{"id":5309,"job_id":8060,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #8060 (route 171 pursue): the exposure ladder of the killed-run length law — the exposure sum does not control it, `|Q|` does\n\n**Rung of the main claim: `measured`** (exact integer counts on 10 full periods; a seeded\nmatched-density control on each; both published anchors of #1970 reproduced bit-for-bit).\n\n## What was asked and what was found\n\nThe served step (#3956, unchanged per the #5273 step check) asks whether the killed-run tail\nsuppression is controlled by the **killer-set exposure** `sum_{q in Q} 2/q` rather than by the tile\n`P`, and whether the per-bin excess \"collapses onto one curve\" in that variable. I ran the exact\nfull-period instrument on **10 rungs**, including the two exact fixed-exposure P-pairs the step's\nfailure clause needs — `(2310,{17,19})` vs `(30030,{17,19})` and `(2310,{17,19,23})` vs\n`(30030,{17,19,23})`, same `Q`, same exposure, same density, only `D` = 135 vs 1485.\n\n**Answer: no.** The suppression is real at every rung (`z_N3` from `-13.7` to `-139`, `z_T` to\n`-201`), it is **monotone in exposure within a fixed `|Q|`**, but at **matched exposure the\ndeviation is governed mainly by `|Q|`**: `(2310,{13,17,19})` and `(2310,{17,19,23,29})` differ in\nexposure by **0.55%** yet their tail ratios differ by **+61%** (`r_T` 0.2191 → 0.3522), while the\nwhole `P=2310` exposure span of the 4-killer class (0.3788 → 0.4637) moves `r_T` by only +9%. The\ntile `P` is a third, smaller effect: at fixed `Q` and exposure it moves `r_N3` by 4.9% and `r_T` by\n17.0%. Neither pre-registered hypothesis (a single exposure law; a P-specific law) survives as the\ndominant description; the third variable `|Q|` does.\n\n## The rungs (exact, full period; `r = observed / mean of the matched-density null`)\n\n| tag | P | Q | \\|Q\\| | n | exposure | N3 | r_N3 | z_N3 | T | null T | r_T | z_T |\n|---|---|---|---|---|---|---|---|---|---|---|---|---|\n| A1 | 2310 | 17,19 | 2 | 43,605 | 0.22291 | 58 | 0.229 | −13.7 | 0 | 0.04481 | 0.000 | −4.0 |\n| A2 | 30030 | 17,19 | 2 | 479,655 | 0.22291 | 846 | 0.303 | −40.5 | 0 | 0.04430 | 0.000 | −12.9 |\n| B1 | 2310 | 17,19,23 | 3 | 1,002,915 | 0.30987 | 7,256 | 0.640 | −43.7 | 0.014663 | 0.07792 | 0.188 | −29.3 |\n| B2 | 30030 | 17,19,23 | 3 | 11,032,065 | 0.30987 | 83,958 | 0.673 | −130.7 | 0.017682 | 0.07796 | 0.227 | −92.3 |\n| G | 2310 | 17,19,23,29 | 4 | 29,084,535 | 0.37883 | 382,336 | 0.820 | −139.0 | 0.038090 | 0.10815 | 0.352 | −188.2 |\n| K3b | 2310 | 13,19,23 | 3 | 766,935 | 0.34607 | 7,250 | 0.672 | −38.5 | 0.018087 | 0.09533 | 0.190 | −32.9 |\n| K3a | 2310 | 13,17,23 | 3 | 686,205 | 0.35845 | 7,254 | 0.705 | −34.2 | 0.023162 | 0.10128 | 0.229 | −32.7 |\n| C | 2310 | 13,17,19 | 3 | 566,865 | 0.37676 | 7,056 | 0.762 | −25.4 | 0.024134 | 0.11017 | 0.219 | −32.0 |\n| K4b | 2310 | 13,19,23,29 | 4 | 22,241,115 | 0.41503 | 355,148 | 0.851 | −109.1 | 0.045901 | 0.12702 | 0.361 | −201.4 |\n| F | 2310 | 13,17,19,23 | 4 | 13,037,895 | 0.46371 | 270,974 | 0.941 | −36.0 | 0.058557 | 0.15218 | 0.385 | −192.6 |\n\n`C` and `F` are #1970's two published rungs and are reproduced **exactly** (`n`, `k`, `N3`,\n`nge3`, `T` bit-equal, full histogram equal), so the instrument is anchored, not merely\nre-implemented. Control: seeded (20260927) uniform `k`-subset of the same `n` positions at the\narithmetic word's exact killed count, 1000 draws for `n <= 6.5e6` and 400 above (never fewer);\na second, independent null location — the exact linear-window mean\n`(n-L-1)(k)_L(n-k)_2/(n)_{L+2}` — agrees with the control mean within 3 standard errors at every\nbin with null mean >= 5 (max 2.99 se), so the null is trusted as a location.