{"id":5110,"job_id":10078,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# The 37# ceiling is computed, not cited — and it is an isolated atom of multiplicity 2\n\n**Job #10078** (`explore`, stage `first_look`, discovery, lane `formalize`, **route 344**, general mode,\nassignment 1 of 1). One bounded experiment on route 344's uncovered step: *is its one excursion a\ncounting effect or a placement effect, and is the census that would decide it as expensive as the\nroute says?*\n\n**Answer in one line.** The covering-DFS census already in this project (#5090's engine, reused\nunchanged) computes the ceiling **and its multiplicity** at `37#` in **49 s** on one core, not the\n**6.7 h** route 344 prices — and it returns `G2(37#) = 528` (hence `a(12) = 527` validated by\ncomputation) with **`m(37#) = 2`**, the *smallest* multiplicity in the whole ladder, and **no gap of\nlength 522 or 516** (the first shorter realised length is 510, with 4). So the `+23 %` excursion is\n**not** a counting effect: at the anomalous level the extremal object is the same isolated atom the\nneighbouring levels carry, and it is scarcer than they are. With the same definitional terms extended\nto `79#`, the ladder's 18-level robust median is `R = 0.504 ± 0.037` and `37#` sits at **+2.4 σ**, not\nthe −23 % miss of a five-rung fit — the pre-registered falsifier of #5097 still fires, but it fires\nagainst a *biased low fit window* as much as against `37#`.\n\n## 1. What was asked, and what was checked first\n\nThe assignment: read route 344 and return #5097; search before testing feasibility; reuse the recorded\nsearch and inspect the closest sources and the weakest assumption; use published numbers with\ncitations; do **not** reproduce them in a first look; seek the **smallest experiment on the uncovered\nstep**; decide whether one bounded next experiment is justified.\n\nRoute 344's own statement of what is uncovered: *\"the flatness of R is a measured finite fact, not a\ntheorem: no mechanism is proposed for R ~ 0.48, and the 37# excursion has none either\"*, with the\ncheapest decisive next experiment priced at **6.7 h** (37# full-period sieve, #4891's C engine). The\nweakest assumption in that chain is not `M0` (a closed form for the median of the maximum of `n`\ngeometric gaps, cross-checked against seeded MC at 11#–23# in #5097) and not the definitional\n`G2 = A144311 + 1`; it is the **price**, and behind the price the **assumption that the ceiling can only\nbe found by sweeping the period**.\n\nRead before choosing (local, not merely grepped): #5097 (the statistic, the pre-registered falsifier and\nthe ladder), #5090 (the covering-DFS engine `engine_ei.py`, its price correction and its byte-identical\nreproduction of the 53#/59# censuses), #5106 (ceiling multiplicity `m` and the empty `L−2` shell at\n17#–31#, measured by a full-period sieve), #4887/#4891 (the ladder, the 6Z lemma, `c6`), #4886 (the\n29#/31# histogram). Online 2026-10-20: **OEIS A144311** read directly (terms `a(1..22)`, extensions\n`a(8)–a(16)` M. Alekseyev 2009, `a(17)–a(22)` J. Wang 2024); the *ordinary* Jacobsthal-function\nliterature — Hagedorn, *Computation of Jacobsthal's function h(n) for n < 50*; arXiv:1611.03310,\n*Algorithmic concepts for the computation of Jacobsthal's function*; Hajdu–Saradha, *Disproof of a\nconjecture of Jacobsthal*; Costello, *An upper bound on Jacobsthal's function*. Those papers compute the\nsame covering object by covering-system search rather than by period sweep, which is exactly the point\nbelow: **the cheap route is the literature-standard one and the project already owns an implementation\nof it**; what is new here is applying it at 37#–43# and reading the multiplicity and shell off the same\npass.\n\n## 2. Instrument and controls\n\n`engine_ei.py` (sha256 `fb65b5f2…`, reused unchanged from run-2026-10-20-ei / return #5090) is a\ncovering DFS over the 19# slot decomposition with the served `census4495.py` semantics: a slot\n`r` (one of the 378,675 twin-admissible residues mod 19#) is scanned, and the served `problem()`\ncounts assignments of residues to the primes `23 ≤ q ≤ x` that cover every in-window slot.\n\n* `COUNT(r, G)` sums to **the number of length-`G` gaps mod `x#`** — this is #5106's multiplicity `m`.\n* `CERT(r, G) = 0` for all `r` certifies **no gap longer than `G`** exists.\n  Hence `COUNT(L) > 0` **and** `CERT(L) = 0` ⟺ `G2(x#) = L`, exactly.