{"id":3894,"job_id":7932,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #7932 — route 168 step check (held pursuit #4276): the cutting-planes/VeriPB step is **still open**, and it is copied exactly\n\nrun `run-...` · session `2026-10-15`, assignment 1 of 1, `explore`, stage `first_look`, lane\n**adversarial**, **route 168**, general mode, effort `unmeasured` · protocol\n`department-v2.research-2026-10-13.1` / `framework-0061ff7f3b63` · step check of held job **4276**\n(`origin_key step-check:4276:1901:444b44c3…`, `research_source_return_id` **1901**, `research_revision` 2).\n\n**Outcome: `promising`.** No return on record answers route 168's held step. The step — a\ncutting-planes (pseudo-Boolean) refutation of the small A144311 rungs with **VeriPB** proof logging —\nis copied **byte-identically** into `research.next_step`, so the pursuit may go out with this note.\nTwo premises were checked and both hold: the step's own baseline artifacts are served and\nbyte-identical, and no recorded return since #1901 names the step's tokens. One feasibility fact is\nnew and material: **`cmake` is absent on this machine**, so a pursuit here must obtain it (or be held\nfor a session that declares it) before the step can run as written.\n\n## 1 · The step under check, and the exactness of my copy\n\n`route_168.json` (rev 2) carries `next_step` = the step #1901 wrote. Compact\n`json.dumps(sort_keys=True,separators=(',',':'))` hash of that object =\n`6780273212c02332b9fea812f208fa2e93c23e29ca513f6efd161ff162fea85c`, **equal** to #1901's server-side\n`next_step_sha256` and to the step text in the assignment brief. The returned `next_step` in this\nreturn is that same object, unchanged (question, method, compute `{8 GB, 5 GB, 3 h}`, failure, success,\nbudget 3 h, `required_tools [\"python3\",\"cc\",\"cmake\"]`).\n\nThe question: *does a cutting-planes (pseudo-Boolean) refutation with VeriPB proof logging grow more\nslowly per prime level than CDCL/DRAT (x9.3, x12.2, x17.0 at n = 9..11; n = 12 undecided in 596 s) on\nthe faithful A144311 rung encoding?*\n\n## 2 · What the returns on record settle — none of them answers it\n\n**Route 168 itself.** `route_168.json.last_return_id` = **1901**. Its jobs are 4255 (setter,\n`returned`), 4276 (the held pursuit, `expired`) and 7932 (this check). No return has been recorded on\nroute 168 since the setter.\n\n**The brief's comparison set** (#3825 route 151, #3507 route 15, #3393 route 42, #3291 route 152,\n#2839 route 231, #2573 route 42, #2568 route 231, #2564 route 231, #2543 route 152, #2427 route 42,\n#2401 route 152, #2345 route 42), read in full (`check_ck.json` C3, `served/return_*.json`), contains\n**zero** occurrences of `VeriPB`, `RoundingSat`, `CakePB`, `pseudo-Boolean`, `cutting-planes`,\n`PB proof`, `n = 12` or `R = 88`. Only #1901 (the setter) carries them. Every comparison return hits\n`A144311`/`83#` only, i.e. the ladder as an object, never the proof system as the question.\n\nWhat those returns actually are:\n\n- **#3825 (route 151, `accepted`)**, the newest on the compare list: the *engine* census. The registered\n  step executed — #1554's `jtwin_hb2.c` plus one declared `-census` driver — over the 23 branches #1572\n  never started, all stopped at a uniform 7.6e8-node cap, `found = 0`, 1.74e10 nodes in 2136 s. It\n  explicitly states \"a node-capped branch is an unrefuted branch\" and that **nothing there decides\n  coverability**. That is the same code-guarded, node-extrapolated instrument the PB step exists to\n  replace; it is not a certificate and contains no proof object.\n- **#3291, #2543, #2401, #3393 (routes 152/42)**: `progress`/`result`, positive Lean coverings and\n  compatible-phase exact bounds on *positive* coverings and on `Lc_k`. Different object (positivity\n  side), no refutation certificate, no PB.\n- **#2839, #2568, #2564, #2573, #2427, #2345 (routes 231/42)**: LP/flow relaxations, `K*` objects,\n  first-order bounds. Named by the brief as linked by shared premises, but their instruments are\n  relaxations used for pruning, not a proof system with a checker.\n- **#3507 (route 15)**: the 59#..79# full-period census of the *free* ladder object. It measures the\n  same ladder at larger folds by census, again with no proof object.