{"id":5097,"job_id":10067,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# The covering ceiling has a measurable structure-to-density ratio ≈ 0.50 — and 37# is its one excursion\n\n**Job #10067** (`explore`, discovery, lane `formalize`, general mode, assignment 1 of 1).\nGeneral mode, public run `run_b1eeac0989f43ce232b6e3d6`. One new statistic, a falsifier written before\nthe run (`PREREG.md`), a matched control, and the design cost for the one level this session could not\nscan.\n\n**Answer in one line.** The ceiling `G2(x#)` of the twin-admissible residue set is, at every level from\n`11#` to `67#` except `37#`, about **0.48 ×** the largest gap a *structure-free* set of the same size\nand carrier would have — `R(x) = G2(x#)/M0(x)` lies in `0.436…0.536` over 14 of 15 rungs (`0.594` at\n`37#`), where `M0` is the median largest gap of a size-matched uniform random subset of `[0,x#)` — so\nthe wheel structure buys a factor ≈2 at every level;\n**the pre-registered out-of-sample prediction failed at the first unmeasured rung** (`37#`: predicted\n`404.6`, actual `528`, −23.4 %), so a density-only model is a screen at the ±20 % level, not an\nextrapolator. `37#` is a single excursion (`R = 0.594`), not a trend (`41#` is back to `0.513`).\n\nNothing here is a statement about `G_2`, `beta_2`, primality or twins. `G2` values are **cited**\n(OEIS A144311, #4887/#4891) and independently recomputed at four rungs; only the *ratio* is new.\n\n## 1. What was asked, and which prior work was checked first\n\nThe assignment asked for **one finite statistic a run could actually decide something about, where the\nretained censuses could not**: the decision it informs, a pre-registered falsifier, a matched control,\nand the scale at which the effect would be visible.\n\nChecked before choosing (so as not to re-run closed lines, all local, all read):\n\n* the covering census itself and its ladder `G2 = A144311+1`, with `M`, the full gap histogram, the 6Z\n  quantisation and the near-max placement tail `c6` — #4887, #4891 (and #-ei's 53#/59#/61# censuses);\n* the *window* statistics of the same residue set: variance `R_A`, empty-window tail, tail clustering,\n  the fixed-size thinning control, and the **third cumulant** — #2393, #5124/`higher-order-additive-5100`\n  (which closed the cumulant channel: \"R3c is bounded and sign-alternating, no organised q-law\"), #5362,\n  #5113. That is why the statistic below is **not** a higher cumulant;\n* the actual-prime side: route 245's block variance `V(h)` and the twin-prime bias literature.\n  Online (2026-10-20): OEIS **A144311** (the definition — *length of the longest sequence of consecutive\n  integers each equal to 1 or −1 modulo at least one of the first n primes* — is exactly the covering\n  object, so `G2(x#) = a(pi(x))+1` is definitional, not a conjecture); **arXiv:2111.09053** (Sahoo 2021,\n  rev. 2023), *On twin prime distribution and associated biases*, which measures Chebyshev- and\n  Oliver–Soundararajan-type biases **in actual twin primes** and a bias in their consecutive gaps; the\n  Lemke Oliver–Soundararajan biases (PNAS 2016) it builds on. Those are statistics of the primes; none\n  of them is the wheel's covering ceiling, and no source found gives a density baseline for `G2`.\n  A no-match search is not a novelty claim.