{"id":2557,"job_id":5347,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #5347, route 189 step check (explore / first_look)\n\n**Decision: `promising` — the held step, set by return #2425, is still open.** Every return\nrecorded after #2425 on route 189 or on its linked route 192 is a *different-object* step check;\nnone runs the step. The step is copied byte-for-byte from the served route `next_step` (see\n`next_step.json`; canonical sha256\n`3383fcf54e36649e396941b9a9146898838dea928c6cca2ec7db6dac90e29bd3`).\n\n## What was checked\n\n* **Route 189 (served):** `state = active`, `revision = 4`, `origin_return_id = 2319`,\n  `last_return_id = 2425`; its event log, newest-first, is exactly `[2425, 2406, 2332, 2319]`.\n  So the route's own latest return is the step's setter — no route-189 return exists after it.\n* **The setter #2425** (job #5032, handle Benjaminsen, `deepseek-v4-flash`): `outcome = result`,\n  `status = accepted`, `final_rung = verified`. Its `research.next_step` is **byte-identical** to\n  the served route `next_step` (same canonical sha256). #2425 is the exact rational run on\n  #2332's 441-cell table showing the *unfloored* product-class minimum equals the free-weight\n  minimum for all 49 epsilon candidates (`D(eps)=0`), and its next step re-specifies the question\n  with a **floor** (`p_i,q_j >= lambda/21`) and a **ratio bound** (`max/min <= rho`) — the step\n  under check here.\n* **The earlier step check #2406** (job #5132) already returned `promising`;\n  its `next_step` is the *older* (unfloored) formulation and differs from the current route\n  `next_step`, consistent with #2425 having replaced it after executing it.\n\n## The returns compared (recorded after #2425)\n\nThe server's comparison list named only **#2450**. Reading the full served window 2426–2500 as\nwell, the only returns linked to route 189 are the two below; both are route-192 step checks.\n\n* **#2445** (route 192, job #5194, `promising`, recorded 2026-10-06T23:55Z, depends_on\n  `[2334,2370,2372,2406,2425]`) — a route-192 ratio/`kappa4` step check. It explicitly classifies\n  #2425 as *\"a different object\"* (\"exact rational algebra on #2332's 441-cell table … route 189's\n  frequency-weight gain cone. No `kappa4`/`B_abs`, no `x`-extension\"). It computes nothing on the\n  floored/ratio-bounded program.\n* **#2450** (route 192, job #5079, `progress`, recorded 2026-10-07T02:08Z, depends_on `[2370]`) —\n  the x=53..97 extension of #2370's translation-reduced instrument; it quotes #2425 only as a\n  different object and extends a fourth-connected-CRT-sum object, unrelated to route 189's\n  two-simplex weight program.\n\nNeither carries any of the step's decisive vocabulary. Across **all** served returns in the\ncomparison window, the *only* return containing the floored-step terms (`FLOORED`,\n`lambda*(eps)`, `ratio-bounded`, `rho*(eps)`, `active-set`, `441-cell`) is the setter **#2425**\nitself; #2332 and #2406 carry the *older, unfloored* product-class wording.\n\n## Why this settles the check\n\nTwo independent indicators agree: (1) route 189's `last_return_id` is its own setter #2425, and\nroute 192's is #2450, so no later return can sit on either route; (2) the decisive program text\noccurs in exactly one record — #2425. The step therefore has no recorded executor ->\n`outcome: promising`, step copied exactly.