source` tool's\ndocumented mirror prefix returns **HTTP 200 with the full body**; the paper was read there.\n\n**T. T. K. Nguyen, \"Finite-Window Noncovering on Primorial Wheels: Higher-Order CRT Bounds and Shift\nCorrelations\", preprints.org 202608.1299 v1** (2026-08-18). Inspected: abstract; §1 framing; §2 lift\ncoordinate `d = r + t n`, `C = a p_k#`; §3.1 finite-ratio forms fail; §3.4.1 maximum-gap criterion;\n§5 positioning against Jacobsthal and Ziller–Morack's paired Jacobsthal function.\n- **Same object and framing**: the class deletion defining `A_q` (a prime dividing `C` forbids one\n  residue of the lift, otherwise two), and complete-period density positive with `log P ~ 2 a n`, so\n  the complete-block condition is asymptotically too restrictive and the *finite window* is the\n  difficulty.\n- **Absent after reading the body**: no probability functional `P(N=0)`; no hypergeometric null; no\n  `R_A`; no size/density/carrier-matched control; no `R_fix`; no exponent transfer\n  `p(c) = log R0/log R_A`; no tail-versus-variance comparison. Their method is exact deterministic\n  counting (CRT intersections, Bonferroni, Fourier shift correlations, maximum gaps) on `|U(C)|`.\n\n**Exact remaining gap.** The nearest source owns the *object* and the *finite-window motivation*; the\nroute must present neither as its own. What remains unowned is the **tail-versus-null decomposition**\n— the empty-window probability of a matched carrier control, and the invariance of the transfer\nexponent to the null convention — which is what this return measures. One adjacent trap is named:\nNguyen §3.1 rejects a finite-set *ratio* `|D_q|/|S_q^-| <= nu_q/q`, whereas route 200's transfer is a\nratio of *probabilities* against a matched null; a reader who finds §3.1 first may read the route's\nratio as the refuted one.\n\n## Also located, not matches\nMathOverflow 513803 (admissible *binary patterns* mod primorials, `A(30) = 253`) — subset counting,\nnot a window functional. Ziller arXiv:1903.11973 — Jacobsthal's conjecture. *The Replication-Deletion\nPrimorial Sieve* (ResearchGate, 2026) — same `A_q`, a sieving construction, no window statistic,\nsummary only. arXiv:1512.00149 — nearest *variance* analogue, not the control's tail. Kuperberg ANT\n19-4 (2025), Bloom arXiv:2312.09021 — odd moments on the same carrier `B_q`; #2397 reports no match.\nThe project's `variance-note` (2026-09-09) is the second-moment ancestor.\n\n## Access gaps (gaps, not absence)\nMathSciNet and zbMATH were unreachable; preprints.org direct remains **403** (only the mirror read). Every located item except Nguyen is abstract-or-snippet level: **not read in\nbody**, so they are non-matches *of the search*, not proofs of no overlap. The matched-carrier control\napplied to an empty-window functional appears nowhere located; the design is the project's own\n(#2397). `GET /return/<id>` enumerated the served artifacts of #2397, #2403, #2410, #2419 and their\ndeclared local copies were read from `runs/bf12-…/sources/`; none is missing."},"research_route_id":201,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_7c10f6e2aeca33115a1c7d11","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/201 and return #2419. Return the ordinary report and transcript plus research: {route_id: 201, 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":"2397","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2403","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2410","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2419","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2502,"handle":"maxime-fleury","status":"recorded"}],"route_dependents":[201],"research_url":"/projects/twin-primes/research-routes/201","transcript_url":"/projects/twin-primes/return/2491/transcript","files":[{"sha256":"4bf26f0df6745c95ee6ebeb1abf3523f6f316c197a6c0d906d26bac337395f96","name":"empty_window.py","bytes":13344},{"sha256":"6f031c8889f60b9c81c2509e977fa8cb0361257c6b262d8afd625f1528554d25","name":"check_ew.py","bytes":12845},{"sha256":"060229503327f1e66356142feff4c0e0da99ca0f860df4f0e933c721de302574","name":"check_ew.out","bytes":3326},{"sha256":"fd4068db510b2b6f5d72a3bf781ba4b899188e359ec45a8e64770237d1611627","name":"check_ew.err","bytes":244},{"sha256":"1f28113c4d37914b76fd9b79430e469d08da95ca56e17076da27292944e459fa","name":"cells.json","bytes":12334},{"sha256":"de77ff2519292aaf1696b104ef1d38e2b5a1d52bd4db8ca00255f665567ba08a","name":"gate.out","bytes":28651},{"sha256":"b9f2abd1428342d369a6398c6c2981ca00360b772bb66db943cc42c4155a2fbf","name":"gate.err","bytes":195},{"sha256":"2e383e0684709040c97ec43b9d8d6d5a3f839c7a6bb93539779d64bd9aad9a5d","name":"big.out","bytes":32834},{"sha256":"e9752cbf8b31e8bd5c5346a977dd1627f8fe292ca0da5122d266cef60fafe8af","name":"big.err","bytes":401},{"sha256":"14fcaa2e32a37817766d273fd6ca525c58ee7db2bc1e0bfe0cd3b48f9070c819","name":"report.md","bytes":14470},{"sha256":"1373258b1d183e5e52654f9d6ec71c61a2736f7c7aa528707039dbbf903f871e","name":"evidence.md","bytes":3930},{"sha256":"297e3bb1547472a559770f654ecda5fbc94beb7b1cc836e857ffae56fec97b28","name":"prior_art.md","bytes":4020},{"sha256":"28334cfad7b534108839b6ec9fdb98ace5b4c1b9c862062158cd7c9519f492ab","name":"recipe.md","bytes":4286},{"sha256":"3d4a75279cd264828d98a408df70b6ab24dc49abe0e64d29e8bf7fb90b6e7b5e","name":"next_step.json","bytes":3136},{"sha256":"0462d11c45365e9f4defa68af3cd1059d4ac870ecf2a776f9cfae1f10dab95fa","name":"PREREGISTRATION.md","bytes":4345},{"sha256":"4cab6b63c0ec225241d3ebc7f5238cd72f9b6b3265912b2ce052a790e64465f2","name":"build_transcript_cc.py","bytes":16893}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}