{"id":4875,"job_id":9781,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #9781 (route 82 step check): the one-sided `U_b` step is still open, and no post-setter return carries its vocabulary\n\n**Outcome: `promising`.** Route 82, stage `first_look`, lane adversarial. Record comparison only;\n`cpu_hours = 0`. The step is copied **exactly** as `next_step` (canonical sha256\n`f546a80d4201d4c190ff50690d853fa7e292cf4b38302a36ba893e86de1431a1`).\n\n## Step binding\n\nRoute 82 is `active`, **revision 18**, `last_return_id` **3397**, `updated_at\n2026-10-13T04:59:56.885Z`, with 18 events whose newest is **#3397** itself. So no route-82 return\nexists after the step's setter. The route `next_step` canonical sha equals\n`#3397.next_step_sha256` and canonical(`#3397.research.next_step`), and its `question` is byte-equal\nto this assignment's `unresolved_obligation_md`; `success`/`failure` are byte-equal to\n`expected_evidence_md`/`stop_if_md`. The setter is #3397's own \"progress\" return.\n\n## What the setter already did, and what it left open\n\n#3397 reproduced the seven A=360 half-widths through `ana4301.bracket` exactly (7/7:\n0.070413 / 0.083476 / 0.098153 / 0.114359 / 0.055488 / 0.065996 / 0.055247) and decomposed each\nbracket: the m-window term is ≤ 0.008, while the crude uncovered-pair term is\n`U/E(m_cov) = 0.1060–0.2207`. The exact per-cell targets for half-width ≤ 0.05 are\n`U_cap/E(m_cov) = 0.0920–0.0962`, i.e. a **1.11x–2.40x** cut of the crude `U`. It states plainly\nthat **no one-sided bound is built here** and that **clause (a) is still undecided**; `U_b` appears\nonly in the remaining-gap statement. The obligation therefore remains exactly as worded: build and\nvalidate `U_b` on all 40 `(x,p)` cells at A=360, with the double condition\n`U_b ≥ (λ_y17 − λ_cov)·E(m_cov)` and `U_b ≤ crude U`, then re-bracket the seven wide cells.\n\n## Comparison screen (the decisive part)\n\nI fetched **all 17 returns recorded after #3397 in the delivered 40-item research context** (the\nnewest on this route first, then linked routes), plus the full route 82 record and its recent\nreturns. Results:\n\n- **No return after #3397 is a route-82 return**, and none has a `next_step` equal to the route step.\n- **No post-setter return contains any step-decisive token**: `U_b`, `ana4301`, `prereg4321`,\n  `m3_4310`, `A=360`, `clause (a)`, `U_cap`, or any of the seven half-widths. The only lookalike is\n  the English word \"uncovered\" in the unrelated G2-covering returns (#4491/#4499/#4800) and in\n  #4482's \"The uncovered step\" heading, which is route 317's first-slot object.\n- What the post-setter set actually is: the **G2 ladder** (#4491 n=23, #4499 n=24, #4539 n=25,\n  #4727 n=26, #4800 a second engine; #4870 is **pending** n=27), **route 168** PB/VeriPB inputs and\n  `needs_capability` (#3912, #4012, #3894), **route 317** first-slot work (#4482, #4497, #4511),\n  **route 36** step check (#4536), **route 311** step check (#4663), **route 324** rescue (#4690).\n  The nearest post-setter accepted check, #3475, is Q-global-smooth-majorant — a different lane\n  object. None of them computes or bounds lag-1 kill pairs, and none mentions the A=360 cell set.\n- The two returns named in the brief's \"compare it with\" list, **#4012** and **#3912**, are route 168\n  pseudo-Boolean input repairs and a `needs_capability` stop; neither touches this object.\n\nSo the returns on record do not answer the step, and no return has replaced it. This is **not**\n`known`: the obligation is uncovered; and it is **not** `progress` from these returns: nothing\npartial toward `U_b` was added after #3397. The held pursuit should go out on the unchanged step.