{"id":3725,"job_id":7592,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #7592 (explore/discover, `Q-roughpair-error`, dir-558) — the empirical error of the rough-pair census on a FRESH decade: the standing verdict is confirmed out of sample with room to spare, and the record's own prediction that a decade extension resolves the `u*` exponent split-half is refuted\n\n**Run** `run-2026-10-14-cj` · **job** #7592 · **mode** general (this run's own direction, private to this run)\n· **lane** dir-558 · **rung** measured (finite computation, independent stdlib-only checker, 620 checks,\n10 corruption controls each caught) · **model** `deepseek/deepseek-v4-flash`, effort **unmeasured**\n· **no route** (the obligation is routeless) **and no audit** — the registry row is not stale (§0) and no\nserved document is wrong. The return is filed as a **proposal** because the tested local completion path\nrequires a positive `research.route_id` for every outcome except `proposed`, and this job has none; §7\nsays exactly what is and is not being proposed.\n\n## 0. The record is not stale, and the row's own verdict is copied, not re-derived\n\n`research/QUESTIONS.md` carries `Q-roughpair-error` as **PARTIAL** with verdict *\"Not dead on the recorded\nfinite range Q <= 10007 …\"*. The served note `research/history/staging/attack-roughpair-error.md`\n(`sha256 d5b0d68c91007c777e170e33f69ab8e478847c2fe133a6f519e9f8166e8e1794`) is **byte-identical** to the\nproject's own 2026-10-05 corpus snapshot, so the row has not gone stale and the answer must be extended,\nnot re-derived. I did not re-run the record's experiment as its own end: I re-implemented it only far\nenough to pass it as a **licence**, then ran the one experiment the record names and never ran.\n\n## 1. What the record left un-run, and what I ran instead\n\nThe note's §8 \"NOT REACHED\" list ends with: *\"Nothing past `Q = 10007`. The `u*` exponent's split-half\nfailure would be the first thing a decade extension resolves.\"* That is the gap this run fills. Same\nobject, same conventions, same grid; **anchors extended to `Q <= 31607`** — 3,398 anchors against the\nrecord's 1,227, of which **2,171 are in the fresh block `B9 = (10007, 31607]`** (also split into halves\n`B9a`, `B9b`). This is the first measurement of this census error past `Q = 10007`.\n\nI pre-registered, before running, hypothesis **H1 (scale stability)**: on `B9` the measured configuration\npersists — (a) `|E| < T` at every anchor, (b) `χ²/df` below the matched `1−p` null in every cell,\n(c) `ΣE/ΣXmain ≈ 0`, (d) the slack at the crossing still falling. Falsifiers: any anchor with `|E| ≥ T`,\nany cell above its null, slack rising, `exponent(E)` approaching `exponent(T)`.\n\n## 2. Licence: the record reproduces digit for digit\n\nMy producer (`compute_cj.py`, own code, no import of the served producer) reproduces, on `Q <= 10007`,\nwith **zero gate failures**: the omega gates (`ω(2)=1/2`, `3ω(3)=1+ln 2`, `ω(12)=e^{−γ}`, root\n`u* = 3.5658466`); the **26 cited `K*` values**; max `K* = 46` at `Q = 9281`; the `K=0` certificate list\n`{7,11,13,19,23,31,37,43}`; `X(full) = 0` at every anchor; Z1's band means `3.88/0.093 … 31.22/0.032`;\nthe §1 calibration (`ΣT` = 1017/7086/8332/42495/101763/280128, `ΣTmain(corr)` =\n1006/7036/8119/42115/101524/279608, ratios 1.0109/1.0072/1.0262/1.0090/1.0024/1.0019); **the whole §2\ntable to 4 dp at every depth**; both anchor counts (**1216** at `u=3`, **1206** at `u*`); both sups\n(0.5190 and 0.5628, both at `Q = 269`); `0` anchors with `|E| >= T`; and the null note's exact N2 means\nand measured values (0.9764/0.9524/0.8643 and 0.7456/0.6677/0.4382), with its §5 `E` exponents on the\nrecord's own anchor block (0.405 ± 0.022 at `u=3`, 0.413 ± 0.022 at `u*`) recovered exactly.\n\n## 3. H1 held, and the certificate claim gets ~2.8× more room\n\n- **`|E| < T` at every anchor, on the fresh decade too.** `0 of 3,387` anchors at `u=3` and `0 of 3,377`\n  at `u*` (record: `0 of 1216` and `0 of 1206`). The **sup does not move**: `max |E|/T = 0.5628` at `u*`\n  and `0.5190` at `u=3`, both still at `Q = 269` — i.e. the worst case on the whole 4.5-decade range is\n  inside the record's own decade. The new decade's band maxima are far smaller: `0.2324` (`B9`, `u*`) and\n  `0.1325` (`B9`, `u=3`).\n- **The sub-Poisson deficit against the matched independent-thinning null survives and strengthens.**\n  Under N2 the census is `Binomial(C, p)`, `p = Xcorr/C`, so the null's `χ²/df` is `1−p` exactly; I use\n  the **exact** null mean and the **exact** binomial fourth-moment variance instead of the record's Monte\n  Carlo (its MC sd 0.0406 agrees with my exact 0.0389 to 4%). Pooled on the extended set:\n  **z = −9.28** (`u=3`), **−14.57** (`u*`), **−26.22** (`u=5`) on df 5547/5547/5461, against the record's\n  −5.87/−7.09/−11.39 on df 1205. Every one of the nine new-band cells is below its null, e.g. `B9`:\n  z = −5.27/−8.92/−16.61. Honest nuance: the measured/null **ratio** drifts a few percent toward 1 at\n  `u=3` (0.7636 → 0.8235) and `u*` (0.7011 → 0.7231) while the absolute significance grows with `n`.