{"id":3380,"job_id":7014,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 233 first-look step check — the x=17 / coordinate-subset-equality step is still OPEN (`cpu_hours` 0; no experiment run)\n\nVerdict: **promising**. Route 233's held step is still open and is copied exactly as `next_step`\n(sha256 `9934be992e09d7a51a6b0880925c2cc5c1737b11d0bf62b97963c40c1a2fdd98`); the held pursuit #5402\nthen goes out with this note. No experiment (`tau_set(17)`, a Z/510510 cover, or a product\nconstruction) was run — this is the required read-and-compare first look.\n\n## Step binding (exact)\nRoute 233 `active`, **revision 2**, `last_return_id` **2592**, `updated_at` 2026-10-09T12:53:09.575Z.\nIts **2 events are exactly its own returns {2592, 2587}**; the newest event is **#2592**, which equals\n`last_return_id` => **no route-233 return after the setter**. The step's canonical (sorted-key) sha256\nis `9934be…fd98` == route `next_step` == **#2592**'s `next_step_sha256` == `#2592.research.next_step`\n== the brief's inline \"The step:\" object == `work/step_ai.json`. The assignment's\n`unresolved_obligation_md` is byte-equal to the step `question`.\n\n## The setter leaves it open\n**#2592** (route 233, `explore`, `accepted`, rung `proven`; trusted review #697) pins `tau_set(11)=36`\nand `tau_set(13)=54` and the coordinate-subset bound **attained at x = 5, 7, 11, 13**. Its own report\nstates the limit verbatim: \"Only four rungs; tightness beyond `x = 13` is a conjecture. The projection\ninequality supplies only `>=`; a cover-construction (CRT/product assembly) would be needed to prove\nequality in general.\" Its `next_step` *is* the held step: extend to x=17 (Z/510510) and attack the\nproduct-assembly question. **#2587** (route 233, `proposed`, rung `recorded`) already tabulates an x=17\n*best lower bound* **51.333** from exact sub-product `kappa_C`'s — a lower bound, not a pin, and it\ncarries no x=17 cover and no assembly construction.\n\n## The returns on record do not answer it\n- No route-233 return exists after #2592 (route `events`/`last_return_id`, the authoritative membership\n  test).\n- **Lane SOTA `dir-558`** (served `GET /state/dir-558`), version 2 (2026-10-12, wrote #3208), current\n  through **#3181**, `accepted_since` 1, `unverified` empty: its verified line still reads \"the\n  coordinate-subset bound ATTAINED at x = 5, 7, 11, 13 (#2592)\" — **no `tau_set(17)`, no `510510`,\n  no product construction**. If the step had been answered on the lane it would appear here.\n- The newest returns checked carry **0** step tokens: **#3181** (route 198, accepted), **#3269** (route\n  230, the one accepted since; 0 `tau_set(17)`/`510510`/assembly), **#3208** (the SOTA upkeep return).\n- **Bounded probe** `work/probe_ai.jsonl`: 20 journaled `GET /return/<id>` for ids **2593–2612** — **0**\n  route-233 returns, **0** step tokens. Bounded because this server answers each `GET /return/<id>` in\n  ~5 s; the route record's `last_return_id`/`events` is the decisive membership test.\n\n## Why not `known` / `progress`\n`known` needs a named return that settles the step — an exact/verified `tau_set(17)`, a Z/510510 cover\nmeeting `max_C kappa_C/eps`, or a product-assembly construction. None exists. `progress` needs a new\npartial answer that replaces the step; the newest route position is the setter itself (#2592), and\n#2587's 51.333 predates it without closing it. So the brief's `promising` clause applies: route 233\nhands out the step unchanged with this note, and these returns never hold it again.\n\nChecks: offline checker `check_ai.py` **32 checks / 0 FAIL, exit 0**; `--corrupt` **6 FAIL, exit 1**.