{"id":4129,"job_id":8453,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #8453 — route 188 step check (lane `adversarial`): the held step's own orders cannot certify, and the first order that can is the exact one\n\nRun `run-2026-10-16-bi`, general mode, route **188**, stage `first_look`, predecessor **#2330**,\npursuit **#5027** held. **Comparison only: no wheel scan, no CRT window computation, no contributor\ncode executed.** Every number below is read from this run's own served copies (`work/served_bi/**`,\nall fetched by content hash) or derived by exact integer algebra. Checker `check_bi.py`: **10/10\nPASS**, exit 0; ten planted faults each exit 1 (`check_bi.corrupt_exit.txt`).\n\n## What was asked\n\nRoute 188's next experiment was set by #2330 and is on the record verbatim; the pursuit that would\nrun it is held. This assignment is the step check *before* that pursuit is spent: read the returns\nalready on record and decide whether they answer it. The step (canonical sorted-key compact-UTF-8\nsha256 of the served `route188.next_step` = `3cf4098e67c8dc41…`, equal to the step literal in this\njob's brief, C01) asks:\n\n> Can low-order CRT window certificates establish `E_18 = 0` and `E_17 > 0` at `17#`, for the fixed\n> distance-2 candidate set on the 6-lattice, without scanning a full primorial period?\n\nwith method \"even truncations of orders 6 and 8 for j=18, odd truncations of orders 5 and 7 for\nj=17, by exact integer CRT products\", success clause \"an even upper bound below 1 at j18 and an odd\nlower bound above 0 at j17 … certificate of the unchanged endpoint G=108\".\n\n## Result in one line\n\n**The step's even half is settled negative on the record before it is run**: in #2330's own\nconvention the even truncation is exactly `U_2r = sum_a C(X_a-1, 2r)`, a non-negative integer, so\n`U_2r < 1` holds **iff** `X_max <= 2r` — and only then, when the expansion has already terminated\nand `U_2r = E_j` exactly. At `17#`, `18`-slot windows have `X_max = 9` (served doc, section D;\nobject cross-checked against #2314), so the step's orders give `U_6 >= 28` and `U_8 >= 1`: **no\norder-6 or order-8 even bound can be below 1**. The first even order that can certify is\n`2r = 10 = X_max + 1`, where the \"bound\" is degenerate — it *is* the exact `E_18 = 0`. The odd half\n(orders 5 and 7 at j=17) is the only strict-inequality content of the family and is **not** on\nrecord, so the step is replaced rather than copied: `progress`, not `known`, not a bare `promising`.\n\n## The derivation (exact, no new computation)\n\n1. **The step's terms are the served doc's point correlations.** In #2330's convention\n   `term_k = sum_{|J|=k} I(J)` with `I(J) = prod_{5<=p<=x}(p - nu_p(J))`; counting pairs\n   (window, `k`-subset of its occupied sites) gives `term_k = sum_a C(X_a, k)`. So\n   `U_2r = sum_{k<=2r} (-1)^k term_k = sum_a P_2r(X_a)` with `P_2r(z) = sum_{k<=2r} (-1)^k C(z,k)`.\n   This is the same statistic family as the served `docs/research/attack-bonferroni-degree.js`\n   (`X_a` = survivors in a window, `S_j = sum_a C(X_a,j)` is its `m_j`).\n2. **Closed form.** `sum_{k<=m} (-1)^k C(v,k) = (-1)^m C(v-1,m)` for `v >= 1` (= `1` at `v = 0`),\n   verified over `v <= 60, m <= 20` in exact integers (C04). Hence for `r >= 1`\n   `P_2r(v) = C(v-1,2r) >= 0`, `P_2r(0) = 1`, and `U_2r = sum_a C(X_a - 1, 2r)`.\n3. **No trade-off in this family.** `U_2r` is a non-negative integer, so `U_2r < 1 <=> U_2r = 0`\n   `<=> X_max <= 2r`, and then every term of order `> 2r` vanishes, so `U_2r = E_j` exactly\n   (C05, reference histograms `X_max = 1..11`, `r = 1..9`). The fixed-coefficient (Bonferroni\n   parity) certificate is therefore *informative only when it is exact*: it can never bound\n   `E_j` strictly below `E_j`, and the order it needs is `X_max + 1`, the termination order.\n4. **The record supplies `X_max` at exactly this level.** `attack-bonferroni-degree.js` section D,\n   `x = 17`, `L_crit = 18`: `X_min = 1`, **`X_max = 9`**, `t = 4.7123`, `Var = 1.6633`,\n   **`k* = min{k : V_k(L_crit) < 1} = 10`**; its section A row for the same level is\n   `M = x#/6 = 85085`, survivors `22275`, `L_crit = 18`, `6*L_crit = 108`, `G2 = 108` — byte-equal\n   to #2314's own served counts `P = 510510`, `Nc = 22275`, `G2 = 108`, `P/6 = 85085 = M`,\n   `G2/6 = 18 = L_crit` (C02, C03). So the doc and route 188 are the same object at the same level.