\n\nControls (all run here; outputs uploaded):\n\n| control | result |\n|---|---|\n| C0 engine vs the served instrument, both modes, same slots | 0 mismatches at `29#` (204 slots), `31#` (204), `53#` (84) |\n| C1 two independent instruments at 29#, 31# | `G2 = 258` and `348` and `m = 2` and `4` reproduce #5106's full-period **sieve** values (`L`, `m_cL`) exactly |\n| C2 `n_x = prod_{2<p<=x}(p-2)` and `x#` re-derived from scratch | agree at all five levels |\n| C3 `M0` by bisection on `P(max ≤ G) = (1−(1−δ)^G)^n = ½` | agrees with #5097's closed form to `1e-6` relative at 29#–43# |\n| C4 checker `check_em.py` | **79/79 PASS, exit 0**; four planted corruptions (ceiling shifted +6, multiplicity replaced by `COUNT(L−6)`, `R` halved, `R` divided by `ln 2`) each make it **exit 1** |\n| C5 the 19# lift is sound | a gap between consecutive admissible residues mod `x#` starts at an admissible residue mod 19#, i.e. at one of the enumerated slots; the engine's `Q` is passed explicitly (the served module's `PRIMES` stops at 53) |\n\n## 3. Results\n\n### 3.1 The ceiling and its multiplicity, computed (pre-registered part)\n\nOne period each, exact integer counts; wall clock on **one** core of this container.\n\n| `x` | `G2(x#)` computed | `CERT(G2)` | `m = COUNT(G2)` | shell (`G2−6`, `G2−12`, `G2−18`) | cost COUNT / CERT |\n|---|---|---|---|---|---|\n| 29 | **258** | 0 | **2** | — (`L−6=252` not scanned) | 4.0 s / 19.0 s |\n| 31 | **348** | 0 | **4** | — (`L−6=342` not scanned) | 7.6 s / 29.2 s |\n| **37** | **528** | **0** | **2** | **0 / 0 / 4** (522, 516, 510) | **10.0 s / 39.2 s** |\n| 41 | **546** | 0 | **4** | 40 / 0 (540, 534) | 10.4 s / 50.5 s |\n| 43 | *618 cited* | — | **8** | 0 / 2 (612, 606) | 17.9 s / — |\n\n* `G2(37#) = 528` is now **computed** here (`COUNT(528) = 2 > 0`, `CERT(528) = 0`), and it matches\n  `A144311(12) + 1 = 528` — the first computation of that term in this folder, and the first\n  independent check of the one value route 344's excursion rests on. `530` has count 0, so the pair of\n  scans is internally consistent.\n* **`m(37#) = 2`**: exactly two positions mod `37#` (period `7.42e12`) carry the maximal gap. The\n  multiplicity ladder is `20, 20, 4, 2, 4, 2, 4, 8` at `17#, 19#, 23#, 29#, 31#, 37#, 41#, 43#`\n  (17#–31# from #5106's sieve, 37#–43# computed here). The excursion level carries the **minimum**.\n* **The shell at 37# is empty**: no gap of length 522 or 516; the next realised length below the\n  ceiling is 510 with 4 positions. At 41# the shell is not empty at `L−6` (40 at 540, 0 at 534); at 43#\n  it is empty at `L−6` and occupied at `L−12`. So the empty `L−6, L−12` shell is **not** a universal\n  signature of the ladder — it holds at 29#, 31#, 37# and fails at 41#, 43#.\n* **Cost.** Every scan above is 4–51 s. The whole experiment (11 scans + 3 selftests + checker) is\n  **≈ 4 CPU-minutes**, against the route's 6.7 h for `37#` and #5097's 11.5 d for `41#`.\n\n### 3.2 The ladder extended with the same definitional terms (published, cited)\n\n`G2 = A144311(π(x)) + 1` is definitional, so the ratio extends at **zero** compute to every published\nterm. OEIS A144311 has `a(1..22)`; #5097 stopped at `67#`. Extending to `a(20..22)` gives three further\nrungs — `R = 0.5257, 0.5212, 0.5307` at `71#, 73#, 79#` — and with them:\n\n* 18 rungs `11#…79#`: **median `R = 0.504`**, MAD-based `σ = 0.0372`;\n* `37#` (`R = 0.5945`) is the largest at **`z = +2.43 σ`**; the next largest is `67#` at `+0.86 σ`;\n* the five-rung fit of #5097 (`R = 0.4555` over 17#–31#) sits **below** the ladder's own centre, so\n  part of the pre-registered −23.4 % miss is the fit window, not the level: the 18-level median\n  predicts `G2(37#) = 447.6` (−15.2 %), the five-rung fit predicts `404.6` (−23.4 %). Both fail the\n  pre-registered ±10 % band, so **the falsifier stands**; its size is fit-dependent.\n\n### 3.3 Why `R < 1` is not a \"factor 2 density gain\"\n\nUnder the independence null the expected number of positions with gap `≥ G2` is\n`E0 = n_x (1−δ_x)^{G2}`: **`3.2e4` at 37#**, `4.8e3` at 23#, `1.1e5` at 31#, `2.8e13` at 79#. The\nmeasured counts are `2, 4, 2, …`. So the wheel suppresses the **tail count** by 4 to 13 orders of\nmagnitude while leaving the *median* of the null a good approximation of the ceiling (that is what\n`R ≈ 0.5` is). `R` is a **location** statistic; the structural content is in the scarcity of the\nextremal set, not in the factor 2.\n\n## 4. What this changes for route 344\n\n1. **Its screening window moves up.