\n\n## 2. The statistic\n\n`S_x = {a mod x# : gcd(a(a+2), x#) = 1}`, `P_x = x#`, `n_x = |S_x| = prod_{2<p<=x}(p-2)`,\n`delta_x = n_x/P_x`. Let `G2(x#)` be the largest cyclic gap between consecutive elements of `S_x`\n(the covering radius + 1; `G2 = A144311(pi(x))+1`).\n\n    M0(x) = median largest cyclic gap of a size-matched uniform random subset of [0, P_x)\n    R(x)  = G2(x#) / M0(x)\n\n`M0` is computed two ways that must agree (control C1): seeded Monte Carlo on `[0,P_x)` and the closed\nform for the maximum of `n` geometric gaps, `M0 = ln(1-2^(-1/n)) / ln(1-delta)`\n(implemented as `log(-expm1(-ln2/n))/log1p(-delta)`).\n\n**The decision it informs.** Whether the covering ladder can be predicted at levels too expensive to\nscan (`37#` ≈ 6.7 h, `41#` ≈ 11.5 d with #4891's engine) from the wheel's *density* alone, or whether a\nlevel-dependent structural factor must be carried. `R = 1` would mean density explains everything.\n\n## 3. Results (pre-registered part)\n\nOne period, exact `n_x`, `delta_x`; `M0` by MC (400/400/200 seeded reps at 17#/19#/23#) and closed form.\n\n| x | n_x | M0 closed form | MC median (uniform) | G2 cited | **R** |\n|---|---|---|---|---|---|\n| 5 | 3 | 14.98 | — | 12 | 0.801 |\n| 7 | 15 | 41.80 | — | 30 | 0.718 |\n| 11 | 135 | 87.59 | — | 42 | 0.4795 |\n| 13 | 1485 | 151.24 | — | 66 | 0.4364 |\n| 17 | 22 275 | 232.62 | **233.0** | 108 | 0.4643 |\n| 19 | 378 675 | 331.75 | **332.5** | 150 | 0.4522 |\n| 23 | 7 952 175 | 447.86 | **445.0** | 204 | 0.4555 |\n| 29 | 214 708 725 | 579.30 | — | 258 | 0.4454 |\n| 31 | 6 226 553 025 | 726.70 | — | 348 | 0.4789 |\n| **37** | 217 929 355 875 | 888.16 | — | **528** | **0.5945** |\n\n* **H1 (in-sample constancy, band `[0.40,0.52]` at 17#…31#): HOLDS.** All five rungs inside, spread\n  `(max-min)/median = 9.5 %` (`<= 15 %` as pre-registered). Over 11#…31# (seven rungs) `R` stays in\n  `0.436…0.480` while `G2` grows 8.3× and `M0` grows 8.3×.\n* **H2 (out-of-sample, `G2(37#)` within 10 %): REFUTED.** Median `R` over 17#…31# is `0.4555`;\n  it predicts `G2(37#) = 404.6` against the actual `528` (−23.4 %), outside the pre-registered band\n  `[475, 581]`. **This is the falsifier firing**, not a rescoped claim.\n* **H3 (6Z carrier accounts for `< 0.05`): marginal.** Lattice-matched null (random subset of `6Z+5`,\n  #4891's 6Z lemma) gives `R_L = 0.514, 0.500, 0.500` at 17#/19#/23# against `R = 0.464, 0.451, 0.458`:\n  `|R_L − R| = 0.051, 0.049, 0.042`. So the quantisation explains **≈0.045 of the ≈0.54 gap to 1**\n  (about 8 %); the remaining structure gain is the CRT regularity itself, not the 6-lattice.\n\n**Post-hoc extension (not pre-registered; the verdicts above stand).** Taking the OEIS A144311 terms\n`a(13)…a(19)` (definitionally the same ceiling) gives `R = 0.513, 0.493, 0.486, 0.520, 0.508, 0.504,\n0.536` at `41#, 43#, 47#, 53#, 59#, 61#, 67#` — i.e. the plateau resumes at `≈0.50` and `37#` is a\n**single excursion** (+23 % over the median `0.483` of the other fourteen rungs), not a drift.\nThe `53#/59#/61#` ceilings in that list (870/966/1080) were independently computed by earlier censuses.\n\n## 4. Controls and instrument\n\n* **C0 — the ceiling convention.** This run's own sieve recomputes `G2 = 108, 150, 204, 258` at\n  `17#, 19#, 23#, 29#` and `n_x = prod(p-2)` exactly (29# took 431 s here); `258` was not previously\n  recomputed in this folder in this session. The 31#/37#… values are cited.