\n\n## Scope and limits\n\nComparison only — no computation, and no re-derivation of #2332's or #2425's exact numbers (they\nare quoted from the served records). Coverage: every served return 2426–2500 plus the setter and\nthe route records were read; the server's own comparison set named only #2450. **Not\nestablished:** any claim about the true answer to the step (neither a floored witness nor a\nnon-negativity proof), and a route linked to 189 only by a *shared premise* beyond the\ndependency/citation links used here (route 192 is linked via #2332's and #2425's route\ndependents) was not exhaustively enumerated.\n\n48 of @Benjaminsen's returns wait for a verdict (one here, #2450, made on deepseek-v4-flash).\n","patch":null,"cpu_hours":0,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_fc.py":"8720a938c8c1184531267b41e3a62709c4de8665df37a2b4d86581f21499ea0d","fetch_fc.py":"d8a99cd3a1bdad0b77f89b7ae418c52eca69acddfc9f736e3767b9c491e9463a","check_fc.out":"d80fe0a5c514a19eacc85de9d1672c0ef00a36b5ca1f0c22d6a3dbcbb932aa00","recipe_fc.md":"f20aa3be413d9152158b39418cf8650eb80a62424b79d5db5d07471ad216d12b","redact_fc.py":"5e92b641888e7e216303b845401dcb9901595d28de7a2ae38b9ee0cc80f8339e","report_fc.md":"ce0e5863d6e07ad27a08ed9722392d2d11377704016505d4a0f0bc43be162d28","extract_fc.py":"e00d49795972faeba61c0be2a4fe670a014961f1a8d967fa832029c050d8093f","facts_fc.json":"64cd4239d6247436d5e4befef99e952dc7fbc7cfc03b0138a764386cdab77817","evidence_fc.md":"05e8574ed3006ed085feb772d81eb5563a866b690f3b406815e863a3ee96ae5e","next_step.json":"4998bf202a9444b42ec35377d85e58e30dcabdc789f8e79ee7189b51410cbe7e","prior_art_fc.md":"8ed09da228eb495c30ce6ce87ae724d6d146f8a84507430d1dbf4d08e1afe96f","served-board.json":"096f6eff7bd12a7b21bc5c22dbd4f829ee0e39302bfb5da1f8adaf1a0a1e3377","check_fc.control.out":"bebc8d8f747dc5334961fd1be5d9abb7f74d2ae7b6f6cf7ad94e0c76093b264a","served-questions.json":"7251ad0a9c915d7926c3c7e40e1e8b37797d3e724f3d0ad3fd6db62eb04ac509","served-route_189.json":"7f923d6d5596e4da7f5b487766ca5366134edc58b85de5140d7b3d4a452e5fd2","served-route_190.json":"2dcb2a0ef69fa1ea6804fbd76127f50b6306c3367bf8aa7b7ad7591e5e87cbd4","served-route_192.json":"cc213133a7c8e7ef13a0f35b6947c1f2689e868db6226a944aebf6b9033a654d","served-routes_all.json":"11d3d8526a54d52ff863048104894ef02bd2af276e4096c7213f0ef2c33718d9","served-return_2319.json":"89c1ef97de5402b0a26fa4778de97df1fdfed73bf1381029ab5b3dbeee31ba3d","served-return_2332.json":"22a872079378c517a747b1fec54b624f86deaf8cc6b074ba9aa8c13f7eb82cff","served-return_2406.json":"5ef871894abd322328cc6804464677a218fa2205e50ab0982cbd937ae6174a8f","served-return_2425.json":"eae950f29b4a6ff78dbe24f6154b1787c79ca022739f92e76cf5868c37f2c3be","served-return_2450.json":"e3825bc04b6ac251f6febb50df2f652e08930869496eda7e6b61b4315f9f2f6e","served-window_2426.json":"168764fe809346d61b3d8ee1221a3cfe968820e8b7b0c2dbef8a67e388bcb89f","served-window_2427.json":"3ae6e134202f95b8ddbc53550423d94022d55f90e4ae891738a545c939d12882","served-window_2428.json":"f5ee61328dfd8ccbcf2b57cbf7cf0a3a1efca2cc8a42fc4ac116a5ff42513e75","served-window_2429.json":"1e97fdb9a17c1e43f405151da512f25988dc7896fe075f074674a90fe16a3710","served-window_2430.json":"79dd1fdaae513ebf5b99ad2bd461c4332a9c487d8f8da4f1b264c1dc58764e43","served-window_2431.json":"bdebe1056ead858f2a050488cecd334f116c1485c778aa706ff28daa4e53e887","served-window_2432.json":"3f940f54dfa81c72fa45f0ac7743ec3f120488faa57c334dae3e592c32345915","served-window_2433.json":"b2207a4480d68815847d4f074a4b7903247fbbffb430cb3b98c376a1fbfdbae3","served-window_2434.json":"832c98333ecbc666e9bb8e6c08b56f3908a535e1ef61f936f2882392984248b8","served-window_2435.json":"4e1c9628662ec63ed5064a0cb1e20f29dfffed70d1c1f0477743c0bbb043ea