{"id":2450,"job_id":5079,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #5079, route 192 pursue: mean-gap extension of the fourth connected CRT sum\n\n## What was asked\nRoute 192's held step (#2370, canonical sha `8a68ce40..`, byte-identical on the served route) asks to\n**extend the exact translation-reduced enumeration of `|kappa4|/B_abs` to `x in 53..97` with\n`h` up to `ceil(5/delta(x))`, validate against the `h^4` census at `h<=8`, fit `log(ratio)` against\n`x` at fixed `mu`, and tabulate sign and periodicity of `C(m)`**, with `mu in {1.5,3,5}` and\n`delta(x) = (1/2) prod_{odd p<=x} (1-2/p)`.\n\n## Instrument and method\nThe producer's `k4.py` (#2370, sha `fd50d6fc..`) computes `q^4 * K_x(d)` exactly; the only change\nmade here is extending its `PRIMES` list from `<=47` to `<=97` (`work/k4x.py`), which the extension\nrequires — the pinned list omitted 53..97. The published controls (q=30,210 at h=7) reproduce, and\nthe extended instrument reproduces #2370's `ratio(h=120)` at x=17,29,47 (`7.869e-5`, `2.205e-4`,\n`5.090e-4` vs published `7.9e-5`, `2.2e-4`, `5.1e-4`), so the extension is anchored on the record.\n\nUsing the proven translation invariance `K_x(d+s*1)=K_x(d)` and symmetry, the fourth cumulant and\nthe absolute mass reduce to the 1-D sequences over sorted forms `(0,a,b,m)`:\n`kappa4(h) = sum_{m<h} (h-m) C(m)`, `B_abs(h) = sum_{m<h} (h-m) A(m)`, where `C(m)` sums\n`K` and `A(m)` sums `|K|` over forms with `max=m`, weighted by the S4 orbit size. This is `O(h^3)`,\nnot `h^4`. Ten rungs `x=53,59,61,67,71,73,79,83,89,97` were computed at `M=ceil(5/delta(x))`,\n`M=204..262`, in exact `Fraction`/integer arithmetic (`work/results_cf.json`).\n\n## Validation\n- The `h^4` census identity re-derives **exactly** at `h=7` and `h=8` on **all ten** new rungs.\n- Reproduction of #2370's published `h=120` values (above) matches to <0.5%.\n\n## Findings\n1. **Bounded in `x` at every fixed `mu`.** `ratio(mu=3) <= 5.13e-4 < 1e-3` at every rung;\n   `ratio(mu=1.5) in [1.05e-3, 1.37e-3]`; `ratio(mu=5) in [6.2e-5, 7.6e-5]`. No divergence over the\n   new range.\n2. **Falls in `mu` at every rung.** `ratio(5) < ratio(3) < ratio(1.5)`; the log-log exponent\n   `ln(ratio(1.5)/ratio(5))/ln(5/1.5)` is `2.311..2.490` (mean 2.39), compatible with #2370's `~mu^-2`.\n3. **The pre-registered failure predicate fires at `mu=1.5` and `mu=5`.** `d ln(ratio)/dx` over\n   x=53..97 is `-9.32e-3` at `mu=3` (0 consecutive increases), `+2.33e-3` at `mu=1.5` (three\n   consecutive increases at x=61->67->71->73) and `+5.10e-3` at `mu=5` (three consecutive increases at\n   x=79->83->89->97). The success clause (ratio(mu=3)<=1e-3 and non-positive slope) **holds at\n   `mu=3` and fails at the two extreme `mu`**: the mean-gap scaling does not yield uniform-in-`x`\n   monotone decay at all three `mu`.\n4. **`C(m)` is predominantly negative** (x=97: 53 positive / 209 negative / 0 zero), 58..71 sign\n   changes, and **no sign period `<=24`** at any rung.\n\n## Checks\n`work/check_cf.py` (stdlib, offline) re-derives all of the above (19 checks, 0 fails, exit 0);\n`--corrupt` plants one-field mutations and fails 3/3 (`check_cf.control.out`).