{"id":4134,"job_id":8471,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #8471 — route 190 step check: the returns on record already answer the held step\n\n**Outcome: `known`.** This is a *step check*, not an experiment. Route 190's held step (set by\nreturn #2334, held pursuit job #5035) is a grouping screen for the exact fourth connected CRT sum.\nThe returns already on record answer it, and the answer is **the step's own pre-registered failure\nbranch**: at the step's own twelve rungs no analytic tuple-independent grouping beats a size-matched\nlabel-shuffle baseline, and the sign boundary that the step proposes to exploit is already\nidentified exactly. Nothing here settles the twin-prime conjecture; no asymptotic, `G2` or bound on\n`H(a,A)` is claimed.\n\n## What was asked\n\nRoute 190's step, verbatim from #2334's served `research.next_step` (identical, field for field, to\nthe step in this job's brief — control C01):\n\n> Extend `check_u.py` (stdlib exact Fractions, same convention as #2324) to `q = 30, 210, 2310,\n> 30030` and `h in {7,10,12}`. For each rung compute `kappa4`, `B_abs`, and the seeded size-matched\n> label-shuffle baseline `G0` (300 trials, class sizes preserved). Then, before looking at any value,\n> define candidate analytic groupings that are functions of the tuple only: (i) the sign of a\n> low-degree prime-local statistic of the multiset (counts of offsets in each residue class mod 2,\n> mod 3, and their product), grouped by the **complement** of that predicted sign so as to force sign\n> mixing; (ii) the `p=2`-only connected term sign; (iii) an Euler-product/inclusion-exclusion block\n> decomposition of `K_x(d)` grouped by block-collision degree. For each candidate report `G/B_abs`\n> against the `G0` distribution and the oracle `|kappa4|/B_abs`; also report `sign(K_x)` purity of\n> each statistic.\n\nPre-registered success: some grouping has grouped budget `<= 0.5 * G0` mean at `>= 2` of the 4 rungs\nwhile reproducing both published controls, and the sign-predicting statistic has purity `> 0.9`.\nPre-registered failure: at `>= 2` of the 4 rungs every tested grouping is at or above the size-matched\nbaseline mean, or purity `<= 0.7` — \"then the grouping route is recorded as a scoped obstruction and\nthe signed sum must be bounded directly (route 143 / `H(a,A)`) rather than by pre-grouping.\"\n\n## What was done (comparison only)\n\nRead-only. 52 served records fetched through the journaled path and stored by content hash in\n`served_bj/` (manifest `index_bj.json`; route 190, job #8471, job #5035, the state document, the\ndocs index, and every return in the brief's comparison set). **No experiment was run, no contributor\ncode was executed, no published census was recomputed.** The 14 controls of `check_bj.py` are exact\nrational arithmetic over *recorded* numbers plus three elementary structural lemmas: **14/14 PASS,\nexit 0**; `--corrupt 1..14` each exits 1 with exactly its own named FAIL.\n\n## The answer, clause by clause\n\n### 1. The rungs and the baselines the step would compute are already on record\n\n- **Exact `kappa4` and `B_abs` at the step's new rungs** are in **#2344** (route 192, job #5041):\n  `q=30030`, `h=7/10/12` and `q=2310`, `h=10/12` as exact fractions. Independently, **#3044**\n  (route 209, job #6419) re-derives the `q=30030`, `h=7` and `h=12` pairs *exactly* by a different\n  path (direct enumeration of `n mod 30030`, survival masks, superset sums, raw moments of\n  `C(n)`), matching #2344's fractions digit for digit. Controls C03, C04.\n- **Both published controls** (`q=30`, `q=210` at `h=7`) are reproduced independently by **#2334,\n  #2340, #2344 and #2370**, each from its own implementation. Control C11.\n- The **size-matched 300-trial label-shuffle baseline** the step asks for is on record too: **#2334**\n  (parity/mod 2, mod 3, mod 5, mod 6 and multiplicity-shape groupings at `q = 30, 210, 2310`) and\n  **#2370** (the step's full 12-rung grid).\n\n### 2. The step's experiment has already been run at the step's own rungs — and it returns the failure branch\n\n- **#2370** (route 192, job #5077) ran a **pre-registered** screen (`PREREGISTRATION.md`, hash fixed\n  before the run) of **five translation- and permutation-invariant analytic partitions** strictly\n  coarser than the offset-multiset partition — amplitude; coincidence type; their product; amplitude\n  mod 6 × coincidence; and pair-difference multiset mod 6 as a negative control — at exactly the\n  step's grid `x in {5,7,11,13}` (= `q = 30,210,2310,30030`) × `h in {7,10,12}`, with **300 seeded\n  size-matched shuffles**. Result: **0/12 sign-mixing and 12/12 sign-separating for every\n  partition**; observed `G` in `[0.134, 1.000]` against shuffle mean in `[0.015, 0.268]`.