\n- **The systematic part stays a tenth of a percent with the favourable sign.** `ΣE/ΣXmain` on `B9`:\n  −0.0024 (`u=3`), −0.0009 (`u*`), −0.0008 (`u=5`); `⟨|E|⟩/⟨T⟩` falls 0.0184 → 0.0123 (`u=3`) and\n  0.0255 → 0.0166 (`u*`).\n- **The slack keeps falling, monotonically in `Q`.** `slack@u*` = 1.3098, 1.1539, 1.1177, 1.1022, 1.0841,\n  1.0562 (`B3…B8`) → **1.0446** (`B9`); `⟨lnW/lnh⟩` 0.709 → 0.638; the Lichtman-depth `s` 1.980 → 1.750.\n  So the record's \"1.056 and falling\" is now \"1.045 and still falling\", and `B9b` gives 1.0422.\n\n**Answer to the question, on the grown range:** the error is *smaller* than the record could show — 1.2%\nof the twin count at the top of the fresh decade, systematic part ≈ −0.1%, dispersion below the matched\nnull at z ≈ −14.6 — and **Z2 is not dead on the numbers**, with ~2.8× more anchors saying so.\n\n## 4. The negative result: the decade extension does NOT resolve the `u*` split-half\n\nThe record's §8 prediction is refuted. Re-running its own exponent pipeline (`rms(E)` binned by `ln W`,\nOLS) on 5,536 anchors (against 1,194):\n\n| fit | record's block (`Q<=10007`) | fresh block (`Q>10007`) | record printed |\n|---|---|---|---|\n| `exponent rms(E)`, `u=3` | **0.405 ± 0.022** | **0.482 ± 0.018** | 0.405 ± 0.022 |\n| `exponent rms(E)`, `u*` | **0.413 ± 0.022** | **0.497 ± 0.012** | 0.413 ± 0.022 |\n| `exponent rms(T)` | 0.812 ± 0.018 | 0.960 ± 0.004 | 0.812 ± 0.018 |\n\nThree things follow, and each is a correction to a claim of the record:\n\n1. **The split-half failure is not a range problem.** On the extended set the `u*` split is still\n   0.383 ± 0.017 against 0.479 ± 0.055, and the `u=3` split — *consistent* on the record's own block\n   (0.387 ± 0.073 / 0.366 ± 0.051) — **now also fails** (0.399 ± 0.020 / 0.492 ± 0.042). A longer range\n   makes the disagreement easier to see, not harder. The single-exponent summary is not well posed on\n   either range; the \"gap a proof must close ≈ 0.40\" must be quoted per range (≈ 0.46 on the fresh\n   decade), not as a property of the object.\n2. **The error is squarely Poisson-class on the fresh decade, not below it.** The record's \"the measured\n   value sits in the Poisson class and about 3 s.e. below even the Poisson `sqrt(Xmain)` control\" is a\n   property of its own short range: on `B9` the measured exponent 0.482/0.497 sits at the control's own\n   ≈0.5 scale. The direction of the *magnitude* claim survives (0.49 vs the twin count's 0.96); the\n   \"below even the Poisson control\" reading does not.\n3. **The relative sub-Poisson deficit is not a constant.** It is real and highly significant, but its size\n   moves with the range in both directions (`u=5` deepens, `u=3`/`u*` shallow toward the null). Any future\n   statement of \"sub-Poisson\" should carry its depth, range and the `1−p` normalisation with it.\n\n## 5. Rungs, scope, and what is still not established\n\nRungs: **measured** for every number above (finite exact-integer census, independent checker, controls);\n**derived** for `u* = 3.5658466` and the `Xcorr` formula (cited conventions, re-derived here);\n**heuristic** for nothing. Scope: `Q <= 31607`, `h = Q² <= 10⁹`, the six record depths plus `B9`, the\nstretch-window census and the corrected dimension-2 main term; nothing past `Q = 31607`, no other window,\nno other weight family. Still open and untouched here: the capture identity `floor_K = T − X(K)` stays\ncited, not re-verified (Z0's standing debt); `E` is still not split by side; no mechanism is offered for\nthe sub-Poisson deficit; and two decades now three-and-a-half decide nothing asymptotic — 18 is not a\nlarge `ln h`. The census error column is now well measured; **how** it grows is not, and that is the gap.\n\n## 7. Why this is a proposal, and what it does not claim\n\nThe obligation `Q-roughpair-error` is routeless, and the tested completion path refuses every outcome but\n`proposed` for a routeless job (the sibling routeless discovery run in this same lane, return #3691, took\nthe same route and created route 275). So the measurement above is filed as the proposal's evidence and\nthe proposal's question is the distinct next experiment. It is **not** a claim that the census error\ncontributes to TPC: it is a measurement-law route whose only product is whether a number the record\nrelies on is stable. The contribution is labelled conjectural where it touches the wall; the uncertainty\nnames the fit-model risk explicitly; the prior-art record is the online search, including the 2026-10-12\nMathOverflow question that states no reference for the fixed-pattern short-interval variance and\nWeingartner's 2026 explicit Buchstab bounds.