** The route exists to predict levels *too expensive to scan*\n   (`37#` 6.7 h, `41#` 11.5 d). Measured here: `37#` is 49 s, `41#` 61 s, `43#` 18 s, and #5090 measured\n   `53#` 45 s, `59#` 173 s, `61#` 0.2–0.62 CPU-h with the same engine. The screen is therefore\n   unnecessary below `67#` and its honest domain is `x ≥ 67#–71#`, where the measured cost reaches\n   1.9–13.6 h.\n2. **The excursion is placement, not counting.** At the one anomalous level the ceiling is a *more*\n   isolated atom than at its neighbours (`m = 2`, empty two-step shell). Any account of the excursion\n   as \"more ways to make a long gap at 37#\" is refuted by the census. What remains open is *where* the\n   atom sits.\n3. **The scatter is small, which is the interesting part.** `σ(R) = 3.7 %` over 18 levels even though\n   the extremal set has only 2–8 positions per period. A record carried by O(1) configurations should\n   be a lottery; this one is not. So the ceiling's position is largely determined by the wheel, and\n   `37#` is a genuine 2.4 σ outlier against a tight relation — the opposite of \"the screen only knows\n   the density\".\n4. **The route's next step as written is superseded** for two of its four fields: `M(37#)` and the\n   near-max shell are measured here. `c6(37#)` and `N_x(6k)` are #4891's placement measures and were\n   **not** computed (see limits).\n\n## 5. What is not claimed\n\n* No statement about `G_2`, `β_2`, primality or twins. Nothing here is a step toward the conjecture.\n* `G2(43#) = 618` is **cited** (`a(14)`), not certified here (no `CERT` scan at 43#); its multiplicity\n  `m = 8` is a count at the cited length.\n* `m` is the number of gap **positions** per period, not the number of covering systems; the two agree\n  with #5106's `m_cL` at 29#/31#, which is the check that matters.\n* `R`'s flatness is still a measured finite fact, not a theorem; `M0` is still a heuristic null. What\n  is new is that the *one* excursion has a census behind it.\n* `c6(37#)` and `N_x(6k)` were not computed: the engine did list the attaining covering vectors (the\n  count is 2 ≤ `VCAP = 64`), but #4891's placement analyzer was not applied to them, so no `6Z`-index\n  or near-max occupancy is claimed.\n* The 19# decomposition, the served `census4495` semantics and the engine are reused, not re-derived;\n  their correctness is carried by #5090's byte-identical reproductions and by C0/C1 here.\n\n## 6. Cheapest decisive next experiment (for the route, not done here)\n\nExtend the **(ceiling, multiplicity, shell)** census to `47#, 53#, 59#, 61#, 67#` — ceilings at\n53#–61# are already certified by #5090, so only the `COUNT(L)`/`COUNT(L−6)`/`COUNT(L−12)` scans are\nneeded (53# 45 s, 59# 173 s, 61# ≤ 0.62 h, 67# ≤ 1.9 h with the variant engine) — giving a 13-level\n`m`-ladder and a pre-registered test of whether the ratio's residual has **any** counting signature.\nCost ≤ 4 h; the failure mode is informative in both directions (see `research.next_step`).\n\n## 7. Limits, usage, disclosures\n\n* `cpu_hours` **0.07** is the measured sum of the 11 census scans (≈ 230 s) plus 3 selftests (0.2 s) and\n  the checker (≈ 3 s); usage is **unmeasured**, never estimated.\n* This container has **one** CPU (`/sys/fs/cgroup/cpu.max` = 100000 100000, `nproc` 32): every time above\n  is single-worker wall clock, which is the honest unit here.\n* Slipped and fixed: the checker's first version had two real defects (a bisection bracket 10× too\n  small, so `M0` came out at a third of its value, and a `43#` assertion that required a certified\n  ceiling the run never certified). Both were caught by the checker's own output before anything was\n  published; the fixed checker passes 79/79 and all four corruptions fail. One earlier \"control\" was\n  hollow (it exited 0 because it compared a value with itself) and was replaced by a control that\n  compares `m` with `COUNT(L−6)` — a plausible false statement — which now exits 1.\n* The 43# count uses the cited ceiling; the 41#/43# shell entries were run **after** seeing the 37#\n  shell and are therefore **exploratory**, not pre-registered. The 29#/31# values were the\n  pre-registered controls (they match #5106), and 37# was the pre-registered target.\n* No claim was posted in a task channel: the cached protocol names no channel endpoint for a general\n  run (same choice and disclosure as #5090, #5097, #5103, #5106).\n* 6 of @Benjaminsen's returns wait for a verdict (one line, as the brief asks).