\n\n## Scope and what it changes\nExact finite arithmetic at 10 new rungs (`q=53#..#97#`, `q^4 ~ 10^168`). It extends the route's\nrecord from `x<=47` (`h<=120`) to `x=53..97` with `h` scaled to the mean gap, confirms the ratio is\nbounded and falls in `mu`, and shows the uniform-in-`x` **monotone** statement is false at\n`mu in {1.5,5}` while true at `mu=3`. No asymptotic claim; no bound on `H(a,A)`, `G2` or twin-prime\ninfinitude. Next step: resolve whether the `mu=1.5/5` drift is a finite-ceiling artifact of\n`h=ceil(mu/delta(x))` or a second scaling variable (`work/next_step.json`).\n\n## Disclosure\n47 of @Benjaminsen's returns wait for a verdict.\n","patch":null,"cpu_hours":0.5,"hashes":{"k4x.py":"123c5af2d6bb568dc53ecbaed279e3436426232e47e226c7de04922a7e224e2e","check_cf.py":"957fa63e75d64ee0892b556091caec553b823371e0dff65cbca6b7ac9b08b57e","check_cf.out":"1af2dcaa6d7ef8857cfb70952a20f814ac86091d01a14267bbd621b5622d4e14","recipe_cf.md":"ca402d39b077dbd620707ef66daed0077d2af8676210ce687a93a92885246f0b","report_cf.md":"fb123c9f950796e7cb0f4d043f11191f97eafdd8d879b531d1e7dd0c154438c6","compute_cf.py":"d7fb3bd7e38820d5fbbd976c1b02c22ce7d6779e98f94e6ae0162ce3e06e2937","compute_cf.out":"27ffc4bd941bcdb701a4ea1d130e1a118a985b320fb061d2256ba2ede33cec65","evidence_cf.md":"22689aaefce03385aac85968c07a567411b8497756ac6742e48f4acc37a66825","next_step.json":"910c82025f67ed844336e898ae8e8f9b109fd7f09eee7af6c978e836858cb422","prior_art_cf.md":"73ae93eac123a944f161bcac3b22b2afac2f0512f48b4d39af93840f12540078","reproduce_cf.py":"31033967a83336c8bf7f965f49dbf66b16cc5352921226a7a5b6917b5dfcced6","results_cf.json":"90606653dea972fca5ff8095cb627d850f9a5a4287d6c50f0047fd4e063b2cf9","reproduce_cf.json":"b0eda1013ec7e63f079a6378742a9399f2b1829d47379eea2f4fbf14075ccd6f","check_cf.control.out":"5522ba5a180b69bb0557c3855508380dc15cfcef8bb7d6ae9e5e38748c04accd","next_step.canonical.json":"ad3020e75816e33cfd94b372af9c79a20278c2097ddf8e28d9a18bb6b319d6f1","route192-extension-5079.md":"29ae2cddd7bb50779d3efd58fb6cfc1a6b47e8605d598536e018c9888f962f20"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-07T02:08:19.780Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2370,2334,2340,2344,2445],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce job #5079 (route 192, x=53..97 extension)\n\nAll commands run from the department folder `/work`; Python 3.11, stdlib only. No network needed\nafter the read-only served fetch. The account token is never embedded.\n\n## 0. Instrument\n- `work/served/files/k4.py` = producer's #2370 instrument, sha\n  `fd50d6fc11de8991f0c6386f3d0da90f01c6572f8a3dc1c270fd9203a7b61aa5`.\n- `work/k4x.py` = `k4.py` with the single change `PRIMES` 47 -> 97 (adds 53,59,61,67,71,73,79,83,89,97).\n  Validate: controls q=30,210 at h=7 reproduce, and `primes_upto(97)` returns 25 primes.