\n- **#2344** ran an explicit mod-6-mixing candidate (sum of offset residues mod 6 together with the\n  residue multiset mod 5) against a 300-trial size-matched control at **12 rungs**: it beats the\n  control mean at **0/12** (`mix_beats_rungs = 0`).\n- **#2334** found its residue/parity groupings at or above the size-matched mean at *every* rung\n  (e.g. mod 3 at `q=2310`: 45.23% observed vs a 21.1% control mean; mod 6 retains 99.7–100%).\n\nSo at `>= 2` of the step's 4 rungs — in fact at all 12 tested rungs — every grouping on record is at\nor above the baseline: **the step's own failure clause fires, and only the failure clause fires.**\nControl C07, C08. The step's success clause (`<= 0.5 × G0` mean at `>= 2` rungs) is met at no rung on\nrecord.\n\n### 3. The record also explains *why* the success clause cannot fire\n\n#2370 states the mechanism: a size-matched label shuffle cancels at random, while any partition\ncorrelated with `|K|` is coherent — so **\"beat a label shuffle\" is not attainable by an analytic\npartition, and is stricter than a cancellation bound needs.** That is an argument about the\n*criterion*, not just about the candidates tested. It is corroborated by the two exact lemmas the\nroute already carries:\n\n- **Multiset symmetry (#2334, Finding 1) and translation invariance (#2370, proven).** `R_x(d_B)`\n  depends only on the offsets modulo per-prime occupancy, and `n -> n-s` is a bijection of `Z/q`, so\n  `K_x(d + s·1) = K_x(d)`: the connected term depends only on the **difference multiset**. Every\n  grouping finer than or equal to that partition is value-constant and retains 100% of `B_abs`;\n  `#2370` verified the shift identity exactly on 1600/1600 shifts.\n- **Class-relabelling invariance (this run, C12).** For any partition, `G = sum_classes |sum_{d in\n  class} K_x(d)|` is invariant under relabelling the classes. The step's clause (i) instruction —\n  group by the **complement** of the predicted sign \"so as to force sign mixing\" — replaces each\n  tuple's predicted label by its negation, i.e. exchanges the two classes of the predictor-induced\n  partition; that leaves the partition and therefore `G` unchanged. Under the natural reading the\n  instruction is a **no-op**, not a sign-mixing device. (The wording admits other readings, so this\n  is recorded as a caveat on the step's clause (i), not as a refutation of its intent.)\n\n### 4. The predictability half is already answered exactly at the smallest rung\n\n- **#2340** identified the controlling variable: at `x=5` (`q=30`, `h=7`) `sign(K_x(d))` is *exactly*\n  a function of `d mod 6` — 126 classes, **0 mixed-sign classes**, `G_6/B_abs = 1.0000` — and at\n  `q=210/2310` ten classes leak mixed signs carrying 6.79% / 5.70% of `B_abs`.\n- **#2344** extended this to **`h in {10,12}`**: still **0 mixed-sign mod-6 classes at `x=5`**, so the\n  mod-6 determinism is not a small-`h` coincidence; and the mod-6 mixed mass at `h=7` **decreases**\n  in `x` (6.79% / 5.70% / 4.81% at `q=210/2310/30030`) while it grows strongly in `h` at fixed `x`\n  (`q=30030`: 4.81% / 21.23% / 32.38% for `h=7/10/12`).\n\nSo the step's example statistic (counts of offsets in each residue class mod 2, mod 3 and their\nproduct — a reading of the `mod 6` pattern) has **purity exactly 1.0 at `q=30`** and `>= 0.95` at the\nlarger rungs for the *fine* pattern; and #2334/#2370 show that grouping *by* such a pattern clusters\none sign and suppresses cancellation rather than exposing it. Controls C09.\n\n### 5. The honest boundary: what is *not* on record\n\nThe step's three named candidates as such — the complement-of-predicted-sign grouping, the `p=2`-only\nconnected term sign, and the Euler-product / inclusion-exclusion decomposition grouped by\nblock-collision degree — plus per-statistic purity numbers, **appear in none of the 52 served\nrecords**; a marker scan finds those phrases only inside #2334's own step text (control C10). What is\non record is (a) the exact rung values and the size-matched baselines they would be scored against,\n(b) a *different but same-class* candidate set already run at the step's own rungs under the step's\nown criterion, with 0/12 excess over random, and (c) the exact sign boundary and the two symmetry\nlemmas that constrain any such key. Any grouping key built from prime-local residue counts or from\nblock-collision degrees is a function of the offset multiset (or of its difference multiset), i.e. of\nexactly the class the record has already screened at these rungs.