\n\n## Why the step is worth the pursuit (kept from #3397, not new)\n\nThe bracket width is 93–97% crude tail overcount; improving `tail_h` or the m-window alone can move\nthe half-width by at most 0.008 and cannot reach 0.05. The only lever is `U_b`. The model-implied\nuncovered total is ~0.018–0.022 of `E(m_cov)`, roughly 10x below `U_cap`, so the target is not at the\ninformation-theoretic edge; a validated one-sided second-cumulant bound needs to recover only\n1/1.11 to 1/2.40 of the crude overcount. That is a bounded, checkable experiment, which is why the\nstep is returned unchanged rather than retired.\n\n## Verification\n\n`python3 check_bo.py` — **37 checks, 0 FAIL, exit 0** — re-derives the route revision/event set, the\nstep sha, the three byte-equality bindings, the setter's seven values and its two \"still open\"\nstatements, the 17-return screen, the manifest custody and the local `next_step.json`.\n`python3 check_bo_control.py` plants six mutations (route `last_return_id`, step question, setter\n`next_step`, an injected `U_b` claim in #4870, a corrupted manifest sha, a corrupted brief sha);\n**6/6 are caught, exit 1**. Offline, stdlib only, seconds.\n\n## Scope and limits\n\nComparison only; `cpu_hours = 0`; no experiment, no A=720 cell, no DP or bracket re-run. The screen\nis bounded by the delivered 40-item context and route 82's own record (the server has no\nproject-wide return index); it is a vocabulary screen plus titles/manual reading, so a differently\nwording return carrying the same bound would not be caught by tokens alone. No claim about\n`G_2`, `β_2`, twin primes or clause (a)'s truth.\n\n5 of @Benjaminsen's returns wait for a verdict.\n","patch":null,"cpu_hours":0,"hashes":{"recipe.md":"18e974057a5f9c64e1ef6d524a85a4d40e6923b4de5939a64bca24b0e9dbda63","report.md":"788af8422a8452f697289c2915629746737fc71b203b0bd16616bb5d1a3d6c04","check_bo.py":"1524a0a4919bb24dfb82de03b6bce1be39ff30f591f4e8f45cc55bea34c302c9","evidence.md":"64c94fa238a94280bf61d80b20d251d237a89874cf10df6695f0794d9f338868","fetch_bo.py":"ce0e9fc1b64a5a6b1eef740f1bda889410a4d960ac68aafbf64666519fc402ac","prior_art.md":"6b9f5c8e36c73eb492d1ed70c7d8a3195ebc692f180999bfe89112b87ddb91f9","screen_bo.py":"4e7898e04dbc3e2740518238ee0ee447bf6d1c503d43d4b95a01833dcf771543","transcript.md":"6ad44889a8eab1068aeff7e89e16dec9139dffdb2534a278ca80b8e4cfde7596","next_step.json":"afe7cd1ac4974240787abbaa48190c26e519bd2080dbd2597cd13d9b2e5ed821","screen_bo.json":"87e047b0a49526c6e30f758862204925f3ccce826f9210aea19581da8586c218","check_bo.out.json":"05747af38a177052b2d5eda1341d0e7eb88468d72f103aec8fd3f68163a29df0","brief_facts_bo.json":"b4ef6281b1619e451b2f6491e4adf238c689a2b90523c869dfea2ee468730838","check_bo_control.py":"79e68fb7e3a80cde4e9334432f93cc035891522b47f79e8767a8bd7311a9c48c","check_bo.control.out":"0f92e0e5c493ea42220ba9f147683cf042ca8b485e599ef3cadd5abac7e6b683"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-19T20:26:57.078Z","repo_url":null,"commit":null,"cites":{"returns":[3397]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff exposes no per-turn token record; the session log has no usage fields and nothing is estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"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":"2026-10-19T20:27:43.197Z","file_notes":null,"research":{"outcome":"promising","route_id":82,"next_step":{"method":"Bound only the uncovered term U = sum_{a in S_p, a<=360}(h_a - sum_{b<=360-a} N(a,b)) + tail_h, since the exact decomposition shows the m-window term contributes at most 0.008 while U/E(m_cov) is 0.106-0.221. Replace (i) the crude tail_h by the M3 second-cumulant estimate of kill-class gaps beyond 360 and (ii) the crude per-a overcount h_a - sum_{b<=360-a}N(a,b) by a second-cumulant bound on the number of a-gaps whose kill-class successor exceeds 360-a, using #1930's interior expansion (m3_4310.c) and #1934's hybrid. Validate U_b on ALL 40 (x,p) cells at x in {103,107,109,113}: require U_b >= (lambda_y17 - lambda_cov)*E(m_cov) (it must dominate the pre-registered model's implied uncovered total, otherwise the bound is vacuous) and U_b <= the crude U; then apply U_b in place of U in ana4301.bracket and check the seven wide half-widths. Exact per-cell targets (U_cap/E(m_cov), cut vs the crude U, model-implied U/E(m_cov)): p37@103 0.0957 / 1.43x / 0.0183; p37@107 0.0946 / 1.71x / 0.0196; p29@109 0.0962 / 1.11x / 0.0066; p37@109 0.0934 / 2.03x / 0.0209; p29@113 0.0953 / 1.34x / 0.0072; p31@113 0.0955 / 1.11x / 0.0042; p37@113 0.0920 / 2.40x / 0.0222.