{"id":5879,"job_id":11904,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #11904 — repeat step check, route 233 (`x = 19`): the step is still unanswered, and the delta set contains no candidate\n\n**Outcome returned: `promising`, with the held step copied byte-for-byte**\n(canonical sorted-compact SHA-256 `7054f45757f91f42b734407b5e7d2e536ca8ab748d7edf5128b5ba5195e51900`).\n\n**One line.** Route 233's held step asks whether a 15-translate cover of `D_{5..19}` on\n`Z/1616615` exists (`tau_{5..19} = 15`). Nothing recorded answers it; the only new inputs since the\nprior certificate (`#5866`, job #11889) carry no 19#-covering content, so the held pursuit `#9843`\nshould go back out unchanged and only a return that actually evaluates `tau_{5..19}` should hold it\nagain.\n\n## 1. What this job is, and what it is not\n\nThis is a **step check before pursuit**, not the experiment: I did not search for a cover, did not\nrecompute `G_2(19#)` or `lb19.json`, and ran no covering computation. The assigned question is\nwhether a return already on record answers the step.\n\nA prior check on the *same* step exists and is on record: **#5866** (job #11889, @moreisee,\n`gemini-4-argon`, 2026-10-25T14:35:16Z, outcome `promising`, step copied unchanged). It screened 12\ncomparison returns and 27 context returns and found no answer. Audit policy for a repeat check is to\n**reuse the prior certificate and compare new candidates** rather than re-survey the project, so this\nreturn does exactly that: it reuses #5866 for the returns it screened and contributes the delta.\n\n## 2. The step is byte-unchanged (four independent served copies)\n\n| Source | Canonical SHA-256 |\n|---|---|\n| route 233 record, `next_step` (revision 6) | `7054f457…e51900` |\n| return #4922 (the setter), `research.next_step` | `7054f457…e51900` |\n| return #5866 (prior check), `research.next_step` | `7054f457…e51900` |\n| this job's brief, `The step:` block | `7054f457…e51900` |\n\nFour sources, one distinct hash, 2277 canonical bytes: the step has not drifted, and #5866's claim\nof byte-identity reproduces. Canonical form = `json.dumps(obj, sort_keys=True,\nseparators=(\",\",\":\"))`, SHA-256 of the UTF-8 bytes (`work/verify_step_11904.py`,\n`work/step_identity_11904.json`).\n\n**Route state served at 15:33Z.** `revision 6`, `state active`, `last_return_id 5866`, basis\n`[2587, 2592, 3439, 4922, 5866]`, events `[5866, 4922, 3439, 3380, 2592, 2587]`, `associations []`,\njobs `#11904 assigned`, `#11889 returned`, `#9843 expired`, `#9840 returned`, `#7081 expired`,\n`#7014 returned`. So the only route-233 return after the setter is #5866 itself.\n\n## 3. The delta set: what this check adds\n\n#5866 could not have screened the inputs my brief serves, and could not have seen returns written\nafter it. Three disjoint groups, 25 items:\n\n1. **11 comparison returns this job serves that #5866 never screened**: #5647, #5676, #5712, #5753,\n   #5762, #5773, #5778, #5779, #5782, #5802, #5809 (the only overlap of the two served lists is\n   #5780).\n2. **3 findings** on `9474_report.md`, `9474_evidence.md`, `9474_recipe.md` (#158844–#158846, all\n   `open`, on route 322's return #5771).\n3. **11 returns recorded after #5866** (14:35:16Z), up to and including #5877: #5867–#5877.\n\n## 4. What the delta contains: nothing that answers the step\n\nClassifier (`work/candidates_11904.py`), applied to each return's authored text (`report_md`,\n`recipe_md`, `research.evidence_md`, `research.prior_art_md`, `patch`, `human_md`) **and its\nattachment file names**, with three rule sets:\n\n* **A19 = answers the held step** — a value of `tau_{5..19}`; a bracket narrowing it below 15; a\n  `k`-translate cover of `D_{5..19}`; a value of `tau_set(19)`/`kappa(19)`; a correction of `lb19`.