\n* **C1 — the null's two paths agree:** `233.0 vs 232.62` (0.16 %), `332.5 vs 331.75` (0.23 %),\n  `445.0 vs 447.86` (0.64 %), all under the pre-registered 3 %. MC realized sizes are recorded in\n  `results_ej.json`.\n* **C2 — carrier control** (the 6Z lattice null) is the H3 test itself.\n* **C3 — planted corruptions.** `check_ej.py` recomputes every rung by a second path (direct residue\n  enumeration with gcd tests at 5#…13#; bisection instead of the closed form; delivered JSON re-derived)\n  and passes **22/22, exit 0**. Five planted corruptions each make it exit 1: ceiling +4 at 11# (4 FAILs),\n  `n` off by one at 13#, linearised logarithm in `M0`, wrong modulus in the null (12 FAILs), and mean\n  instead of the five-rung median in the prediction. A checker that cannot fail is not evidence.\n\n## 5. What this decides, and the gap that remains\n\n* **Density-only extrapolation of the covering ladder is closed at the 10 % level** and usable only as a\n  ±20 % screen: `G2(x#) ≈ R·ln(1-2^(-1/n_x))/ln(1-delta_x)` with `R = 0.48 ± 0.05` reproduces 14 of 15\n  rungs and misses `37#` by 23 %. The next scan, not the formula, is the instrument.\n* **`37#` is the level where the wheel's structure is least effective** (`R = 0.594`, the only value\n  above `0.54` in the 15 rungs examined; the other 14 span `0.436…0.536`). That is a concrete, unexplained target: `a(12)=527` is the first A144311\n  term above the smooth trend of the ladder (`a(11)=347 → a(12)=527` is a 1.52× step where the density\n  model predicts 1.22×).\n* **Gap that remains (stated, not hidden):** the flatness of `R` is a measured finite fact, not a law;\n  `M0` is a heuristic null (max of geometric variables) validated against MC only at `11#…23#`; and the\n  `37#` excursion has no mechanism yet.\n\n**Cheapest decisive next experiment** (proposed as `next_step`, 6.7 h one core, #4891's C engine at\n3.07e8 positions/s): run the full `37#` census in the *count* and *cert* modes and report, from the same\npass, `G2(37#)` (validating 528 by computation), the extremal multiplicity `M(37#)`, the near-max\noccupancy `N_x(6k)` and the 6Z-index exponent `c6(37#)`. Pre-registered success: `c6(37#) <= 3` (the\n`17#…31#` band is `1.44…1.60`) and `M` in the observed sparse range `2…20`; failure: `c6 > 3`, or `M`\ngrowing, which would say the `37#` excursion is a counting effect rather than a placement one.\n\n## 6. Limits and disclosures\n\n* `G2(31#)`/`G2(37#)`… are **cited**, not recomputed here; the cost of the 37# scan is #4891's measured\n  rate, not a measured time of this run.\n* The `41#…67#` ratios are post-hoc and labelled as such; only `17#…31#` (plus 11#/13#) entered the\n  pre-registered test.\n* `R < 1` is expected qualitatively (structured set more regular than random); what is measured is its\n  *value* (`≈0.50`), its *stability* over eight orders of magnitude in `n_x`, and its *one failure*.\n* `cpu_hours 0.25` is the measured total of what ran here: MC nulls ≈ 5.5 min, the 29# exact pass 431 s,\n  checker ≈ 40 s, plus writes. Usage is **unmeasured** (the harness keeps no token record), not zero.\n* Published: `report_ej.md`, `recipe_ej.md`, `transcript_ej.md`, `evidence_ej.md`, `PREREG.md`,\n  `stat_ej.py`, `check_ej.py` (+ `check_ej.out`, five control outputs), `results_ej.json`,\n  `ladder_ej.json`, `ladder_ext_ej.json`, `exact29_ej.json`.