\n","patch":null,"cpu_hours":0,"hashes":{"check_ai.py":"1a0c1d087ad66753bebe71c0d7cd2bf56ff80ffbaacfc7219d4ecef7f88c3c45","fetch_ai.py":"54ba013a9e200bfbb49d2ba4a499b50d7c3b61ed123fec2eb10b7281525413cc","probe_ai.py":"116c00094fd81ca96e8444ca1a236d9da735aaacdd2a123d33311b0b41652d97","check_ai.out":"d3d02034da92c3b73cf6c460a25e600ca9bbb7c0ed0c230e22844b7db3a41862","recipe_ai.md":"f3c94d2ee17157e7f062930fe4717aca93fb84ea5a6f385a55add1115b4df128","report_ai.md":"d2ad15b62f91c7ce0a969b2381e458dd1deed3e0a49af3ed2d54242863628499","step_ai.json":"4b12bb89038f5ba56d2275cba5be4baeb45c1df7034a9cfec4822d4a37ea2b37","analyze_ai.py":"4856c4ca2699c9842b709937552e2a0e68a628929eee956e0dd0f82f006adfe0","facts_ai.json":"147f5cbbd34541c69fa385f8135d17a8bdbf09ed8e5447c5f82245f658e20232","evidence_ai.md":"500c4c7416a78e4214c0aef6b9329acfaf8b1131505e14dd887344606dfcde0a","probe_ai.jsonl":"3449254b1173f232166a136c67f24908ea416cbed455c41ef25a9e4564caf0eb","prior_art_ai.md":"4440aef257a559b09bf19009f04057465483aad1e5c8b34c7b0cea05ec436983","transcript_ai.md":"c24856c12fd79946c7f50944c5b4a1103c4688c4366d45bc9c8502d0871df919","next_step_ai.json":"4b12bb89038f5ba56d2275cba5be4baeb45c1df7034a9cfec4822d4a37ea2b37","check_ai.control.out":"c2464b00740a89855f81a52b94af3b456c315a4356aa33f29511603c41ee9b6f","served-return-2587.json":"3187bb4768a3a7a4eff8816c37f3038b11ade680ed601e0c3e59a91e41dbd32b","served-return-2592.json":"3bb43e8f093d8002e1f360b02d7c951ed11d025c0b3f35ed0aafed8d99577422","served-return-3181.json":"37f8fa35a80167c62dbb5eb9c8e0dc5d35655188bf1890b55ec2b336c146d8c8","served-return-3208.json":"9d13a6932c4e44c52ec9818ca90f8f86c9b2113945442df1283a7786243ee4a7","served-return-3269.json":"5b1ffdbd87ebdf516676d227e25527de3ef71f130288e77b30f57710c35a2db1","served-state-dir-558.json":"ccc52a977f9bf47f52b0202c12baa3ae8089f00494dfa9575f6e271f827ef495","served-research-routes-233.json":"624e5831b0b61c85c7f8b2e59eadb495080b762f9998861a295198a80d42a845"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-13T03:49:33.440Z","repo_url":null,"commit":null,"cites":{"files":["d2ad15b62f91c7ce0a969b2381e458dd1deed3e0a49af3ed2d54242863628499","500c4c7416a78e4214c0aef6b9329acfaf8b1131505e14dd887344606dfcde0a","4440aef257a559b09bf19009f04057465483aad1e5c8b34c7b0cea05ec436983","f3c94d2ee17157e7f062930fe4717aca93fb84ea5a6f385a55add1115b4df128","c24856c12fd79946c7f50944c5b4a1103c4688c4366d45bc9c8502d0871df919","4b12bb89038f5ba56d2275cba5be4baeb45c1df7034a9cfec4822d4a37ea2b37","4b12bb89038f5ba56d2275cba5be4baeb45c1df7034a9cfec4822d4a37ea2b37","1a0c1d087ad66753bebe71c0d7cd2bf56ff80ffbaacfc7219d4ecef7f88c3c45","d3d02034da92c3b73cf6c460a25e600ca9bbb7c0ed0c230e22844b7db3a41862","c2464b00740a89855f81a52b94af3b456c315a4356aa33f29511603c41ee9b6f","54ba013a9e200bfbb49d2ba4a499b50d7c3b61ed123fec2eb10b7281525413cc","4856c4ca2699c9842b709937552e2a0e68a628929eee956e0dd0f82f006adfe0","116c00094fd81ca96e8444ca1a236d9da735aaacdd2a123d33311b0b41652d97","3449254b1173f232166a136c67f24908ea416cbed455c41ef25a9e4564caf0eb","147f5cbbd34541c69fa385f8135d17a8bdbf09ed8e5447c5f82245f658e20232","624e5831b0b61c85c7f8b2e59eadb495080b762f9998861a295198a80d42a845","3bb43e8f093d8002e1f360b02d7c951ed11d025c0b3f35ed0aafed8d99577422","3187bb4768a3a7a4eff8816c37f3038b11ade680ed601e0c3e59a91e41dbd32b","ccc52a977f9bf47f52b0202c12baa3ae8089f00494dfa9575f6e271f827ef495","37f8fa35a80167c62dbb5eb9c8e0dc5d35655188bf1890b55ec2b336c146d8c8","5b1ffdbd87ebdf516676d227e25527de3ef71f130288e77b30f57710c35a2db1","9d13a6932c4e44c52ec9818ca90f8f86c9b2113945442df1283a7786243ee4a7"],"handles":[],"returns":[2592,2587],"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":null,"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":"promising","route_id":233,"next_step":{"method":"Extend the exact ladder to x=17 (Z/510510). Pin every sub-product kappa_C exactly by the projection lower bound against a verified cover (the LB is now a running maximum over sub-sets; greedy is not reliable, so keep explicit covers for the sub-products where greedy overshoots), then, if max_C kappa_C/eps meets a constructed cover of Z/17#, tau_set(17) is pinned. In parallel, attack the structural question: the projection bound is an equality iff an optimal full cover can be assembled from optimal coordinate covers, so search for (or rule out) a CRT/product assembly construction.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The x=17 gap stays open and no product construction is found; tightness then remains a four-rung observation and the decisive experiment is a proof that no product construction can be optimal.","success":"max_C == cover size at x=17 (pins tau_set(17) and continues the tightness pattern), or an explicit product construction that builds a full optimal cover from coordinate covers at general x.","question":"Is kappa(D,G_x) = max_C kappa(D_C,G_C) for every x (the coordinate-subset family attains the full covering multiplicity), or does the product strictly exceed it at some rung?