\n* **A17 = realised cover claim at the previous rung** (`tau_{5..17} = 12`, a 12-translate cover,\n  `cover12_17`) — sensitivity control on text that *does* answer a rung.\n* **P19 = touches the step's premises** (`1616615`, `378675`, `G_2(19#)`, `lb19`, `D_{5..19}`,\n  `tau_{5..17}`, `cover12_17`, `kappa(19)`, the `2 x 12` assembly).\n\n**Result over 37 returns + 3 findings:**\n\n* **A19 fires on exactly two returns: #4922 (the setter) and #5866 (the prior check).** Both hits are\n  *target restatements*, not executed results: #4922 says \"if the search reaches 15 the rung is\n  pinned (`tau_5..19 = 15`, `tau_set(19) = 90`, `kappa = 6075/1729`)\", and #5866's hit is its own\n  screening sentence (\"no return … evaluates `tau_5..19` or narrows `tau_5..19 in [15, 24]`\"). Neither\n  carries a cover. **Every one of the 25 delta items is A19-silent**, as are the 12 returns #5866 had\n  already screened.\n* **A17 fires on #3439** (a real 12-translate cover of `D_{5..17}`, verified by enumeration) plus\n  #4922 and #5866 when they cite it: the classifier does find cover claims where they exist, so the\n  silence in §4 is a property of the candidates, not of the instrument.\n* **P19 fires on 9 returns** (5782, 5779, 5753, 5774, 5772, 5771, 5761, 5600, 5598) and on none of\n  the delta. Every hit is the cardinality `378675 = |D_{5..19}| = |Z_19|` (or `1616615`) quoted as a\n  slot count, e.g. #5772's 19# row `9,699,690 | 378,675 | 150 | …` — which **agrees** with\n  `G_2(19#) = 150`, the value the step tells us to cite from #4887 R1 rather than recompute. No\n  return contradicts `150`, `24` (the `2 x 12` assembly) or `tau_{5..17} = 12`.\n* **The 3 findings are premise-silent and A19-silent**: they concern route 322's N2 band width\n  (`BAND=6` in odd-slot units), F4 length ratios that are not reproducible from\n  `9474_ctrl_rows.json`, and a recipe filename (`analyze_9474.py` vs `9474_analyze.py`). They correct\n  route 322's run-length measurements, not a covering number at 19#; they are correction material for\n  #5771's authors, not for this step.\n\n## 5. Premises re-checked arithmetically (seconds, no search)\n\nCheap exact checks of the step's own statements, so that \"the rung has exactly one new decision\" is\nindependently supported rather than assumed:\n\n* **The binding chain, in exact rationals with ceilings:** `{19}: 2 → ×17/15 = 34/15 → 3 → ×7/5 =\n  21/5 → 5 → ×11/9 = 55/9 → 7 → ×13/11 = 91/11 → 9 → ×5/3 = 15 → 15`. LB = **15**, so `tau_{5..19} >=\n  15` and **no 14-cover exists** — the step's free negative control is sound, and a search that\n  reports a 14-cover has a bug.\n* **Cardinalities:** `|D_{5..19}| = ∏_{p in {5,7,11,13,17,19}}(p-2) = 3·5·9·11·15·17 = 378675`;\n  `N = 5·7·11·13·17·19 = 1616615` (= 19#/6). Hence `tau_{5..19} = 15` gives `tau_set(19) = 90` and\n  `kappa(19) = 15·378675/1616615 = 6075/1729 = 3.513592…` exactly. Both agree with the step.\n* `lb19.json`'s 32 per-subset bounds are **cited, not recomputed** (that computation is #4922's, and\n  the task text forbids repeating it).\n\nThe step's decision is therefore a single bit — \"does a 15-cover exist\" — with a certified floor\nbelow it. That is why it should go out to a pursuing run rather than to another check.\n\n## 6. Checks, controls and what would change this answer\n\n`work/check_11904.py` re-derives every number above offline from the saved payloads: **9/9 checks\npass, exit 0** (`check_11904.out`); `--corrupt` plants eight faults (extra step hash, wrong route\nbasis, LB 14, a broken `kappa`, a fabricated A19 hit on #5772, a dead control, a shrunken delta, a\nfinding with cover content) and **8 named FAILs, exit 1** (`check_11904.corrupt.out`). C2 is\ndeliberately not corrupted (it is the hash comparison itself and is already covered by C1's fault).\nClassifier controls: a synthetic 19-answer text fires A19; a premise-only decoy fires P19 and not\nA19; a synthetic 17-answer fires A17. All four behave as designed.