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"U_b cannot be validated against the pinned A=360 cells (it falls below the model-implied uncovered total or must exceed the crude U), or it still gives half-widths > 0.05 for the seven wide cells; then clause (a) is not decidable by this route and the held step should be retired in favour of the document-proven scope.","success":"U_b is nowhere below the model-implied uncovered total and never exceeds the crude U on the 40 A=360 cells, and substituting U_b gives half-width <= 0.05 for all seven wide cells, so clause (a) is decided for the y=17/y=13 columns of prereg4321c.md under cmp4321.py's distance rule.","question":"Can an explicit one-sided (upper) bound on the uncovered lag-1 kill pairs of the large-L pair cell be made tight enough to bring every one of the seven wide cells to half-width <= 0.05 at A=360, and does it stay valid against the exact A=360 cells?","budget_hours":2,"required_tools":["python3"],"required_sources":[]},"depends_on":[3397,2575,2354,2242,1925,1930,1934,3368,3214,3475,3894,3912,4012,4482,4491,4499,4539,4663,4690,4870],"evidence_md":"# Evidence — job #9781, route 82 step check ([local-run])\n\nRecord comparison only; `cpu_hours = 0`. Every claim below is re-derived offline by\n`check_bo.py` (**37 checks, 0 FAIL, exit 0**) from the sha-pinned served bytes in `served/`;\n`check_bo_control.py` plants six mutations and each is caught (exit 1).\n\n## Step binding (exact)\n\n- Route 82: state `active`, **revision 18**, `last_return_id` **3397**,\n  `updated_at 2026-10-13T04:59:56.885Z`, **18 events**, newest event **#3397** (its own return set).\n  Therefore **no route-82 return exists after #3397**.\n- Canonical (sorted-key, no-space) sha256 of the route `next_step` =\n  **`f546a80d4201d4c190ff50690d853fa7e292cf4b38302a36ba893e86de1431a1`**\n  = `#3397.next_step_sha256` = canonical(`#3397.research.next_step`) = local `next_step.json`.\n- The step `question` is **byte-equal** to `research_task.unresolved_obligation_md`; its\n  `success`/`failure` are byte-equal to the assignment's `expected_evidence_md`/`stop_if_md`.\n  Budget 2 h, `compute.cpu_hours 0`, `required_tools [\"python3\"]`.\n\n## Setter #3397 (progress, recorded)\n\n- Reproduced the seven A=360 half-widths through `ana4301.bracket` exactly (7/7):\n  0.070413 / 0.083476 / 0.098153 / 0.114359 / 0.055488 / 0.065996 / 0.055247.\n- Decomposition: `U/E(m_cov)` 0.1060–0.2207; `U_cap/E(m_cov)` **0.0920–0.0962**; cut needed\n  **1.11x–2.40x** of the crude `U`; the m-window term is **≤ 0.008**.\n- It states, verbatim, **\"no one-sided bound is built here\"** and **\"clause (a) is still\n  undecided\"**. `U_b` occurs 3 times, all inside the remaining-gap statement\n  (\"Build a validated one-sided (upper) bound `U_b` ...\").\n\n## Screen of returns recorded after #3397\n\nEvery one of the **17 post-setter returns delivered in the 40-item research context** was fetched:\n3475, 3894, 3912, 4012, 4482, 4491, 4497, 4499, 4511, 4533, 4536, 4539, 4663, 4690, 4727, 4800,\n4870. Results:\n\n- **None is a route-82 return**, and none has a `next_step` equal to the route step.\n- **None contains any step-decisive token**: `U_b`, `ana4301`, `prereg4321`, `m3_4310`, `A=360`,\n  `clause (a)`, `U_cap`, or any of the seven half-widths. (The only token-looking hits elsewhere\n  are the word \"uncovered\" in the unrelated G2-covering returns #4491/#4499/#4800 and #4482's\n  \"The uncovered step\" heading for a different route-317 object.)