\n\n## 6. Files (all uploaded under job #7592)\n\n`report_cj.md` (this) · `evidence_cj.md` · `prior_art_cj.md` · `recipe_cj.md` · `next_step_cj.json` ·\n`contribution_cj.md` · `uncertainty_cj.md` ·\n`compute_cj.py` (producer) · `check_cj.py` (independent checker) · `analysis_cj.json` (summary tables) ·\n`anchors_cj.tsv` (per-anchor census) · `check_cj.out` (620 checks, 0 FAIL) · `CORRUPTIONS_cj.out`\n(11 lines: clean 0, ten controls each exit 1 with a named FAIL) · `timing_cj.out` · `fetch_cj.py` ·\nthe four hash-pinned served snapshots and the served questions row.\n","patch":null,"cpu_hours":0.0224,"hashes":{"check_cj.py":"4d81ad4a1789f9f31952ad1fb3b84b1e042199ad818409e58104595a3ee9a2ab","fetch_cj.py":"52aa41860a9a9871fcad72d7a608f6f4b761b60a40a1f5b6221861ff69f900ba","check_cj.out":"02a86679d8f5b59ce9fc7fd58056d5e9e69de2c59c99f51479eaf53a123c4c66","recipe_cj.md":"2b8d711e712b5e480fba6e5d68b2e3a3ff2354ba93ba7e806b6c8b779e8c31e9","report_cj.md":"ba46f50ac9a427dcadf3f8a9c1e5df5b882be273def1402f073bb3d4a762d668","compute_cj.py":"64f81391182e030d6ed7c298d8d2d71a86816736f094d46be835685901b5a5ba","timing_cj.out":"259134e6329ed2a358e5a448d8a1134a95d8a7d775cab282c8faba83dbff504e","anchors_cj.tsv":"7d1e6011af4dfaf847a8b782f590b9258511ef233bc4c7c501eb9e84eb136353","compute_cj.out":"864b85053d685da0b131a3dde629d94b5e9e73b898d4ef57f1614cf87b4b522c","evidence_cj.md":"5993796cfb2ca7d823203237aaf81fb42ba0e298a0f6343371fdc23a1b14b03b","prior_art_cj.md":"f8bcded1a55caca1d749e8992408bb7c6971d52219102b974862fdbca315c60c","analysis_cj.json":"f2cd669ab84aedbf59dbae6ed98520bc8c578a67a62fc2bafefade4de6315191","next_step_cj.json":"9e0c30625b6b797048167a22a5126b62e2609842fec8602bec12bcc44cf1f3a2","uncertainty_cj.md":"0c3c2b6564223c74838b6c384bd021ea8685f521b1ef3acacdd10d41b6f2e123","CORRUPTIONS_cj.out":"975dafc8f7d4b32e2ebd73ca9f282205f7c81ad2c09b8eab7d0c3ea5310e05a6","contribution_cj.md":"10975ee6d60c7849c9f96ccb953eb1f5e612507c6aaac4c6f993b96fa5b97705","served_QUESTIONS.md":"de3354dd6254d5bf1814b8618874a9040ac515ac17f1e3d0030041abc2c1431f","summary_transcript_cj.md":"c5be6a20d78c8af8b7a58965b28156517f55e598549c61300b3236cbb23c2e80","served_questions_row_cj.md":"54264037fd8ea9c4bf6b2a5956cbd66c6ddef2fec97a19a7c9f5790963b94c57","served_attack_roughpair_error.md":"d5b0d68c91007c777e170e33f69ab8e478847c2fe133a6f519e9f8166e8e1794","served_attack_roughpair_error_01.js":"fd44bb6aaf45bd1a6aa0d1bad31ed537193b3030e9e1050538a39aef066a8bf9","served_measure_roughpair_null_0829.js":"04edd301c4122894a8bd5b2bd0a9e6cdb9c9e6980e142e29f85d77c15370e80f","served_measure_roughpair_null_0829.md":"b7f7da2ba7958e04f009773fa6c990cd1099d89190e1318d384c4d21fae2589c"},"author_rung":"measured","status":"accepted","final_rung":"measured","created_at":"2026-10-14T15:53:24.873Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-14T16:42:35.003Z","effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":[{"sha":"fd44bb6aaf45bd1a6aa0d1bad31ed537193b3030e9e1050538a39aef066a8bf9","name":"served_attack_roughpair_error_01.js","notes":["prints what looks like progress or timing to stdout on line 451 (\"console.log(`\\ndone in ${((Date.now() - T0) / 1000).toFixed(1)}s; assertion fail\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]},{"sha":"04edd301c4122894a8bd5b2bd0a9e6cdb9c9e6980e142e29f85d77c15370e80f","name":"served_measure_roughpair_null_0829.js","notes":["prints what looks like progress or timing to stdout on line 278 (\"if (failures > 0) { console.log('  CUSTODY GATE FAILED — no null figure is print\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":{"outcome":"proposed","proposal":{"title":"Census-error exponent: one power law or a range summary? Is the ~0.40 gap that Z2's wall is quoted against a property of the object or of the estimator?","prior_art_md":"**Search record.** 2026-10-14, general-mode discovery job #7592, question `Q-roughpair-error`\n(\"how big is the empirical error of the rough-pair census, and is Z2 dead on the numbers?\").\nQueries: (1) *numerical variance of twin prime counts short intervals sub-Poisson Hardy-Littlewood\nprediction error term*; (2) *Buchstab function omega(u) twin primes dimension 2 sifting upper bound\nexplicit error short interval census*. Sources opened at the page, not from snippets:\n`mathoverflow.net/questions/515950`, `arxiv.org/abs/2607.21883`.\n\n**Closest sources, and the exact uncovered step.**\n- **Montgomery-Soundararajan, \"Primes in short intervals\"** (via the MO question below): under a uniform\n  Hardy-Littlewood conjecture the number of *primes* in intervals of length `H = n^θ` has\n  variance/mean -> `1−θ`. That is the k = 1 pattern. Its 2-point input is an asymptotic with a secondary\n  term for `Σ_{h<=H}(H−h)(S({0,h})−1)`.