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[2592,2587],"evidence_md":"# Evidence — route 233 first look: the x=17 / coordinate-subset-equality step is still open\n\n## Object and conventions (unchanged from the route record; re-read, not recomputed)\n`G_x = Z/x#`, `D_x = prod_{p<=x} S_p` with `S_p = Z_p \\ {0, p-2}`, `eps_x = |D_x|/|G_x|`,\n`kappa(S,G) = tau(S,G)|S|/|G|`, `tau_set(x) = tau(D_x,G_x)`, `G_2(x#)` = max gap of `D_x`. Source:\nroute 233 `prior_art_md` / corpus `history/staging/import-vc-nets.md` §4.1.\n\n## Binding (server records fetched read-only this run)\n- `GET /research-routes/233`: `state: active`, `revision: 2`, `last_return_id: 2592`,\n  `updated_at 2026-10-09T12:53:09.575Z`, `events = [2592, 2587]`, `lane_id 32`, `origin_return_id 2587`.\n  Newest event == `last_return_id` => no route-233 return after the setter.\n- Held step canonical sha256 (sorted keys, compact): `9934be992e09d7a51a6b0880925c2cc5c1737b11d0bf62b97963c40c1a2fdd98`.\n  Equal to route `next_step` sha, to `#2592.next_step_sha256`, and to the brief's inline step object.\n- `#2592`: route 233, type `explore`, status `accepted`, `final_rung proven`, trusted review #697;\n  its `research.next_step` == the held step (it is the setter).\n- Assignment `research_task.unresolved_obligation_md` == the step `question` (byte-equal).\n\n## The setter's own limits (quoted from #2592's evidence)\n\"At every computable rung the max_C bound is attained: `kappa(D,G_x) = max_C kappa(D_C,G_C)` exactly\nfor `x = 5,7,11,13`\" … \"Only four rungs; tightness beyond `x = 13` is a conjecture. The projection\ninequality supplies only `>=`; a cover-construction (CRT/product assembly) would be needed to prove\nequality in general.\" The step asks for exactly those two missing items (x=17 pin, or the assembly).\n\n## #2587 on record (x=17 lower bound, not a pin)\n#2587's ladder row: `x=17`: `1/eps 22.919 / best LB 51.333` (from exact sub-product `kappa_C`'s), with\nno x=17 cover meeting the bound and no assembly construction. So the step's success clause is unmet.\n\n## The lane has not moved the step\n`GET /state/dir-558` (SOTA), `version 2`, `through_return_id 3181`, `accepted_since 1`,\n`unverified: []`, updated 2026-10-12 by #3208. Its verified line: \"the coordinate-subset bound\nATTAINED at x = 5, 7, 11, 13 (#2592)\". No `tau_set(17)`, no `510510`, no product-assembly entry.\nNewest returns checked by token scan: #3181 (route 198), #3269 (route 230; the one accepted since),\n#3208 (SOTA upkeep) — all **0** for `tau_set(17)`/`510510`/assembly. #3268, #3270 are 404.\n\n## Bounded probe (bounded on purpose)\n20 sequential journaled `GET /return/<id>` for **2593–2612** (`probe_ai.jsonl`): **0** route-233\nreturns, **0** step-token hits. Each such GET costs ~5 s on this server, so a full sweep of\n2593–3360 (~770 ids) is ~65 min and is not what a comparison-only first look needs; the route\nrecord's `last_return_id`/`events` plus the SOTA (`through_return_id`) is the authoritative test.\n\n## Independent checker\n`check_ai.py` (stdlib, reads only this run's local records, no network): **32 checks / 0 FAIL, exit 0**;\n`--corrupt` plants a moved `last_return_id`, a `tau_set(17)` line and a probe hit => **6 FAIL, exit 1**.\n\n## Limits / scope\nComparison only: no experiment, no computation of `tau_set(17)`, no cover, no product construction,\n`cpu_hours 0`. Verdict applies only to the unchanged held step; a distinction between tightness and\nnon-tightness at x=17 is not made here, and none of #2587's or #2592's accepted finite statements are\nmodified.","prior_art_md":"# Prior-art / on-record comparison — route 233, `tau_set` at x=17 and the product-equality question\n\nThis is a step check (compare existing returns), not a novelty search. The relevant prior work is the\nroute's own record and the lane state of the art; both were read at source this run.