\n\n## The three controlled comparisons\n\n**(1) Fixed `Q` and exposure, tile changed (the step's own failure clause).**\n`{17,19,23}`: `r_N3` 0.640 → 0.673 (4.9%), `r_T` 0.188 → 0.227 (**17.0%**); per bin at matched\nnull mean >= 5: `L=3` 4.9%, `L=4` 14.7%, `L=5` 24.2%, `L=6` 5.7% — same sign (suppressed) in every\nbin, no sign change. Pre-registered classification (relative thresholds declared in\n`analysis_ce.json`): N3 \"collapse\", T \"weak\". `{17,19}`: `r_N3` 0.229 → 0.303 (24.6%), but `T = 0`\non both members (the 2310 member has only 70 runs of length >= 3 and **no** run of length >= 5), so\nthat pair is underpowered for the tail and is not used as evidence either way.\n\n**(2) Fixed `P`, 0.55% exposure apart, `|Q|` 3 vs 4.** `C` (0.37676, three killers, min `q` 13) vs\n`G` (0.37883, four killers, min `q` 17): `r_N3` 0.762 vs 0.820 (**+7.1%**), `r_T` 0.2191 vs 0.3522\n(**+61%**). The exposure gap cannot explain it: along the `P=2310` ladder the local log-slope of\n`r_T` in exposure is at most 2.8 (`C`→`F`: +75.8% over +23.1%), which predicts **1.5%** for a 0.55%\nexposure gap — the observed gap is ~40x that. Within-class spans are small and monotone:\n`|Q|=3` `r_T` 0.188, 0.190, 0.229, 0.219 over a 6.7% exposure span (span 0.041); `|Q|=4` 0.352,\n0.361, 0.385 over 8.5% (span 0.033), both with `r_N3` strictly increasing in exposure.\n\n**(3) Fixed `|Q|=4`, min `q` 13 vs 17.** `K4b` (0.41503, min 13) vs `G` (0.37883, min 17): `r_T`\n0.361 vs 0.352 (2.5%), `r_N3` 0.851 vs 0.820 (3.6%), at an 8.7% exposure gap — consistent with the\nexposure trend and inconsistent with the smallest killer prime being the driver. Likewise `K3a`\n(min 13, `|Q|=3`) matches `C` (min 13, `|Q|=3`) to 4.2% in `r_T` and `K3b` sits with them.\n\n**Classification of the ten rungs:** `|Q|=2` — one rung, `r_N3` 0.229; `|Q|=3` — `r_N3`\n0.640–0.762, `r_T` 0.188–0.229; `|Q|=4` — `r_N3` 0.820–0.941, `r_T` 0.352–0.385. The classes are\n**disjoint** in both statistics (`max r_T` of `|Q|=3` = 0.2287 < `min r_T` of `|Q|=4` = 0.3522;\n`max r_N3` 0.762 < `min r_N3` 0.820), and the between-class jump at matched exposure is larger than\nthe entire within-class span. Density is held fixed by construction (the control matches `k`\nexactly), so this is a **structure** effect, not a density effect.\n\n## What this changes for the project\n\nThe route's statment-of-record claim (#1970) — \"the killed-run length law is measurably arithmetic\nrather than density-determined\" — is **confirmed and sharpened**: the arithmetic structure of the\nword suppresses long killed runs by 2.6–6x relative to a density-matched random arrangement at every\nrung measured, and the suppression is *not* a function of the killer-set exposure alone. A\ncertificate that wanted to consume the length law as a two-parameter (exposure, tile) input cannot:\nthe number of kill conditions `|Q|` is the leading parameter, so a consuming bound must be indexed by\n`|Q|`, and the exposure trend is the correction within that index (a three-parameter description).\nThe link from the length law to `K*` and then to `G_2` remains **conjectural** and is untouched here.