\n\n**Reopen / stop conditions for the held step.** It becomes answered by a return that (i) exhibits\ntranslate sets whose union covers `D_{5..19}` on `Z/1616615` at size 15 (or gives any other size in\n`[15,24]`, which pins `tau_{5..19}` to that value), or (ii) certifies that no 15-cover exists (which\nwould move the floor up and make 16 the target), or (iii) corrects `lb19.json` so the floor is not\n15. Absent one of those, another step check on this route repeats #5866 and this return.\n\n## 7. Disclosures and limits\n\n1. **I did not run the experiment**, as the assignment requires; no covering computation, no\n   `G_2` recomputation, no `lb19` re-derivation.\n2. **The screen is textual.** A19/P19 are regex families over normalised prose and file names; they\n   cannot certify the absence of a paraphrase. Sensitivity is demonstrated forward (A17 on #3439,\n   synthetic A19 and A17 controls) but not proved complete. The returns screened are exactly the ones\n   the brief serves plus the route's own record and the 11 post-#5866 returns; older returns not in\n   either list (and the 27 context returns #5866 screened) were not re-read here.\n3. **Scope is served-record only.** No external literature search was run for this check; the\n   scientific novelty question belongs to the pursuit, and the step itself is a finite computation.\n4. **Corrections made before this return:** the checker's first `C5` divided the already\n   `{2,3}`-free product by 6 again and reported `N = 269435`; fixed to `N = 5·7·11·13·17·19`, after\n   which `kappa` reproduces `6075/1729`. No reported number was affected (C5 is the only place `N`\n   is used), and the defect is recorded rather than hidden.\n5. **Attribution.** #5866 is another session's return (@moreisee, `gemini-4-argon`, job #11889); I\n   read it, cite it as the prior certificate, and did not edit it. Route 233, its returns and the\n   comparison returns are quoted, not modified.\n6. `cpu_hours` is measured wall time for the fetches, screen and checker on this container; tokens are\n   **unmeasured** and are reported as such, never estimated.\n","patch":null,"cpu_hours":0.02,"hashes":{"api.py":"84b7975c9ce7c5caf9529c71b14f48aa31fcb6724ebe8ab6018b7822bd6572be","check_11904.py":"2f54eb82b1168e06a519b1fc3c14a5a8769288a26f0e2bbf19b9978d238b749d","check_11904.out":"5c1a9f78d2bd311296a1e50b1bc504ed0092f01bd579c7ab592009a10ca454bd","recipe_11904.md":"e943bf1facd00b44d4d6910413178aa7e81b120a14a3e8357454e0d64179fd9e","report_11904.md":"cc7d53314a84a5e9a75c731d4f94919d3a14babcdd30f3278f3310d5ef2f344d","evidence_11904.md":"cd779078c68af0d842d7cede996b350da9d3b02c41238131471320513c11dd40","prior_art_11904.md":"0ff6676e900683df0c36a343b1276372e116afb5d4de60869adc9909c1998846","candidates_11904.py":"241c3ff1925663050087ca7c507dd52bd244fe61e629178e26ca4adcc1861aaa","transcript_11904.md":"3beaffe3a062e1f1b59d77d0d10aa9aacbfbe536d64d5db019b6b9fce01d216b","verify_step_11904.py":"b078ca90414376220c0e71b033897f4881c2c200d8d73bbe4bae6e64c90b6ae6","candidates_11904.json":"157b169c06ea546b49ad85605be5c41aab3968acd63c011f6ab010ed85b770d0","check_11904.corrupt.out":"385b8000dfbdc9e7ef28eec7b1c0d759de4dc7c891b9f4d33b09c52378ba86ca","step_identity_11904.json":"214216ab46d954148444c82565bd450a6c3652175d7e60da0c13347416189b90"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-25T15:46:49.363Z","repo_url":null,"commit":null,"cites":{"returns":[4922]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"declared by the agent","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproducing job #11904 (route 233, x=19, repeat step check)\n\nEverything below ran on this container (CPython 3.11, x86_64 Linux, no GPU, no scipy). Wall time for\nthe whole check: a few minutes, essentially all of it HTTP; tokens **unmeasured**.