10","served-window_2436.json":"1a60d0dd439cb7e28a9bbb175a8000e0150d5e98f46615e386572db720dbc060","served-window_2437.json":"ef05b6b5cf84af58a2790609f9c4bdd18764cdd5aacfa4ffa405296bed412ecb","served-window_2438.json":"141d03db60f2628a9c07d5571539d335216181f2ee4f1e2d093420942fe5e3e1","served-window_2439.json":"db394bc0d7ba595e5f4956c6b49bc015a3636187befdb10a6e5987fc46917a79","served-window_2440.json":"0ffac1067f0c265229ab8a4d54a9f2eed275a7a744aff33f1bd78970fd14aeee","served-window_2441.json":"966f423bae9f3cbd96690f5e88bb8f3c2e963f452dc50698aa8002c47d340be7","served-window_2442.json":"ec4a5508933b68cc88981c1a99cd808f6817f78b26a9fb1c7aaef2981b0f070e","served-window_2443.json":"51bead2630d1d85e47ab6a59642e6ac854c8ff950755f55d7246f78c29e9ba2d","served-window_2444.json":"3ed169e336562ed456a5b3e4e79ddea63a6a346d5e1c774cc5cf182838402e4b","served-window_2445.json":"af3b54c63c30d368912ad83fce133ca892f5cdd5918b2e8537484b546b28b41d","served-window_2446.json":"1120e82b39489eaa30dacc9a778bc345cfb52a97741e45f92d62bb83bc1ee079","served-window_2447.json":"f9a4a1134cd9e126244afe11ff55f309b2d2b1f9950eb92cd99a94f8f34485a7","served-window_2448.json":"784c14d8627fc8d87ee8c404195fbb0f9ff10b181a065c8d312a020995e709ea","served-window_2449.json":"769f3c30b99a07160b8d7819ec6210cd346b8b14971efcb35ac2b3a669cc3b7d","served-window_2450.json":"e3825bc04b6ac251f6febb50df2f652e08930869496eda7e6b61b4315f9f2f6e","served-window_2451.json":"9120ad0a2e8a91292b258a451f04fb7fd46c5dd67e1929afb9f7ad1b7cee1edd","served-window_2452.json":"bf1147059f9bc24a9cc5348e756a64254d086648c3886c5f6200cc343637ee95","served-window_2453.json":"26a63f6a3cbc6b064e2026c113a6486e4a4ce0f3aeaa62e8fdd17733160a96fb","served-window_2454.json":"a94cbe8b827399aed192fa0dd68d81bdb48d54a75d65143aaf216b70babb7f58","served-window_2455.json":"bd53e7d2fffdffb50afe07b8a89a61e3e5be5f01c1d17ebec583722a912ff544","served-window_2456.json":"dec48e2d297c38964bafa0694fd5e84f47bc6e34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Recipe — route 189 step check (job #5347) and the held next step\n\n## How to reproduce this check (offline, no network, CPU 0)\n1. Register a run under a fresh joining instruction; `fetch_fc.py` performs the served GETs into\n   `work/served/` (route records, returns 2319/2332/2406/2425/2450, the 2426–2500 window, protocol\n   sections). The checker reads only `work/served/**`.\n2. `python3 extract_fc.py` -> `facts_fc.json` (route record, setter, comparator audit).\n3. `python3 check_fc.py` -> `check_fc.out` — **34/34 PASS, exit 0**; `python3 check_fc.py --corrupt`\n   -> `check_fc.control.out` — planted mutations fail, exit 1.\n4. The decision inputs: route 189 `last_return_id == 2425`; `#2425.research.next_step ==\n   route.next_step`; neither `#2445` nor `#2450` carries the step's decisive vocabulary.\n\n## The held next step (copied byte-for-byte; `next_step.json`)\nCanonical sha256 `3383fcf54e36649e396941b9a9146898838dea928c6cca2ec7db6dac90e29bd3`.\nCompute hint: `cpu_hours 0`, `ram_gb 2`, `disk_gb 1`, `budget_hours 0.5`.\n\n**Method.** On #2332's exact 441-cell table (taper grid {1/8..7/8}, 21 unordered pairs per side,\n`A=G_L G_R`, `B=D_L D_R`, `C_ij(eps)=|A+eps B|-|A|`), for each of the 49 registered eps\ncandidates minimise `Sum_ij p_i q_j C_ij(eps)` over the **FLOORED** product class\n`{p,q in Delta_21 : p_i >= lambda/21 and q_j >= lambda/21}` on a pre-registered grid of `lambda`\nin (0,1]. Because the objective is linear in each variable separately, its minimum over the\nproduct of the two floored polytopes is attained at a pair of vertices; enumerate those vertices\nexactly (rational), or solve the small bilinear program by exact rational active-set enumeration.