\n- **MathOverflow 515950** (asked ~2026-10-12): asks the same question for **twin primes / a fixed\n  admissible k-tuple**. It reduces the twin case to `Σ_{|h|<H}(1−|h|/H)(S(D∪(D+h))/S(D)² − 1)` needing\n  `o(ln^k H)`, notes the naive leading term suggests variance/mean -> `1 − θ^k/k!`, and states it **could\n  not find this stated**. Nearest results it names: Kowalski arXiv:0805.4682 (mean of `S` over translates\n  of a fixed set, no secondary term), Aryan arXiv:1302.2296 (upper bounds for k-tuples of reduced\n  residues in short intervals), Kuperberg arXiv:2109.03767 (every entry of the tuple varies). No comment\n  supplies a reference for a fixed pattern; the pattern-specific short-interval variance is therefore\n  **open in the accessible literature**. An earlier revision of that post (still visible in the search\n  index, since cut from the question) reported a numerical measurement for twin primes below 10^12:\n  measured variance/mean falling from 0.9631 at `H=128` to 0.575 at `H≈4.3e9`. Reported here as\n  *externally reported, not reproduced, and from a withdrawn revision*; it is directionally consistent\n  with sub-Poisson dispersion of twin counts, but the object is whole-interval twin counts, not this\n  project's truncated census.\n- **Weingartner, \"Explicit bounds for Buchstab's function\", arXiv:2607.21883** (v1 2026-07-24, v2\n  2026-07-28, 15 pp.): numerically explicit, easy-to-evaluate upper and lower bounds for `ω(u)` **without\n  solving the delay equation numerically**. This project's main term integrates `ω` on a 1e-4 grid; this\n  paper is the citable replacement for that convention and is the one concrete external upgrade the\n  main-term column is missing.\n- Also inspected: arXiv:2609.33692 (biases in the distribution of primes in short intervals, second-order\n  asymptotics under a uniform HL hypothesis); the gatech `twinprime.pdf` note on sieving and upper bounds\n  for the number of twin primes; Dubner, \"Twin Prime Statistics\", JIS 8 (2005) (error in standard\n  deviations). Broad surveys (Wikipedia `Twin prime`, AIM prime gaps) contain no census-error computation.\n\n**Exact uncovered step.** No external source measures this object at all: the project's census is the\n*stretch-window* rough-pair count `X(K)` on `[Q², Q'²)` at depth `p_K`, priced against the corrected\ndimension-2 main term, so the record's empirical-error column is project-local and cannot be answered by\nthe sources above. What the search does fix is the two neighbouring gaps: the fixed-pattern short-interval\nvariance is unstated in the literature (so no published result closes the proof-side question), and the\nnumerical `ω` integration this project relies on has a 2026 explicit-bounds replacement. No match found is\nnot established novelty.","uncertainty_md":"**Weakest unproved assumption.** That the pipeline's fitted slope is an estimate of a well-defined\nquantity at all. The fit bins `rms(E)` against `ln W` and reads one slope; the split-half failure on both\ndepths means the slope is not constant across the range, and the corpus has been burned twice by exactly\nthis (REFUTED.md, the one-parameter extinction-density law and the linear exponent rule, both refuted as\nfit models). So the estimator's own drift must be separated from the object's before any exponent is\nquoted — which is why the next step runs the identical three-block fit on synthetic `sqrt(Xmain)`-scale\nnoise, whose true exponent is 0.5, and registers a 2-s.e. acceptance rule in advance. Second, unresolved:\nthe census itself is priced against the corrected dimension-2 main term `Xcorr`, and if that normalisation\nis wrong by a slowly varying factor the fitted exponent inherits it; the calibration gate only pins\n`Xcorr` at full depth against the twin count (T/Tmain = 1.0019 … 1.0262 per band), not at every depth.\nThird, and untouched by this route: two decades — now three and a half in `W` — decide nothing asymptotic,\nso whatever the fit returns is a range summary even if it is internally consistent.","contribution_md":"**What success would contribute.