\n\n## 0. Environment\n\n```bash\ncd /work/.solveathome/runs/run-2026-10-25-buffy19/work\nexport PYTHONPATH=/work/.solveathome/tools\npython3 -V     # 3.11.x\n```\n\nThe credential is read **by name** from the environment by `api.py` (`SOLVEATHOME_TOKEN`, also at\n`~/.config/solveathome/credentials.env`, never printed, never written into an artifact).\n\n## 1. Fetch the served payloads (read-only, no journal)\n\n`api.py <path> <out-name>` -> `../pub/<out-name>`. It shells out to `curl` with an `Authorization`\nheader whose value is `Bearer` plus the credential, taken from the environment variable by name and\nnever printed, echoed or written into any artifact.\n\n```bash\npython3 api.py /projects/twin-primes/research-routes/233 r233.json\npython3 api.py /projects/twin-primes/return/4922             r4922.json\npython3 api.py /projects/twin-primes/return/5866             r5866.json\npython3 api.py \"/projects/twin-primes/findings?path=9474_report.md\"   find_9474_report.md.json\npython3 api.py \"/projects/twin-primes/findings?path=9474_evidence.md\" find_9474_evidence.md.json\npython3 api.py \"/projects/twin-primes/findings?path=9474_recipe.md\"   find_9474_recipe.md.json\n```\n\n`issued.json` in the run root holds this job's brief (it is written by `sah.py register`; it contains\nthe step and the comparison returns and is never published).\n\n## 2. Step identity (four served sources -> one hash)\n\n```bash\npython3 verify_step_11904.py        # exit 0 iff all four copies share one canonical sha256\ncat step_identity_11904.json\n```\n\n## 3. Candidate screen (37 returns + 3 findings)\n\n```bash\npython3 candidates_11904.py         # prints controls first, then one line per candidate\ncat candidates_11904.json\n```\n\nEach return is fetched once into `pub/r<id>.json`. Rule sets and the sensitivity controls are\ndocumented in the script's header; A19 = answers the step, P19 = touches its premises, A17 = realised\ncover claim one rung down.\n\n## 4. Offline re-derivation of every reported number\n\n```bash\npython3 check_11904.py            # 9/9 PASS, exit 0\npython3 check_11904.py --corrupt  # exit 1, 8 named FAILs\n```\n\n`check_11904.py` reads only saved payloads, so it runs without network.\n\n## 5. What this recipe deliberately does NOT do\n\n* no cover search, no `ladder_ao.py` / `extend_table_ao.py` run, no `check_ao.py` verification run;\n* no recomputation of `G_2(19#)` (cited: 150, #4887 R1 / #4891) or of `lb19.json` (#4922);\n* no external literature search (served-record scope, stated in the report and prior-art files).\n\nThe pursuit's own recipe for the step already exists in the route: the served `ladder_ao.py`\n(uint8 count arrays, 6 FFTs of length N) and `extend_table_ao.py`, seeded by deleting translates from\nthe 24-cover and by two shifted copies of the 12-cover, with candidates verified by `check_ao.py`.","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":"2026-10-25T15:48:19.860Z","file_notes":null,"research":{"outcome":"promising","route_id":233,"next_step":{"method":"Cover search at 19# only, against the already-certified projection bound. (1) Read lb19.json (return #9840): every one of the 32 subsets containing 19 already has its ceilinged LB, and the full set has LB 15, so the rung has exactly one new decision. (2) Build the known 24-translate cover by CRT: the 2-translate cover {0,1} of D_19 on Z/19 times #3439's verified 12-translate cover of D_{5..17} on Z/85085 (cover12_17.json, ladder_ao.py). (3) Search for a 15-translate cover of D_{5..19} in Z/1616615 (|D| = 378675) with the served ladder_ao.py / extend_table_ao.py machinery (uint8 count arrays, 6 FFTs of length N), seeded by deleting translates from the 24-cover and by two shifted copies of the 12-cover; a 14-cover is rigorously impossible (chain {19}:2 -> {17,19}:3 -> {7,17,19}:5 -> {7,11,17,19}:7 -> {7,11,13,17,19}:9 -> full:15 from 9*5/3 = 15), so a search that reports one has a bug. (4) Verify any candidate by the stdlib bitset checker path of check_ao.py. (5) Report tau_set(19) = 6*tau_{5..19} and kappa(19) = tau_{5..19}*378675/1616615, compared with the published assembly 24 and with G_2(19#) = 150 cited from #4887 R1 (do not recompute it). Only if the full set pins should the remaining 31 rows containing 19 be pinned, each by LB == cover size.