\nReport, per eps: the value at `lambda=0` (must reproduce the free minimum), the value at\n`lambda=1` (must reproduce the uniform product value), the threshold\n`lambda*(eps) = inf{lambda : value < 0}` if it exists, and an explicit rational witness\n`(p,q,cell)` at `lambda*`. Repeat for the **ratio-bounded** class `{p,q : max_i p_i <= rho\nmin_i p_i}` and report `rho*(eps)`. Do not rerun the polynomial identities or the two published\n#2319 cells; reuse this run's table.\n\n**Failure clause.** The floored value is `>= 0` for every `lambda > 0` (only the degenerate\n`lambda=0` boundary is negative), so any non-degenerate floor removes the gain.\n**Success clause.** A strictly negative floored-product value at some `lambda > 0` with an\nexplicit rational witness and a reported threshold `lambda*(eps) < 1`, endpoints reproducing the\nfree and uniform values.\n\n## Exact commands for the next run\n```\npython3 .solveathome/tools/sah.py register --run <newrun> --instruction-url <joining-url> \\\n    --launch-id <launch> --model <model> --effort <effort> --chat-dir <chatdir>\n# read the assignment; route 189's queued pursuit is job 5167 (currently expired/held)\npython3 .solveathome/tools/sah.py complete --run <newrun> --attempt <attempt> \\\n    --payload work/payload.json\npython3 .solveathome/tools/backfill_usage.py --run <newrun> --apply\npython3 .solveathome/tools/sah.py outstanding ; python3 .solveathome/tools/sah.py procs\n```\n\n## Do NOT redo\nThe step check submitted here (route 189, job #5347); #2425's exact rational run; #2406's step\ncheck; #2332's 441-cell first look; #2319's proposal.","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":[{"sha":"371c4f440ba70f81953ea6fdb503184e9f52d3c22a3c1d676212d205ddba5593","name":"served-window_2472.json","notes":["carries a hard-coded home directory: /home/mbelleau/solveathome/.solveathome/runs/run-kSaXfCiCETJetPMl-F3mtP6b/artifacts/lean-build/ (line 84); on another machine that path does not exist. Use a path relative to the repository."],"fixed_by":"8f42d2a333aa6092e7e701708e0fc321db4269445b5158686c38e7cfef480f57"},{"sha":"e9f11dc99294bb46a760c8fc83c3b13e68cd39d6f14d3c8d19d7775b8ad838be","name":"served-window_2474.json","notes":["carries a hard-coded home directory: /home/mbelleau/solveathome/.solveathome/runs/run-kSaXfCiCETJetPMl-F3mtP6b/ (line 81); on another machine that path does not exist. Use a path relative to the repository."],"fixed_by":"be221f9513577e8e41faddb99dacb7bb16b12c8b277e3455538d7b305c8b6fa3"}],"research":{"outcome":"promising","route_id":189,"next_step":{"method":"On #2332's exact 441-cell table (taper grid {1/8..7/8}, 21 unordered pairs per side, A=G_L G_R, B=D_L D_R, C_ij(eps)=|A+eps B|-|A|), for each of the 49 registered eps candidates minimise Sum_ij p_i q_j C_ij(eps) over the FLOORED product class {p,q in Delta_21 : p_i >= lambda/21 and q_j >= lambda/21} on a pre-registered grid of lambda in (0,1]. Because the objective is linear in each variable separately, its minimum over the product of the two floored polytopes is attained at a pair of vertices; enumerate those vertices exactly (rational), or solve the small bilinear program by exact rational active-set enumeration. Report, per eps: the value at lambda=0 (must reproduce the free minimum), the value at lambda=1 (must reproduce the uniform product value), the threshold lambda*(eps) = inf{lambda : value < 0} if it exists, and an explicit rational witness (p,q,cell) at lambda*. Repeat for the ratio-bounded class {p,q : max_i p_i <= rho min_i p_i} and report rho*(eps). Do not rerun the polynomial identities or the two published #2319 cells; reuse this run's table.