\n\n## Owning source (already on the route record, not re-searched)\nBollobás–Janson–Riordan, *On covering by translates of a set*, RSA 38 (2011) 33–67, arXiv:0910.3815v2.\nTheir **Lemma 3.6** `max(kappa(S1),kappa(S2)) <= kappa(S1 x S2)` is the two-factor product inequality\niterated over CRT coordinates; their **Cor 3.2** is the greedy/Rogers–Stein bound `tau <= (n/k)(log k+1)`\n(not optimal here: x=11 greedy 42 vs optimum 36). The prior-art record and search are #2592/#2587's and\nwere not re-run for this comparison.\n\n## On-record positions that bear on the step\n- **#2592** (route 233, accepted/proven): `tau_set(11)=36`, `tau_set(13)=54`; coordinate-subset bound\n  **attained at x = 5, 7, 11, 13**; explicitly leaves x=17 and the general equality open.\n- **#2587** (route 233, proposed/recorded): proven `tau_set(x) >= pi(x)+1`; corpus §4.4 table reproduced;\n  x=17 **best lower bound 51.333** — no x=17 upper bound.\n- **Lane SOTA `dir-558`** v2 (through #3181, +1 accepted since): same attainment list (x ≤ 13); x=17 not\n  present; `unverified` empty. Nothing on the lane claims `tau_set(17)` or an assembly construction.\n- Recent lane returns scanned for the step's tokens (#3181, #3269, #3208) and the bounded probe\n  2593–2612: **no** `tau_set(17)`, `510510`, Z/17# cover, or \"assemble a cover from coordinate covers\".\n\n## Exact difference this first look records\nNothing new is claimed mathematically. The comparison establishes only that (i) the held step is still\nopen, (ii) the two decisive objects — a verified Z/510510 cover meeting `max_C kappa_C/eps`, or a\nCRT/product-assembly construction proving `kappa(D,G_x) = max_C kappa(D_C,G_C)` in general — are\nabsent from the record, and (iii) the newest route position is the setter itself (#2592). The route\ntherefore hands the unchanged step to a pursuit, with this note.\n\n## Access gaps / residuals\nNo literature-absence or novelty claim is made. The comparison is bounded: it relies on the route\nrecord (`events`/`last_return_id`), the served lane SOTA (`through_return_id`), the newest lane returns\nby token, and a 20-id probe; it does not exhaustively re-read every return id on the server. The\nAhlswede product-space appendix (Bielefeld preprint 74) remains an access gap recorded by #2592, not\nevidence of absence."},"research_route_id":233,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_790aaf8fc3229ab84595a29a","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,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #233's next experiment was set by return #2592, and it has waited since 2026-10-09, and the record may have moved on. 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\":\"Extend the exact ladder to x=17 (Z/510510). Pin every sub-product kappa_C exactly by the projection lower bound against a verified cover (the LB is now a running maximum over sub-sets; greedy is not reliable, so keep explicit covers for the sub-products where greedy overshoots), then, if max_C kappa_C/eps meets a constructed cover of Z/17#, tau_set(17) is pinned. In parallel, attack the structural question: the projection bound is an equality iff an optimal full cover can be assembled from optimal coordinate covers, so search for (or rule out) a CRT/product assembly construction.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The x=17 gap stays open and no product construction is found; tightness then remains a four-rung observation and the decisive experiment is a proof that no product construction can be optimal.\",\"success\":\"max_C == cover size at x=17 (pins tau_set(17) and continues the tightness pattern), or an explicit product construction that builds a full optimal cover from coordinate covers at general x.\",\"question\":\"Is kappa(D,G_x) = max_C kappa(D_C,G_C) for every x (the coordinate-subset family attains the full covering multiplicity), or does the product strictly exceed it at some rung?\",\"budget_hours\":3,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2587, #2592 (GET <project base>/return/<id>).\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. 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