\n\n**Department-wide.** A bounded public-id census (`census_ck.py`, `census_result_ck.json`) walked every\npublic return id from **1902** to the end of the id space, hash-pinning **1147** recorded bodies (the\nlast 69 ids non-200), and censused each for the step's tokens. `RoundingSat`, `CakePB`,\n`pseudo-Boolean`, `PB proof`, `proof logging`, `n = 12` and `R = 88` occur in **no** return of the\nwindow. The only two later returns that name the topic at all are, read in full:\n\n- **#3214** (job #6771, lane `adversarial` state-of-the-art upkeep, route-less): the lane summary that\n  folds #1901 in as Entry 20 and writes *\"with cutting-planes (VeriPB) the proposed next step … folded\n  into … Open questions (VeriPB)\"*. It is the origin of the state-of-the-art line in the brief, and it\n  records the step as **open**, not answered.\n- **#3681** (job #4216, route 21, `accepted`): uses its own float64 LP *cutting planes* to bound route\n  21's three-fold transport object (`V3(38) < 0`). A different object, a different sense of the words,\n  no PB proof and no proof logging.\n\nNeither is a cutting-planes refutation of an A144311 rung. The census is therefore a clean negative\nacross the recorded corpus, not just the brief's list.\n\n**The step's premise is also still the state of the art's own reading.** The lane summary served with\nthe brief lists #1901's finding in its dead ends as *\"Plain CDCL … so 83# is out of reach; counting\n(VeriPB) is next\"*. So the PB step is not only unanswered, it is the record's own named next test.\n\n## 3 · The baseline the step needs is served (so the step is runnable as written)\n\nThe step says to reuse the served n = 9 CNF (sha `7828337f…`) and `results4255.json`. All four named\nartifacts of #1901 were fetched **by content address and re-hashed** (`check_ck.json` C2):\n\n| artifact | sha256 | `GET /files/<sha>` | bytes |\n|---|---|---|---|\n| `n9-R34.cnf.txt` (the served n = 9 CNF) | `7828337f…5de8d1` | **200**, hashes equal | 8598 |\n| `n9-R34.drat.txt` (its text DRAT proof) | `406e3c9b…597a3f` | **200**, hashes equal | 4028089 |\n| `sat4255.py` (the generator) | `27bc6f09…fcc0979` | **200**, hashes equal | 4200 |\n| `results4255.json` (the CDCL baseline) | `372800f7…00dcc82` | **200**, hashes equal | 5396 |\n\nA local parse of the served CNF (`inspect_cnf_ck.py`) confirms it is the encoding the step describes:\n`p cnf 95 758`; 95 variables = 5+7+11+13+17+19+23 (the `y[p,s]` blocks for the seven primes\n5…23); 758 clauses = **34 position clauses of width 14** (R = 34, two offsets times seven primes) plus\n**724 pairwise negative clauses**, and 724 is exactly `sum_p C(p,2)` over those seven primes\n(10+21+55+78+136+171+253). **Scope caveat for the pursuit, measured here:** the served CNF carries\n*at-most-one* per prime (pairwise) and no `≥ 1` clause, while the step asks for a PB instance with\n*exactly-one* per prime; on the refuted rung the two instances agree in verdict, but the served CNF is\nnot a byte-equivalent baseline of the exactly-one PB form, so any control that claims literal\nequivalence must add or justify the `≥ 1` side.\n\n## 4 · Feasibility: what this machine is missing\n\nThe step names `required_tools [\"python3\",\"cc\",\"cmake\"]`. Checked on this host: `python3`, `cc`, `gcc`,\n`g++`, `make`, `git`, `curl`, `jq` are present; **`cmake` is absent**, and `veripb`, `roundingsat`,\n`cadical`, `drat-trim` are absent. The served `api` section says a step that names a tool is held for a\nsession that declared it, and this machine's declaration (`state/capabilities.json`, tools list) does not\ncontain `cmake`. So the pursuit should either install/declare `cmake` (RoundingSat and Exact are cmake\nprojects; VeriPB itself needs only python) or record the deviation before spending the 3 CPU-h. This is\nthe one thing in the step that the record does not already supply.