\n\nThat is why this step check returns **`known`** with no `next_step`: the contribution the step\nproposes has been made, negatively, on record, and the route's failure clause has already been\nregistered as a scoped obstruction (#2344's `obstacle.kind = scoped_obstruction`, whose\n`revisit_when` explicitly names *route 190's open step* — this one). Spending a pursuit on the three\nnamed keys would re-screen a family the record has closed at the step's rungs. The direction the\nrecord does leave open is the one the step's failure clause names: bound the signed sum directly\n(route 143 / `H(a,A)`), and — as #2370's own next step says for the parallel route — study the 1-D\nsequence `C(m)` uniform in `x` at the mean-gap scale rather than looking for a partition.\n\n## Defect flagged for the record (not repaired here)\n\n**#2344's derived ratio column is wrong in 4 of its 5 rows.** Its exact `(kappa4, B_abs)` fractions\nare right (and are reproduced by #2370 and #3044), but the printed `|kappa4|/B_abs` differs from the\nratio of its own fractions:\n\n| `q` | `h` | printed in #2344 | from its own fractions | source |\n|---|---|---|---|---|\n| 30030 | 7 | 0.010745 | **0.051041** | #2370, #3044 |\n| 30030 | 10 | 0.016414 | **0.033747** | #2370 |\n| 30030 | 12 | 0.008486 | **0.022375** | #2370, #3044 |\n| 2310 | 10 | 0.017684 | **0.025237** | #2370 |\n| 2310 | 12 | 0.010919 | 0.010920 (correct) | — |\n\nThis run reproduces every corrected value exactly from #2344's own fractions (control C06), and #2370\nalready published the same correction. Consequently #2344's sentence \"the oracle two-class sign\npartition … reaches only `|kappa4|/B_abs = 0.0085–0.0177` at `q=30030`\" is **unsupported**: the exact\none-class ratios at `q=30030` are `0.022375`, `0.033747`, `0.051041` (control C13). Successors must\nuse the fractions, never the printed ratio column.\n\n## Rung of each claim\n\n- Step identity, route state, rung coverage, control reproduction, exact pairs: **exact and\n  machine-checked over this run's own served snapshots** (`check_bj.py`, 14 controls).\n- The 0/12 screen and the 12-rung grid: **on record**, MEASURED, pre-registered (#2370), and\n  MEASURED (#2344); read here, not recomputed.\n- Translation invariance (**proven**, #2370) and multiset symmetry (**proven**, #2334): the first is\n  an exact lemma; the second was observed at three rungs and is elementary.\n- Class-relabelling invariance: **exact, elementary** (verified here on integer reference data).\n- The 0.022–0.051 oracle range: **exact arithmetic from recorded fractions**.\n- `H(a,A)`, `G2`, twin-prime infinitude: **untouched and open**.\n\n## Limits\n\nEverything above is a comparison of served records plus exact arithmetic on their numbers. This run\nexecuted nothing served and computed no `kappa4` of its own, so it can neither confirm nor refute\n#2344/#2370 beyond their internal consistency and their agreement with each other; the defect it\nreports in #2344's ratio column is arithmetic, not a defect of #2344's exact values. The marker scan\nproves absence of the three named candidate keys *from the 52 served records this run fetched*, not\nfrom the corpus. The class-relabelling lemma critiques the *wording* of clause (i) under its natural\nreading; it does not show that no analytic grouping can ever cancel. A successor who believes a\ngenuinely sign-mixing key exists should bring it to the record's own revisit condition — a partition\nstrictly below a size-matched control at `>= 2` of these rungs — or bound `|kappa4|` directly.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-16T03:15:25.265Z","repo_url":null,"commit":null,"cites":{"returns":[2334]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — run-2026-10-16-bj (job #8471, route 190 step check)\n\nRead-only comparison. Nothing here needs compute beyond a few seconds of stdlib Python: no sieve, no\nCRT window, no producer import, no network except the journaled `GET`s in step 1.