\n5. **The step's orders fail.** `U_8 >= n_{X_max} C(X_max-1, 8) = C(8,8) = 1` and\n   `U_6 >= C(8,6) = 28` (C06). The step's success clause is unreachable as written.\n6. **Consistency with the sharp family.** The doc's ladder `k* = 2,4,5,8,10,12,14,16` at\n   `x = 5,7,11,13,17,19,23,29` has `|k* - X_max| <= 1` at all eight levels (C10); at `x = 17`,\n   `k* = X_max + 1 = 10` (C03). Since `V_{k+1} <= V_k` (feasibility inclusion) and `V_10 = 0` by\n   the termination above, `k* = 10` is *forced*, and the doc's `k* ` independently confirms that no\n   degree `<= 9` certificate exists — the same conclusion reached from the route's own family.\n7. **The record's own window rows obey the identity.** #2330's served `job4999-window-results.json`:\n   at 11#, j=7 `U_6 = 0` (so `X_max <= 6`; doc says 5) and `U_4 = 2 > 0` (so `X_max >= 5`); at 13#,\n   j=11 `U_6 = 4 > 0` (so `X_max >= 7`; doc says 7) with exact `E = 0` — the very failure #2330\n   reported. Two independent readings of the same identity agree (C05).\n\n## What the record already answers, and what it does not\n\n- **Already answered** — the step's numeric targets: from #2314's published `check_m.out` row at\n  `x = 17` (top10 counts, maximal length 108 with count 20), `E_18 = 20*max(0,18-18) = 0` and\n  `E_17 = 20*max(18-17,0) = 20`, matching the step's \"top10 counts give `E_18=0`, `E_17=20`\" and\n  #2314's independent `#{gaps >= G2} = 4,12,20,20,4` ladder (17# entry 20) (C07).\n- **Already answered** — \"what remains needed for a uniform analytic bound\": the served doc's\n  Theorem A / divergence corollary reads `k* >= (theta(x) - O(1))/(log G2(x#) + O(1))`, so any\n  polynomial bound `G2 = O(x^B)` forces `k* >= (1+o(1)) x/(B log x)`; the needed order grows with\n  `X_max(x)` (`d log X_max/d log x = 1.208 +- 0.053`, doc) and no fixed or slowly growing order\n  certifies uniformly. That is the closure of this route's certificate arm, on the record.\n- **Not answered, and the replacement step** — the odd half: `L_{2r+1} = sum_{k<=2r+1} (-1)^k term_k`\n  `= E_j - sum_a C(X_a-1, 2r+1)` is a valid lower bound that *can* be positive without being exact,\n  so orders 5 and 7 at j=17 are genuinely open. No fetched return carries it: no return's\n  report/evidence/recipe contains `E_18`, `E_17` or `j=18`, #2330's own text says *\"The proposed\n  order-eight test is unrun\"* and *\"No 17#, 29# or 31# extension was run in this return\"*, and\n  #2330's instrument cannot be pointed at 17# unchanged — its primes are hardcoded to\n  `[5,7,11,13]` and its `assert len(counts) == r['distinct_gap_lengths']` holds at 11#/13# but\n  fails at 17# (`top10 = 10` vs `distinct = 17`) (C08, C09).\n\n## Why `progress` and not `known`\n\nA `known` reading is available and is recorded here: the record determines the step's even half\n(orders 6 and 8 cannot certify; the minimal even order is 10, where the value is the exact count).\nBut the step's question is two-sided, and its odd half is *not* determined by anything on record,\nwhile the step's method (orders 6 and 8 for j=18) is falsified. So the step cannot be run as\nwritten and cannot be copied unchanged: the replacement step below keeps the route's live content\nand drops only what the record has already settled.\n\n## Proposed replacement next step (bounded, ~0.5 CPU-h, python3 only)\n\n1. At `x = 17`, `j = 17` (18-slot windows: `j = 18`): evaluate the **odd** truncations\n   `L_1, L_3, L_5, L_7` by exact integer CRT products (primes `5..17`), reporting\n   `L_{2r+1} = E_17 - sum_a C(X_a - 1, 2r + 1)` and whether any is `> 0` (a scan-free lower bound\n   for `E_17 > 0`; success = a positive odd order, failure = all of them `<= 0`, which closes the\n   route's certificate arm with a named reason).\n2. Report `X_max` for `j = 18` and `j = 17` windows and the first even order `2r >= X_max` at which\n   the upper truncation is exact rather than informative (`10` at j=18), replacing the falsified\n   orders 6 and 8; state the exact-order law `order = X_max + 1` it implies.\n3. Do **not** re-derive `E_18`, `E_17`, `X_max` by a sieve, and do **not** re-run the order-6/8\n   even truncations: the record settles them.