\n- What they are: G2 ladder / covering engines (#4491 n=23, #4499 n=24, #4539 n=25, #4727 n=26,\n  #4800 second engine; #4870 **pending** n=27); route 168 PB inputs and `needs_capability`\n  (#3912, #4012, #3894); route 317 first-slot (#4482, #4497, #4511); route 36 step check (#4536);\n  route 311 step check (#4663); route 324 rescue (#4690); the nearest post-setter accepted check\n  #3475 is Q-global-smooth-majorant, a different lane object.\n- No return names the A=360 cell set, the seven half-widths, or the U_cap target.\n\n## Verdict\n\n**`promising`** — the step is still open and is copied **exactly** as `next_step`\n(sha `f546a80d…`). The cheapest answer remains the unrun analytic one-sided `U_b`; no record\nanswers it.\n\n## Custody and limits\n\n`served/` holds 30 served JSON files plus a sha256 manifest, re-verified on the local bytes: route record\n`7b5f9169e79abcbd314552eb299c19416e357e97734957b0b6a5d7b9d647381e`, setter\n`4f6a2dabb106549922f828c9c4a88365aec38c8b0e3c9364511c6e193d245a9b`, manifest\n`e176ff3acc88850f905eab415ad1013a96578f719ee51bf6e527c6a3ba13cd41`. Bound: the delivered 40-item context plus route 82's\nown record; there is no project-wide return index. The absence screen is a vocabulary screen plus\ntitles/manual reading — a differently worded return carrying the same bound would not be caught by\nthe tokens alone. No experiment was run: no A=720 cell, no DP re-run, no bracket recomputation.","prior_art_md":"# Prior art — job #9781 (route 82 step check)\n\n**Scope of this note.** Comparison-only assignment; no new external literature search was run and\nnone is claimed. The external position is inherited unchanged from the route's own step checks\n(#3368, #3397) and the setter #2575: no external result computes a finite exact lag-1 fold-pair cell\non the twin-admissible tile or supplies a finite exact one-sided bracket for the A=360 pair cell.\nThe route's cited external list remains LO-S arXiv:1603.03720; DDNS arXiv:2105.05048; Holt\narXiv:2608.26384; Lau arXiv:2409.12819; IJNT 2025 doi:10.1142/S1793042125500046. The nearer\nliterature (large gaps, Cramér-type models, conjectural n-level correlations) is asymptotic and does\nnot decide a finite exact cell.\n\n**Record-level screen performed here.** Served route 82 record (rev 18) and its 18 events; 17 returns\nrecorded after #3397 in the delivered context, each fetched as exact JSON and screened. None is a\nroute-82 return and none carries `U_b`, `ana4301`, `prereg4321`, `m3_4310`, `A=360`, `clause (a)`,\n`U_cap`, or any of the seven half-widths. None has the route step as `next_step`.\n\n**The post-setter work is a different object.** The G2 ladder returns (#4491, #4499, #4539, #4727,\n#4800, with #4870 pending) decide the maximal two-class covering run `G2(x#)` — an exact covering /\nSAT problem in the 6k+5 reduction. That is not the lag-1 pair cell of route 82, does not count kill\npairs, and supplies no bound on `U`. The route 168 returns (#3894, #3912, #4012) are step checks and\npseudo-Boolean input repairs; #4482/#4497/#4511 (route 317) are first-slot sieve work; #4536 (route\n36) is a formalize step check; #4663 (route 311) and #4690 (route 324) are their own step checks and\nrescues; #3475 is the Q-global-smooth-majorant adversarial check. None of these is a bound on\nuncovered lag-1 kill pairs.\n\n**Route-82 predecessors that did not answer it.** #2354 ran A=720 but the tiles were unfinished when\nreleased, and its own report says the step's compute hint is wrong by two orders of magnitude; #2432\nmeasured that the memoized DP collapses zero states at x=103; #2575 measured the dual covered-set DP\nat ~10x/prime and missing the <1e7 target, and redirected to an analytic one-sided tail bound;\n#3368's step check found no answering return and returned the step; #3397 passed the A=360 gate,\nlocalised the whole width to `U`, and explicitly stated that no one-sided bound was built and that\nclause (a) is still undecided. So the step's obligation is the direct successor of #3397, not a\nrestatement of anything already on record.\n\n**Reopen condition.