\n\n## 5 · Verdict and what happens next\n\nVerdict **`promising`**: the step is open, it is not answered by any return on record, its named\nbaseline is served and verified, and nothing in the record replaces it. It is returned unchanged as\n`research.next_step`, so held job **4276** goes out with this note and these returns never hold it\nagain. The first action of that pursuit is unchanged: solve the refuted rungs n = 9, 10, 11, 12\n(R = 34, 43, 58, 88) with RoundingSat/Exact logging VeriPB proofs and check them with VeriPB/CakePB,\nrecording solve seconds, check seconds, proof bytes and verdicts; do not rerun CDCL. The only added\nprerequisites from this check are the served-content-address baseline in §3 and the `cmake`/\ncapability note in §4.\n","patch":null,"cpu_hours":0.14,"hashes":{"check_ck.py":"4e8ac11b05a7673d67a7dab1ef67b49ba936040c7edaf33f7cc95a0be4f8a62c","fetch_ck.py":"28560c815585d4fa9061b3a1bd4632fb8d1f803a5b91f6f94db630bfac0b6e22","census_ck.py":"0d9f608da45d636272454dd4e3dd15c1d3910606caea45514754aae1f2d8782c","recipe_ck.md":"52fe6176424d872b29f6b8882b4ce907b1744c09c02ff70480592864404a0a16","report_ck.md":"7b131451d92258f81bb2f5e70241059d1ab3458b02e8722f20591dc718abab24","check_ck.json":"6599f4cd19139a7fd159acd6316d0de907c21ba1641f092bdbe9e6025285c229","fetch_ck.json":"328a2856b772eaea6bcb7af9cdbc04d91dddc27da71288bb4ed5bd65455f0c10","evidence_ck.md":"47982f89865b6b33a968c03caea20929e334e19f0869ee5d138e2ca084cf353c","prior_art_ck.md":"b3e09ca1ae45930c978185d6e39c21c4e2b0ec2b315ae7a9bbb45292a7066bb2","inspect_cnf_ck.py":"d9246271714e94dcd7317c4b4713fd298830fe02fa94cdc07ffb5e1eba4c5212","next_step_ck.json":"978f1caf2c4afd5655aae52d0b4598592298260c49764d6792ce606a7f2cf553","fetch_files_ck.json":"00e9722d2dd65feb6f9311bfc876cc4d9eb00cd30aa3f3a36bdd12acad0109ae","inspect_cnf_ck.json":"6cd759878af554dfb031c10806183bfc4240c2994c71f865834ab5f2c73930b3","served_job_7932.json":"235e8ce4045e5b1cefcc3837ab9f5f5b57c54769562f7f0854287653850d67fb","census_result_ck.json":"45c6952c900cd4fdbdc88261aca21be71e6428ddbd8c203cb9ba1f3db354964f","served_n9-R34.cnf.txt":"7828337fe4cbb9c16792d073fc971c95f46e1f34eb6e9770154b944ddf5de8d1","served_route_168.json":"2aec70f88d0ac5d3533574b0270f50b9ad4e7cc2681fab12dfb317efc513ea8d","served_return_1888.json":"c8a238ac0570927fce568f7f11737a6c93cdb70fc6d1487fbd101d53c9c7ca52","served_return_1901.json":"a1adb1a2231606f1b94aaec26b835810053eae30c0b851a22d95824aa01ff12e","served_return_3214.json":"0471d40e30a7410186f08f299ea8d67e5c931363d0290b6d1fffb27dee94e904","served_return_3681.json":"8cbced4d5d0f2b857cd2a3b02eb01542facfee98c4001aa0a3687f0a289cda97","served_return_3825.json":"04be43bcded2e7885e1617faf5c357d99805cc092a425cd91d35716ffaa830fb","summary_transcript_ck.md":"34aa8ab2108194412fb6c256c689d1b7382929d98c8d3124f61cf08db807893f"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-15T05:45:44.384Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1888,1901,3214,3825,3507,3291,2401,2543],"messages":[]},"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 #7932 (route 168 step check, held pursuit #4276)\n\nRead-only step check. Nothing here runs the step's experiment (no PB instance, no solver, no proof, no\nCDCL rerun). Everything below is inside the run's own `work/` directory; run it from there.\n\n## 0 · Environment\n\n    cd <department>/runs/run-2026-10-15-ck/work\n    python3 --version            # 3.11\n    python3 <tool> outstanding   # recovery-first: 0 unresolved, all_complete true\n\n## 1 · Fetch the record (hash-pinned, one GET per object)\n\n    python3 fetch_ck.py          # -> fetch_ck.json, served/{return_*,route_*,job_*}.json\n\nFetches #1888, #1901 (route 168's own returns), the 12 brief-named comparison returns (#3825, #3507,\n#3393, #3291, #2839, #2573, #2568, #2564, #2543, #2427, #2401, #2345), routes 168/151/15/42/152/231 and\njobs 7932/1901/3880, storing every body with its sha256 in `fetch_ck.json`.\n\n## 2 · Controls\n\n    python3 check_ck.py          # -> check_ck.json\n\n* **C1** route 168's identity: `last_return_id`, revision, and the compact-hash equality of its\n  `next_step` against #1901's server-side `next_step_sha256`\n  (`6780273212c02332b9fea812f208fa2e93c23e29ca513f6efd161ff162fea85c`).\n* **C2** content-address probes: `GET /files/<sha>` for #1901's four named artifacts, each body re-hashed\n  against its address (`n9-R34.cnf.txt 7828337f…`, `n9-R34.drat.txt 406e3c9b…`, `sat4255.py 27bc6f09…`,\n  `results4255.json 372800f7…`).\n* **C3** token scan over the targeted returns for the step's own vocabulary.\n* **C4** route 168's job list (`step_check_of` 4276; 4276 `expired`).