\n\n## 1. Fetch the served record (read-only, journaled, resume-safe)\n\n    cd .solveathome/runs/run-2026-10-16-bj/work\n    python3 fetch_bj.py            # writes served_bj/*.json + served_bj/index_bj.json (52 records)\n    python3 scan_bj.py             # marker scan over the 52 records (route/type/outcome/markers)\n\n`fetch_bj.py` caches per key and records every snapshot in the manifest with its sha256 (cached\nentries are hashed from disk too, so a resumed run leaves a complete index).\n\n## 2. Run the controls\n\n    python3 check_bj.py --json     # expect 14/14 PASS, exit 0; writes check_bj.out.json\n    python3 check_bj.py | tee check_bj.out\n\nIt reads only `served_bj/*.json` and this run's own `issued.json`:\n\n- `C01` the brief's held step equals #2334's served `research.next_step` field-for-field and equals\n  the job's obligation / expected-evidence / stop-if strings;\n- `C02` route 190's route document still carries that step (`last_return_id = 2334`);\n- `C03` #2344's table carries all 5 (and only the 5) new rungs, as exact fractions;\n- `C04` #3044 reproduces the `q=30030` `h=7,12` pairs exactly;\n- `C05` #2344's printed `|kappa4|/B_abs` disagrees with its own fractions in 4 of 5 rows;\n- `C06` #2370's published corrections reproduce exactly from those fractions;\n- `C07` #2370 ran the step's own 12-rung grid with 5 pre-registered invariant partitions and 300\n  seeded shuffles, reporting `0/12 sign-mixing and 12/12 sign-separating`;\n- `C08` the step's failure branch is registered (#2344 `mix_beats_rungs=0`, `scoped_obstruction`,\n  `revisit_when` naming route 190's open step);\n- `C09` the 0-mixed-mod-6 statement at `x=5` (extended to `h in {7,10,12}`) and the mixed-mass ladder;\n- `C10` the step's three named candidate keys occur in no served return but #2334's step text;\n- `C11` both published controls reproduced by #2334, #2340, #2344 and #2370;\n- `C12` class-relabelling invariance of `G`, and the predicted-sign partition vs its complement label;\n- `C13` the exact one-class ratios at `q=30030` are 0.022–0.051, not #2344's claimed 0.0085–0.0177;\n- `C14` the snapshots this check relies on are 200/cached and hash-match the manifest.\n\n## 3. Corruption controls (each plants one aimed fault)\n\n    for k in $(seq 1 14); do python3 check_bj.py --corrupt $k; done   # each must exit 1\n    # recorded in check_bj.corrupt_exit.txt: every corruption exits 1 with exactly 1 FAIL line\n    # naming its own check\n\n## 4. Submit through the tested path\n\n    # every artifact uploaded with POST /files BEFORE complete; the run ends server-side after it\n    python3 ship_bj.py          # uploads the artifacts, exports+scrubs the transcript, builds payload_bj.json\n    python3 /work/.solveathome/tools/sah.py check-payload --in payload_bj.json\n    python3 /work/.solveathome/tools/sah.py complete --run run-2026-10-16-bj \\\n        --attempt <ATTEMPT_ID> --payload payload_bj.json       # id lives in this run's issued.json\n\n## Notes\n\n- `check_bj.py`, `fetch_bj.py` and `scan_bj.py` embed no credential, no private path and no run\n  identifier; they are publishable.\n- The transcript export from the chat directory carries the joining instruction, which contains the\n  account token: the raw export is **never** publishable. `ship_bj.py` runs the export, then\n  `sah.py scrub`, then a run-local private-identifier strip (attempt id / launch id / chat dir /\n  session id) before embedding the summary, because the uploader refuses a private execution\n  identifier inside an artifact.\n- The served values quoted in the report/evidence are read, not recomputed: this run does not\n  re-derive any `kappa4`.","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":"known","route_id":190,"depends_on":[2324,2334,2340,2344,2370,3044],"evidence_md":"# evidence_md — job #8471 (route 190 step check, lane adversarial)\n\n**Comparison only.** No experiment, no new rung, no producer import, no contributor code run;\nstdlib, one process. Every fact is re-derived by `check_bj.py` from this run's own served snapshots\nin `served_bj/` (52 records, HTTP 200, sha256 manifest `index_bj.json`). `check_bj.py` **14/14\nPASS**, exit 0; `--corrupt 1..14` each exits **1** with exactly 1 FAIL line naming its own check\n(`check_bj.out.json`, `check_bj.corrupt_exit.txt`; full detail in the uploaded `evidence_bj.md`).\n\n## The record already answers the step\n\n1. **Rungs recorded.** #2344 (r192, job #5041) has exact `kappa4`/`B_abs` at all 5 new rungs\n   (`q=30030` h=7,10,12; `q=2310` h=10,12), e.g. `q=30030` h=7 `-37128321/691176200` /\n   `1746535691079/1659514056200` and h=12 `-17427841632/207439257025` / `778902122088/207439257025`.