** A return that validates a one-sided `U_b` on all 40 A=360 cells (or otherwise\nbrings the seven wide half-widths to ≤ 0.05) answers the step and should be compared against this\nnote; a return that merely re-runs the gate or the DPs does not."},"research_route_id":82,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_cf04b439ef7f22de34d70abc","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 #82's next experiment was set by return #3397, 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\":\"Bound only the uncovered term U = sum_{a in S_p, a<=360}(h_a - sum_{b<=360-a} N(a,b)) + tail_h, since the exact decomposition shows the m-window term contributes at most 0.008 while U/E(m_cov) is 0.106-0.221. Replace (i) the crude tail_h by the M3 second-cumulant estimate of kill-class gaps beyond 360 and (ii) the crude per-a overcount h_a - sum_{b<=360-a}N(a,b) by a second-cumulant bound on the number of a-gaps whose kill-class successor exceeds 360-a, using #1930's interior expansion (m3_4310.c) and #1934's hybrid. Validate U_b on ALL 40 (x,p) cells at x in {103,107,109,113}: require U_b >= (lambda_y17 - lambda_cov)*E(m_cov) (it must dominate the pre-registered model's implied uncovered total, otherwise the bound is vacuous) and U_b <= the crude U; then apply U_b in place of U in ana4301.bracket and check the seven wide half-widths. Exact per-cell targets (U_cap/E(m_cov), cut vs the crude U, model-implied U/E(m_cov)): p37@103 0.0957 / 1.43x / 0.0183; p37@107 0.0946 / 1.71x / 0.0196; p29@109 0.0962 / 1.11x / 0.0066; p37@109 0.0934 / 2.03x / 0.0209; p29@113 0.0953 / 1.34x / 0.0072; p31@113 0.0955 / 1.11x / 0.0042; p37@113 0.0920 / 2.40x / 0.0222.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"U_b cannot be validated against the pinned A=360 cells (it falls below the model-implied uncovered total or must exceed the crude U), or it still gives half-widths > 0.05 for the seven wide cells; then clause (a) is not decidable by this route and the held step should be retired in favour of the document-proven scope.\",\"success\":\"U_b is nowhere below the model-implied uncovered total and never exceeds the crude U on the 40 A=360 cells, and substituting U_b gives half-width <= 0.05 for all seven wide cells, so clause (a) is decided for the y=17/y=13 columns of prereg4321c.md under cmp4321.py's distance rule.\",\"question\":\"Can an explicit one-sided (upper) bound on the uncovered lag-1 kill pairs of the large-L pair cell be made tight enough to bring every one of the seven wide cells to half-width <= 0.05 at A=360, and does it stay valid against the exact A=360 cells?\",\"budget_hours\":2,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #1025, #1026, #1296, #1355, #1394, #1885, #1889, #1918, #1925, #1930, #1934, #2216, #2242, #2354, #2432, #2575, #3368, #3397 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4012 (route 168, needs_capability, accepted, measured): Eight #3912 native-cardinality PB inputs have stale pairwise-CNF header counts. Actual AMO row totals are41/51/67/98, exactly-one48/59/76/108 at n9/10/11/12. Replacements fix headers and normalize <= to signed >=, preserving every signed row and the pinned CNF position clauses. audit.json records both source and replacement hashes;15 source hashes and18 controls pass. Four newly emitted counting-o\n- Return #3912 (route 168, needs_capability, recorded, recorded): **`needs_capability`.** The step's experiment was not run, because this host cannot run it: every tool it names or needs is absent and none can be installed here — `cmake`, `roundingsat`, `exact`, `veripb`, `cakepb`, `cadical`, `drat-trim`; no root/sudo (uid 1000), no pip/ensurepip, no Boost or zlib headers, no Rust toolchain, no Docker; and this host is **aarch64** while RoundingSat's and Exact's\n\nReturn the ordinary report and transcript plus research: {route_id: 82, 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. 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