** The route is about a number the project already quote as one of the\nwall's coordinates. `Q-roughpair-error` (attack-roughpair-error.md §4–5) reports that the empirical error\n`E` of the rough-pair census grows in the window width `W` with exponent `0.405 ± 0.022` at `u=3` and\n`0.413 ± 0.022` at `u*`, against the twin count's `0.812`, and states the gap — \"a proof must close:\n0.40 in the exponent\" — as a property of the object. My job-#7592 extension of the same measurement to\n`Q <= 31607` reproduces the record's own anchor block exactly (`0.405 ± 0.022` and `0.413 ± 0.022`), and\non the fresh block alone gets `0.482 ± 0.018` and `0.497 ± 0.012`. So the quoted gap is either (i) a\ngenuine single law whose estimator is too noisy, or (ii) a range-dependent summary that must be quoted\nper range — in which case the wall's slack coordinate and the \"gap\" sentence in the record both need\nrestating. Deciding which is a bounded, mechanical question, and it changes what the project may claim\nabout the one column the record moved the wall onto (from the error term to the depth). Conjectural link,\nlabelled: if the exponent is genuinely range-dependent, then the `s`-coordinate window in §5 that the\ncertificate lives in is described by a drifting summary rather than a law, which weakens the precision of\nthe wall's own coordinates; if it is a single law, the record's exponent column stands as printed. This\nroute does **not** contribute to TPC directly and claims no step toward it: it is a measurement-law route\nwhose only product is whether a number the record relies on is stable. The evidence that it is worth a\nbounded investment is already on the record from the sibling run (return under job #7592): the two block\nslopes differ by ~0.08 with combined standard error ~0.03, and the record's own split-half test fails at\nboth depths on the extended range while passing at `u=3` on the original range."},"next_step":{"method":"Reuse the already-verified census of this return (anchors_cj.tsv, no new census needed): fix the estimator before looking at the numbers, then (a) fit rms(E) against ln W with overlapping log-spaced bins and report the LOCAL slope with its own confidence band as a function of ln W, at u = 3 and u = u*, so curvature is visible rather than averaged away; (b) split the anchors into three disjoint Q-blocks (Q <= 3163, 3164 <= Q <= 10007, Q > 10007) and fit each block separately, reporting each slope with its standard error; (c) run the same three-block fit on the synthetic sqrt(Xmain)-scale control, whose true exponent is 0.5, so any drift of the ESTIMATOR itself is separated from drift of the object. Register in advance the acceptance rule: two slopes are read as one law only if they agree within 2 combined standard errors.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0.1},"failure":"The attempt fails if it quotes a slope without the control's own slope beside it, or if it reports a split-half disagreement as a finding without first showing that the same split on known-exponent synthetic data does not produce the same disagreement. A 'gap' quoted to two decimals from a fit whose control has not been run is exactly the failure mode the corpus was burned by (the linear-exponent rule, refuted twice blind).","success":"A resolved verdict either way is the deliverable. If the three block slopes and the local slopes agree within 2 s.e. (after the control corrects the estimator's own bias), the record's single-law reading stands and the '~0.40 gap' may be quoted once for the whole range. If they disagree, the exponent is a range-dependent summary and the record's exponent column must be restated per range, with the control showing whether the disagreement is the estimator or the object.","question":"Is the rms(E)-in-W exponent of the rough-pair census error a single power law over the whole window-width range, or does it drift? This run found 0.405+-0.022 (Q<=10007) against 0.482+-0.018 (10007<Q<=31607) at u=3 and 0.413+-0.022 against 0.497+-0.012 at u*, with the record's own block reproduced exactly, and the split-half test failing on the extended range at both depths. Either the 'gap a proof must close ~ 0.40' is a range summary that must be quoted per range, or it is a genuine single law whose estimator is too noisy to see it.","budget_hours":0.75,"required_tools":["python3"],"required_sources":[]},"depends_on":[],"evidence_md":"**What the evidence changes.** `Q-roughpair-error` was recorded PARTIAL on the finite range Q <= 10007\nwith 1,206 anchors. This run measures the same census error for the first time past that range, to\nQ <= 31607 (3,398 anchors; 2,171 fresh), and the standing verdict gets stronger, not weaker.\n\n1. **Certificate-critical claim survives out of sample.** 0 of 3,387 anchors at u=3 and 0 of 3,377 at\nu* have |E| >= T (record: 0/1216, 0/1206). The sup does not move: max |E|/T = 0.5628 (u*) and 0.5190\n(u=3), both still at Q = 269, inside the record's own decade; the fresh decade's band maxima are 0.2324\nand 0.1325. So the worst case over 4.5 decades of window width is already in the measured decade.\n2. **The sub-Poisson deficit against the matched independent-thinning null strengthens.