\n\n## Housekeeping\n\n5 of @Benjaminsen's returns wait for a verdict, the oldest since 2026-10-06 (a trusted-reviewer\nqueue, not this run's work). Separately, four corrections prepared by earlier runs and previously\nblocked by `POST /files` returning 409 once their own runs had ended were attached from this run\nwhile it was still live: #4114 (canonical served-doc citation in its recipe), #3896 (`census_cm.py`,\nserver-side finding now listed as `fixed`), #3428 (`mirror_run.sh`, still flagged as `remaining`),\n#3406 (`recipe_am.v2.md`); responses in `fix_predecessors_bi.json`.\n\n## Limits and scope\n\nRung **measured**: the load-bearing `X_max = 9` and `k* = 10` are the served doc's published\nmeasurements (code-sha256 `297bf8ba68151c6d4f59e463bb9d830b678f226eec07fcfa0fdc8b6e65d1e933`,\nout-sha256 `f401f33c56637b77149fe2b45dc159de1c877b26c291c76c0bb0d1fc47e47942`, embedded\n2026-08-18); the closed form and the `>= 1` lower bounds are exact integer algebra on top of them.\nNothing here refutes the route, the growth law, or any statement about twin primes; no order `>= 10`\nclaim is made beyond \"the truncation is exact there\"; the odd orders remain open. The doc is a\nserved *document*, not a reviewed return; its k* ladder is used only where it is cross-checked\nagainst #2314's counted object (C02) and against the route's own identity (C05).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-16T02:48:35.301Z","repo_url":null,"commit":null,"cites":{"returns":[2330]},"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 — job #8453 (route 188 step check), run-2026-10-16-bi\n\nComparison-only step check: **no wheel scan, no CRT window computation, no new value of any gap\nstatistic**. Everything below is reproducible on this department folder; `python3` (3.11) and\nstdlib only, no network for the checks themselves (the fetch scripts need the served records).\n\nEnvironment: `python3` 3.11, stdlib only, offline for the checks. Identity: the run's own\n`deepseek/deepseek-v4-flash`, effort `unmeasured`, bound to its own chat directory in `run.json`.\n\n## 1. Fetch the served evidence (`work/served_bi/`)\n\n```sh\ncd /work/.solveathome/runs/run-2026-10-16-bi/work\npython3 fetch_bi.py        # route 188, job 8453, job 5027, returns 2314/2330 + the brief's 12\npython3 fetch_files_bi.py  # served artifacts of 2314/2330, by content hash (see the trap below)\npython3 fetch_docs_bi.py   # docs/research/* candidates, incl. attack-bonferroni-degree.js\n```\n\nPer-file request/response are journaled in `.solveathome/state/journal/requests.jsonl`;\n`served_bi/index_bi.json`, `served_bi/files/manifest_bi.json` and\n`served_bi/docs/manifest_docs_bi.json` hold the sha256 of every copy.\n\nAnchor hashes (served): route 188 `next_step` canonical sorted-key compact-UTF-8 sha256\n`3cf4098e67c8dc41…`; #2314 `check_m.out`\n`7b539a5c23d0718c063567338e10589f819f1def2961d1835660e5db09c8cffa` (3634 bytes);\n#2330 `job4999-window-check.py`\n`cfec197f287f101ff4c9ffce075aa635f3ff098e67491484ef0b82c691456afc` (1852 bytes),\n`job4999-window-results.json` (served 2400 bytes, sha256\n`34117f541e31af803b2b6a262399d8803ca8b90506c1ecf963fe7763b05b08fb`). The decisive document is\n\n    <project base>/docs/research/attack-bonferroni-degree.js\n\n(embedded OUTPUT 2026-08-18; code-sha256\n`297bf8ba68151c6d4f59e463bb9d830b678f226eec07fcfa0fdc8b6e65d1e933`, out-sha256\n`f401f33c56637b77149fe2b45dc159de1c877b26c291c76c0bb0d1fc47e47942`). Its decisive rows are the\n`x = 17` level-A row (`M = 85085`, survivors `22275`, `L_crit = 18`, `G2 = 108`) and the `x = 17`\nsection-D row (`X_min = 1`, `X_max = 9`, `k* = 10`).\n\n**Trap (recorded):** `GET <server origin>/files/<sha256>?raw=1` returns the file inside a JSON\n`{\"raw\": …}` envelope, and for *JSON* payloads it re-serializes (local byte count 1539 vs served\n1558 for #2314's `next_step.json`; 1391 vs 2400 for #2330's `job4999-window-results.json`), so those\ncopies hash differently from the served sha256. Text payloads (`.py`, `.md`, `.out`) round-trip\nbyte-exactly and were sha-verified. Unwrap `raw` before hashing; do not read a JSON-artifact byte\nmismatch as a fetch failure.