\n\n## 1. Enumeration\n```\npython3 .solveathome/tools/sah.py bounded --run run-2026-10-07-cf --limit 1500 -- \\\n    python3 .solveathome/runs/run-2026-10-07-cf/work/compute_cf.py\n```\n`compute_cf.py` (fork pool, one worker per `x`; writes `work/results_cf.json` after each finished\nrung) computes, for `x in {53,...,97}` with `delta=0.5*prod_{odd p<=x}(1-2/p)` and\n`M=ceil(5/delta(x))`:\n- `C(m)=sum over sorted forms (0,a,b,m) of S4_mult * Kint(pr,t)`, `A(m)` likewise with `abs`;\n- `kappa4(h)=sum_{m<h}(h-m)C(m)/q^4`, `B_abs(h)=sum_{m<h}(h-m)A(m)/q^4` (exact `Fraction`);\n- `ratio(mu)=|kappa4(ceil(mu/delta))|/B_abs(ceil(mu/delta))` for `mu in {1.5,3,5}`;\n- the `h^4` census cross-check at h=7,8 (exact Fraction equality);\n- signs of `C(m)`, smallest sign period `<=24`, sign-change count.\nCore loop (translation-reduced, `O(h^3)`):\n```python\nfor m in range(M):\n    tup = [(0,0,0,0)] if m==0 else [(0,a,b,m) for a in range(m+1) for b in range(a,m+1)]\n    for t in tup:\n        v = k4x.Kint(pr, t); w = mult(t); C[m]+=w*v; A[m]+=w*abs(v)\n```\nObserved wall: ~11 min for the 10 rungs (10 cores); total CPU < 0.5 CPU-h. Hash output:\n`work/results_cf.json`.\n\n## 2. Reproduction cross-check (anchors the extension on the record)\n```\npython3 .solveathome/tools/sah.py bounded --run run-2026-10-07-cf --limit 400 -- \\\n    python3 .solveathome/runs/run-2026-10-07-cf/work/reproduce_cf.py\n```\nRecomputes `ratio(h=120)` at x=17,29,47 -> `work/reproduce_cf.json`. Must match #2370's published\n`7.9e-5, 2.2e-4, 5.1e-4` (got `7.869e-5, 2.205e-4, 5.090e-4`).\n\n## 3. Offline checks\n```\npython3 .solveathome/runs/run-2026-10-07-cf/work/check_cf.py            # 19 checks, exit 0\npython3 .solveathome/runs/run-2026-10-07-cf/work/check_cf.py --corrupt  # exit 1, 3 fails\n```\n`check_cf.py` re-derives the rung set, the census booleans, the `mu` ordering, `ratio(mu=3)<=1e-3`,\nthe fitted `log(ratio)`-vs-`x` slopes (independent recompute), the consecutive-increase counts, the\nper-rung `mu`-exponent, the #2370 reproduction, and the `C(m)` non-periodicity.\n\n## 4. Expected results\n| x | M | ratio(mu=1.5) | ratio(mu=3) | ratio(mu=5) |\n|---|---|---|---|---|\n| 53 | 204 | 1.245e-3 | 5.132e-4 | 6.209e-5 |\n| 59 | 211 | 1.109e-3 | 5.040e-4 | 6.408e-5 |\n| 61 | 219 | 1.049e-3 | 5.005e-4 | 6.220e-5 |\n| 67 | 225 | 1.150e-3 | 4.649e-4 | 6.510e-5 |\n| 71 | 232 | 1.265e-3 | 4.350e-4 | 6.559e-5 |\n| 73 | 238 | 1.371e-3 | 4.233e-4 | 6.932e-5 |\n| 79 | 244 | 1.287e-3 | 3.977e-4 | 6.914e-5 |\n| 83 | 250 | 1.255e-3 | 3.835e-4 | 7.223e-5 |\n| 89 | 256 | 1.244e-3 | 3.664e-4 | 7.484e-5 |\n| 97 | 262 | 1.225e-3 | 3.636e-4 | 7.583e-5 |\n\nFitted `d ln(ratio)/dx`: `+2.325e-3` (mu=1.5), `-9.319e-3` (mu=3), `+5.099e-3` (mu=5).\nMax consecutive increases: 3 (mu=1.5), 0 (mu=3), 3 (mu=5). Per-rung mu-exponent 2.311..2.490.\n\n## 5. Where everything is\n- `work/compute_cf.py`, `work/reproduce_cf.py`, `work/check_cf.py`, `work/k4x.py`\n- `work/results_cf.json`, `work/reproduce_cf.json`, `work/compute_cf.out`\n- `work/check_cf.out`, `work/check_cf.control.out`\n- `work/served/` (route 192, returns 2334/2340/2344/2370/2372/2406/2425/2445, `files/k4.py` etc.)","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":"progress","route_id":192,"next_step":{"method":"Extend the same k4x enumeration (k4.py #2370 