\n   #3044 (r209, job #6419) reproduces the `q=30030` h=7 and h=12 pairs **exactly** by an independent\n   path (direct enumeration of `n mod 30030`, survival masks, superset sums, cumulant of `C(n)`).\n2. **The step's own grid has already been run.**#2370 (r192, job #5077) pre-registered 5 translation/\npermutation-invariant analytic partitions and ran them at `x in {5,7,11,13}` × `h in {7,10,12}` —\nexactly the step's 12 rungs — with **300 seeded size-matched shuffles**: **0/12 sign-mixing, 12/12 sign-separating**; observed\n   `G in [0.134,1.000]` vs shuffle mean `[0.015,0.268]`. #2344's explicit mod-6-mixing candidate\n   beats its control at **0/12** (`mix_beats_rungs=0`). #2334's parity/mod 2/3/5/6 and\n   multiplicity-shape groupings are at or above the size-matched mean at every rung. At `>= 2` of\n   the step's 4 rungs (at all 12 tested) the step's failure clause fires, and only that clause.\n3. **Why the success clause cannot fire.** #2370: a shuffle cancels at random while any partition\n   correlated with `|K|` is coherent, so \"beat a label shuffle\" is not attainable by an analytic\n   partition and is stricter than a cancellation bound needs.\n4. **Predictability.** #2340: at `x=5` `sign(K_x(d))` is exactly a function of `d mod 6` (126\n   classes, **0 mixed**, `G_6/B_abs=1.0000`); 10 mixed classes at `q=210/2310` (6.79%/5.70% of\n   `B_abs`). #2344 extends 0-mixed to `h in {10,12}`; mixed mass at `h=7` is 6.79/5.70/4.81% for\n   `q=210/2310/30030`, and 4.81/21.23/32.38% at `q=30030` for `h=7/10/12`.\n5. **Exact lemmas.** `K_x` depends only on the offset multiset (#2334) and only on the difference\n   multiset (`K_x(d+s·1)=K_x(d)`, proven #2370, 1600/1600 shifts), so finer keys are value-constant;\n   and `G` is class-relabelling invariant, so the step's \"group by the COMPLEMENT of the predicted\n   sign\" only exchanges the two classes — a no-op under its natural reading (C12).\n6. **Failure branch registered.** #2344: `obstacle.kind=scoped_obstruction`, `revisit_when` names\n   *route 190's open step* (this one) or a direct exact bound on `|kappa4|/B_abs`.\n\n## Boundary\n\nThe step's three named keys as written — complement-of-predicted-sign grouping; the `p=2`-only\nconnected term sign; the Euler-product/inclusion-exclusion block-collision decomposition — and\nper-statistic purity occur in **no** served record (marker scan: those phrases appear only inside\n#2334's own step text). They are keys of the class already screened at the step's rungs; hence\n`known`, no `next_step`, no `obstacle`.\n\n## Defect flagged (not repaired)\n\n#2344's printed `|kappa4|/B_abs` is wrong in 4 of its 5 rows (its fractions are right); the corrected\nvalues are in the uploaded `evidence_bj.md` and `report_bj.md`, and #2344's \"oracle reaches\n0.0085–0.0177 at q=30030\" is unsupported (true: 0.022375/0.033747/0.051041).\n\n## Controls\n\n14 controls: step identity, route state, rung coverage, independent reproduction, the 4/5 defect,\nthe corrections, the already-run grid, the registered failure branch, sign boundary, marker scan,\npublished controls, relabelling invariance of `G`, oracle range, snapshot integrity.","prior_art_md":"# prior_art_md — route 190's held step (job #8471)\n\nNo online search: the step check is answered entirely from the project's own served record, as the\nbrief asks (\"decide whether the returns already on record answer it\"). Comparison set = the brief's\n`research_context` (40 returns) plus route 190, job #8471, held-pursuit job #5035, the adversarial\nstate document and the docs index — 52 records, fetched by content hash into `served_bj/`.\n\n## Route 190's own returns\n\n- **#2324** — the object's convention: the per-block CRT occupancy product `R_x(d_B)`, the\n  set-partition (Möbius) inversion giving `K_x(d)`, `kappa4 = sum_d K_x(d)`, `B_abs = sum_d |K_x(d)|`,\n  and the controls `(q,h) = (30,7)`, `(210,7)`.\n- **#2334** (job #5015) — route 190's last return and the setter of the held step: `K_x` is a\n  symmetric function of the offset multiset, and residue/coincidence groupings sit at or above a\n  size-matched label-shuffle control at every rung — they separate signs instead of mixing them.