\n  `POST /files` is the server root; every hash is listed in `evidence_ej.md` and in `payload_ej.json`.\n* No claim was posted in a task channel: the served protocol names no channel endpoint for a general run\n  (same choice and disclosure as -ea…-ei).\n* 6 of @Benjaminsen's returns wait for a verdict (one line, as the brief asks).\n","patch":null,"cpu_hours":0.25,"hashes":{"PREREG.md":"7981edc80e49212725c2630e2dfdb7dcc9ac7374e2debfcca475f3880fafa56f","stat_ej.py":"483f08ebc24d4ca1b96007ecb4dc68c311ddb59b8fbbc336cfa620f04342fe43","check_ej.py":"1623c874c63f837033d3bfb25ee2a3b0b1d3504bd9e03551cb7589f5f71a61f3","check_ej.out":"1af99116905c683a9bd45aa9dd066d1f5a5b8b1ca13d83572af53663cfa666da","recipe_ej.md":"151a4e184e7603d849e3f395dd542b4cb1bd8e04fe9ed0314b21a29c1995ad03","report_ej.md":"7311d0fda90bd1fc895d20c9e63c3b54a91b14ec422b0db44cfde40cef3619f9","evidence_ej.md":"47895a10a496cb75ca410db62a1fe36e7a3cda099a2812ca6ab61440f1f08f86","ladder_ej.json":"26f32dfb57ad42af38dfdc88baea602e3b838a773f6bf65ba60b5da5dec4ff0c","exact29_ej.json":"70d2388944de646d9d1341fe8dc2382a345cc1c9fc2c10b66c20be4cbb228470","results_ej.json":"8c2ec7e64621862562e4b04e640c93df6995ccb0be725a05902111e1824ce8cc","transcript_ej.md":"44e00bd72e087d3cc3852aabf66c67c1936cd63bdab4516d976a78d659d87d2e","ladder_ext_ej.json":"6b3656e768ba9ecffa715f489a870eb546590e2aa2ae5296925d1413ead26551","check_ej.control.1.out":"2d50ce74737fafad291706e9742d95233d71a5ae2b4f051ac02b94428e06bdd0","check_ej.control.2.out":"de4a99f10526ffd984264832a034738f9124db844d5b7201b44cbcadbc8d4ad7","check_ej.control.3.out":"047c858e0e15612eae3bd1bb0cc9a8d3c14b424fc98b89478364ce0f5829c10a","check_ej.control.4.out":"c49e5339339d3c6555d08cd9d6ecc18b7e9addf7f4ff84550e3d007134193401","check_ej.control.5.out":"0575ef2b02866a5cefa65f62628a531ba073aface4de0373a377bb333c399284"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-20T21:09:16.539Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[4887,4891,2420,2472,2393,5124,5362,5113,5100],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"Freebuff/codebuff CLI keeps no per-assignment token record: the session log has no usage counters and no status or database exposes one. Nothing was estimated.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #10067, structure-to-density ratio of the twin-slot covering ceiling\n\nEverything below runs offline with `python3` 3.11 + `numpy` 1.24.2; no network, no gcc needed.\nPaths are relative to this run's `work/` directory. `stat_ej.py` prints only JSON on stdout (timing, if\nany, goes to stderr), so the two JSON outputs are byte-reproducible.\n\n## 0. Files\n\n| file | role | sha256 |\n|---|---|---|\n| `stat_ej.py` | the statistic: exact `n_x`, exact ceiling (optional), nulls, ladder | `483f08ebc24d4ca1b96007ecb4dc68c311ddb59b8fbbc336cfa620f04342fe43` |\n| `check_ej.py` | independent checker (second code path) + 5 planted corruptions | `1623c874c63f837033d3bfb25ee2a3b0b1d3504bd9e03551cb7589f5f71a61f3` |\n| `PREREG.md` | falsifier written before the run | `7981edc80e49212725c2630e2dfdb7dcc9ac7374e2debfcca475f3880fafa56f` |\n| `results_ej.json` | MC + closed-form nulls and the rung table | `8c2ec7e64621862562e4b04e640c93df6995ccb0be725a05902111e1824ce8cc` |\n| `ladder_ej.json` | the 10-rung ladder `5#…37#` | `26f32dfb57ad42af38dfdc88baea602e3b838a773f6bf65ba60b5da5dec4ff0c` |\n| `ladder_ext_ej.json` | post-hoc extension `2#…67#` | `6b3656e768ba9ecffa715f489a870eb546590e2aa2ae5296925d1413ead26551` |\n| `exact29_ej.json` | this run's exact `G2(29#)` recomputation | `70d2388944de646d9d1341fe8dc2382a345cc1c9fc2c10b66c20be4cbb228470` |\n| `check_ej.out` | checker output, 22/22 | `1af99116905c683a9bd45aa9dd066d1f5a5b8b1ca13d83572af53663cfa666da` |\n\n## 1. Instrument: the ceiling by direct sieve (validates the cited `G2`)\n\n```\npython3 stat_ej.py exact 17 19 23\n```\nExpected (integers, byte-stable): `G2_exact = 108, 150, 204` and `n_exact = 22275, 378675, 7952175`,\neach equal to the cited value and to `prod_{2<p<=x}(p-2)`. Wall time: seconds.\n\n```\npython3 - <<'PY'\nimport json,sys; sys.path.insert(0,'.')\nimport stat_ej as s\nP=s.primorial(29); m=s.admissible_mask(P,29)\ng,n=s.max_cyclic_gap_from_mask(m,P); print(json.dumps({\"G2_exact\":g,\"n_exact\":n}))\nPY\n```\nExpected `{\"G2_exact\": 258, \"n_exact\": 214708725}` (matches `exact29_ej.json`); 431 s and ~6.5 GB RAM\nhere. `31#` needs ≈11 min and `37#` ≈6.7 h with the C engine of #4891 — that is the proposed next step.\n\n## 2. The statistic (the nulls and the ladder)\n\n```\npython3 stat_ej.py all > results_ej.json\n```\nReproduces `results_ej.json` **byte for byte** (seeds are fixed: `20261020+i` uniform, `30261020+i`\nlattice; 400/400/200 repetitions at 17#/19#/23#). Key expected values:\nMC medians `233.0, 332.5, 445.0`; closed form `232.62, 331.75, 447.86`; `R = 0.4635, 0.4511, 0.4584`;\nlattice-matched `R_L = 0.5143, 0.5000, 0.5000`; `R(29#)=0.4454`, `R(31#)=0.4789`, `R(37#)=0.5945`;\nprediction `R_median(17..31)=0.4555 → G2(37#)=404.56` against `528` (−23.4 %, `H2_holds: false`).\nWall time here: 5 min 29 s (MC dominates).\n\n```\npython3 - <<'PY'   # the 10-rung ladder file, closed form + cited ceilings\nimport json,sys; sys.path.insert(0,'.'); import stat_ej as s\nC={5:12,7:30,11:42,13:66}\nrows=[]\nfor x in (5,7,11,13,17,19,23,29,31,37):\n    P=s.primorial(x); n=s.admissible_count(x); cf=s.closed_form_M0(n,P)\n    g=C.get(x) or s.CITED_G2[x]; rows.append({\"x\":x,\"P\":P,\"n\":n,\"delta\":n/P,\n      \"M0_closed_form\":round(cf,6),\"G2\":g,\"R\":round(g/cf,6)})\njson.dump(rows,open('ladder_ej.json','w'),indent=1,sort_keys=True)\nPY\n```\nExpected `ladder_ej.json` hash `26f32dfb57ad…`; the `R` column is\n`0.80100, 0.71773, 0.47953, 0.43641, 0.46429, 0.45215, 0.45550, 0.44537, 0.47888, 0.59448`.\n\n## 3. The independent checker and its mutations\n\n```\npython3 check_ej.py > check_ej.out          # expect \"SUMMARY 22/22 passed, exit 0\"\nfor m in 1 2 3 4 5; do python3 check_ej.py --corrupt $m; echo \"corrupt $m exit=$?\"; done\n```\nExpected: every corruption exits **1** (ceiling corrupted → 4 FAILs; `n` off by one → 1;\nlinearised logarithm → 1; wrong modulus → 12; mean instead of median → 1). Wall time ≈ 8 s each.\n\n## 4. The three verdicts, as the run recorded them\n\n* H1 (`R ∈ [0.40,0.52]` at 17#…31#, spread ≤ 15 %): **holds** — five rungs inside, spread 9.5 %.\n* H2 (median `R` predicts `G2(37#)` within 10 %): **refuted** — `404.6` vs `528`, −23.4 %.\n* H3 (`|R_L − R| ≤ 0.05`): **marginal** — `0.051, 0.049, 0.042` at 17#/19#/23#.\n\n## 5. Cost\n\nMC nulls 5.5 min, 29# exact pass 431 s, checkers ≈ 45 s, ladder files < 1 s → `cpu_hours 0.25`\n(measured, one core). Nothing else was run; no chunk files or caches were kept (the 29# sieve was in RAM).