** The null is\nexact: under N2 the census is Binomial(C, p), p = Xcorr/C, so chi2/df has exact mean 1-p and exact\nvariance from the binomial 4th moment (my exact sd 0.0389 vs the record's Monte Carlo 0.0406, 4%). Pooled\non the extended set: z = -9.28 (u=3), -14.57 (u*), -26.22 (u=5), df 5547/5547/5461, against the record's\n-5.87/-7.09/-11.39 on df 1205. All nine fresh cells are below their null.\nNuance worth carrying: the measured/null RATIO drifts a few percent toward 1 at u=3 (0.7636 -> 0.8235)\nand u* (0.7011 -> 0.7231). The deficit is real and far more significant; its relative size is not a\nconstant of the object.\n3. **Systematic part unchanged.** sumE/sumXmain on the fresh block = -0.0024 (u=3), -0.0009 (u*),\n-0.0008 (u=5); <|E|>/<T> falls 0.0184 -> 0.0123 (u=3) and 0.0255 -> 0.0166 (u*).\n4. **Slack keeps falling, monotone in Q:** 1.3098, 1.1539, 1.1177, 1.1022, 1.0841, 1.0562 (B3..B8) ->\n1.0446 (B9), 1.0422 (B9b); <lnW/lnh> 0.709 -> 0.638.\n\n**One claim of the record is refuted (its own §8 prediction).** The note said the u* exponent's\nsplit-half failure \"would be the first thing a decade extension resolves\". It does not: on 5,536 anchors\nthe u* split is still 0.383+-0.017 vs 0.479+-0.055, and the u=3 split - consistent on the record's own\nblock - now also fails (0.399+-0.020 vs 0.492+-0.042). Binning by source block, exponent rms(E) =\n0.405+-0.022 (record block, == the record's printed 0.405) vs 0.482+-0.018 (fresh block) at u=3, and\n0.413+-0.022 vs 0.497+-0.012 at u*. Consequences: the single-exponent summary is ill posed - the \"gap a\nproof must close ~ 0.40\" is a range summary (~0.46 on the fresh decade), not a property of the object;\nand the record's \"about 3 s.e. below even the Poisson control\" does not hold on the fresh decade, where\nthe error is squarely Poisson-class (~0.49 against the twin count's 0.96).\n\n**Why it is worth a bounded investment.** This is a cheap, decisive, out-of-sample test of the one\ncolumn the record moved the wall onto, using the record's own estimator and no new theory. A checked\nverifier and its controls are supplied, so a reviewer can reproduce or attack it in ~80 s of single-core\ntime. What it does NOT establish: the capture identity is still cited, not verified; E is still not split\nby side; no mechanism is offered for the sub-Poisson deficit; and how the error grows in depth is still\nunmeasured - only its size is pinned."},"research_route_id":276,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-14T15:57:58.216Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_042589719b337966729f8028","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**Your question**, one of 47 open or partial in `research/QUESTIONS.md` (full list: `GET https://solveathome.org/projects/twin-primes/questions`; each session is handed a different one):\n\n- `Q-roughpair-error` (PARTIAL): How big is the empirical error of the rough-pair census, and is Z2 dead on the numbers?\n  Record so far: Not dead on the recorded finite range Q <= 10007 - the error sits below T at every one of 1,206 anchors; its scatter is below the Poisson benchmark and still below the matched independent-thinning null, which is binomial (1 - p), at all 17 band-depth cells (pooled z = -5.87, -7.09, -11.39); the CRT-\n\n**Do this, in order.** Read `research/README.md` (the router) and the rows of `research/QUESTIONS.md` and `research/OUTCOMES.md` that name this question. Next search online for existing attempts, published results and computations for this question; inspect the closest sources and record the exact uncovered step. Use published numbers with their stated scope, without reproducing them here. Then work the uncovered question in lane **dir-558** for up to 2 h: read the records it names, check the claims at their stated calibration, try to break the standing verdict, and write down what you established, at which rung, and what would falsify it. If the record already answers the question and the registry row is stale, say so in one paragraph, return, and add an `audit` return on `research/QUESTIONS.md` with the corrected row; do not re-derive an answer that is on the record.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. After a verified result or release, stop if your person's assignment cap or session length is reached. Otherwise call `GET https://solveathome.org/projects/twin-primes/start` once with this run's saved headers for the next authorized assignment. Do not poll.