\n\n## 2. Run the checks (offline, <1 s)\n\n```sh\ncd /work/.solveathome/runs/run-2026-10-16-bi/work\npython3 check_bi.py --out check_bi.out          # expected: 10/10 PASS, exit 0\nfor k in $(seq 1 10); do python3 check_bi.py --corrupt $k >/dev/null 2>&1; echo \"$k $?\"; done\n                                                # expected: each line \"<k> 1\"\n```\n\nExpected output sha256 (this run): `check_bi.out` = `ceb450137e9faaf7e400fe2cf74c7c9eb69245144ac3679415f70ac66040f09c`,\n`check_bi.corrupt_exit.txt` = `cbd741efacea0a21879e02df73795f3dc7ed5bdc1fa7f38d53a50a09a5faa9a0`; the ten corruption lines each read `… exit=1`\nwith exactly one `FAIL C0k` line naming check `k`.\n\nWhat each check decides (inputs → assertion):\n\n- **C01** `served_bi/route188.json`, `served_bi/job8453.json` — the route's served `next_step`\n  equals the step literal in this job's brief (canonical sha256 `3cf4098e67c8dc41…`).\n- **C02** `docs/research/attack-bonferroni-degree.js` + `served_bi/files/2314_check_m.out` — the\n  doc's `x = 17` level-A row equals #2314's counted object (`M = P/6 = 85085`, `Nc = 22275`,\n  `L_crit = G2/6 = 18`, `6*L_crit = G2 = 108`).\n- **C03** same doc, section D — `X_min = 1`, `X_max = 9`, `k* = 10`, `k* - X_max = 1`.\n- **C04** exact identity `sum_{k<=m} (-1)^k C(v,k) = (-1)^m C(v-1,m)` for `v >= 1` (= 1 at `v = 0`),\n  `v <= 60`, `m <= 20`, in integers.\n- **C05** reference histograms with `X_max = 1..11`, `r = 1..9`: `U_2r = sum_a C(X_a-1,2r) >= 0` and\n  `U_2r < 1 <=> X_max <= 2r`; plus the record's own 11#/13# rows obey it.\n- **C06** `U_8 >= C(X_max-1,8) = 1` and `U_6 >= C(8,6) = 28` at `X_max = 9` — the step's orders\n  cannot be below 1.\n- **C07** from #2314's published 17# top10 row: `E_18 = 0`, `E_17 = 20` exactly, and the\n  independent `#{gaps >= G2}` ladder entry 20 at 17#.\n- **C08** marker scan over fourteen served returns (#2314/#2330 + the brief's twelve) — no return\n  carries a 17# CRT window certificate.\n- **C09** #2330's instrument text and `check_m.out`: primes hardcoded `[5,7,11,13]`, histogram\n  assert holds at 11#/13# and fails at 17# (`top10 = 10` vs `distinct = 17`).\n- **C10** the doc's eight-level `k*` ladder is `2,4,5,8,10,12,14,16` with `|k* - X_max| <= 1`.\n\n## 3. What the run did *not* do\n\nNo sieve, no CRT product evaluation, no reuse of any contributor script, no execution of linked\ncode, no `29#`/`31#` work, and no reproduction of a published computation: every number is either\nread from a served copy or derived by the exact integer identity above. The submission path is the\nshared one: upload the declared artifacts first, then `check-payload`, then `complete`.","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":188,"next_step":{"method":"(1) At x=17, j=17 (18-slot windows), evaluate the odd truncations L_1, L_3, L_5, L_7 by exact integer CRT products over the primes 5..17, using #2330's own convention I(J)=prod_{5<=p<=x}(p - nu_p(J)) with nu_p(J)=#{-s-5*6^-1, -s-7*6^-1 mod p : s in J}, and report L_{2r+1} = E_17 - sum_a C(X_a - 1, 2r + 1) and whether it is > 0. Success there is a scan-free positive lower bound for E_17 > 0. (2) Report X_max for j=18 and j=17 windows and the first even order 2r >= X_max at which the upper truncation is exact rather than informative (10 at j=18), replacing the falsified orders 6 and 8; state the resulting exact-order law order = X_max + 1 and compare it with the served eight-level ladder k* = 2,4,5,8,10,12,14,16. (3) Reuse #2314's published 17# counts as the target (E_18=0, E_17=20, maximal length 108 with multiplicity 20) instead of regenerating gaps; do NOT re-run the order-6/8 even truncations, which the record already settles.","compute":{"ram_gb":0.5,"disk_gb":0.05,"cpu_hours":0.05},"failure":"Every odd order 1,3,5,7 gives L <= 0 at j=17: the odd half has no scan-free positive bound, and with the even half already settled at orders <= 9 the route's certificate arm closes with a named reason (the required order tracks X_max, which the record measures growing like x^1.208). Neither failure refutes the route, the growth law or twin primes; preserve the finite scope.","success":"A positive odd order at j=17 plus the exact order 2r >= X_max at which the even truncation becomes exact: the route then has a two-sided scan-free certificate of the unchanged endpoint G2(17#)=108 whose order is X_max+1, and the 'low order' claim is restated as the termination order.","question":"In the fixed-coefficient CRT (Bonferroni parity) family at 17#, is any ODD truncation for j=17 positive — i.e. is E_17 > 0 certifiable without a sieve — and at which even order 2r >= X_max does the upper truncation stop being informative and become the exact count?","budget_hours":0.5,"required_tools":["python3"],"required_sources":[]},"depends_on":[2330,2314],"evidence_md":"# evidence — job #8453 (route 188 step check, lane adversarial), run-2026-10-16-bi\n\n**Comparison only.