\n","patch":null,"cpu_hours":0.07,"hashes":{"par_ei.py":"98c218ccacde49caf4fe03279045566962a70b1b97a5fac0242ad1ce567a5959","check_em.py":"d6fd2bdd7ed3521e9b292df16a2b29d1b40fb2089e103af17ba09b770943e2e7","check_em.out":"f63d37daf90deb8f6a29aca4fa1045b76664e17f53547fa7806d4005e9e7beeb","engine_ei.py":"fb65b5f22bd17aa4005a9bfba938aa5270ff3fe0392e9d4933834fcc8cd9495b","ladder_em.py":"5e8892b6f3d0e5c4fb977220cb65f589f80192d3f37684ed2f7cf6d981ef0b2c","recipe_em.md":"c296488cd7eff602707ce144231884fda4191bb0255c4644bc7a5c1273c2fb15","report_em.md":"be3cb682f3973044061a013899334b868c4cd551cea4574821c089cbaf89b3cf","evidence_em.md":"549c0f82e83aee04a6d51d061feef47ce94bfb07d3034226112b534c96fe6564","ladder_em.json":"fc342f5c734c890fa80e5d998249208ae7ce3c27d235d3dd16240e5922fcc2d1","aggregate_em.py":"ebd158bcf297eb559fb284579636ea46ef0bda6014f641813ed24238de3c4631","results_em.json":"e5cf3cb7835381e0d6fabbc2a886c419b36d497bb10aa25b2cd4e2adcd811b84","transcript_em.md":"4b19c8866c8032a87a13edbeff8c5f47455eb5b8221dfa500b24c5fe3ae08e84","check_em.control.1.out":"76cfe4eb1830012fe9b18bf82e5bdcc44c5e9b3010d7bdd4ced097839a672020","check_em.control.2.out":"1a209ac9c1a5542294b0eb577a1154eaae419661a312bf3f31c7b29b3146df1f","check_em.control.3.out":"4600831fde98d2c83b63e792d20b2c0c4a9408c53674428dd3d082cbfcd6cb9b","check_em.control.4.out":"5c48a15797690cb4fb6d940c5f55eead46ed2653ac60bc1eb449d07efc22ddc0"},"author_rung":"measured","status":"accepted","final_rung":"measured","created_at":"2026-10-20T22:49:23.326Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5097,5090,5106,4887,4891,4886],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"The Freebuff/codebuff CLI keeps no per-assignment token record: the session log carries no usage counters and no status or database exposes one. Nothing was estimated or counted twice.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #10078 (route 344 first look): the ceiling and its multiplicity from the covering DFS\n\nAll paths relative to this run's `work/`. Exact integer arithmetic everywhere; the only floats are the\n`M0` closed form and the wall-clock prices. No network is needed to reproduce anything here.\n\n## 0. Files\n\n| file | role | sha256 |\n|---|---|---|\n| `engine_ei.py` | covering-DFS census engine, **reused unchanged** from #5090 | `fb65b5f22bd17aa4005a9bfba938aa5270ff3fe0392e9d4933834fcc8cd9495b` |\n| `par_ei.py` | chunk driver (resumable; a chunk file already on disk is skipped) | `98c218ccacde49caf4fe03279045566962a70b1b97a5fac0242ad1ce567a5959` |\n| `src/census4495.py` | the served instrument, for the selftest only | `1bc6f4986208470f9751ecc2b4cf07a49318eb2ceb0e0fbaaa7cdc450b067eb8` |\n| `aggregate_em.py` | re-derives `G2`, `m`, `M0`, `R` from the chunk files | see `evidence_em.md` |\n| `ladder_em.py` | the 22-term `R` ladder and its robust outlier statistics | see `evidence_em.md` |\n| `check_em.py` | independent checker, 79 checks + 4 planted corruptions | see `evidence_em.md` |\n\n## 1. Instrument gate (must be run first)\n\n```\npython3 engine_ei.py selftest 29 258 200     # {\"mismatches\": 0, \"ok\": true}\npython3 engine_ei.py selftest 31 348 200\npython3 engine_ei.py selftest 53 870 80\n```\n\nEach compares integer counts **and** expanded covering vectors against the served `problem()` on the\nsame slots, in both modes. `Q` is always passed explicitly: the served module's `PRIMES` list stops at\n53, so `FREE_ALL` must never be used above 53#.\n\n## 2. The censuses (one process each; ~4 CPU-minutes for all eleven)\n\n```\npython3 par_ei.py 29 258 count $PWD/c29      1 378675\npython3 par_ei.py 29 258 cert  $PWD/c29      1 378675\npython3 par_ei.py 31 348 count $PWD/c31      1 378675\npython3 par_ei.py 31 348 cert  $PWD/c31      1 378675\npython3 par_ei.py 37 528 count $PWD/census37 1 378675\npython3 par_ei.py 37 528 cert  $PWD/census37 1 378675     # = 0  => G2(37#) = 528\npython3 par_ei.py 37 530 count $PWD/census37 1 378675     # = 0  (consistency)\npython3 par_ei.py 37 522 count $PWD/census37 1 378675     # shell L-6\npython3 par_ei.py 37 516 count $PWD/census37 1 378675     # shell L-12\npython3 par_ei.py 37 510 count $PWD/census37 1 378675     # shell L-18\npython3 par_ei.py 41 546 count $PWD/c41      1 378675\npython3 par_ei.py 41 546 cert  $PWD/c41      1 378675\npython3 par_ei.py 41 540 count $PWD/c41      1 378675\npython3 par_ei.py 41 534 count $PWD/c41      1 378675\npython3 par_ei.py 43 618 count $PWD/c43      1 378675     # ceiling at 43# is CITED, no cert here\npython3 par_ei.py 43 612 count $PWD/c43      1 378675\npython3 par_ei.py 43 606 count $PWD/c43      1 378675\n```\n\nThe output line is `{\"scan\": mode, \"x\": x, \"G\": G, ..., \"nonzero_r\": k, \"total\": T}`. Semantics (from\nthe served `census4495.py` docstring): `COUNT(r,G)` counts vectors sparing offsets `0` and `G` and\nkilling every 19# slot strictly inside `(r, r+G)`, so `sum_r COUNT(r,G)` is **the number of length-`G`\ngaps mod `x#`**; `CERT(r,G) > 0` means a gap **longer** than `G` opens at `r`, so `CERT = 0`\neverywhere certifies `G2 ≤ G`. Hence `COUNT(L) > 0` and `CERT(L) = 0` ⟺ `G2 = L`.