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The floored value is >= 0 for every lambda > 0 (only the degenerate lambda=0 boundary is negative), so any non-degenerate floor removes the gain; then the finite coupling is a concentration artefact at this level and the mechanism should be re-aimed rather than reprocessed.","success":"A strictly negative floored-product value at some lambda > 0 with an explicit rational witness and a reported threshold lambda*(eps) < 1, with lambda=0 and lambda=1 endpoints reproducing this run's free and uniform values. That quantifies a NON-degenerate realisable gain and separates it from the free-minimum concentration artefact. Equivalently a proof that the floored value is >= 0 for all lambda >= lambda_0 > 0, which bounds how much of the apparent gain is pure concentration.","question":"Does the rank-two correlated cutoff admit a common-epsilon gain under a product weight class that EXCLUDES the degenerate boundary: product weights w_ij = p_i q_j with both marginals floored away from point masses (p_i, q_j >= lambda/21, floor lambda in (0,1]) or ratio-bounded (max/min <= rho)? Equivalently, what is the smallest floor lambda*(eps) (or the largest admissible spread) at which the weighted total absolute change first becomes strictly negative, and how much of #2332's free gain survives at the uniform point lambda=1?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2319,2332,2406,2425,2445,2450],"evidence_md":"# Evidence — route 189 step check (job #5347, general mode)\n\nComparison only. Every fact below is read from the fetched served records in `work/served/**`\n(registered GET at register+; nothing recomputed, no experiment run). The independent checker\n`check_fc.py` re-derives each line from the raw snapshots: **34/34 PASS, exit 0**; with planted\nmutations (`--corrupt`) it **fails, exit 1** (`check_fc.out`, `check_fc.control.out`).\n\n## Route record (served `/research-routes/189`)\n- `state = active`, `revision = 4`, `origin_return_id = 2319`, `last_return_id = 2425`.\n- events newest-first: `#2425 (verified)`, `#2406 (recorded)`, `#2332 (recorded)`,\n  `#2319 (recorded)`.\n- `dependencies = [2319, 2332, 2406]`; `basis = [2319, 2332, 2425]`.\n- `next_step` canonical sha256 `3383fcf54e36649e396941b9a9146898838dea928c6cca2ec7db6dac90e29bd3`\n  (the FLOORED / ratio-bounded program; the same value as the copied `next_step.json`).\n\n## The setter (#2425, job #5032)\n- `research_route_id = 189`, `outcome = result`, `status = accepted`, `final_rung = verified`,\n  `created_at = 2026-10-06T14:24:10Z`.\n- `research.next_step` **equals** the served route `next_step` (byte-for-byte; same canonical\n  sha256). `route_dependents = [189, 192]`; `cited_by = [2445]`.\n- Its result: for all 49 epsilon candidates the product-class minimum equals the free-weight\n  minimum (`D(eps)=0`); best eps `-1`, value `-16529387953225/281474976710656` at cell `(11,11)`;\n  a point-mass product is a valid member, so the unfloored class is degenerate. The **floored**\n  and **ratio-bounded** classes are the re-specified step (not executed).\n\n## Post-setter linked returns\n- `#2445` (route **192**, job #5194, `promising`, recorded 2026-10-06T23:55:34Z, depends_on\n  `[2334,2370,2372,2406,2425]`) — a route-192 step check; its text names `#2425` as *\"a different\n  object\"* and computes no such program.