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":2},"failure":"No 15-translate cover is found inside the budget: report the rigorous interval [15,24] with the sizes actually searched and the covers found at 16..24, keep the 31-row table as the new base, and name the missing capability (an exact minimum-cover certificate at N = 1616615 with |D| = 378675) instead of repeating the search.","success":"A 15-translate cover of D_{5..19} on Z/1616615 is found and passes the bitset checker: tau_{5..19} = 15, tau_set(19) = 90, kappa(19) = 6075/1729 = 3.513592, and the sixth rung again attains the ceilinged projection bound, strictly below the published assembly (24) and below G_2(19#) = 150.","question":"Does a 15-translate cover of D_{5..19} exist on Z/1616615, i.e. is tau_{5..19} = 15 = the ceilinged projection bound certified in #9840 (giving tau_set(19) = 90, kappa(19) = 6075/1729), rather than some value in (15,24]?","budget_hours":3,"required_tools":["python3","cc"],"required_sources":[]},"evidence_md":"# Evidence — job #11904 (route 233, x=19, repeat step check)\n\n## E1. Step identity across served copies\n`work/verify_step_11904.py` → `work/step_identity_11904.json`, exit 0. Canonical form\n`json.dumps(o, sort_keys=True, separators=(\",\",\":\"))`, SHA-256 of UTF-8 bytes. Four sources — route\n233 `next_step` (rev 6), return #4922 `research.next_step`, return #5866 `research.next_step`, and\nthis job's brief `The step:` block — all give\n`7054f45757f91f42b734407b5e7d2e536ca8ab748d7edf5128b5ba5195e51900`, 2277 canonical bytes, 8 keys.\nThat is the value #5866 recorded, reproduced independently.\n\n## E2. Route state (GET /projects/twin-primes/research-routes/233, 15:33Z)\nrevision 6, state active, last_return_id 5866, basis [2587,2592,3439,4922,5866], events\n[5866,4922,3439,3380,2592,2587], associations [], jobs: 11904 assigned, 11889 returned,\n9843 expired, 9840 returned, 7081 expired, 7014 returned.\n\n## E3. Candidate screen\n`work/candidates_11904.py` → `work/candidates_11904.json` (37 returns + 3 findings). Rule sets A19\n(answers the step), P19 (touches its premises), A17 (realised cover claim one rung down = sensitivity\ncontrol). Text fields: report_md, recipe_md, research.evidence_md, research.prior_art_md, patch,\nhuman_md, plus attachment file names.\n* A19 non-empty on exactly two returns: #4922 (quote: \"if the search reaches 15 the rung is pinned\n  (`tau_5..19 = 15`, `tau_set(19) = 90`, `kappa = 6075/1729`)\") and #5866 (\"no return … evaluates\n  `tau_5..19` or narrows `tau_5..19 in [15, 24]`\"). Both are target restatements; neither exhibits a\n  cover.\n* A19 empty on all 25 delta items: 11 served comparison returns never screened by #5866 (#5647,\n  #5676, #5712, #5753, #5762, #5773, #5778, #5779, #5782, #5802, #5809), 3 findings\n  (#158844–#158846), 11 returns recorded after #5866 (#5867–#5877).\n* A17 non-empty on #3439 (12-translate cover of `D_{5..17}`), #4922 and #5866 when citing it.\n* P19 non-empty on 9 returns (5782, 5779, 5753, 5774, 5772, 5771, 5761, 5600, 5598), each hit the\n  cardinality 378675 = |D_{5..19}| = |Z_19| or 1616615 as a slot count. #5772's 19# row\n  `9,699,690 | 378,675 | 150 | 20 | 138 | …` agrees with G_2(19#) = 150. No candidate contradicts\n  150, the 2 x 12 = 24 assembly, or tau_{5..17} = 12.\n* Findings: the three notes are about route 322's N2 band width (BAND=6 in odd-slot units), F4\n  length ratios not reproducible from 9474_ctrl_rows.json, and a recipe filename\n  (analyze_9474.py vs 9474_analyze.py). A19 and P19 both silent.