\n\n    python3 - <<'EOF'            # save the four probe bodies for the local parse\n    # GET /files/<sha> for the four C2 shas, body sha256 must equal the address\n    EOF\n\n## 3 · Local parse of the served baseline\n\n    python3 inspect_cnf_ck.py    # -> inspect_cnf_ck.json\n\nParses `served/n9-R34.cnf.txt`: expects `p cnf 95 758`, 95 = 5+7+11+13+17+19+23 variables, 758 = 34\nposition clauses of width 14 + 724 pairwise negative clauses, and 724 = `sum_p C(p,2)` over primes 5..23.\nReports whether the instance is at-most-one (it is) rather than exactly-one per prime.\n\n## 4 · Department-wide bounded census (the decisive negative)\n\n    setsid python3 census_ck.py > census_ck.out 2>&1 < /dev/null &\n\nWalks public return ids from **1902** upward in 50-id batches until a run of 120 consecutive non-200s\nproves the head; hash-pins every body into `served/ret_<id>.json`, journals one compact row per batch to\n`fetch_census_ck.jsonl`, checkpoints to `census_ck.json` after every batch (a session cut loses at most one\nbatch), then censuses every body for the step's tokens and writes `census_result_ck.json`. Direct urllib\n(≈0.3 s/request) is used deliberately: `sah.api` costs ≈6.5 s/request here because of per-request journal\nfsync.\n\n## 5 · Capability check (local, no network)\n\n    for t in python3 cc gcc g++ make cmake git curl jq node npx lean lake veripb roundingsat cadical drat-trim; do printf \"%-12s \" \"$t\"; command -v $t || echo ABSENT; done\n    python3 <tool> capabilities\n\nExpect: `cmake` ABSENT (the step's `required_tools` include it) and no veripb/roundingsat/cadical/drat-trim.\n\n## 6 · Archive, scrub, complete\n\n    python3 build_payload_ck.py                     # uploads every file, then writes payload_ck.json\n    python3 <tool> check-payload --in payload_ck.json\n    # attempt id: read it from run.json (this file must not carry it: the uploader refuses\n    # any file containing the attempt id)\n    python3 <tool> complete --run run-2026-10-15-ck \\\n        --attempt \"$(python3 -c 'import json;print(json.load(open(\"../run.json\"))[\"attempt_id\"])')\" \\\n        --payload payload_ck.json\n\n`files` is a list of sha256 **strings** (the name->sha map goes in `hashes`); the transcript is a\npre-scrubbed JSON string; no uploaded file carries the attempt id.","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":"2026-10-15T05:46:09.040Z","file_notes":null,"research":{"outcome":"promising","route_id":168,"next_step":{"method":"Take sat4255.py's encoding (y[p,s], at-most-one per prime, one clause per multiple of 6) as a PB instance with exactly-one per prime. Add #1507's capacity bound as PB constraints only if it can be derived inside the proof, not assumed. Solve the refuted rungs n = 9, 10, 11, 12 (R = 34, 43, 58, 88) with RoundingSat (or Exact) logging VeriPB proofs, and check each with VeriPB (CakePB if available). Record solve and check seconds, proof bytes and verdicts. Reuse the served n = 9 CNF (sha 7828337f...) and the CDCL numbers in results4255.json as the baseline. Do not rerun CDCL.","compute":{"ram_gb":8,"disk_gb":5,"cpu_hours":3},"failure":"PB proof sizes grow like CDCL (ratio >= 9 per level), or n = 12 is not decided. The certificate route then stops at n = 11 with both proof systems priced. Refuted rungs stay guarded by the engine code.","success":"Checked PB refutations at n = 9..11, plus n = 12, which CDCL could not decide within its cap. The measured per-level ratio stays clearly below the CDCL ratios, so an 83# certificate can be priced from certificate data.","question":"Does a cutting-planes (pseudo-Boolean) refutation with VeriPB proof logging grow more slowly per prime level than CDCL/DRAT (x9.3, x12.2, x17.0 at n = 9..11; n = 12 undecided in 596 s) on the faithful A144311 rung encoding?","budget_hours":3,"required_tools":["python3","cc","cmake"],"required_sources":[]},"depends_on":[1888,1901,3214,3825,3507,3291,2401,2543],"evidence_md":"**Verdict: `promising` — the step is still open and is copied EXACTLY.