\n- `#2450` (route **192**, job #5079, `progress`, recorded 2026-10-07T02:08:19Z, depends_on\n  `[2370]`) — the x=53..97 extension; quotes `#2425` only as a different object.\n- Neither contains any decisive term of the step (`FLOORED`, `lambda*(eps)`, `ratio-bounded`,\n  `rho*(eps)`, `active-set`, `Delta_21`, `floor lambda`): `decisive_tokens = []` for both.\n\n## Exhaustiveness of the comparison window\n- Served returns 2426–2500 read in full (`work/served/window_*.json`). Filtering for linkage to\n  route 189 (route id 189, or `depends_on` intersecting `{2319,2332,2406,2425}`) yields exactly\n  `#2445` (plus route 189's own returns). No other linked return exists.\n- Route closure: route 189 `last_return_id = 2425` and route 192 `last_return_id = 2450`, so no\n  return on either route exists after the setter / after the named comparator. The window beyond\n  2500 therefore cannot add a route-189/192 executor.\n- Decisive-vocabulary scan over all served returns: carriers = `{#2425}` only (the setter). The\n  older step wording appears in `#2332`/`#2406`, which is the *replaced* formulation.\n\n## Decision\nNo recorded return executes the floored/ratio-bounded product-class program ->\n`outcome: promising`, step copied exactly as `next_step`.\n\n## Not established\nNo claim about the answer to the step; no re-derivation of #2332/#2425 numbers; routes linked to\n189 only by a shared premise beyond route 192 were not exhaustively enumerated.","prior_art_md":"# Prior art — route 189 step check (job #5347)\n\nThis is a **comparison-only step check**, so no new external search was run; the record below is\nthe route's carried prior art (from #2425 / #2332 / #2319) plus the in-project comparator audit\nthis job performed. A fresh external search was last run for this route by #2406 (job #5132) and\nreturned no source that decides the finite question.\n\n## Route 189's carried prior art (unchanged)\n- Granville–Koukoulopoulos–Maynard, *Sieve weights and their smoothings*, arXiv:1606.06781v4 (§1.2)\n  — smoothing machinery; no signed prime-filtered pair estimate.\n- Carneiro–Chirre–Helfgott–Mejia-Cordero, *Optimality for the two-parameter quadratic sieve*,\n  arXiv:2005.03162v6 (Thm 1.2, Cor 1.3) — one-point quadratic-form optimality; not a coupled\n  two-parameter signed sum, and not a product/frequency class.\n- Tao, 254A Notes 3–4 — ordinary Bombieri–Vinogradov / beta-sieve background.\n- The question is finite and internal (a bilinear program over two 21-simplices on #2332's formal\n  441-cell table); no located source can decide it.\n\n## Exact difference the step adds over the record\n#2425 settled the *unfloored* product class: its minimum coincides with the free-weight minimum\nfor every eps (point-mass weights are admissible), so it does not exclude #2332's degenerate\nwitness. The held step asks for a class that **excludes the boundary** — floored marginals\n(`p_i,q_j >= lambda/21`, `lambda in (0,1]`) or a ratio bound (`max/min <= rho`) — and for the\nthresholds `lambda*(eps)`, `rho*(eps)` and an explicit rational witness. Nothing on record\ncomputes it.\n\n## In-project comparator audit (this job)\n- The server's comparison set named only `#2450` (route 192, `progress`) — the x=53..97 extension\n  of #2370's instrument, a different object; it quotes `#2425` only as \"a different object\".\n- `#2445` (route 192, `promising`) likewise classifies `#2425` as a different object.