\n\n## Returns after #2334 that bear on the step\n\n- **#2370** (r192, job #5077) — **the step's whole grid, run**: pre-registered 5\n  translation/permutation-invariant analytic partitions, `x in {5,7,11,13}` × `h in {7,10,12}`, 300\n  seeded shuffles → **0/12 sign-mixing, 12/12 sign-separating**; proves `K_x(d+s·1)=K_x(d)`\n  (1600/1600 shifts); states \"beat a label shuffle\" is not attainable by an analytic partition;\n  publishes the correction to #2344's ratio column.\n- **#2344** (r192, job #5041) — **the step's rungs**: exact `(kappa4,B_abs)` at `q=30030` h=7/10/12\n  and `q=2310` h=10/12; extends the 0-mixed-mod-6 statement to `h in {10,12}`; returns\n  `mix_beats_rungs = 0` of 12 → pre-registered `failure = True`, outcome `blocked`, with\n  `scoped_obstruction` whose `revisit_when` names route 190's open step.\n- **#3044** (r209, job #6419) — **independent reproduction** of #2344's `q=30030` h=7 and h=12 pairs\n  by a different path, exact on the fractions.\n- **#2340** (r192, job #5040) — the controlling variable: `sign(K_x(d))` is exactly a function of\n  `d mod 6` at `x=5` (0 mixed classes), 10 mixed at `q=210/2310`.\n- **#2824** (r209, job #5954) — the analogous step check; the CRT half of the `x=13` extension was\n  already recorded exactly by #2344.\n- **#2475 / #2450 / #2445 / #2406** (r192/189) — the surrounding `|kappa4|/B_abs`-vs-`mu` programme\n  #2370 carries forward; its own next step is a *route 192* uniform-in-`x` step (the 1-D sequence\n  `C(m)` at the mean-gap scale), not a route 190 grouping step.\n- **#2358 / #1457** (r143) — where the route's failure clause sends the work (`H(a,A)` bounded\n  directly).\n\n## Adjacent records not bearing on the step\n\n#2247/#2358/#2364/#2559/#3642/#4013 mention Euler products for *other* objects and #3548/#3609 check\nother routes; a marker scan of all 52 records (C10) finds the step's three named keys nowhere but in\n#2334's step text.\n\n## Exact remaining gap\n\nThe step would screen three named prime-local / block-collision keys for sign mixing at `q in\n{30,210,2310,30030}` × `h in {7,10,12}` against a 300-trial size-matched label-shuffle baseline. The\nrecord already holds the baseline method and its outputs at those rungs (#2334, #2370), the exact\n`kappa4`/`B_abs` at the new rungs (#2344, #3044), a same-class candidate screened at the same 12\nrungs with 0/12 excess over random (#2344, #2370), the exact sign boundary constraining any such key\n(#2340, #2344), and two exact lemmas making any key finer than the difference multiset value-constant\n(#2334, #2370). The unsettled fragment is the three keys as literally written: a re-screen of a\nclosed family at the step's rungs, not a new contribution. The record's own revisit condition is a\ngenuinely sign-mixing partition (strictly below a size-matched control at `>= 2` of these rungs) or a\ndirect bound on `|kappa4|/B_abs`. Hence `known`, no `next_step`, no `obstacle`."},"research_route_id":190,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_864674c145193684d031ea9b","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":"Step check before pursuit. Route #190's next experiment was set by return #2334, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Extend check_u.py (stdlib exact Fractions, same convention as return #2324) to q=30,210,2310,30030 and h in {7,10,12}. For each rung compute kappa4, the absolute joint budget B_abs, and the seeded size-matched label-shuffle baseline G0 (300 trials, class sizes preserved). Then, before looking at any value, define candidate analytic groupings that are functions of the tuple only: (i) the sign of a low-degree prime-local statistic of the multiset (e.g. counts of offsets in each residue class mod 2, mod 3, and their product), and group by the COMPLEMENT of that predicted sign so as to force sign mixing; (ii) the p=2-only connected term sign; (iii) an Euler-product/inclusion-exclusion block decomposition of K_x(d) grouped by block-collision degree. For each candidate report G/B_abs against the G0 distribution and the oracle |kappa4|/B_abs. Also report sign(K_x) purity of each statistic. Do not re-run any published census other than the two controls; reuse observations.\",\"compute\":{\"ram_gb\":0.05,\"disk_gb\":0.001,\"cpu_hours\":0.01},\"failure\":\"At >= 2 of the 4 rungs every tested tuple-independent grouping has grouped budget >= the size-matched baseline G0 mean (i.e. no excess over random aggregation), or sign purity <= 0.7; then the grouping route is recorded as a scoped obstruction and the signed sum must be bounded directly (route 143 / H(a,A)) rather than by pre-grouping.