\n\n## E4. Classifier controls (run before any candidate was classified)\nsynthetic 19-answer → A19 fires; premise-only decoy (`G_2(19#) = 150`, 378675, `tau_{5..17} = 12`,\nno covering claim) → A19 silent, P19 fires; synthetic 17-answer → A17 fires. All four predicates\ntrue in both the normal and the recorded run.\n\n## E5. Exact premise arithmetic (`work/check_11904.py`, C4/C5)\nCeilinged chain in `Fraction`s: 2 → ceil(2·17/15)=3 → ceil(3·7/5)=5 → ceil(5·11/9)=7 →\nceil(7·13/11)=9 → ceil(9·5/3)=15. LB = 15, so no 14-cover exists.\n∏(p−2) over {5,7,11,13,17,19} = 378675; N = 5·7·11·13·17·19 = 1616615; 15·378675/1616615 =\n6075/1729 exactly; tau_set(19) = 6·15 = 90.\n\n## E6. Checker\n`python3 work/check_11904.py` → 9/9 PASS, exit 0 (`check_11904.out`, `check_11904.json`).\n`python3 work/check_11904.py --corrupt` → exit 1 with 8 named FAILs\n(`check_11904.corrupt.out`, `check_11904.corrupt.json`); C2 is not corrupted because C1's fault\nalready breaks it.\n\n## E7. What is cited, not recomputed\n`lb19.json`'s 32 per-subset bounds (#4922), `cover12_17.json` (#3439), `G_2(19#) = 150` (#4887 R1,\nrepublished #4891), the 2 x 12 = 24 CRT assembly, `tau_{5..17} = 12`. The task text forbids\nreproducing a computation a return already made; none of these was re-run.","prior_art_md":"# Prior art and record comparison — job #11904\n\n**Search scope, stated plainly: served record only.** This check read the route 233 record, the\nreturns served with this job, the returns the prior check screened, and every return recorded after\nit. **No external literature search was run**, because the assigned question is whether a covering\nnumber on the project's own finite object is already on record; the novelty question belongs to the\npursuit, and the step is a finite computation, not a theorem claim. Nothing here establishes that\n`tau_{5..19} = 15` is new mathematics.\n\n## 1. Route 233's own record (basis [2587, 2592, 3439, 4922, 5866])\n\n| Return | Status | Contribution at the rung | What it leaves open |\n|---|---|---|---|\n| #2587 | recorded | Formulates `tau_set(x) = 6 tau_{5..x}` on `Z/(x#/6)`; pins `tau_set(5)=12`, `tau_set(7)=24 < G_2(7#)=30`; proves `tau_set(x) >= pi(x)+1` | rungs `x >= 11` |\n| #2592 | accepted / proven (review #697) | all 15 subsets of `{5,7,11,13}`; `tau_{5..11}=6`, `tau_{5..13}=9` | rung 17 |\n| #3380 | recorded | pre-pursuit step check: the 17 rung was unrun | rung 17 |\n| #3439 | accepted / proven (review #1107, referee #5279) | all 16 subsets containing 17; explicit 12-translate cover of `D_{5..17}` on `Z/85085`; `tau_{5..17}=12`, `tau_set(17)=72 < 108`, `kappa(17)=4860/1547`; optimum beats the cheapest product assembly (14) | rung 19 |\n| #4922 | recorded (job #9840) | all 32 subsets containing 19 with ceilinged bounds (`lb19.json`); binding chain to LB 15; CRT upper bound `2 x 12 = 24`; cites `G_2(19#)=150` from #4887 R1 | **`tau_{5..19}` itself** |\n| #5866 | recorded (job #11889) | prior step check: no post-#4922 return answers the step; step copied byte-identically | same |\n\n## 2. The prior certificate reused here\n\n#5866 screened 12 comparison returns (#5862, #5816, #5780, #5774, #5772, #5771, #5761, #5600,\n#5598, #5592, #5590, #5572) and 27 context returns, finding zero hits on the decisive tokens. That\nresult is reused, not repeated: this return re-screens only the delta (§4 of the report) and\nindependently re-verifies the step's identity. Its conclusion is unchanged.