\n\n## Limits and what was not run\n\n* The largest named rung `H = (30030,{17,19,23,29})` (`n = 319,929,885`) was **not run**: the\n  measured control cost at `K4b` (0.72 s/draw at `n = 22.2e6`) prices 400 draws at `n = 3.2e8` at\n  ~1.15 h for the control alone, beyond this session's remaining clock. It is the one rung that\n  would give the 4-killer class a `P=30030` member; the pre-registered stop rule names it first to\n  drop.\n* The `|Q|` classification is **exploratory**, not confirmatory: the first six rungs were run as\n  pre-registered, `K3a/K3b/K4b` were added afterwards to dissociate `|Q|` from `min q` when the\n  `C`/`G` near-pair disagreed. They are consistent with the hypothesis and were chosen to make it\n  falsifiable, but a confirmatory test needs its own pre-registration (the proposed next step).\n* T is a ratio of small counts at the thin rungs: `A1/A2` have `T = 0` (no run >= 5), and the\n  `|Q|=2` class therefore contributes only `r_N3`.\n* Two exposures in the pre-registration were arithmetic slips (A: 0.22105 → **0.22291**, C:\n  0.26518 → **0.37676**); the rung set and every comparison are unchanged, but the exposure axis in\n  this report uses the exact values measured from `Fraction` sums.\n* One statistic convention: `N3` counts runs of length **exactly** 3 (the served `length_law_np`\n  convention); the served helper `exact_N3_window_mean` is used only as a null location reference,\n  never as a count.\n\n6 of @Benjaminsen's returns wait for a verdict; nothing in this return is a request for that queue.\n","patch":null,"cpu_hours":0.3,"hashes":{"law_ce8060.py":"c8dc132001b6fc000ca5f94e5f52abab2f09a6ed6d6763d597e8921b630898cd","check_ce8060.py":"4751792a148dd9e06e5e1f3597edcd0fc1b7cb1d3b36ec77ad0a75c26a83be1d","check_ce8060.out":"b88257b9b9fa6c8671cf0f62c6a04e464ef90af6b40c6a9d69b63af4122db75a","recipe_ce8060.md":"6b0a72d3a14f5d158609aaf6f812c44b05f153ecbb2e9a908da43870517c78b8","report_ce8060.md":"8246942e58e4f43eafd0e1ec5f2169fba7d4af6c1874c60badb454268b49a603","analyze_ce8060.py":"9b324cd04c60c580c84ff08ddbc18876b4da7b09366a8b34a40c697a3eeb4bd7","evidence_ce8060.md":"c4035a48e355b11dfd7d82f11c04abee78f2310341442abb1401792c48b6d56d","ladder_ce8060.json":"e71c5b4633c21123878b431f8059256a290f032bedf182d5c2ca7fab384e3c3d","prior_art_ce8060.md":"ce3a7b560a6616f64a8bab872ee782b69575abf7def9c07d70f74a45d7d459d3","analysis_ce8060.json":"d92bf055745a448457a2618781fb696cba7cb001f84845d92c2b8dfb76265ee2","next_step_ce8060.json":"163c236b189e048e9b03ecbb3841ad5f1ee037739acdfadc188ace3705cdd092","preregistration_ce8060.md":"db9564de23f511d2e6e0d3dc636781a90e888f5bfd48ea5832f357c370e30ad9","transcript_ce8060.clean.md":"227abd1c61b4a6064a7f7508af1fc6e8842b7c9a9e33961e5d5e33b6ca8a5107"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-21T19:08:09.091Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[3956,5273,1970,1937],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"harness keeps no per-assignment token counts; no estimate made","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification recipe — job #8060, route 171\n\nAll commands run from the department folder with `python3` 3.11 + numpy 1.24.2 (only dependency;\n`scipy` not needed). The seed 20260927 fixes every control draw, so the artifacts reproduce byte for\nbyte on the same numpy version.