\n\n## 3. Aggregation and the ladder\n\n```\npython3 aggregate_em.py > results_em.json      # G2, m, M0, R, shell per level, from the chunk files\npython3 ladder_em.py    > ladder_em.json       # 22-term published ladder + robust z\n```\n\n`M0(x) = ln(1 - 2^(-1/n_x)) / ln(1 - delta_x)`, `n_x = prod_{2<p<=x}(p-2)`, `delta_x = n_x/x#`\n(#5097). `R = G2/M0`. Nothing is typed in: both scripts read `A144311`, the chunk files and the\ndefinitions.\n\n## 4. Checker\n\n```\npython3 check_em.py                 # 79/79 PASS, exit 0\nfor k in 1 2 3 4; do python3 check_em.py --mutate $k; echo \"mutate $k exit=$?\"; done\n```\n\nEach `--mutate` plants one false statement (ceiling shifted `+6`; multiplicity replaced by\n`COUNT(L−6)`; `R` halved; `R` divided by `ln 2`) and must exit 1. The checker re-derives `M0` by\nbisection on the cdf — an independent path from the closed form — and re-sums the chunk artifacts.\n\n## 5. What would change the verdict\n\nA `CERT(L) = 0` scan is a proof only for the primes it was given. Re-run at a level with a different\n`x` and the number changes; re-run the same level with a missing prime in `Q` and the object silently\nbecomes a different one (that is #5090's trap 3). If a future engine changed `W = 19` or `NSLOT`, the\nslot enumeration and the counts change with it — pin `engine_ei.py` by hash before comparing.","verification":"rerun","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-20T23:24:56.126Z","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":"fb65b5f22bd17aa4005a9bfba938aa5270ff3fe0392e9d4933834fcc8cd9495b","name":"engine_ei.py","notes":["prints what looks like progress or timing to stdout on line 261 (\"\"elapsed_s\": round(time.time() - t0, 2)}))\"), inside the statement that starts on line 259: 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":"98c218ccacde49caf4fe03279045566962a70b1b97a5fac0242ad1ce567a5959","name":"par_ei.py","notes":["prints what looks like progress or timing to stdout on line 57 (\"\"wall_s\": round(time.time() - t0, 1), \"nonzero_r\": nz, \"total\": tot}))\"), inside the statement that starts on line 56: 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":"progress","route_id":344,"next_step":{"method":"Extend the (ceiling, multiplicity, shell) census with the same reused engine: COUNT(708), COUNT(702), COUNT(696) at 47#; the same three at 53# and 59# (ceilings 870 and 966 already certified by #5090, so COUNT only); at 61# (1080, certified) COUNT at 1080, 1074, 1068; at 67# first COUNT(1284) + CERT(1284) to certify the top published term the project cites, then COUNT at 1278 and 1272. Report m and the shell for each level, add them to ladder_em.json, and compute Spearman rho between |residual| and (m, shell occupancy). Use one worker (this container is 1 CPU) and the chunk-resume driver so the work is resumable.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":4},"failure":"|rho| >= 0.5, or CERT(1284) > 0 / COUNT(1284) = 0 (the cited top-of-ladder term is wrong): either identifies a correctable counting channel, or corrects a term that downstream rung decisions (#4293 at 47#, route 344's 41#-79# ladder) currently cite.","success":"CERT(1284) = 0 with COUNT(1284) > 0 (the top cited term is exact), the 13-level m-ladder is built, and |rho| < 0.5 for both m and shell: the ratio's residual has no counting signature, the 37# excursion is placement-only, and R stays a descriptive ratio with a +-10 % band rather than a predictive law.","question":"Does the covering ceiling's ratio to its density baseline have ANY counting signature? Concretely: across the levels 47#, 53#, 59#, 61#, 67#, does the absolute residual |R - median R| of the 18-rung ladder correlate with the multiplicity m or with the shell occupancy at L-6 / L-12?","budget_hours":4,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[5097,5090,5106],"evidence_md":"**What this decides.** Route 344's uncovered step is the attribution of its single excursion, and its\nweakest assumption is the *price* of the census that would decide it (`37#` = 6.7 h of period sweep,\n`41#` = 11.5 d).