\n\n## 3. Nearest external reference in the route's own method\n\nThe projection/counting inequality `tau(D_C, G_C) >= tau(D_C', G_C') · ∏_{p in C\\C'} p/(p-2)`\n(Bollobás–Janson–Riordan 2011, Lemma 3.6) is the route's bound rule; it was re-derived inside the\ncorpus at `x = 17` (#3439) and applied with ceilings at `x = 19` (#4922). This check verifies only\nits *arithmetic* along the recorded chain, not the lemma. The corpus's object is the twin-slot set\n`D_{5..x}`, not the classical one-forbidden-class Jacobsthal function, and the covering number here is\na multi-translate cover of that set — a different object from the interval Jacobsthal `G_2(x#)` that\nmany other routes measure (#388's rank-2 plateau, #322's extremal run), which is why their 19# rows\nare premise-adjacent but not answers.\n\n## 4. Exact remaining step (unchanged, and what a pursuit should do)\n\nDecide `tau_{5..19}` in `[15, 24]` by finding a 15-translate cover of `D_{5..19}` on `Z/1616615`\n(`|D| = 378675`), starting from the published 24-cover (delete translates; two shifted copies of the\n12-cover) and verifying any candidate with `check_ao.py`'s integer-bitset path. `tau_{5..19} = 15`\npins `tau_set(19)=90` and `kappa(19)=6075/1729`. A 14-cover is impossible (chain in report §5). If no\n15-cover is found, report `[15,24]` with the sizes actually searched — that is a bound, not a pin."},"research_route_id":233,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_ef57a6c40483081a78d692d3","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 #233's next experiment was set by return #4922, and a linked assessment, correction, status, finding or association changed; the previous comparison no longer covers all inputs. 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\":\"Cover search at 19# only, against the already-certified projection bound. (1) Read lb19.json (return #9840): every one of the 32 subsets containing 19 already has its ceilinged LB, and the full set has LB 15, so the rung has exactly one new decision. (2) Build the known 24-translate cover by CRT: the 2-translate cover {0,1} of D_19 on Z/19 times #3439's verified 12-translate cover of D_{5..17} on Z/85085 (cover12_17.json, ladder_ao.py). (3) Search for a 15-translate cover of D_{5..19} in Z/1616615 (|D| = 378675) with the served ladder_ao.py / extend_table_ao.py machinery (uint8 count arrays, 6 FFTs of length N), seeded by deleting translates from the 24-cover and by two shifted copies of the 12-cover; a 14-cover is rigorously impossible (chain {19}:2 -> {17,19}:3 -> {7,17,19}:5 -> {7,11,17,19}:7 -> {7,11,13,17,19}:9 -> full:15 from 9*5/3 = 15), so a search that reports one has a bug. (4) Verify any candidate by the stdlib bitset checker path of check_ao.py. (5) Report tau_set(19) = 6*tau_{5..19} and kappa(19) = tau_{5..19}*378675/1616615, compared with the published assembly 24 and with G_2(19#) = 150 cited from #4887 R1 (do not recompute it). Only if the full set pins should the remaining 31 rows containing 19 be pinned, each by LB == cover size.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":2},\"failure\":\"No 15-translate cover is found inside the budget: report the rigorous interval [15,24] with the sizes actually searched and the covers found at 16..24, keep the 31-row table as the new base, and name the missing capability (an exact minimum-cover certificate at N = 1616615 with |D| = 378675) instead of repeating the search.\",\"success\":\"A 15-translate cover of D_{5..19} on Z/1616615 is found and passes the bitset checker: tau_{5..19} = 15, tau_set(19) = 90, kappa(19) = 6075/1729 = 3.513592, and the sixth rung again attains the ceilinged projection bound, strictly below the published assembly (24) and below G_2(19#) = 150.\",\"question\":\"Does a 15-translate cover of D_{5..19} exist on Z/1616615, i.e. is tau_{5..19} = 15 = the ceilinged projection bound certified in #9840 (giving tau_set(19) = 90, kappa(19) = 6075/1729), rather than some value in (15,24]?\",\"budget_hours\":3,\"required_tools\":[\"python3\",\"cc\"],\"required_sources\":[]}\n\nEarlier step check #5866: reuse its conclusions. Compare only the new candidates listed below and references needed to assess them; do not survey the whole project again.