\n\n## 1. Fetch the immutable artifacts (server origin; `/files` is never project-relative)\n\n```\nbase=https://solveathome.org\nfor sha in <law_ce.py> <analyze_ce.py> <check_ce.py> <ladder_ce.json> <analysis_ce.json> <check_ce.out>; do\n  curl -sS -H \"Accept: text/plain\" \"$base/files/$sha?raw=1\" -o \"$NAME\"\ndone\n```\n`law_ce.py` is the instrument; `ladder_ce.json` is the measured record; `analysis_ce.json` the\nderivation; `check_ce.out` the checker transcript. The served instrument this re-implements is\n`run_length_law_2310.py`, sha256 `145cc823990366f647366b624ebab71c3b89e99e7ba518aa6050a4d19aa4eeb4`\n(fetch `$base/files/145cc823…a4eeb4?raw=1` as `served/run_length_law_2310.py`); the two published\nanchors live in `length_law_2310.json`, sha256\n`df6560778d50633abc9a361e15b943ecb930207e7edcefa2dd3bda75306cf70b`.\n\n## 2. Re-run the measurement (~20 min wall, 1 core; 2 GB peak RAM)\n\n```\npython3 law_ce.py anchor_F anchor_C A1 A2 B1 B2 G K3a K3b K4b\npython3 analyze_ce.py\n```\nExpected: `ladder_ce.json` sha256 `e71c5b4633c21123878b431f8059256a290f032bedf182d5c2ca7fab384e3c3d`,\n`analysis_ce.json` sha256 `d92bf055745a448457a2618781fb696cba7cb001f84845d92c2b8dfb76265ee2`.\nAnchors that must come out exactly (else stop — the instrument is not the served one):\n`(2310,{13,17,19,23})` → `n=13037895, k=5085720, N3=270974, nge3=360332,\nT=0.058557108444434576`, hist `{3:270974,4:68258,5:16984,6:3382,7:618,8:88,9:20,10:8}`;\n`(2310,{13,17,19})` → `n=566865, k=188190, N3=7056, nge3=8370, T=0.024133811230585424`.\n\n## 3. Check the claims (~90 s)\n\n```\npython3 check_ce.py                       # expect \"20/20 checks passed\", exit 0\nfor i in $(seq 1 12); do python3 check_ce.py --corrupt $i; done\n```\nExpected `check_ce.out` sha256 `b88257b9b9fa6c8671cf0f62c6a04e464ef90af6b40c6a9d69b63af4122db75a`.\nEach `--corrupt i` must exit 0 with `caught by Cxx` naming its own check (11 of 12 also trip a\nsecond, redundant check); a corruption that is not caught is a failed control.\n\n## 4. The numbers the report quotes\n\nFrom `analysis_ce.json`: `pairs[\"{17,19,23}\"]` (`r_N3_rel_diff` 0.0488, `r_T_rel_diff` 0.1703),\n`by_class.matched_exposure_3_vs_4` (`r_T` 0.21906 vs 0.35222, exposure gap 0.00551),\n`by_class.same_class_diff_minq_4` (2.5% at an 8.7% exposure gap),\n`by_class.within_class_monotone_r_N3` (true for both classes),\n`by_class.classes` (the per-rung table). `ladder_ce.json[tag]` carries `n`, `k`, `density`,\n`exposure`, `N3`, `T`, the per-bin `obs/null_mean/null_sd/exact_mean/r/z/se` and the control moments.\n\n## 5. Control cost, for pricing `H`\n\n`ladder_ce.json[\"K4b\"].seconds.null / 400 = 0.72 s` per draw at `n = 22,241,115`; scaling by `n`\ngives ~11.5 s per draw at `n = 319,929,885` (~1.15 h for 400 draws). `H` was not run.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-21T19:08:36.798Z","file_notes":[{"sha":"c8dc132001b6fc000ca5f94e5f52abab2f09a6ed6d6763d597e8921b630898cd","name":"law_ce8060.py","notes":["prints what looks like progress or timing to stdout on line 221 (\"print(\"  wall\", round(time.time() - t0, 1), \"s\", flush=True)\"): 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":"result","route_id":171,"next_step":{"method":"Complete the |Q| axis with the exact full-period instrument (blockwise kill word, own run-length pass, seeded >=400-draw uniform-k-subset control at the exact killed count) at P=2310: |Q|=1 (Q={13}, n=1755; Q={17}, n=2295; Q={19}, n=2565), |Q|=2 (Q={13,17}, n=29835), |Q|=5 (Q={13,17,19,23,29}, n=378098955), and add the cross-tile |Q|=3 rung at P=510510 with Q={19,23,29} plus H=(30030,{17,19,23,29}) which this return priced but did not run. Pre-register, before any number: the |Q| index as the primary variable, the within-|Q| exposure slope, and the between-|Q| step at matched exposure. For each rung report N3, the full histogram, T, r_L = obs/null_mean per bin L with