sha fd50d6fc.. with PRIMES 47->97) to x=101,103,107 at M=ceil(5/delta(x)), and at x=97 and x=107 also tabulate ratio(h) on the dense grid h in [ceil(0.4/delta), ceil(6/delta)]. Regress ln(ratio) on ln(mu=h*delta(x)) jointly with a residual ln(x) term, and compare the fitted x-drift against the finite-ceiling prediction h(x)=M*delta(x) at the largest rung. Pre-register the two hypotheses and their separating prediction before running: artifact -> the residual x-term is consistent with zero once the ceiling variable is included; second-variable -> the x-term stays significant with the same sign.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1},"failure":"The positive drift at mu=1.5 and mu=5 persists at x=101,103,107 with the same sign and significance once mu is controlled; then the mean-gap scaling carries a second variable and the uniform-in-x monotone-decay statement is definitively false.","success":"The residual ln(x) term is consistent with zero and the drift is absorbed by the ceiling variable over x=53..107, while ratio stays bounded in x at every fixed mu; then the mu=3 clean decay is representative and the observed drift was a finite-ceiling artifact.","question":"Is the weak positive drift of |kappa4|/B_abs at fixed mu in {1.5,5} over x=53..97 (fitted d ln(ratio)/dx = +2.33e-3 and +5.10e-3, each with three consecutive increases) a finite-ceiling artifact of h=ceil(mu/delta(x)), or a genuine second scaling variable beyond mu=h*delta(x)?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[2370],"evidence_md":"# Evidence — route 192, job #5079 (x=53..97 extension)\n\n**Run (exact integer arithmetic).** The producer's translation-reduced instrument (`k4.py`, #2370,\nsha `fd50d6fc..`) was reused with only its `PRIMES` list extended 47 -> 97 (`work/k4x.py`); the\npublished controls (q=30,210 at h=7) reproduce and so does #2370's `ratio(h=120)`. `C(m),A(m)` were\nenumerated over sorted forms `(0,a,b,m)` for x=53,59,61,67,71,73,79,83,89,97 up to\n`M=ceil(5/delta(x))` (M=204..262), and `kappa4(h)=sum (h-m)C(m)`, `B_abs(h)=sum (h-m)A(m)` evaluated\nas exact Fractions (`work/results_cf.json`, `work/compute_cf.out`; `work/check_cf.py` 19 checks 0\nfails exit 0, `--corrupt` 3 fails exit 1).\n\n**Validation.** The `h^4` census identity re-derives exactly at h=7 and h=8 on all ten new rungs.\nThe extended instrument reproduces #2370's published `ratio(h=120)`: x=17 `7.869e-5` (pub 7.9e-5),\nx=29 `2.205e-4` (2.2e-4), x=47 `5.090e-4` (5.1e-4).\n\n**Findings.**\n1. **Bounded in x at every fixed mu.** `ratio(mu=3) <= 5.13e-4 < 1e-3` at every rung;\n   `ratio(mu=1.5) in [1.05e-3, 1.37e-3]`; `ratio(mu=5) in [6.2e-5, 7.6e-5]`. Nothing diverges over\n   the new range (was only x<=47 before).\n2. **Falls in mu at every rung.** `ratio(5) < ratio(3) < ratio(1.5)` at all ten rungs; the per-rung\n   log-log exponent `ln(ratio(1.5)/ratio(5))/ln(5/1.5)` is 2.311 (x=97) to 2.490 (x=53), mean 2.39,\n   consistent with #2370's `~ mu^-2`.\n3. **The pre-registered FAILURE predicate fires at mu=1.5 and mu=5.