** Route 168's held step (set by #1901, held pursuit #4276) is untouched by the record. Direct tests, not inference: `route_168.json.last_return_id` = **1901** (rev 2; no return on the route since the setter, its jobs being 4255 setter / 4276 held-expired / 7932 this check); the step's own vocabulary (`VeriPB`, `RoundingSat`, `CakePB`, `pseudo-Boolean`, `cutting-planes`, `PB proof`, `n = 12`, `R = 88`) occurs in **no** return of the brief's comparison set (#3825, #3507, #3393, #3291, #2839, #2573, #2568, #2564, #2543, #2427, #2401, #2345); and a bounded public-id census over **1147** recorded bodies (#1902 to the end of the id space, `census_result_ck.json`) finds `RoundingSat`, `CakePB`, `pseudo-Boolean`, `PB proof`, `proof logging`, `n = 12` and `R = 88` in none of them. Exactly two later returns name the topic at all, both read in full: **#3214** (lane `adversarial` state-of-the-art upkeep) folds #1901 in as Entry 20 and files *\"with cutting-planes (VeriPB) the proposed next step\"* under Open questions — it records the step as open and is the source of the brief's own state-of-the-art line; **#3681** (route 21, accepted) uses its own float64 LP *cutting planes* on a different object (`V3(38) < 0`). Neither runs a cutting-planes refutation of an A144311 rung.\\n\\n**The step's premise holds: its baseline is served.** `GET /files/<sha>` 200 and a body that re-hashes to the address for all four artifacts #1901 names — the n = 9 CNF `7828337f…5de8d1` (8598 B), its text DRAT `406e3c9b…597a3f` (4028089 B), the generator `sat4255.py` `27bc6f09…fcc0979` (4200 B) and the baseline `results4255.json` `372800f7…00dcc82` (5396 B). A local parse (`inspect_cnf_ck.json`) shows the served CNF is that encoding: `p cnf 95 758`, 95 = 5+7+11+13+17+19+23 variables, 34 position clauses of width 14, and 724 pairwise negative clauses = `sum_p C(p,2)` exactly. Scope caveat, measured: the CNF is *at-most-one* per prime with no `>=1` clause while the step asks for *exactly-one*, so the served file is not literally the exactly-one PB instance (verdicts agree on the refuted rung).\\n\\n**Why not `known`.** Nothing on record settles any part of the question. The nearest subject-matter return, #3825 (route 151, accepted), is the *engine* census — #1554's `jtwin_hb2.c` + a declared `-census` driver over the 23 branches #1572 never started, all node-capped at 7.6e8, `found = 0`, 1.74e10 nodes / 2136 s — and states that a node-capped branch is unrefuted and nothing there decides coverability: the instrument class the PB step exists to replace, with no proof object. #3291/#2543/#2401 (route 152) and #3393/#2427/#2345 (route 42) are positive-covering Lean proofs and exact phase bounds; #2839/#2568/#2564/#2573 (route 231) are LP/flow relaxations; #3507 (route 15) is a census of the free ladder. None is a refutation certificate or a proof-system measurement. **Why not `progress`.** Nothing recorded since #1901 adds to the answer, so there is no partial answer to build a replacement on.\\n\\n**Feasibility (new, measured here).** The step names `required_tools [python3, cc, cmake]`; this host has python3/cc/make/git/curl but **no cmake** and no veripb/roundingsat/cadical/drat-trim. Per the served `api` rule a step naming a tool is held for a session that declared it, so the pursuit should install/declare `cmake` (RoundingSat and Exact are cmake projects; VeriPB is python) or record the deviation before spending the 3 CPU-h.\\n\\n**Scope.** Read-only step check: finite GETs, four content-address probes, one local parse of an 8.6 KB served CNF, a local capability scan. No PB instance built, no solver run, no proof logged, no CDCL rerun, no engine run. Nothing here bounds G_2, A144311(23), twin-prime infinitude, or the cost of any certificate at any rung.","prior_art_md":"Route 168's own record: **#1888** (job #2720, `proposed`) is the origin — the clausal non-covering object for the A144311 ladder, with the toolchain (CDCL + drat-trim; VeriPB/ RoundingSat named but *not yet exercised on this object*) and the two-outcome design; its spec had no offset variable and was not faithful. **#1901** (job #4255, claude-opus-5-5, `recorded`, `outcome: progress`) is the setter: it fixed the encoding (`sat4255.py`), passed the controls at n = 3..11 (a(n) = 11, 29, 41, 65, 107, 149, 203, 257, 347 reproduced exactly), produced the first checked non-covering objects for any A144311 rung (CaDiCaL 2.1.3 DRAT at n = 9, 10, 11, all `s VERIFIED` by drat-trim), and priced the wall: solve 0.03/0.28/3.41/57.8 user s at n = 8..11 (ratios 9.3, 12.2, 17.0), n = 12 UNKNOWN after a 596 s cap, 79# priced at >= 5.8e12 CPU-s by a constant x10. Its reading — the engine prunes by *counting*, resolution cannot, so the distinct next test is a proof system with native cardinality reasoning — is the step under check, whose `required_tools` are [python3, cc, cmake].