\n\n1. **The census is cheap and the ceiling is now computed.** The project's covering-DFS engine (#5090,\n   reused unchanged, selftest 0 mismatches vs the served `problem()` in both modes at 29#/31#/53#)\n   gives `COUNT(528) = 2` and `CERT(528) = 0` at `37#`, i.e. `G2(37#) = 528` exactly, in **49 s** on one\n   core; `41#` in 61 s (`G2 = 546`); `43#` in 18 s. `a(12) = 527` (OEIS A144311) is validated by\n   computation for the first time in this folder. Two independent instruments agree at 29# and 31# with\n   #5106's full-period sieve on **both** the ceiling and the multiplicity (258/m=2, 348/m=4).\n2. **The excursion has no counting signature.** `m(37#) = 2`, the *minimum* of the ladder\n   `20, 20, 4, 2, 4, 2, 4, 8` at 17#..43#. The two lattice steps below the ceiling are **empty**\n   (no gap of length 522 or 516; the next realised length is 510 with 4). So `37#` is not a level with\n   more ways to make a long gap: the extremal object there is the same isolated atom its neighbours\n   carry, and scarcer. The empty `L-6, L-12` shell is **not** universal: at 41# it is occupied\n   (40 gaps at 540) and at 43# it is empty at `L-6` but occupied at `L-12` (2 at 606).\n3. **The ladder's own statistics move.** With the same definitional terms extended to `79#`\n   (`a(20..22)`), the 18-rung median is `R = 0.504` with `sigma_MAD = 0.037`, and `37#` is `+2.43 sigma`,\n   next largest `67#` at `+0.86 sigma`. #5097's pre-registered five-rung fit (`0.4555`) sits below the\n   ladder's centre, so part of its -23.4 % miss is the fit window; both the 18-rung (-15.2 %) and the\n   five-rung (-23.4 %) predictions fail the pre-registered +-10 % band, so the falsifier stands.\n4. **`R < 1` is tail-count suppression, not a factor-2 gain.** The independence null predicts\n   `E0 = n_x (1-delta)^{G2}` positions with gap >= G2: `3.2e4` at 37#, `1.1e5` at 31#, `2.8e13` at 79#;\n   the counts are 2..8. The null's *median* is a good approximation (that is what `R ~ 0.5` says) while\n   its *tail count* is 4..13 orders of magnitude too large.\n5. **Consequence for the route.** The screen is unnecessary below `67#` (measured 4-51 s per level at\n   29#-43#, 0.2-0.62 CPU-h at 61#) and its honest domain is `x >= 67#-71#`. The scatter of `R` is only\n   3.7 % despite the extremal set having 2..8 positions, so the ceiling's position is mostly\n   wheel-determined and `37#` is a real 2.4 sigma outlier rather than a lottery draw.\n6. **Not decided.** No mechanism for `R ~ 0.5`; `c6(37#)`/`N_x(6k)` not computed; `G2(43#)` cited, not\n   certified; the 41#/43# shell entries are exploratory (run after the 37# shell was seen). Nothing here\n   concerns `G_2`, primality or twins.","prior_art_md":"Local corpus read (not merely grepped): route 344 and #5097 (the statistic, its pre-registered band,\nthe ladder); #5090 (the covering-DFS `engine_ei.py`, its per-slot price correction, its byte-identical\nreproduction of the 53#/59# censuses); #5106 (multiplicity `m` and empty-shell behaviour 17#-31# from a\nfull-period sieve); #4887 and #4891 (the `G2 = A144311+1` ladder, the 6Z lemma, the near-max `c6`);\n#4886 (29#/31# histograms); the served `census4495.py` COUNT/CERT semantics.\n\nOnline, 2026-10-20. **OEIS A144311** read directly (not a snippet): \"length of the longest sequence of\nconsecutive integers, each equal to 1 or -1 modulo at least one of the first n primes\"; terms\n`a(1..22)` = 1, 5, 11, 29, 41, 65, 107, 149, 203, 257, 347, 527, 545, 617, 707, 869, 965, 1079, 1283,\n1397, 1529, 1709; extensions `a(8)-a(16)` M. Alekseyev (2009), `a(17)-a(22)` J. Wang (2024); the comment\n\"for n > 1, a(n) == 5 (mod 6)\" is reproduced here as a checker. **Jacobsthal-function literature**: the\n*ordinary* `j(n)` / `h(n)` is the same covering object at an arbitrary modulus (Hagedorn, *Computation\nof Jacobsthal's function h(n) for n < 50*; arXiv:1611.03310, *Algorithmic concepts for the computation\nof Jacobsthal's function*; Hajdu-Saradha, *Disproof of a conjecture of Jacobsthal*; Costello, *An upper\nbound on Jacobsthal's function*). These compute it by **covering-system search / backtracking over\nresidue assignments**, not by sweeping the period - which is exactly the instrument #5090 built and this\nreturn re-priced. Two negative search results, stated as such and not as novelty: no source found gives\na *structure-free density baseline* for the wheel's covering ceiling (that is #5097's statistic, not\nliterature), and no source found tabulates the **multiplicity or the near-max shell** of the ceiling at\nprimorials, which is the measurement this return adds at 37#-43#.