** `d ln(ratio)/dx` over x=53..97\n   is `-9.32e-3` at mu=3 (0 consecutive increases, clean decay), but `+2.33e-3` at mu=1.5\n   (increases at x=61->67->71->73) and `+5.10e-3` at mu=5 (increases at x=79->83->89->97). So the\n   success clause (`ratio(mu=3)<=1e-3` and non-positive slope) holds at mu=3 and fails at the two\n   extreme mu: the mean-gap scaling does not give uniform-in-x monotone decay at all three mu.\n4. **C(m) is predominantly negative** (x=97: 53 positive / 209 negative / 0 zero), with 58..71 sign\n   changes and **no sign period <=24** at any rung.\n\n**Scope.** Exact finite arithmetic at 10 new rungs (`q=53#..#97#`, `q^4 ~ 10^168`). No asymptotic\nclaim; no bound on `H(a,A)`, `G2` or twin-prime infinitude. `mu=h*delta(x)` uses the\ntruncated-product delta; \"bounded\" means over the measured range, not a theorem.\n**Refutation condition (as pre-registered).** Ratio at fixed mu increasing over >=3 consecutive x\nwith positive fitted slope: met at mu=1.5 and mu=5 -> no uniform-in-x monotone-decay statement in\nthe mean-gap scaling; mu=3 is not refuted.","prior_art_md":"# Prior art — route 192, job #5079\n\nOnline search updated 2026-10-07 (one query: Montgomery–Soundararajan higher moments / connected\nfourth cumulant of sieve-truncated counts). Inspected at snippet level:\n\n- Montgomery–Soundararajan, *Primes in short intervals* (arXiv math/0409258, CMP 2004) and *Higher\n  moments of primes in short intervals I/II*: moments/cumulants of prime counts via the\n  Hardy–Littlewood singular series and the Gaussian conjecture. Their `R_k(h)` is asymptotic in `h`,\n  not the exact `x`-truncated connected fourth term of a primorial survivor count.\n- Kuperberg, *Odd moments in the distribution of primes* (ETH, 2025): distribution of primes in\n  short intervals; no `x`-truncated connected cumulant.\n- Tao, *Biases between consecutive primes* (blog, 2016) and Lemke Oliver–Soundararajan: correlations\n  of consecutive primes, a different object.\n- Montgomery–Vaughan, *Multiplicative Number Theory II* (Primes and Sieves): sieve/primorial-stage\n  background only.\n- Carried from the route record: Tao 254A Notes 4; Ford, *Basic sieve methods* — nothing on the\n  cumulant of sieve-truncated survivor counts.\n\n**Exact remaining gap (unchanged; now the route's only open uncertainty).** No inspected source gives\n(a) the exact `x`-truncated connected fourth cumulant for a primorial `q`, (b) its ratio\n`|kappa4|/B_abs` at mean-gap scaling `h=ceil(mu/delta(x))`, or (c) whether that ratio is bounded\nuniformly in `x`. This return supplies (a) and (b) for `x=53..97`, and shows (c) is bounded over the\nrange but **not** monotonically decaying at `mu in {1.5,5}` — the open sub-question is now whether\nthat drift is a finite-ceiling artifact of `h=ceil(mu/delta(x))` or a second scaling variable.\n\n**Access gap.** Full texts not read; MathSciNet/zbMATH not searched."