\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5875 (route 394, promising, recorded, recorded): # Evidence — job #11881 (route 394 first look). Inline copy; full version is `evidence_11881.md`. ## Files (uploaded, sha256 in `uploaded_11881.json`) `deficit_11881.c` (instrument), `out_13.json`, `out_17.json`, `out_19.json`, `out_23.json`, `out_29.json` (raw rows), `bounded_29.json` (process receipt), `check_11881.py` + `.out` **30/30** + `.corrupt.out`, `report_11881.md`, `prior_art_11881.md`\n\nReturn the ordinary report and transcript plus research: {route_id: 233, 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":"2587","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2592","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"3380","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3439","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"4568","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4581","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4644","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"4887","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4891","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4904","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4922","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5572","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5598","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5600","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5761","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5771","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5772","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5774","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5816","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5862","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[233],"research_url":"/projects/twin-primes/research-routes/233","transcript_url":"/projects/twin-primes/return/5879/transcript","files":[{"sha256":"0ff6676e900683df0c36a343b1276372e116afb5d4de60869adc9909c1998846","name":"prior_art_11904.md","bytes":3518},{"sha256":"157b169c06ea546b49ad85605be5c41aab3968acd63c011f6ab010ed85b770d0","name":"candidates_11904.json","bytes":31406},{"sha256":"214216ab46d954148444c82565bd450a6c3652175d7e60da0c13347416189b90","name":"step_identity_11904.json","bytes":2088},{"sha256":"241c3ff1925663050087ca7c507dd52bd244fe61e629178e26ca4adcc1861aaa","name":"candidates_11904.py","bytes":9104},{"sha256":"2f54eb82b1168e06a519b1fc3c14a5a8769288a26f0e2bbf19b9978d238b749d","name":"check_11904.py","bytes":7213},{"sha256":"385b8000dfbdc9e7ef28eec7b1c0d759de4dc7c891b9f4d33b09c52378ba86ca","name":"check_11904.corrupt.out","bytes":1072},{"sha256":"3beaffe3a062e1f1b59d77d0d10aa9aacbfbe536d64d5db019b6b9fce01d216b","name":"transcript_11904.md","bytes":4715},{"sha256":"5c1a9f78d2bd311296a1e50b1bc504ed0092f01bd579c7ab592009a10ca454bd","name":"check_11904.out","bytes":1034},{"sha256":"84b7975c9ce7c5caf9529c71b14f48aa31fcb6724ebe8ab6018b7822bd6572be","name":"api.py","bytes":1519},{"sha256":"b078ca90414376220c0e71b033897f4881c2c200d8d73bbe4bae6e64c90b6ae6","name":"verify_step_11904.py","bytes":4218},{"sha256":"cc7d53314a84a5e9a75c731d4f94919d3a14babcdd30f3278f3310d5ef2f344d","name":"report_11904.md","bytes":9733},{"sha256":"cd779078c68af0d842d7cede996b350da9d3b02c41238131471320513c11dd40","name":"evidence_11904.md","bytes":3803},{"sha256":"e943bf1facd00b44d4d6910413178aa7e81b120a14a3e8357454e0d64179fd9e","name":"recipe_11904.md","bytes":3059}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"cc7d53314a84a5e9a75c731d4f94919d3a14babcdd30f3278f3310d5ef2f344d","next_step_sha256":"7054f45757f91f42b734407b5e7d2e536ca8ab748d7edf5128b5ba5195e51900","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}