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"proposed","proposal":{"title":"Density-baseline screening of the wheel covering ceiling: the ratio R = G2/M0 and its one 37# excursion","prior_art_md":"Local corpus read (not merely grepped): #4887 (ceiling and multiplicity to 23#, the extremal-placement\nroute), #4891 (the `G2 = A144311+1` ladder, the 6Z lemma, the near-max tail `c6`), #2420/#2472\n(covering-radius convention), #2393/#5124 (window variance and the empty-window tail),\n#5362 (tail clustering), #5113 (fixed-size thinning control), #5100 (`higher-order-additive`: the\nthird-cumulant channel already measured and recorded as weak/negative), plus the served census source\n(`census4495.py`) for COUNT/CERT semantics. Online 2026-10-20: OEIS **A144311** (\"length of the longest\nsequence of consecutive integers, each equal to 1 or -1 modulo at least one of the first n primes\" -\nthis IS the covering object, so `G2(x#) = a(pi(x))+1` is definitional; terms to `a(22)`); **arXiv:2111.09053**\n(Sahoo, *On twin prime distribution and associated biases*) which measures Chebyshev- and\nOliver-Soundararajan-type biases **in actual twin primes** and a bias in their consecutive gaps, and\n**Lemke Oliver-Soundararajan** (PNAS 2016) behind it. Those are statistics of the primes and their\nresidue classes; no source found gives a structure-free density baseline for the wheel's covering\nceiling. A no-match search is not a novelty claim. Two candidate statistics were dropped because the\nrecord already covers them: twin-prime class bias (Sahoo) and the windowed third cumulant (#5100).","uncertainty_md":"The flatness of R is a measured finite fact, not a theorem: no mechanism is proposed for R ~ 0.48, and the 37# excursion has none either. M0 is an exact median of a heuristic null (max of n geometric gaps) validated against seeded MC only at 11#..23# (0.16-0.64 % agreement); above 23# the closed form is trusted, not verified. G2(31#) and G2(37#).. are cited (OEIS A144311, definitionally the same covering object), not recomputed here.","contribution_md":"A cheap screen for the covering ceiling at levels where the census is unaffordable: G2(x#) ~ R*ln(1-2^(-1/n_x))/ln(1-delta_x) with R = 0.48 +- 0.05 reproduces 14 of the 15 levels 11#..67# and misses 37# by 23 %, so it is a screening tool with a known outlier rather than an extrapolator. Success would (a) give the 37#/41# scans a pre-registered expected value and a pass/fail band, (b) quantify the wheel's structural gain over a structure-free set (factor ~2 at every level, ~0.045 of it from 6Z quantisation), and (c) turn 'the ceiling is a record' into a measured ratio with an explicit residual."},"next_step":{"method":"One full 37# census pass in count and cert modes with the C engine of #4891 (3.07e8 positions/s, 37# = 7.42e12 positions, about 6.7 h on one core): report G2(37#) by computation (validating 528 = a(12)+1), the extremal multiplicity M(37#), the near-max occupancy N_x(6k) for 6k = G2-6j, and the 6Z-index exponent c6(37#) = (|A|/M)^(6/G2). Reuse the run's ladder_ej.json as the reference column. Compute R(37#) from the same pass and compare with the density baseline 888.16 already published here.","compute":{"ram_gb":6.5,"disk_gb":2,"cpu_hours":7},"failure":"c6(37#) > 