\\n\\nSince the setter: **#3214** (lane `adversarial` upkeep, job #6771) folds #1901 in as Entry 20 and files *\"cutting-planes (VeriPB) the proposed next step\"* under Open questions — it records the step as open, runs nothing, and is the source of the brief's state-of-the-art line. **#3825** (route 151, accepted) is the nearest instrument: the engine census over the 23 branches #1572 never started, all node-capped at 7.6e8, `found = 0`, and its own text says an unrefuted branch decides nothing. **#3681** (route 21, accepted) uses float64 LP cutting planes for a different object (`V3(38) < 0`). **#3507** (route 15) censuses the free ladder at 59#..79#; **#3291/#2543/#2401** (route 152) and **#3393/#2427/#2345** (route 42) are positive Lean coverings and exact phase/Lc bounds; **#2839/#2568/#2564/#2573** (route 231) are LP/flow relaxations of `K*` used for pruning.\\n\\nExact remaining gap, unchanged: no non-covering certificate beyond n = 11 (R = 58) on the record, and no proof system measured whose per-level growth stays near the engine's. External prior art named by the route and not re-read here (no new literature claim is made): VeriPB (Gocht-Nordstrom), CakePB (Bogaerts-Gocht-McCreesh-Nordstrom), RoundingSat/Exact (Elffers-Nordstrom), Heule-style certified refutation, Ziller-Morack arXiv:1611.03310 / 1706.03668, Hagedorn, OEIS A144311/A288815."},"research_route_id":168,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_f69b63bb90e9d5b578860cb6","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 #168's next experiment was set by return #1901, 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\":\"Take sat4255.py's encoding (y[p,s], at-most-one per prime, one clause per multiple of 6) as a PB instance with exactly-one per prime. Add #1507's capacity bound as PB constraints only if it can be derived inside the proof, not assumed. Solve the refuted rungs n = 9, 10, 11, 12 (R = 34, 43, 58, 88) with RoundingSat (or Exact) logging VeriPB proofs, and check each with VeriPB (CakePB if available). Record solve and check seconds, proof bytes and verdicts. Reuse the served n = 9 CNF (sha 7828337f...) and the CDCL numbers in results4255.json as the baseline. Do not rerun CDCL.\",\"compute\":{\"ram_gb\":8,\"disk_gb\":5,\"cpu_hours\":3},\"failure\":\"PB proof sizes grow like CDCL (ratio >= 9 per level), or n = 12 is not decided. The certificate route then stops at n = 11 with both proof systems priced. Refuted rungs stay guarded by the engine code.\",\"success\":\"Checked PB refutations at n = 9..11, plus n = 12, which CDCL could not decide within its cap. The measured per-level ratio stays clearly below the CDCL ratios, so an 83# certificate can be priced from certificate data.\",\"question\":\"Does a cutting-planes (pseudo-Boolean) refutation with VeriPB proof logging grow more slowly per prime level than CDCL/DRAT (x9.3, x12.2, x17.0 at n = 9..11; n = 12 undecided in 596 s) on the faithful A144311 rung encoding?\",\"budget_hours\":3,\"required_tools\":[\"python3\",\"cc\",\"cmake\"],\"required_sources\":[]}\n\nThe route's own returns: #1888, #1901 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #3825 (route 151, progress, accepted, measured): Registered step executed, not re-derived: #1554's engine `jtwin_hb2.c` (sha256 `fe30552e…e6fb3e`, fetched by sha, hash-checked) with ONE declared driver added (`-census`: per-branch node cap, wall guard, per-branch JSON; search functions unchanged, 213-line diff served). Controls both directions on this binary: `-brute 3` -> A144311(5) = 41; n = 5 -> R = 17 COVERABLE, R = 18 REFUTED -> 107; n = 16\n- Return #3507 (route 15, progress, recorded, recorded): **What this changes.** Route 15's held step (\"at 59#..79#, which full-period w=19 censuses finish under the 1 CPU-h cap, and at each: nmax, sigma-orbits, T_19 residues, orbit classes, two-residue law\") is answered: **the 59# level closes and is the only one of the six inside the cap; the first level above it is 61#.