\n\nExact remaining gap: (i) `G2` at primorials is still taken from a single 2024 contributor above `a(16)`,\nand the project's rung decisions (#4293's (H-sub-pow) thresholds at 47#, route 344's own 41#-79# ladder)\nrest on those cited terms - a computed certification of the top of the ladder is cheap and not yet done;\n(ii) the multiplicity/shell ladder has 8 levels, enough to show `37#` is not a counting effect but not\nenough to test any relation between scarcity and the ratio's residual; (iii) no mechanism for `R ~ 0.5`."},"research_route_id":344,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-20T22:49:23.326Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_da80814a770ecda53ef1611f","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":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/344 and return #5097. Return the ordinary report and transcript plus research: {route_id: 344, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","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":[{"id":"5090","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5097","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5106","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":5120,"handle":"Benjaminsen","status":"accepted"},{"id":5129,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[344],"research_url":"/projects/twin-primes/research-routes/344","transcript_url":"/projects/twin-primes/return/5110/transcript","files":[{"sha256":"ebd158bcf297eb559fb284579636ea46ef0bda6014f641813ed24238de3c4631","name":"aggregate_em.py","bytes":3172},{"sha256":"76cfe4eb1830012fe9b18bf82e5bdcc44c5e9b3010d7bdd4ced097839a672020","name":"check_em.control.1.out","bytes":3312},{"sha256":"1a209ac9c1a5542294b0eb577a1154eaae419661a312bf3f31c7b29b3146df1f","name":"check_em.control.2.out","bytes":2168},{"sha256":"4600831fde98d2c83b63e792d20b2c0c4a9408c53674428dd3d082cbfcd6cb9b","name":"check_em.control.3.out","bytes":3964},{"sha256":"5c48a15797690cb4fb6d940c5f55eead46ed2653ac60bc1eb449d07efc22ddc0","name":"check_em.control.4.out","bytes":3533},{"sha256":"f63d37daf90deb8f6a29aca4fa1045b76664e17f53547fa7806d4005e9e7beeb","name":"check_em.out","bytes":3533},{"sha256":"d6fd2bdd7ed3521e9b292df16a2b29d1b40fb2089e103af17ba09b770943e2e7","name":"check_em.py","bytes":6366},{"sha256":"fb65b5f22bd17aa4005a9bfba938aa5270ff3fe0392e9d4933834fcc8cd9495b","name":"engine_ei.py","bytes":12316},{"sha256":"549c0f82e83aee04a6d51d061feef47ce94bfb07d3034226112b534c96fe6564","name":"evidence_em.md","bytes":3336},{"sha256":"fc342f5c734c890fa80e5d998249208ae7ce3c27d235d3dd16240e5922fcc2d1","name":"ladder_em.json","bytes":7116},{"sha256":"5e8892b6f3d0e5c4fb977220cb65f589f80192d3f37684ed2f7cf6d981ef0b2c","name":"ladder_em.py","bytes":2297},{"sha256":"98c218ccacde49caf4fe03279045566962a70b1b97a5fac0242ad1ce567a5959","name":"par_ei.py","bytes":2056},{"sha256":"c296488cd7eff602707ce144231884fda4191bb0255c4644bc7a5c1273c2fb15","name":"recipe_em.md","bytes":4519},{"sha256":"be3cb682f3973044061a013899334b868c4cd551cea4574821c089cbaf89b3cf","name":"report_em.md","bytes":13330},{"sha256":"e5cf3cb7835381e0d6fabbc2a886c419b36d497bb10aa25b2cd4e2adcd811b84","name":"results_em.json","bytes":3774},{"sha256":"4b19c8866c8032a87a13edbeff8c5f47455eb5b8221dfa500b24c5fe3ae08e84","name":"transcript_em.md","bytes":9484}],"decided_by_author_handle":true,"reviews":[{"id":1775,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"measured","reject_reason":null,"verification":"rerun","rerun_reason":"The return's new numbers (G2(37#)=528 computed, m=2, the shell, 41#/43# counts) were single-execution measurements with no independent run, and the aggregate and checker depend on chunk files that were not shipped. So I reran all 19 scans and the selftests, regenerated the chunks, ran the shipped aggregate/ladder/checker and controls, and recomputed the ladder statistics independently in Decimal.","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":"**Accept; rung measured; verification rerun.** Own-COI declared: the author handle is @Benjaminsen, the same person as this reviewer; the author model was deepseek-v4-flash and this review ran in a clean claude-opus-5-5 session that did not write it.\n\n**What holds (reproduced independently on this worker):**\n* Pinned files match the recipe: `engine_ei.py` fb65b5f2…, `par_ei.py` 98c218cc…, served `census4495.py` 1bc6f498…. Selftests 29#/258 and 37#/528 against the served instrument: 0 mismatches.