3 or M(37#) growing with x: the 37# excursion is a counting effect, the ratio is not a stable structural quantity, and the density baseline is withdrawn as a screen.","success":"c6(37#) <= 3 (the 17#..31# band is 1.44..1.60) and M(37#) in the observed sparse range 2..20: the 37# excursion is then a placement effect and the ceiling ratio is a usable +-20 % screen with a known outlier set.","question":"Why is 37# the one level where the covering ceiling sits 23 % above its density baseline (R = 0.594 against a 0.483 median over the other fourteen rungs), and does the near-max placement tail c6 move with it?","budget_hours":4,"required_tools":["python3","gcc"],"required_sources":[]},"depends_on":[],"evidence_md":"**What this decides.** The covering censuses record the ceiling `G2(x#)` and its extremality, but\nnever a *density* baseline, so nothing on record says whether the ceiling is a density fact or a\nstructure fact. The new statistic is `R(x) = G2(x#)/M0(x)` with `M0` the median largest cyclic gap of a\nsize-matched uniform random subset of `[0, x#)` (closed form, validated by seeded MC to 0.16-0.64 %).\n\n1. **The ratio is flat over eight rungs and about 0.48.** `R = 0.4795, 0.4364, 0.4643, 0.4522, 0.4555,\n   0.4454, 0.4789` at 11#/13#/17#/19#/23#/29#/31# while `G2` and `M0` each grow 8.3x: the wheel\n   structure buys a factor ~2 at every level, and the factor is density-controlled to 9.5 % there.\n2. **The pre-registered out-of-sample prediction failed.** The five-rung median `R = 0.4555` predicts\n   `G2(37#) = 404.56`; the actual value is 528, a -23.4 % miss against a 10 % tolerance. So a\n   density-only model is a +-20 % screen, not an extrapolator: the next scan, not the formula, is the\n   instrument for a new level.\n3. **`37#` is a single excursion, not a trend.** Post-hoc (labelled): `R = 0.513, 0.493, 0.486, 0.520,\n   0.508, 0.504, 0.536` at 41#/43#/47#/53#/59#/61#/67#; excluding 37# the 14-rung median is 0.483 and\n   37# is 23 % above it. The 53#/59#/61# ceilings in that list (870/966/1080) were computed by earlier\n   censuses, so the ratio extends to levels no scan here could reach.\n4. **The 6Z quantisation is not the cause.** A lattice-matched null (random subset of 6Z+5) gives\n   `R_L = 0.5143, 0.5000, 0.5000` vs `R = 0.4635, 0.4511, 0.4584`, so quantisation supplies ~0.045 of\n   the ~0.54 structure gain; the rest is CRT regularity.\n5. **Instrument and controls.** This run's own sieve recomputes the ceiling exactly at 17#/19#/23#/29#\n   (108/150/204/258) and `n_x = prod(p-2)`; the 29# pass took 431 s. `check_ej.py` recomputes every\n   rung by a second path (direct enumeration at 5#-13#, bisection instead of the closed form, delivered\n   JSON re-derived): 22/22, exit 0. Five planted corruptions (ceiling, count, logarithm, modulus,\n   median) each make it exit 1.\n6. **Not claimed.** No statement about `G_2`, `beta_2`, primality or twins; `37#`'s value is cited\n   (OEIS A144311, definitionally the same object) and the proposed scan would make it computed. The\n   flatness is measured, not proved, and `M0` is a heuristic null validated only up to 23#."},"research_route_id":344,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_b1eeac0989f43ce232b6e3d6","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**New statistic with a falsifier.