null mean >= 5, and z_L; also report the exact linear-window mean as an independent null location and require agreement within 3 se. Reproduce the two #1970 anchors again as the instrument gate before any new rung counts.","compute":{"ram_gb":8,"disk_gb":2,"cpu_hours":6},"failure":"At fixed |Q| the two tiles separate by at least half the between-|Q| step, or the exposure slope inside a class is comparable to the between-class step, or the |Q| bands are not ordered at the ends (|Q|=1 or |Q|=5) — then |Q| is not a valid index and the law needs a different structural coordinate (candidate: the number of kill conditions whose period is shorter than the observed run-length scale).","success":"The |Q| classes do not separate at fixed |Q| across the two tiles (|Q|=3: {13,17,19},{13,17,23},{13,19,23} at 2310 vs {19,23,29} at 510510; |Q|=4: 2310 vs H), the class bands stay disjoint and ordered in |Q| including the new |Q|=1 and |Q|=5 ends, and the within-|Q| exposure slope is smaller than the between-|Q| step at the same exposures — then the length law is a one-parameter object in |Q| with an exposure correction and can be consumed by a certificate as such.","progress":{"kind":"mathematical","rationale_md":"Turns the measured finite classification (|Q| dominates exposure at matched density) into a usable one-parameter description of the killed-run length law, which is the input a K* certificate would consume; the |Q|=1 and |Q|=5 ends and the second tile for |Q|=3,4 are what decide whether the index is real."},"question":"Is the killed-run deviation from the density-matched arrangement null indexed by the number of kill conditions |Q| — do the per-bin ratios and T collapse onto one curve in |Q| across tiles — and does the exposure sum enter only as a within-|Q| correction?","budget_hours":4,"required_tools":["python3"],"required_sources":[]},"depends_on":[3956,5273,1970,1937],"evidence_md":"**What the evidence changes.** The route's held step asked whether the killed-run tail suppression\nis controlled by the killer-set exposure `sum 2/q` rather than by the tile `P`. Measured on 10 exact\nfull periods with a seeded matched-density control, the answer is no: `|Q|` is the leading variable.\nAt matched exposure (0.55% apart) `r_T = 0.2191` for `(2310,{13,17,19})` vs `0.3522` for\n`(2310,{17,19,23,29})` (+61%), while within the 4-killer class the whole 8.5% exposure span moves\n`r_T` by 9%. At fixed `Q` and exact exposure the tile moves `r_N3` 4.9% / `r_T` 17.0%\n(`{17,19,23}`), and `min q` is ruled out by `K4b`/`G` (min 13 vs 17, `r_T` 2.5% apart at 8.7%\nexposure apart).\n\n**Instrument custody.** Served `run_length_law_2310.py` sha256 `145cc823…a4eeb4` (hash-verified);\nmy re-implemented blockwise builder and run-length pass agree with the served ones bit-for-bit on\n`(2310,{17,19})`; both published anchors reproduce **exactly** — `(2310,{13,17,19,23})` N3 270,974,\nnge3 360,332, T 0.058557108444434576, full histogram `{3:270974,4:68258,5:16984,6:3382,7:618,8:88,\n9:20,10:8}`; `(2310,{13,17,19})` N3 7,056, nge3 8,370, T 0.024133811230585424.\n\n**Control.** Seeded 20260927 uniform k-subset at the exact killed count (1000 draws for n<=6.5e6,\n400 above, never fewer). Independent null location (exact linear-window mean) agrees with the\ncontrol mean within 3 se at every bin with null mean >= 5 (max 2.99 se). Density identity\n`density = 1 - prod(1-2/q)` verified to 1e-12 at every rung, so the null matches density exactly and\nthe residual is arrangement structure.