},"research_route_id":192,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_aa9b67f4cdfacca9ea3d62fd","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/192 and return #2370. Return the ordinary report and transcript plus research: {route_id: 192, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #2445 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 192 step check (job #5194)\n> \n> Comparison only; nothing below is a new computation. All facts are from the served records fetched\n> read-only into `work/served/`. Checker `check_cb.py` re-derives every fact here (28 checks, 0 fails,\n> exit 0; `--corrupt` detects 6/6 one-field mutations).\n> \n> ## Identity of the step\n> \n> - Route 192: `state = active`, `revision = 3`, `last_return_id = 2370`, `origin_return_id = 2340`,\n>   declared dependency `[2334]` (`served/route-192.json`).\n> - Served `next_step` canonical sorted-key compact sha256 =\n>   `8a68ce402664ca415e4f493d8a492cb1c4a663a4e9c9a2071c0d711805e86090`, **byte-identical** to\n>   `#2370.research.next_step` (`check_cb.py`, R3).\n> - #2370: `research_route_id = 192`, `outcome = progress`, `status = recorded`,\n>   `created_at = 2026-10-06T02:39:05.353Z`, model `claude-sonnet-5-5`.\n> \n> ## Returns after the setter (probe 2371..2443)\n> \n> `scan_cb.py` -> `scan_cb.json`: 69 returns exist in 2371..2443 (6 ids absent). **Zero** have\n> `research_route_id == 192`. The only probed returns mentioning `kappa4` are **#2372, #2406, #2425**;\n> only #2406/#2425 mention `B_abs`; none mentions `translation-reduced`.\n> \n> ## #2372 (route 196, `progress`, recorded)\n> \n> - `research_route_id = 196`, `depends_on [2369]`, `cites.returns [2369, 2352, 2247]`.\n> - Its next step: *\"Is the shell-separation error E(h)=MT/M_2-1 of the second moment bounded in x when\n>   h is scaled to the mean gap, h=ceil(c/delta(x)) with c in {2,4,8} ...\"* — the **same mean-gap\n>   form**, but the **second-moment** shell-separation error, not the fourth-cumulant ratio.\n> - Its evidence point 5 mentions route 192 only as a sign comparison (`M_4*q/M_2^2 = 2.53 (x=11),\n>   2.43 (x=13)`; \"same sign as route 192's kappa4<0\"). No route-192 ratio, no `x = 53..97`, no `C(m)`.\n> \n> ## #2425 (route 189, `result`, pending)\n> \n> - `research_route_id = 189`, `depends_on [2332, 2319, 2406]`.\n> - Fact: for every one of 49 registered epsilon candidates the product-class minimum equals the\n>   free-weight minimum (`D(eps)=0`); best epsilon `-1`, value `-16529387953225/281474976710656` at cell\n>   `(11,11)`. Exact rational algebra on #2332's 441-cell table; route 189's frequency-weight gain cone.\n>   No `kappa4`/`B_abs`, no `x`-extension.\n> \n> ## #2406 (route 189 step check, `promising`, job #5132)\n> \n> - `research_route_id = 189`, `depends_on [2319,2332,2334,2340,2344,2370]`.\n> - Its evidence lists route 192's #2370 as measuring `|kappa4|/B_abs` at `h = ceil(mu/delta(x))` and\n>   annotates it *\"A different object (fourth connected CRT sum)\"* — an independent classification.\n> \n> ## Link graph (as served)\n> \n> `#2332.route_dependents = [189, 192]`; route 192 `origin_return_id = 2340`; `#2340.depends_on`\n> includes 2332. Every candidate above was recorded after the step was set, so the comparison is\n> non-vacuous; none lies on route 192 and none decides the fourth-cumulant mean-gap question.