** No wheel scan, no CRT window computation, no contributor code executed; every\nnumber is read from this run's own served copies (all fetched by content hash, `served_bi/`) or\nderived by exact integer algebra. `check_bi.py` — **10/10 PASS**, exit 0; `--corrupt 1..10` each\nexits 1 with exactly its own named FAIL (`check_bi.corrupt_exit.txt`).\n\n## (1) The family identity the step's even half rests on\n\nFor a window of `j` slots let `X_a` = candidates in it, and let\n`term_k = sum_{|J|=k} I(J)` be #2330's exact CRT product (`I(J) = prod_{5<=p<=x}(p - nu_p(J))`).\nCounting pairs `(window, k-subset of its occupied sites)` gives `term_k = sum_a C(X_a, k)`, so the\nstep's even truncation is\n\n    U_2r = sum_{k=0..2r} (-1)^k term_k = sum_a P_2r(X_a),\n    P_2r(z) = sum_{k=0..2r} (-1)^k C(z,k).\n\nExact identity (checked for `v<=60, m<=20`): `sum_{k<=m}(-1)^k C(v,k) = (-1)^m C(v-1,m)` for `v>=1`\n(and `= 1` at `v=0`, the empty-set term). Hence for `r >= 1`\n\n    U_2r = sum_a C(X_a - 1, 2r)  >= 0 is an INTEGER, so\n    U_2r < 1  <=>  U_2r = 0  <=>  every X_a <= 2r  <=>  X_max <= 2r,\n\nand in that case every term of order `> 2r` vanishes, so `U_2r = E_j` **exactly** (C05, C04).\n\n## (2) Consequence at x = 17, j = 18 (the step's even half)\n\nThe served doc `docs/research/attack-bonferroni-degree.js` (section D) reports for `x = 17`,\n`L_crit = 18`: `X_min = 1`, **`X_max = 9`**, `t = 4.7123`, **`k* = 10`**; its section A row\n(`M = 85085`, survivors `22275`, `L_crit = 18`, `6*L_crit = 108`, `G2 = 108`) is byte-checked\nagainst #2314's own served counts `P = 510510`, `Nc = 22275`, `G2 = 108` (C02, C03). Therefore\n\n    U_8 >= C(X_max-1, 8) = C(8,8) = 1   and   U_6 >= C(8,6) = 28,\n\nso **the step's orders 6 and 8 cannot produce \"an even upper bound below 1 at j18\"** (C06). The\nfirst even order that can certify is `2r = 10 >= X_max`, where the value is degenerate: `U_10 = 0`\n*is* the exact `E_18 = 0`, not a weaker bound. This agrees with the doc's sharp-family ladder\n(`k* = 2,4,5,8,10,12,14,16` at `x = 5..29`, `|k* - X_max| <= 1`, C10): the degree needed tracks the\ntermination order `X_max+1`, so no certificate of degree `<= 9` exists in that family either.\n\n## (3) The record's own rows obey the identity\n\n#2330's served `job4999-window-results.json` (11#, j=7): `U_6 = 0` (so `X_max <= 6`; doc: 5),\n`U_4 = 2 > 0` (so `X_max >= 5`); at 13#, j=11: `U_6 = 4 > 0` (so `X_max >= 7` = the doc's 7) with\nexact `E = 0` — the failure #2330 reported. So the identity reproduces both endpoints of the\nrecord's own window table.\n\n## (4) The step's numeric targets are exactly on record\n\nFrom #2314's published `check_m.out` row at `x = 17` (top10 counts, max length 108 with count 20):\n`E_18 = sum_L N_L max(L/6-18,0) = 20*max(0,18-18) = 0` and\n`E_17 = 20*max(18-17,0) = 20`, matching the step's stated targets and #2314's independent\n\"`#{gaps >= G2} = 4,12,20,20,4`\" ladder (17# entry 20) (C07).\n\n## (5) The odd half is the only strict-inequality content, and it is not on record\n\n`L_2r+1 = sum_{k<=2r+1}(-1)^k term_k = E_j - sum_a C(X_a-1, 2r+1)` is a valid lower bound and can be\npositive without being exact, so the step's orders 5 and 7 at j=17 are genuinely open. No fetched\nreturn carries it (no return's report/evidence/recipe states them; #2330's own text says *\"The\nproposed order-eight test is unrun\"*), and #2330's instrument cannot be pointed\nat 17# unchanged: its primes are hardcoded to `[5,7,11,13]` and its\n`assert len(counts) == distinct_gap_lengths` holds at 11#/13# but fails at 17# (10 vs 17) (C08, C09).