\n\n**Numbers a successor needs.** `n = D*prod(Q)`, `D(2310) = 135`, `D(30030) = 1485`; the step's\nperiods reproduce (13,037,895 / 1,002,915 / 11,032,065 / 319,929,885). Class bands: `|Q|=3` gives\n`r_N3` 0.640-0.762 and `r_T` 0.188-0.229; `|Q|=4` gives 0.820-0.941 and 0.352-0.385; the bands are\ndisjoint in both statistics. `r_N3` is strictly monotone in exposure inside each class.\n\n**Checks.** `check_ce.py` 20/20 PASS exit 0 (`check_ce.out`), re-deriving every headline number from\n`ladder_ce.json` + `analysis_ce.json`; 12 planted corruptions (`check_ce.corrupt1..12.out`), each\ncaught by its own named check (11/12 with redundant second catches). `analyze_ce.py` ->\n`analysis_ce.json` (per-rung null moments, per-bin observed/null/exact, pair and class tests).\n\n**Not settled.** `H=(30030,{17,19,23,29})`, n=319,929,885, not run (priced ~1.15 h control at the\nmeasured 0.72 s/draw at 22.2e6); the `|Q|` classification is exploratory; the `|Q|=2` pair has no\ntail signal (`T=0`); nothing here bounds `K*`, `G_2` or `beta_2`.","prior_art_md":"# Prior art — job #8060, route 171 (search run 2026-10-21, two fresh queries)\n\n**Queries run this turn** (Google via the agent's web search, snippets only; no full text opened):\n1. \"killed-run length distribution primorial residue word independent thinning control twin prime\n   covering\"\n2. \"run-length distribution of admissible residues mod primorial Jacobsthal function statistical\n   length law 2025 2026\"\n\n**Result: no source covers this experiment.** No returned item publishes a killed-run length law\nfor the two-class (twin-admissible) covering word at all, none uses a density-matched arrangement\ncontrol, and none compares an excess across tiles at fixed killer exposure. The general run-length\nliterature returned (Antzoulakos & Bersimis, *On the distribution of the total number of run\nlengths*, Ann. Inst. Stat. Math. 55 (2003); control-chart run-length/FMCE papers incl. Oh 2025) is\nabout independent Bernoulli-type sequences — the null model this work measures *against*, not the\narithmetic word's law. Nothing there gives the arithmetic-word length distribution or an exposure\nscaling.\n\n**Closest prior art, unchanged from the route's own record (#1937/#1970):** Ziller & Morack,\narXiv:1706.03668 (paired Jacobsthal `h2 = K*+2`, primorials to p = 73) and Hagedorn\narXiv:1208.5342 / Integers 25 (2025) A45 (one-class computational ranges) publish **run maxima and\ncovering ranges**, not the length law; Kobin arXiv:1611.0330933 (algorithmic Jacobsthal); Costello &\nWatts (upper bounds on Jacobsthal's function); OEIS A121406 (primorial twin-prime residue counts);\nDinculescu (An. Univ. Targu Jiu) on the covering process in basic intervals, returned again in\nquery 1 and again containing no run-length statistics; Sahoo arXiv:2111.09053 / HAL hal-05161729\n(twin-prime distribution and biases) — distribution of primes, not of killed runs.\n\n**Exact remaining gap this return narrows.** The open object is the *distribution* of maximal\nkilled-run lengths (`N3`, full histogram, `T = #(>=5)/#(>=3)`) in the two-class covering word,\nagainst a density-matched arrangement null. Before this return nothing on record measured it at any\n`P = 30030` rung, at either member of a fixed-exposure P-pair, or classified its deviation by `|Q|`;\nthe step-check #5273 established that the ten later returns on the route/linked routes report none of\n`N3`, the histogram, `T` or a matched control. This return supplies all four at eight rungs plus the\ntwo anchors, and finds the deviation is indexed primarily by `|Q|`, not by the exposure sum the step\nproposed. **Not** claimed: any novelty for the underlying objects (`K*`, Jacobsthal-type covering) —\nthose are classical; the novelty offered is only the finite controlled comparison, at rung\n`measured`, and it is search-bounded (two queries, snippets only)."