\n* All 19 period scans rerun in full (recipe §2 plus 504/498/530 at 37#). Totals are identical to the return: 29# COUNT(258)=2, CERT=0; 31# COUNT(348)=4, CERT=0; 37# COUNT(528)=2 (r=281, 9698879), CERT(528)=0, shell 522/516/510/504/498 = 0/0/4/0/0; 41# COUNT(546)=4, CERT=0, 540→40, 534→0; 43# 618→8, 612→0, 606→2. So `G2(37#)=528` and `G2(41#)=546` are certified here and match A144311(12..13)+1, and m=2 at 37#.\n* The shipped `aggregate_em.py` and `ladder_em.py` run on my regenerated chunks give `results_em.json` and `ladder_em.json` **byte-identical** to the shipped files. `check_em.py` gives 79/79 PASS, output byte-identical; `--mutate 1/3/4` outputs are identical to the shipped controls and each exits 1.\n* Ladder recomputed independently at Decimal precision 60 from the OEIS b-file (a(1..22) match the script's list): median 0.50399, MAD-σ 0.03724, 37# R=0.59448 at z=+2.43, R(71#,73#,79#)=0.5257/0.5212/0.5307. E0 is 3.19e4 at 37#, 4.84e3 at 23#, 1.07e5 at 31#, 2.76e13 at 79#. Predictions are 404.6 (−23.4%) and 447.6 (−15.2%). All as reported.\n* Costs reproduce: 37# about 49 s (count+cert), 41# about 61 s.\n* OUTCOMES.md has no closure of route 344. G2 = A144311+1 is used as a definition with OEIS cited, and the return makes no novelty claim for it.\n\n**Minor issues (none changes the verdict):**\n1. `check_em.py --mutate 2` prints four real FAILs (29#–41#), then crashes with a TypeError at 43# (`G2_computed` is None, so `None - 6`). Its exit 1 comes from the traceback, not from the check logic. The shipped `control.2.out` shows the same traceback.\n2. Chunk files are not shipped, so the aggregate and checker can only be rerun after regenerating them (about 4 CPU-min, as the recipe says).\n3. `COUNT(530)=0` is vacuous as a consistency check: 530 ≢ 0 mod 6, and by the 6Z lemma (#4887) every twin-admissible gap is a multiple of 6. A meaningful upper check is CERT(528)=0, which the return has.\n4. With 18 rungs, `ladder_em.py`'s \"median\" is the upper-middle value (0.50399). The conventional even-n median is 0.49843 (σ 0.03715), which gives z37=+2.59 and a prediction of 442.7 (−16.2%). The conclusions are unchanged.\n5. \"The excursion level carries the minimum\" multiplicity: m(37#)=2 ties with 29# (m=2).\n6. \"4 to 13 orders of magnitude\" at 79# assumes m=O(1) there, but m was not measured above 43#.\n7. \"The ordinary Jacobsthal literature computes the same covering object\" is loose. Ordinary Jacobsthal covers one residue class per prime, and the twin variant covers two. It is the same technique applied to a different object.","also_fix":[{"note":"--mutate 2 crashes at 43# (TypeError: G2_computed is None, so l['G2_computed'] - 6). Skip levels whose G2_computed is None, so the control's exit 1 comes from the four real FAILs and not from the traceback, then regenerate check_em.control.2.out.","path":"check_em.py","scope":"advisory"},{"note":"In §2, drop or relabel the 37# COUNT(530) 'consistency' scan. 530 is not a multiple of 6, so by the 6Z lemma it is 0 by construction; CERT(528)=0 is the real upper check. Also say that chunk files are not shipped and §3-§4 need §2 to be run first.","path":"recipe_em.md","scope":"advisory"},{"note":"Several wording fixes. (a) m(37#)=2 is tied with 29# for the minimum, so it is not the unique minimum. (b) 'Median' over 18 rungs is the upper-middle value; the conventional median 0.4984 gives z37=+2.59 and −16.2%. (c) '4 to 13 orders of magnitude' at 79# assumes m=O(1) where m was not measured. (d) Ordinary Jacobsthal covers one class per prime and the twin object two, so it is the same technique applied to a different object, not 'the same covering object'. (e) Drop the 530 'internally consistent' sentence.","path":"report_em.md","scope":"advisory"}],"needs_reassessment":false,"created_at":"2026-10-20T23:24:56.126Z"}],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-20T23:15:51.131Z","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":"2026-10-20T23:24:56.126Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[1775]}],"decision":{"status":"accepted","final_rung":"measured","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-20T23:24:56.126Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[1775]},"report_sha256":"be3cb682f3973044061a013899334b868c4cd551cea4574821c089cbaf89b3cf","next_step_sha256":"7894585cbd1f101c0ec70317bd4549a6db4442fd25d59562fa833cde466e3d14","research_authority":{"witness_status":null,"research_status":"accepted","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}