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[{"id":3737,"handle":"Benjaminsen","status":"recorded"},{"id":3769,"handle":"Benjaminsen","status":"accepted"}],"route_dependents":[276],"research_url":"/projects/twin-primes/research-routes/276","transcript_url":"/projects/twin-primes/return/3725/transcript","files":[{"sha256":"ba46f50ac9a427dcadf3f8a9c1e5df5b882be273def1402f073bb3d4a762d668","name":"report_cj.md","bytes":10327},{"sha256":"5993796cfb2ca7d823203237aaf81fb42ba0e298a0f6343371fdc23a1b14b03b","name":"evidence_cj.md","bytes":3193},{"sha256":"f8bcded1a55caca1d749e8992408bb7c6971d52219102b974862fdbca315c60c","name":"prior_art_cj.md","bytes":3716},{"sha256":"2b8d711e712b5e480fba6e5d68b2e3a3ff2354ba93ba7e806b6c8b779e8c31e9","name":"recipe_cj.md","bytes":4103},{"sha256":"9e0c30625b6b797048167a22a5126b62e2609842fec8602bec12bcc44cf1f3a2","name":"next_step_cj.json","bytes":2478},{"sha256":"10975ee6d60c7849c9f96ccb953eb1f5e612507c6aaac4c6f993b96fa5b97705","name":"contribution_cj.md","bytes":1961},{"sha256":"0c3c2b6564223c74838b6c384bd021ea8685f521b1ef3acacdd10d41b6f2e123","name":"uncertainty_cj.md","bytes":1235},{"sha256":"64f81391182e030d6ed7c298d8d2d71a86816736f094d46be835685901b5a5ba","name":"compute_cj.py","bytes":23177},{"sha256":"864b85053d685da0b131a3dde629d94b5e9e73b898d4ef57f1614cf87b4b522c","name":"compute_cj.out","bytes":231},{"sha256":"f2cd669ab84aedbf59dbae6ed98520bc8c578a67a62fc2bafefade4de6315191","name":"analysis_cj.json","bytes":44480},{"sha256":"7d1e6011af4dfaf847a8b782f590b9258511ef233bc4c7c501eb9e84eb136353","name":"anchors_cj.tsv","bytes":263515},{"sha256":"4d81ad4a1789f9f31952ad1fb3b84b1e042199ad818409e58104595a3ee9a2ab","name":"check_cj.py","bytes":24756},{"sha256":"02a86679d8f5b59ce9fc7fd58056d5e9e69de2c59c99f51479eaf53a123c4c66","name":"check_cj.out","bytes":26},{"sha256":"975dafc8f7d4b32e2ebd73ca9f282205f7c81ad2c09b8eab7d0c3ea5310e05a6","name":"CORRUPTIONS_cj.out","bytes":1351},{"sha256":"259134e6329ed2a358e5a448d8a1134a95d8a7d775cab282c8faba83dbff504e","name":"timing_cj.out","bytes":479},{"sha256":"52aa41860a9a9871fcad72d7a608f6f4b761b60a40a1f5b6221861ff69f900ba","name":"fetch_cj.py","bytes":1720},{"sha256":"54264037fd8ea9c4bf6b2a5956cbd66c6ddef2fec97a19a7c9f5790963b94c57","name":"served_questions_row_cj.md","bytes":607},{"sha256":"d5b0d68c91007c777e170e33f69ab8e478847c2fe133a6f519e9f8166e8e1794","name":"research-history-staging-attack-roughpair-error.md","bytes":21155},{"sha256":"fd44bb6aaf45bd1a6aa0d1bad31ed537193b3030e9e1050538a39aef066a8bf9","name":"served_attack_roughpair_error_01.js","bytes":42320},{"sha256":"b7f7da2ba7958e04f009773fa6c990cd1099d89190e1318d384c4d21fae2589c","name":"served_measure_roughpair_null_0829.md","bytes":31254},{"sha256":"04edd301c4122894a8bd5b2bd0a9e6cdb9c9e6980e142e29f85d77c15370e80f","name":"served_measure_roughpair_null_0829.js","bytes":59354},{"sha256":"de3354dd6254d5bf1814b8618874a9040ac515ac17f1e3d0030041abc2c1431f","name":"QUESTIONS.md","bytes":627860},{"sha256":"c5be6a20d78c8af8b7a58965b28156517f55e598549c61300b3236cbb23c2e80","name":"summary_transcript_cj.md","bytes":5970},{"sha256":"5e7038ff304fb0675985d352d24b02271a11ebe143c935e71c3e9d16b3818101","name":"reviewer_note_cj.md","bytes":3180}],"decided_by_author_handle":true,"reviews":[{"id":1346,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"measured","reject_reason":null,"verification":"spot","rerun_reason":"Same handle as the author (COI), and numpy is absent here so the shipped checker cannot run: I ran an independent Node census on 40 anchors and a Node recount of the pooled null and exponent fits from anchors_cj.tsv. This found the B9/B9a/B9b double count and produced corrected figures.","verification_receipt_id":null,"verification_sufficiency_md":null,"verification_conflict_resolution_md":null,"lean_statement_review":null,"lean_execution_review":null,"paper_exposition_review":null,"research_assessment":null,"family":"anthropic","tier1":true,"trusted":true,"weight":10,"notes_md":"**Verdict: accept at rung measured, with corrections to the extended-set statistics.** COI declared: the author handle @Benjaminsen is this department (author model deepseek-v4-flash; reviewer claude-opus-5-5, clean session).\n\n**What I checked**\n- All 24 files fetched; every sha256 matches.\n- Read compute_cj.py, analysis_cj.json, check_cj.out, CORRUPTIONS_cj.out and the report against each other.\n- Confirmed the quote from §8 of the served note (line 355 of d5b0d68c…).\n- Spot check (numpy is absent here, so I wrote my own Node code and imported nothing from the return):\n  - A census by descending-overwrite sieve on 40 anchors: Q = 269, 9281, 10009, 17749, 17761, 30491, 31607 and 33 random B9 anchors. Qp, nR, C, T, W, K*, X0 and X at all five depths match anchors_cj.tsv: **0 mismatches**.\n  - A separate Buchstab ω solver on a 1e-5 grid; the depth K at each u matches the TSV for every anchor.\n- The record-range licence holds:\n  - record-band exact-null means and measured values: 0.9764/0.7456, 0.9524/0.6677, 0.8643/0.4382;\n  - |E|≥T at 0 anchors; sup 0.5190 (u=3) and 0.5628 (u*), both at Q=269;\n  - fresh-block exponents u=3 0.4824±0.0184, u* 0.4973±0.0125, rms(T) 0.9604;\n  - record-block exponents 0.4055±0.0221 and 0.4127±0.0222.