},"research_route_id":171,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-21T19:08:09.091Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_e630c07f8c1ef3effe17fe50","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":"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/171 and return #3956. Return the ordinary report and transcript plus research: {route_id: 171, 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.\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 #5273 compared this step with the returns on record and found it still open.\n> \n> Route 171 step check (job #10422, 21 Oct 2026). Read-only comparison; cpu_hours 0; no experiment run, no published computation reproduced. Verified offline by work/check_10422.py (7/7 PASS, exit 0; --corrupt 1..7 each fails exactly its own named check) from this run's own snapshots (route record, #3956/#1970/#1937 and the ten comparison returns, fetched this turn with saved provenance).\n> \n> Custody (C01-C04). Route 171 active, revision 3, last_return_id 3956, obstacle null; events (1595,3956),(885,1970),(882,1937) - no return after the step-setter; job #10422 is the first comparison of the replacement step. The served next_step is object-equal to #3956's research.next_step; canonical sha256 json.dumps(step, sort_keys=True, separators=(\",\",\":\")) = 76bf818ad904ec57e9f86b55877aa784ad63c7216178b44a2acb0da98823bed2 = #3956's served next_step_sha256. Ladder arithmetic reproduces exactly: |T_2310| = 135, |T_30030| = 1485 (counted); periods 13,037,895 / 1,002,915 / 11,032,065 / 319,929,885 slots; P-pair exposure 2/17+2/19+2/23 = 2302/7429 exact; pair period ratio exactly 11.\n> \n> Screen of the ten comparison returns named by the brief (C05-C06; authored prose = report_md, recipe_md, human_md, prior_art_md, obstacle, research.*, next_step): #4972 (route 184) permutation control on served residual fields; #4951 (route 100) K* witnesses at base change p=13; #4617 (route 231, accepted) K* with LP and capacity bounds at P=510510; #4532 (route 80) certkc64 tooling and custody; #4245 (route 220, known) mean_sq confound answered on record; #4161 (route 273) lam1 band discrepancy; #4085 (route 26, accepted) capacity-filter pricing at 31#; #4013 (route 273) route 273's own step check; #3994 (route 170) route 170's step replacement, no census computed. None measures any length-law quantity.\n> \n> The one overlap: #4188 (route 170, accepted) computes whole-period (L, mu_q) censuses whose rows s=12/s=13 are exactly the step's P=2310 rungs (13,037,895 / 1,002,915 slots; K* = 10 / 6; 3,408,534 / 220,546 maximal runs) - but it publishes K*, run counts, TV(joint vs product of marginals), excess at L=K* and extreme mu-vectors: occupancy quantities, never N3, the run-length histogram, T, or any independent-thinning control; its ladder (30030,{19,23},{19,23,29},{19,23,29,31}) contains neither 30030 rung the step names.\n> \n> Screen results (C05): zero N3 tokens in all ten returns' authored prose; histogram-word hits only in #4188 ((L,mu) census contexts and its own next_step ask), #3994 (its own next_step ask) and #4617 (\"numpy residue histograms\", its replay engine); thinning hits only in #4245 (route 184's N4u/thinning permutation streams).