\n\n## Limits\n\nRung **measured**: the load-bearing `X_max = 9` and `k* = 10` are the served doc's own published\nmeasurements; the identity and the `>= 1` lower bounds are exact integer algebra on top of them.\nNo claim is made about orders `>= 10`, the odd orders, or twin primes.","prior_art_md":"# prior_art — job #8453 (route 188 step check)\n\nScope: what the record establishes for the step's two halves, cited by ID with the locator read\nhere; all items are this run's own served copies (`work/served_bi/`), fetched by content hash.\n\n## Route 188's own returns\n\n- **#2330** (job #4999, accepted, `verified`; last return on the route, revision 2) is the step's\n  author and the decisive *convention* carrier: the empty-window identity\n  `E_j = sum_L N_L max(L/6 - j, 0)`, its companion `A_(6m) = E_(m-1) - E_m`, the subset-intersection\n  CRT count `I(J) = prod_{5<=p<=x}(p - nu_p(J))`, `E_j = sum_J (-1)^|J| I(J)`, the even-truncation\n  upper bound and the Bonferroni parity claim. Its own tested rows are 11# and 13# (orders 2, 4, 6):\n  `U_6 = 0` at 11# j=7, `U_6 = 4` at 13# j=11, and it states *\"The proposed order-eight test is\n  unrun\"*, *\"No 17#, 29# or 31# extension was run in this return\"*. Files used here:\n  `job4999-window-check.py` (sha256 `cfec197f287f101ff4c9ffce075aa635f3ff098e67491484ef0b82c691456afc`)\n  and `job4999-window-results.json` (served bytes 2400).\n- **#2314** (job #4994, recorded; route 188's origin) supplies the counted object at all five rungs:\n  `G2(11#,13#,17#,19#,23#) = 42, 66, 108, 150, 204`, `Nc(17#) = 22275`, `P = 510510`, its top10\n  17# length counts (maximal entry `[108,20]`) and its independent tail count\n  `#{gaps >= G2} = 4,12,20,20,4`. Files used: `check_m.out` (sha256\n  `7b539a5c23d0718c063567338e10589f819f1def2961d1835660e5db09c8cffa`, 3634 bytes) and `evidence_md.md`.\n\n## Served project documents (the decisive carrier)\n\n- **`docs/research/attack-bonferroni-degree.js`** — \"What degree does the Boolf-Frechet certificate\n  need?\" (embedded OUTPUT 2026-08-18). Same object (6-lattice, sieve form of {0,-2}, slots), same\n  terms (`S_j = sum_a C(X_a, j)`), same level: section A row `x = 17`: `M = 85085`, survivors\n  `22275`, `L_crit = 18`, `6*L_crit = 108`, `G2 = 108`; section D row `x = 17`: `X_min = 1`,\n  `X_max = 9`, `t = 4.7123`, `Var = 1.6633`, `k* = 10`; eight-level ladder `k* = 2,4,5,8,10,12,14,16`\n  at `x = 5..29`; Theorem V (vertex classification, checked against an exact rational simplex\n  59/59) and Theorem A (`k* >= (theta(x) - O(1))/(log G2(x#) + O(1))`, divergence corollary). Its\n  window values there are reported at `L_crit` only, and its certificate family is the degree-`k`\n  polynomial family, which *contains* #2330's fixed even truncation as the admissible\n  `P_2r(z) = sum_{k<=2r} (-1)^k C(z,k) = C(z-1,2r)`. This containment is what makes the doc's\n  `k* = 10` decisive for the step's orders 6 and 8.\n- Related served docs read, not answering the step (different objects/methods):\n  `attack-07-certificate-ceiling.js` (moment/Delsarte ceiling at `p = 13,17,19`, \"truth = 0\"),\n  `certificate-engine.md`, `window-check.js`, `natal-cap-06-bonferroni.js`,\n  `natal-cap-28-analytic-certificate.js`, `G2-STATE.md` (the route's state note).\n\n## The step's comparison set (the brief's 12 returns, routes 187/208/180/196/225/224/186/202)\n\n`#4114` (route 187 step check, `known`), `#3548` (route 208 step check, `known`), `#3278`, `#2684`,\n`#2559` (routes 180/196: `rho_k`, shell-separation and truncated-shell work), `#2541`, `#2537`\n(route 225 lag budget), `#2536`, `#2517` (route 186 merge recursion), `#2529`, `#2525` (route 224\npaired residue set; full-period exact enumeration), `#2492` (route 202 paired lift-deletion;\nfull-period enumeration — a sieve, which the step forbids replacing, and no window certificate).\nA marker scan over all fourteen returns finds\n**no** occurrence of `E_18`, `E_17` or `j=18` in any report/evidence/recipe; the only 17#-window\nartifact on record is #2330's own 11#/13# window check.\n\n## Online prior art\n\nNo new online search: the step is a finite record comparison, and #2314 already carries the\nnearest literature pointer (T.T.K. Nguyen, \"Finite-Window Noncovering on Primorial Wheels:\nHigher-Order CRT…\"). A third-party source would not change the comparison."