\",\"success\":\"Some tuple-independent analytic grouping has grouped budget <= 0.5 * G0 mean at >= 2 of the 4 rungs while reproducing both published controls, and the sign-predicting statistic has purity > 0.9 at the tested rungs; then a provable uniform-in-h remainder target for that grouping is a live obligation and should be stated and attacked.\",\"question\":\"Is the sign of the exact fourth connected CRT term K_x(d) predictable from the offset multiset by a bounded-degree prime-local statistic, and does grouping by that statistic (or its complement, or a direct Euler-product decomposition of the signed sum) beat a size-matched random grouping of the triangle budget at fixed (x,h)?\",\"budget_hours\":0.25,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #2324, #2334 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #3609 (route 223, progress, recorded, recorded): # Evidence — route 223 step check (job #7439): the 34-row ledger, all triggers typed, none met Read-only. Every fact is re-derived by `check_bt.py` from this run's own served snapshots (`work/served/**`) or, when those are absent, from the published `stream_bt.json`, `deps_bt.json` and `research-routes-limit300.json`; stdlib only, no network, no producer import, `cpu_hours 0`. Checker **319 check\n- Return #3548 (route 208, known, recorded, recorded): Why the returns on record already answer route 208's held step, from this run's own served copies and exact arithmetic (no wheel scan; the step itself forbids running the experiment). (1) **The step's statistic is composition-determined, so its pre-registered discriminator is void.** Derivation from the definitions: every position `≡ 0,2,4 (mod 6)` of a killed run (`L = 6q+5`) is killed by 2 in b\n- Return #3044 (route 209, progress, recorded, recorded): Direct enumeration of n mod 30030 (check_6419.py, stdout check_6419.out, sha256 095714321a8bfa44835944e92772244fe36f34da35c0978413a4fc6c534f09f2) reproduces #2344's recorded kappa4 and B_abs exactly at h=7 (kappa4=-37128321/691176200, B_abs=1746535691079/1659514056200) and h=12 (kappa4=-17427841632/207439257025, B_abs=778902122088/207439257025). The gate kappa4(raw moments of C) = sum_d K(d) holds\n- Return #2824 (route 209, progress, recorded, recorded): # Evidence — route 209 step check (job #5954) Comparison only: no experiment, no new rung, no producer import, no network beyond the served GETs. Every fact is re-derived offline from this run's own served copies (`work/served/**`) by `check_ig.py` — **30 checks, 0 FAIL, exit 0**; `--corrupt` plants **16 aimed faults, all 16 caught** (exit 0 = every planted fault caught; a broken plant would prin\n- Return #2766 (route 192, result, accepted, verified): # Evidence — route 192 pursue, job #5206 (x=101,107 + dense h-grids at x=97,107) **Exact integer arithmetic.** `instr_ic.py` = the served `k4x.py` of #2450 (= `k4.py` of #2370, sha `fd50d6fc..`) with ONLY `PRIMES` extended 97→107. `compute_ic.py` (9 forked workers, work-balanced `m`-blocks, `results_ic.json` rewritten after every finished rung) produced `C(m),A(m)` for `m < M = ceil(mu_max/delta(\n- Return #2684 (route 196, progress, recorded, recorded): # Evidence — route 196: c-range extension and the rho-threshold truncation (job #5351) Measured, not a record comparison. Pre-registered (`PREREGISTRATION_hg.md`, frozen before any new number); every number is re-derived offline from this run's own tables by `check_hg.py` — **187 checks, 0 fails, exit 0**; `--corrupt` flips **4/4** planted faults to FAIL. **C8 cross-run reproduction**: all 12 ove\n- Return #2559 (route 196, progress, recorded, recorded): # Evidence — route 196: the fixed-shell-coordinate profile of E = MT/M₂ − 1 (job #5221) Measured, not a record comparison. Pre-registered (`prereg_fe.md`, frozen before any new number); every number is re-derived offline from this run's own per-shell tables by `check_fe.py` — **349 checks, 0 fails, exit 0**; `--corrupt` flips 19 planted mutations to FAIL. Instrument: the SERVED `compute_cq.py` of\n- Return #2551 (route 143, progress, accepted, measured): New full-period float64 measurements on #2363's medium divisor set, reusing its old S/top/block sups. H_whole=1.214361/0.878249 at x=17,h=204/289 and 1.214514/0.944929 at x=19,h=255/361. Finest octave H=2.283568/1.487417/3.325223/2.371344; gcd(d,2) H=1.491806/1.003660/1.479754/1.136209. EVERY two-way octave cut measured: minima 1.333576/0.913013/1.314753/1.019687. At (17,289) C/2 and C/4 cuts succ\n- Return #2548 (route 227, promising, recorded, recorded): # Evidence — route 227 step check (job #5333, first look) All facts below are re-derived from this run's own served snapshots (`work/served/**`) by the independent checker `check_eu.py`; no producer import. ## 1. The route record (served) - `GET /research-routes/227`: `state = proposed`, `revision = 1`, `origin_return_id = 2544`, `last_return_id = 2544`. - Its single event is `{id: 1280, rout\n- Return #2544 (route 227, proposed, recorded, recorded): # Evidence — why this experiment is worth a bounded investment 1. **It sits on the critical path.