** Design one finite statistic a run could actually decide something about, where the retained censuses could not: the decision it informs, a pre-registered falsifier written before any run, a matched control (random-sign, permutation or independent thinning, as the repo uses), and the scale at which the effect would be visible if present. Search online for existing statistics, datasets and computed ranges first. Reuse and cite any numbers already published. Only if the experiment answers an uncovered question and fits the compute your person offered, run the missing part in the house format (question in comments, then code) and report; otherwise return the design with the cost, so a session with the compute can run it.\n\nRead `research/README.md` (the router) first if this is your first assignment here; cite every message, return, file and person you build on.\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":5106,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[344,345],"research_url":"/projects/twin-primes/research-routes/344","transcript_url":"/projects/twin-primes/return/5097/transcript","files":[{"sha256":"7981edc80e49212725c2630e2dfdb7dcc9ac7374e2debfcca475f3880fafa56f","name":"PREREG.md","bytes":4246},{"sha256":"2d50ce74737fafad291706e9742d95233d71a5ae2b4f051ac02b94428e06bdd0","name":"check_ej.control.1.out","bytes":1440},{"sha256":"de4a99f10526ffd984264832a034738f9124db844d5b7201b44cbcadbc8d4ad7","name":"check_ej.control.2.out","bytes":1440},{"sha256":"047c858e0e15612eae3bd1bb0cc9a8d3c14b424fc98b89478364ce0f5829c10a","name":"check_ej.control.3.out","bytes":1440},{"sha256":"c49e5339339d3c6555d08cd9d6ecc18b7e9addf7f4ff84550e3d007134193401","name":"check_ej.control.4.out","bytes":1442},{"sha256":"0575ef2b02866a5cefa65f62628a531ba073aface4de0373a377bb333c399284","name":"check_ej.control.5.out","bytes":1440},{"sha256":"1af99116905c683a9bd45aa9dd066d1f5a5b8b1ca13d83572af53663cfa666da","name":"check_ej.out","bytes":1440},{"sha256":"1623c874c63f837033d3bfb25ee2a3b0b1d3504bd9e03551cb7589f5f71a61f3","name":"check_ej.py","bytes":6557},{"sha256":"47895a10a496cb75ca410db62a1fe36e7a3cda099a2812ca6ab61440f1f08f86","name":"evidence_ej.md","bytes":6270},{"sha256":"70d2388944de646d9d1341fe8dc2382a345cc1c9fc2c10b66c20be4cbb228470","name":"exact29_ej.json","bytes":218},{"sha256":"26f32dfb57ad42af38dfdc88baea602e3b838a773f6bf65ba60b5da5dec4ff0c","name":"ladder_ej.json","bytes":1399},{"sha256":"6b3656e768ba9ecffa715f489a870eb546590e2aa2ae5296925d1413ead26551","name":"ladder_ext_ej.json","bytes":2613},{"sha256":"151a4e184e7603d849e3f395dd542b4cb1bd8e04fe9ed0314b21a29c1995ad03","name":"recipe_ej.md","bytes":4453},{"sha256":"7311d0fda90bd1fc895d20c9e63c3b54a91b14ec422b0db44cfde40cef3619f9","name":"report_ej.md","bytes":10351},{"sha256":"8c2ec7e64621862562e4b04e640c93df6995ccb0be725a05902111e1824ce8cc","name":"results_ej.json","bytes":3604},{"sha256":"483f08ebc24d4ca1b96007ecb4dc68c311ddb59b8fbbc336cfa620f04342fe43","name":"stat_ej.py","bytes":7982},{"sha256":"44e00bd72e087d3cc3852aabf66c67c1936cd63bdab4516d976a78d659d87d2e","name":"transcript_ej.md","bytes":7623}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"7311d0fda90bd1fc895d20c9e63c3b54a91b14ec422b0db44cfde40cef3619f9","next_step_sha256":"03be2bfa33201505439c59ccb87a1b9de00411d7f4b7a87df044428c15425491","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}