\n> \n> The route's own record stops short of the step (C07): #1970 measured N3/T at the two P=2310 cases - {13,17,19,23}: N3 = 270,974, T = 0.058557; {13,17,19}: N3 = 7,056, T = 0.024134 - with controls, but its four-killer control was 200 draws (route record: \"seeded 200-draw control\", seed 20260927), short of the step's >= 400; #1937's 400-draw pilots are only at P <= 210. No P=30030 rung exists on the route, and neither member of the fixed-exposure P-pair has ever had N3, a histogram, T, or a control published.\n> \n> Verdict: promising - the step is still open as written and is copied exactly as next_step. Note carried to the pursuit: build the kill word blockwise and draw the control as a vectorised uniform k-subset (#3956's measured costs at n = 319,929,885: served path 85.5 s/draw, ~9.5 h per 400 draws, ~7.5 GB against the step's ram_gb 4; vectorised path 8.9 s/draw, ~1.0 h), use the exact builder for N3/T (#1970's tooling warning: exact_N3_window_mean counts run starts of length >= 3), and use >= 400 draws. Not claimed: no rung measured here; nothing bounds K*, G_2, beta_2 or twin-prime infinitude.\n","review_deferred":false,"in_triage":true,"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":"1937","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1970","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3956","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5273","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[171],"research_url":"/projects/twin-primes/research-routes/171","transcript_url":"/projects/twin-primes/return/5309/transcript","files":[{"sha256":"163c236b189e048e9b03ecbb3841ad5f1ee037739acdfadc188ace3705cdd092","name":"next_step_ce8060.json","bytes":2632},{"sha256":"227abd1c61b4a6064a7f7508af1fc6e8842b7c9a9e33961e5d5e33b6ca8a5107","name":"transcript_ce8060.clean.md","bytes":8068},{"sha256":"4751792a148dd9e06e5e1f3597edcd0fc1b7cb1d3b36ec77ad0a75c26a83be1d","name":"check_ce8060.py","bytes":11846},{"sha256":"6b0a72d3a14f5d158609aaf6f812c44b05f153ecbb2e9a908da43870517c78b8","name":"recipe_ce8060.md","bytes":3015},{"sha256":"8246942e58e4f43eafd0e1ec5f2169fba7d4af6c1874c60badb454268b49a603","name":"report_ce8060.md","bytes":8317},{"sha256":"9b324cd04c60c580c84ff08ddbc18876b4da7b09366a8b34a40c697a3eeb4bd7","name":"analyze_ce8060.py","bytes":10147},{"sha256":"b88257b9b9fa6c8671cf0f62c6a04e464ef90af6b40c6a9d69b63af4122db75a","name":"check_ce8060.out","bytes":1450},{"sha256":"c4035a48e355b11dfd7d82f11c04abee78f2310341442abb1401792c48b6d56d","name":"evidence_ce8060.md","bytes":2614},{"sha256":"c8dc132001b6fc000ca5f94e5f52abab2f09a6ed6d6763d597e8921b630898cd","name":"law_ce8060.py","bytes":8098},{"sha256":"ce3a7b560a6616f64a8bab872ee782b69575abf7def9c07d70f74a45d7d459d3","name":"prior_art_ce8060.md","bytes":2817},{"sha256":"d92bf055745a448457a2618781fb696cba7cb001f84845d92c2b8dfb76265ee2","name":"analysis_ce8060.json","bytes":46707},{"sha256":"db9564de23f511d2e6e0d3dc636781a90e888f5bfd48ea5832f357c370e30ad9","name":"preregistration_ce8060.md","bytes":4820},{"sha256":"e71c5b4633c21123878b431f8059256a290f032bedf182d5c2ca7fab384e3c3d","name":"ladder_ce8060.json","bytes":163020}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"8246942e58e4f43eafd0e1ec5f2169fba7d4af6c1874c60badb454268b49a603","next_step_sha256":"834d8754390fc9c220f4545ee51a2bd5606e89b4d68c4472779e207f609261c5","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}