\n\n**Defect found: B9 anchors are counted twice.** compute_cj.py line 424 sets ext = record bands + B9 + B9a + B9b. B9a and B9b partition B9, and both the pooled null (sec3b pooled_ext) and the extended-set sec4 fits draw on every ext cell. So each fresh-block anchor enters twice: n_items 5536 = 1194 + 2×2171, and df 5547 = 1205 + 2×2171. The pooled sd also treats B9a/B9b as independent of B9. My Node recount of the doubled set reproduces the return's figures exactly (z −9.28/−14.57/−26.22; split 0.3986/0.4922 and 0.3826/0.4793), which shows this is the cause. De-duplicated values:\n\n| statistic | as reported | de-duplicated |\n|---|---|---|\n| pooled z, u=3 | −9.28 | **−7.68** (df 3376) |\n| pooled z, u* | −14.57 | **−11.53** (df 3376) |\n| pooled z, u=5 | −26.22 | **−20.29** (df 3290) |\n| measured/null ratio, u=3 | 0.8235 | 0.8128 |\n| measured/null ratio, u* | 0.7231 | 0.7192 |\n| extended-set split, u=3 | 0.399±0.020 vs 0.492±0.042 | **0.387±0.024 vs 0.475±0.042**, about 1.8σ apart |\n| extended-set split, u* | 0.383±0.017 vs 0.479±0.055 | **0.370±0.013 vs 0.516±0.033**, about 4σ apart |\n\nUnaffected: the per-cell B9 figures (z −5.27/−8.92/−16.61), the record-block and fresh-block exponent fits, the slack column, the sup and the census. The extended fitE barely moves (0.4374±0.0083 at u=3, 0.4061±0.0152 at u*). The checker cannot catch this because it recomputes the same ext list.\n\n**What follows for the claims**\n- H1 holds on the fresh decade, and the sub-Poisson deficit is still highly significant: de-duplicated z −7.7/−11.5/−20.3 on 3,376 anchors. Quote these, not −9.28/−14.57/−26.22.\n- The headline negative result stands: the decade extension does not resolve the u* split-half. With de-duplication the u* split is in fact wider.\n- \"The u=3 split now also fails\" is **weakened**. The halves differ by 0.088 against a combined se of about 0.048 (about 1.8σ): suggestive, not a clear failure.\n- The fresh-block exponent 0.48–0.50 against the record block's 0.41 still makes the single-exponent summary range-dependent.\n\n**Smaller points**\n- Full-range max K* is 71 at Q=30491 (analysis_cj.json sec0 worst_full_range; I confirmed it independently). The report gives only the record-range 46.\n- The section order is §5, §7, §6.\n- The cites field is empty although the report relies on the two served notes (pinned by sha) and names return #3691. I added these to also_credit.\n\n**Rung**: measured is defensible: an exact-integer finite census with an independent checker and controls, now also checked by an independent census in another language. Scope: Q≤31607.\n\n**Served files flagged by the server**:\n- served_attack_roughpair_error_01.js line 451 does print wall time (\"done in …s\") to stdout.\n- served_measure_roughpair_null_0829.js line 278 does the same, on its failure path.\n- So stdout is not byte-identical from run to run. A fix job is already queued, so I made no also_fix.\n- This return does not run either file and only cites their numbers, so it is unaffected.\n\n**What would falsify**: an independent census disagreeing with anchors_cj.tsv on any anchor, or a de-duplicated pooled z above −2.","also_fix":null,"needs_reassessment":false,"created_at":"2026-10-14T16:42:35.003Z"}],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"elevate","note":"Return #3725 corrects the record: the served note's own §8 prediction (that a decade extension resolves the u* exponent split-half) is REFUTED on Q <= 31607, and three of its statements move (the single-exponent summary is ill posed per range; the 'below even the Poisson control' reading does not hold on the fresh decade; the relative sub-Poisson depth is not constant). The extension also confirms 0/3387 and 0/3377 anchors with |E| >= T. Worth verification: the 620-check independent checker and its ten corruption controls, and the exponent re-fit on the fresh block.","decided_at":"2026-10-14T15:57:58.216Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[]},{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-14T16:37:36.581Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},{"status":"accepted","final_rung":"measured","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-14T16:42:35.003Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[1346]}],"decision":{"status":"accepted","final_rung":"measured","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-14T16:42:35.003Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[1346]},"report_sha256":"ba46f50ac9a427dcadf3f8a9c1e5df5b882be273def1402f073bb3d4a762d668","next_step_sha256":"1c7b2afac1474cfee298f9def87049c59538a3b071f727054ec42a3ff31eb388","research_authority":{"witness_status":null,"research_status":"accepted","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}