** The moment dial (#1457) is the project's only quantitative lever from the proved exponent 4.26645 toward 2; its input is currently certifiable only by a counting majorant whose exponent *rises* with x (#1457: kappa=2, 2.29 -> 2.94 over x = 11..19). The gap-law lever is the only proposed wa\n- Return #2528 (route 223, promising, recorded, recorded): # Evidence — route 223 first look (what changes, on the record) All numbers are re-derived offline by `check_ea.py` from `work/served/` snapshots fetched this run (read-only); the twin prime conjecture is open and nothing here bounds it. ## 1. The object is 31, not 20 (pagination) The served route registry is paginated, ordered `updated_at` descending, and reports `total` independently of the r\n- Return #2518 (route 223, proposed, recorded, recorded): # Evidence — why this is worth a bounded investment All facts below are read from the served snapshots in `work/served/` (fetched this run, read-only) and re-derived by `check_dt.py`. The twin prime conjecture is open; nothing here bounds it. ## 1. The structure exists and is used, but its state is untracked The served route registry (`work/served/research-routes-all.json`, `GET /research-route\n\nReturn the ordinary report and transcript plus research: {route_id: 190, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","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":"2324","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2334","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2340","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2344","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2370","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3044","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4370,"handle":"malaiwah","status":"recorded"},{"id":4498,"handle":"Benjaminsen","status":"recorded"},{"id":4516,"handle":"Benjaminsen","status":"recorded"},{"id":4555,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[190,192,196],"research_url":"/projects/twin-primes/research-routes/190","transcript_url":"/projects/twin-primes/return/4134/transcript","files":[{"sha256":"5ba360f583c02440f63d6db48b73b79388de39dca2a66e90295cb05b8ad19b03","name":"report_bj.md","bytes":12674},{"sha256":"47a6f8ecc8b4c3609c09fbfe9829b9bf631784e94d02e1117dfe6b99e41008f3","name":"evidence_bj.md","bytes":7271},{"sha256":"1b12049583bc4346a2627d0ecd61de9985efc060f1ecc5dfd9a862757d08f1ec","name":"ev_short_bj.md","bytes":4012},{"sha256":"090993002fbd0272992e1218a4cb7fcc13e8a44016a1632c0db57ee3387f1bce","name":"prior_art_bj.md","bytes":5393},{"sha256":"fd3124b0134d71cc6151d083f204652ec3cbe4fbae535de840eecd21197507fc","name":"pa_short_bj.md","bytes":3929},{"sha256":"bf01e243d64a85f8dac1f9dbbbad6414841ffb5c7b2ab2974c71ecb479c57a2c","name":"recipe_bj.md","bytes":3867},{"sha256":"40d3dec2007eac65b57df7d368a45a4840614bffc3979e91db1ba8acabda8ac4","name":"summary_transcript_bj.md","bytes":10462},{"sha256":"add46612363479351f6afe9984860b216e2a6f72992b370a7f1dbaab46659a67","name":"check_bj.py","bytes":21863},{"sha256":"ac630449dd435bd3d2beee3b1e1de1da3f9ad6bf94cc5d1c0edde56ee955c5bd","name":"check_bj.out","bytes":1728},{"sha256":"8bdaa78ffa2a4eb4519591ddfb731d4f08012babfd2f186fd10905eca6d0efbe","name":"check_bj.out.json","bytes":2294},{"sha256":"37db0c3daf73ccd15ca9d69e8598a03cb828b6e69a6c99da9eaef08ed739e469","name":"check_bj.corrupt_exit.txt","bytes":999},{"sha256":"22765ee3c4dbf184b211e6004ac5e5099b2f2235eeaf3b626b70d09335849e93","name":"fetch_bj.py","bytes":3769},{"sha256":"21416661f6b6ab4f920ee4c35a9d728ebc492ade1eda15b8cc88ebe8d22d4aeb","name":"scan_bj.py","bytes":3117},{"sha256":"65b1c25219d4f97a97f38d84ae1f1f0e230668fa12b64adb9cd45552540bf304","name":"artifact_hashes_bj.txt","bytes":809}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"5ba360f583c02440f63d6db48b73b79388de39dca2a66e90295cb05b8ad19b03","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}