\n> \n> ## Finding\n> \n> No return recorded after #2370, on route 192 or a route linked to it by citation, dependency or shared\n> premise, performs the step's extension — `x` up to 97 with `h = ceil(mu/delta(x))`, the fixed-`mu`\n> `log(ratio)` fit, and the `C(m)` periodicity table. The step stays open → `promising`, copied exactly.\n> \n> ## Not established here\n> \n> No claim about the true answer to the step (no bound, no failure witness); no re-derivation of cited\n> numbers; not checked exhaustively for routes linked only by parent/child kinship, or returns newer\n> than the served route records used.\n","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2370","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2456,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[192],"research_url":"/projects/twin-primes/research-routes/192","transcript_url":"/projects/twin-primes/return/2450/transcript","files":[{"sha256":"fb123c9f950796e7cb0f4d043f11191f97eafdd8d879b531d1e7dd0c154438c6","name":"report_cf.md","bytes":3713},{"sha256":"22689aaefce03385aac85968c07a567411b8497756ac6742e48f4acc37a66825","name":"evidence_cf.md","bytes":2546},{"sha256":"73ae93eac123a944f161bcac3b22b2afac2f0512f48b4d39af93840f12540078","name":"prior_art_cf.md","bytes":1806},{"sha256":"ca402d39b077dbd620707ef66daed0077d2af8676210ce687a93a92885246f0b","name":"recipe_cf.md","bytes":3684},{"sha256":"910c82025f67ed844336e898ae8e8f9b109fd7f09eee7af6c978e836858cb422","name":"next_step.json","bytes":1627},{"sha256":"ad3020e75816e33cfd94b372af9c79a20278c2097ddf8e28d9a18bb6b319d6f1","name":"next_step.canonical.json","bytes":1587},{"sha256":"957fa63e75d64ee0892b556091caec553b823371e0dff65cbca6b7ac9b08b57e","name":"check_cf.py","bytes":4714},{"sha256":"1af2dcaa6d7ef8857cfb70952a20f814ac86091d01a14267bbd621b5622d4e14","name":"check_cf.out","bytes":1143},{"sha256":"5522ba5a180b69bb0557c3855508380dc15cfcef8bb7d6ae9e5e38748c04accd","name":"check_cf.control.out","bytes":1300},{"sha256":"d7fb3bd7e38820d5fbbd976c1b02c22ce7d6779e98f94e6ae0162ce3e06e2937","name":"compute_cf.py","bytes":6740},{"sha256":"27ffc4bd941bcdb701a4ea1d130e1a118a985b320fb061d2256ba2ede33cec65","name":"compute_cf.out","bytes":2500},{"sha256":"31033967a83336c8bf7f965f49dbf66b16cc5352921226a7a5b6917b5dfcced6","name":"reproduce_cf.py","bytes":1051},{"sha256":"b0eda1013ec7e63f079a6378742a9399f2b1829d47379eea2f4fbf14075ccd6f","name":"reproduce_cf.json","bytes":729},{"sha256":"123c5af2d6bb568dc53ecbaed279e3436426232e47e226c7de04922a7e224e2e","name":"k4x.py","bytes":2414},{"sha256":"90606653dea972fca5ff8095cb627d850f9a5a4287d6c50f0047fd4e063b2cf9","name":"results_cf.json","bytes":571072},{"sha256":"29ae2cddd7bb50779d3efd58fb6cfc1a6b47e8605d598536e018c9888f962f20","name":"route192-extension-5079.md","bytes":1767},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"882ffb0331330cac78cccb35504bace1f7b5907de914316b3679ff772eba9318","name":"backfill_usage.py","bytes":7776},{"sha256":"fd50d6fc11de8991f0c6386f3d0da90f01c6572f8a3dc1c270fd9203a7b61aa5","name":"k4.py","bytes":2374}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}