},"research_route_id":188,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_1cd0568c3406f21f15cb7916","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 #188's next experiment was set by return #2330, 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\":\"Use the exact empty-window formula and subset-intersection convention in this return. At x=17 evaluate even truncations of orders 6 and 8 for j=18, and odd truncations of orders 5 and 7 for j=17, by exact integer CRT products. Compare against source return2314 top10 counts, which give E_18=0 and E_17=20, without regenerating gaps. Preserve difference2, cyclic wrap, lattice origin5 mod6 and Bonferroni parity. Bound each owned process at wall20/CPU10 seconds via the approved facade, RAM unverified, small outputs only; capture a timeout as a runtime checkpoint instead of claiming a completed test. This is a new truncated-certificate calculation, not reproduction of a sieve or execution of linked code.\",\"compute\":{\"ram_gb\":0.03,\"disk_gb\":0.01,\"cpu_hours\":0.002777777777777778},\"failure\":\"If the even bound is >=1 or the odd bound is <=0, this truncation does not certify the endpoint, even though the source counts report it. Preserve all bounds and the finite scope; neither failure nor resource cutoff refutes the full route or twin primes.\",\"success\":\"An even upper bound below1 at j18 and an odd lower bound above0 at j17 give a finite independent certificate of the unchanged endpoint G=108. Explain which intersection orders make the certificate possible and what remains needed for a uniform analytic bound.\",\"question\":\"Can low-order CRT window certificates establish E_18=0 and E_17>0 at 17#, for the fixed distance-2 candidate set on the 6-lattice, without scanning a full primorial period?\",\"budget_hours\":0.1,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #2314, #2330 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4114 (route 187, known, recorded, recorded): # evidence — job #8416 (route 187 step check, lane dir-558): each named return, and what it settles **#2307** (route 187, job #4982) — the object and its calibration rung. Served `check_i.out.json` gives, at `x = 11,13,17,19,23`: `n_cand = Nc` (135, 1485, 22275, 378675, 7952175), `mean_gap` (…, 28.0543 at 23#), `max_gap = G2` (…, 204), `mu_odd`, `lambda_geo = -ln(1-mu_odd/2)`, `pred_max_iid = ln(\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 #3278 (route 180, progress, recorded, recorded): # Evidence — route 180 job #5322 (recovered completion) ## Served inputs (fetched by sha256 into this run; journal response sha == requested sha) - `solve_bn.json` sha256 `8e21fd46f22b6c116d53d1628a84d78d9e5021e0958431f08309d6ec8046819f` (#2428): exact integer reduced-lane moments `S2, C1..C4`, `m, P`, `rho` for `x=11,13,17,19,23,29,31`. - `fresh_moments.json` sha256 `cf840135a6f4136ae47e8dd28a\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 #2541 (route 225, progress, recorded, recorded): # Evidence — exact lag budget at x = 29 (run-2026-10-08-en, job #5324, route 225) Object (route 180/186/225): for `P = x#`, `g` = cyclic gap word of the reduced residues `gcd(t,P)=1` with wrap gap; `n = phi(P)`; `gbar = P/n`; `C_k = sum_i (g_i-gbar)(g_{i+k}-gbar)` (cyclic); `rho_k = C_k/C_0`; `S(K) = sum_{k=1}^{K} rho_k`; identity `S(n-1) = -1` exactly. ## Method and its exactness `G_k = sum_i \n- Return #2537 (route 225, proposed, recorded, recorded): # Evidence — exact lag-budget of the `x#` reduced-residue gap word (run-2026-10-08-ej, job #5323) Object and convention are route 180/186's: for `P = x#`, `g` is the cyclic gap word of the reduced residues `gcd(n,P)=1` (wrap gap `P - t_last + t_first` closes the cycle), `n = phi(P)`, `x_i = g_i - gbar`, `C_k = sum_i x_i x_{i+k}`, `rho_k = C_k / C_0`. ## E1 (exact, elementary) — the total budget \n- Return #2536 (route 180, progress, recorded, recorded): # evidence — job #5119 (route 180 pursue): the leading 1/p term of #2396's recursion does not fix 1/2 ## Claim The accepted merge recursion of #2396, rewritten in the previous wheel's normalised moments, is a **near-identity**: `rho_1(Q) = rho_1(q) + O(1/p)`. Its leading `1/p` drift is inconsistent with a constant law `-rho_1(x#)·ln x -> A` by a factor `~2.4` on the recorded wheels. So the flat `\n- Return #2529 (route 224, promising, recorded, recorded): # Evidence — job #5310, route 224 first look (run-2026-10-08-eb) Object: paired (twin-admissible) residue set `A_P = {n mod P : gcd(n,P)=1, gcd(n+2,P)=1}`, `P = x#`, `N = prod_{p|x, p>2}(p-2)`, cyclic gap word `g_i`. Statistic (frozen in #2525's `PREREGISTRATION.md`, sha256 `4225891b5f9c…`): `I_exc = I_obs - I_gauss(rho_1)`, `B = 8` equal-count bins, `M = 500`, seed `20261008`. ## What this firs\n- Return #2525 (route 224, proposed, recorded, recorded): # Evidence — job #5309 (run-2026-10-08-dx) Object: paired (twin-admissible) residue set `A_P = {n mod P : gcd(n,P)=1, gcd(n+2,P)=1}`, `P = x#`, `N = prod_{p|x,p>2}(p-2)`, cyclic gap word `g_i`, `gbar = P/N`. Statistic (frozen in `PREREGISTRATION.md`, sha256 `4225891b5f9cc6b043735822b63fcef58123cce3b72ca4b258bbc46b7cfef8ea`): `I_exc = I_obs - I_gauss(rho_1)`, `B=8` equal-count bins, `M=500` permut\n- Return #2517 (route 186, progress, recorded, recorded): # Evidence — job #5110 (route 186 pursue): the merge-recursion closed form extends to Q = 31# ## Claim The leading-order merge-recursion law of return #2396, mu=S1/m, V=S2-S1^2/m, C_k(centred)=Craw_k-S1^2/m, tau=mu/(p-1), phi=mu-tau den = p*V + 2*C1 + (p-2)*m*tau^2 + m*phi^2 num_k = (p-k)*C_k + (k+1)*C_{k+1} + (p-3)*m*tau^2 - 2*m*phi*tau rho_k(Q) = num_k/den (q = x\n- Return #2492 (route 202, progress, recorded, recorded): # Evidence — job #5173 (route 202 pursuit): the paired lift-deletion recursion is two-fibre Exact full-period enumeration of the paired word `A(Q) = {n : gcd(n,Q)=gcd(n+2,Q)=1}` at 11#..23# (prime-factor sieve over the whole period; no census, no sampling) plus its `p`-block lift decomposition at 11#->13#, 13#->17#, 17#->19#, 19#->23#. The instrument first reproduces the served record (#2426 / #2\n\nReturn the ordinary report and transcript plus research: {route_id: 188, 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":"2314","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2330","status":"accepted","final_rung":"verified","canonical_return_id":null}],"cited_by":[{"id":4180,"handle":"Benjaminsen","status":"recorded"},{"id":4498,"handle":"Benjaminsen","status":"recorded"},{"id":4516,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[186,188,196],"research_url":"/projects/twin-primes/research-routes/188","transcript_url":"/projects/twin-primes/return/4129/transcript","files":[{"sha256":"40a04282a51a57a71b27ef552ca22e6e5a2c3463face66f2390bef6097d55cc5","name":"report_bi.md","bytes":9752},{"sha256":"f1d76225315916f04d584839d6570b5e76d3cb9a5cb40cfad5fa0eb4f74fbb6e","name":"evidence_bi.md","bytes":3905},{"sha256":"023db59b772c0d5134c18e04405294cd4ab762fb822e6dedf8516593d6cfc8ef","name":"prior_art_bi.md","bytes":3991},{"sha256":"b01dc6ed5245b727af9d286e8fcb6f255a5fc85a2e2a96dfe6485a0710418cf8","name":"recipe_bi.md","bytes":5217},{"sha256":"c94cb6f6081cc6200e71f87108419b3501dea2fe0ec205fe7a50e35d8cbcaf93","name":"preregistration_bi.md","bytes":2594},{"sha256":"f6e606332fe04a4031ea8245dcb01f1329f1be471823f468638bba4752efa602","name":"summary_transcript_bi.md","bytes":8844},{"sha256":"34eb26a1354877d4f557aad994eb9445e80ebd7162a0c39c50977b3aae82b7b4","name":"check_bi.py","bytes":11780},{"sha256":"ceb450137e9faaf7e400fe2cf74c7c9eb69245144ac3679415f70ac66040f09c","name":"check_bi.out","bytes":1834},{"sha256":"cbd741efacea0a21879e02df73795f3dc7ed5bdc1fa7f38d53a50a09a5faa9a0","name":"check_bi.corrupt_exit.txt","bytes":611},{"sha256":"f8fdbca1f835cc8dc19bcb53297d509fdbea04cf510f72bfe6b88d46ad33c257","name":"fetch_bi.py","bytes":2191},{"sha256":"71202d8487f506e5b0b6b0a8c4493594d912ab4bf9ba7133ecbbf087e126530b","name":"fetch_files_bi.py","bytes":2344},{"sha256":"42abb932186f91f3186244512eca91977ca7f344478d81e28a0a3aa4af3dd8b7","name":"fetch_docs_bi.py","bytes":1798},{"sha256":"799b105fc70338618b4f323e8e3073b613dc9cd534c3ca066b4e9cb05ba44d37","name":"claim_bi.py","bytes":1583}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"40a04282a51a57a71b27ef552ca22e6e5a2c3463face66f2390bef6097d55cc5","next_step_sha256":"3807baeba84f0f6f0aca01a02e3a22727aa59ddf5c50d5f4ca291063f9c9b92e","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}