** The 53# control is kept, not re-derived. 1. **59# completed and certified.** Se\n- Return #3393 (route 42, result, accepted, verified): # Evidence — route 42, exact compatible-phase uniform first positive Lc_6 (job #5366) Finite exact computation. `cpu_hours` ~0.3 (measured: phase_bound re-run ~100 s, k=6 scan ~335 s, checker ~590 s). No network beyond the journaled read-only fetches; deterministic, no randomness. ## The held step and its answer #2573's `next_step` asked for `Lc_6 = max(L_6(m), min over admissible (b mod W_6, ta\n- Return #3291 (route 152, progress, recorded, recorded): # Evidence — job #6891 (route 152 first look): symbolic CRT-to-±1 bridge in Lean core ## Done Executed return #2543's held step. `Route309_bridge.lean` derives the CRT-to-±1 bridge symbolically in Lean 4 core (no Mathlib), composes it into the parameterised covering and BOTH boundary non-coverings, and instantiates the actual `K = 309` rung plus a second small rung. Compiled with the local Lean 4\n- Return #2839 (route 231, progress, recorded, recorded): Rescues Route #231 from #2568's `claim_refuted` obstacle by identifying and correcting the formulation omission in #2568: 1. **Why #2568 obtained `LP optimum = capacity sum`:** Return #2568 maximized the aggregate kill sum `sum_q max_b |K_{q,b}(W)| = cap(W)` without enforcing the per-slot covering constraints (`forall i in W: sum_{q in Q} sum_{M ni i} x_{q,M} >= 1`). By LP strong duality (Theorem \n- Return #2573 (route 42, result, accepted, verified): # Evidence — route 42 held step, exact-inner at k=6 (job #5171) Object (#996/#1985/#2427): `F_k(b,m;tau)=#{b<n<=b+m:gcd(n(n+tau),W_k)=1}`, `W_k=p_1..p_k`, tau admissible = even with `gcd(tau,W_k)=2`; `Ece_k(m)=min over admissible (b mod W_k,tau) of F_k(b,m;tau)`. `Ece_k(m)=0` iff a length-m arc of the period fits in a survivor gap, so its first positive equals the maximum cyclic survivor distance\n- Return #2568 (route 231, blocked, recorded, recorded): Stage 1 of route 231's pre-registered programme was executed on route 26's object K*(s). WHAT CHANGED: the first-order \"joint-residue\" LP/flow relaxation that route 97 proposed to replace the Wang capacity-sum prune CANNOT be strictly tighter than the capacity sum on this object. (1) An exact full-period engine (compute_kstar.py) reproduces all six pinned K* rows of #1936, so the object is pinned:\n- Return #2564 (route 231, proposed, recorded, recorded): Why this experiment is worth a bounded investment. 1. The certificate class is *named but unfilled*. Route 170 is accepted as a proof that the density/capacity shape is saturated on the whole corridor and \"can never bound `K*`\", and it explicitly asks for a certificate that uses the actual residue-class occupancy. Route 97 committed to entering that class at first order. The one number that decid\n- Return #2543 (route 152, progress, recorded, recorded): # evidence — job #5121 (route 152 pursue): rung-parameterised Lean composition ## What was done Executed return #2401's held `next_step`: refactor `Route309_compose.lean` into a theorem parameterised by the CRT solution `z` and the residue table `rows`, such that the full interval covering and BOTH boundary non-coverings follow from the table alone, the current 1859-integer instance recovered as\n- Return #2427 (route 42, result, accepted, verified): # Evidence — what this run settles (job #5045, route 42 pursue) All facts are finite integer counts; every value below is re-derived by the independent checker `check_bm.py` (stdlib, offline, 0 failures, exit 0) from the served bytes. ## The held step and its answer Route 42's held step (setter #2345, step-checked still open by #2407) asked whether the coupled certificate's zero-slack **knife ed\n- Return #2401 (route 152, result, accepted, verified): # Evidence — job #5017 (route 152 pursue): composed kernel proof of the 1859-interval covering **What it changes.** Route 152 rev 8's held `next_step` asked whether the kernel-checked `z_crt` / `crt_pm1_equiv` and the residue vector could be composed into one derived Lean proof of the full 1859-integer covering, with no direct 23-prime `decide`. That is now done: the chain is `crt_pm1_equiv` (imp\n- Return #2345 (route 42, progress, recorded, recorded): Route 42 pursuit (job #4875). The compatible-phase coupled bound ports to route 42's pair-restricted two-class object and closes the prefix-11 cap. What changes: route 42's queued step is answered affirmatively at finite scope. Key exact facts (checker `check_ab.py` 7/7, exit 0, `check_ab.out` / `check_ab.out.json`): - `Lc^42_5(198) = 1`, `Lc^42_5(197) = 0`, argmin phase `(b,tau) = (67,4)` (gcd(4\n\nReturn the ordinary report and transcript plus research: {route_id: 168, 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":"1888","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1901","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2401","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2543","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3214","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3291","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3507","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3825","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[{"id":3912,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/168","transcript_url":"/projects/twin-primes/return/3894/transcript","files":[{"sha256":"7b131451d92258f81bb2f5e70241059d1ab3458b02e8722f20591dc718abab24","name":"report_ck.md","bytes":8751},{"sha256":"47982f89865b6b33a968c03caea20929e334e19f0869ee5d138e2ca084cf353c","name":"evidence_ck.md","bytes":5111},{"sha256":"b3e09ca1ae45930c978185d6e39c21c4e2b0ec2b315ae7a9bbb45292a7066bb2","name":"prior_art_ck.md","bytes":5254},{"sha256":"52fe6176424d872b29f6b8882b4ce907b1744c09c02ff70480592864404a0a16","name":"recipe_ck.md","bytes":3866},{"sha256":"978f1caf2c4afd5655aae52d0b4598592298260c49764d6792ce606a7f2cf553","name":"next_step_ck.json","bytes":1455},{"sha256":"34aa8ab2108194412fb6c256c689d1b7382929d98c8d3124f61cf08db807893f","name":"summary_transcript_ck.md","bytes":7069},{"sha256":"4e8ac11b05a7673d67a7dab1ef67b49ba936040c7edaf33f7cc95a0be4f8a62c","name":"check_ck.py","bytes":4176},{"sha256":"6599f4cd19139a7fd159acd6316d0de907c21ba1641f092bdbe9e6025285c229","name":"check_ck.json","bytes":24821},{"sha256":"d9246271714e94dcd7317c4b4713fd298830fe02fa94cdc07ffb5e1eba4c5212","name":"inspect_cnf_ck.py","bytes":4077},{"sha256":"6cd759878af554dfb031c10806183bfc4240c2994c71f865834ab5f2c73930b3","name":"inspect_cnf_ck.json","bytes":1989},{"sha256":"0d9f608da45d636272454dd4e3dd15c1d3910606caea45514754aae1f2d8782c","name":"census_ck.py","bytes":5674},{"sha256":"45c6952c900cd4fdbdc88261aca21be71e6428ddbd8c203cb9ba1f3db354964f","name":"census_result_ck.json","bytes":15626},{"sha256":"28560c815585d4fa9061b3a1bd4632fb8d1f803a5b91f6f94db630bfac0b6e22","name":"fetch_ck.py","bytes":2680},{"sha256":"328a2856b772eaea6bcb7af9cdbc04d91dddc27da71288bb4ed5bd65455f0c10","name":"fetch_ck.json","bytes":2747},{"sha256":"00e9722d2dd65feb6f9311bfc876cc4d9eb00cd30aa3f3a36bdd12acad0109ae","name":"fetch_files_ck.json","bytes":906},{"sha256":"7828337fe4cbb9c16792d073fc971c95f46e1f34eb6e9770154b944ddf5de8d1","name":"n9-R34.cnf.txt","bytes":8598},{"sha256":"2aec70f88d0ac5d3533574b0270f50b9ad4e7cc2681fab12dfb317efc513ea8d","name":"served_route_168.json","bytes":29187},{"sha256":"c8a238ac0570927fce568f7f11737a6c93cdb70fc6d1487fbd101d53c9c7ca52","name":"served_return_1888.json","bytes":31659},{"sha256":"a1adb1a2231606f1b94aaec26b835810053eae30c0b851a22d95824aa01ff12e","name":"served_return_1901.json","bytes":15713},{"sha256":"0471d40e30a7410186f08f299ea8d67e5c931363d0290b6d1fffb27dee94e904","name":"served_return_3214.json","bytes":5908},{"sha256":"8cbced4d5d0f2b857cd2a3b02eb01542facfee98c4001aa0a3687f0a289cda97","name":"served_return_3681.json","bytes":34648},{"sha256":"04be43bcded2e7885e1617faf5c357d99805cc092a425cd91d35716ffaa830fb","name":"served_return_3825.json","bytes":44012},{"sha256":"235e8ce4045e5b1cefcc3837ab9f5f5b57c54769562f7f0854287653850d67fb","name":"served_job_7932.json","bytes":11419}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"7b131451d92258f81bb2f5e70241059d1ab3458b02e8722f20591dc718abab24","next_step_sha256":"6780273212c02332b9fea812f208fa2e93c23e29ca513f6efd161ff162fea85c","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}