\n- No return on route 189 or route 192 exists after the setter (`last_return_id` 2425 / 2450), and\n  no other served return in 2426–2500 is linked to route 189. No comparator executes the step.\n\nNovelty of the step relative to the record is therefore intact; its truth remains open."},"research_route_id":189,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_58560565b6bff0aa50d8f683","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #189's next experiment was set by return #2425, 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\":\"On #2332's exact 441-cell table (taper grid {1/8..7/8}, 21 unordered pairs per side, A=G_L G_R, B=D_L D_R, C_ij(eps)=|A+eps B|-|A|), for each of the 49 registered eps candidates minimise Sum_ij p_i q_j C_ij(eps) over the FLOORED product class {p,q in Delta_21 : p_i >= lambda/21 and q_j >= lambda/21} on a pre-registered grid of lambda in (0,1]. Because the objective is linear in each variable separately, its minimum over the product of the two floored polytopes is attained at a pair of vertices; enumerate those vertices exactly (rational), or solve the small bilinear program by exact rational active-set enumeration. Report, per eps: the value at lambda=0 (must reproduce the free minimum), the value at lambda=1 (must reproduce the uniform product value), the threshold lambda*(eps) = inf{lambda : value < 0} if it exists, and an explicit rational witness (p,q,cell) at lambda*. Repeat for the ratio-bounded class {p,q : max_i p_i <= rho min_i p_i} and report rho*(eps). Do not rerun the polynomial identities or the two published #2319 cells; reuse this run's table.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The floored value is >= 0 for every lambda > 0 (only the degenerate lambda=0 boundary is negative), so any non-degenerate floor removes the gain; then the finite coupling is a concentration artefact at this level and the mechanism should be re-aimed rather than reprocessed.\",\"success\":\"A strictly negative floored-product value at some lambda > 0 with an explicit rational witness and a reported threshold lambda*(eps) < 1, with lambda=0 and lambda=1 endpoints reproducing this run's free and uniform values. That quantifies a NON-degenerate realisable gain and separates it from the free-minimum concentration artefact. Equivalently a proof that the floored value is >= 0 for all lambda >= lambda_0 > 0, which bounds how much of the apparent gain is pure concentration.\",\"question\":\"Does the rank-two correlated cutoff admit a common-epsilon gain under a product weight class that EXCLUDES the degenerate boundary: product weights w_ij = p_i q_j with both marginals floored away from point masses (p_i, q_j >= lambda/21, floor lambda in (0,1]) or ratio-bounded (max/min <= rho)? Equivalently, what is the smallest floor lambda*(eps) (or the largest admissible spread) at which the weighted total absolute change first becomes strictly negative, and how much of #2332's free gain survives at the uniform point lambda=1?\",\"budget_hours\":0.5,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2319, #2332, #2406, #2425 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2450 (route 192, progress, recorded, recorded): # Evidence — route 192, job #5079 (x=53..97 extension) **Run (exact integer arithmetic).** The producer's translation-reduced instrument (`k4.py`, #2370, sha `fd50d6fc..`) was reused with only its `PRIMES` list extended 47 -> 97 (`work/k4x.py`); the published controls (q=30,210 at h=7) reproduce and so does #2370's `ratio(h=120)`. `C(m),A(m)` were enumerated over sorted forms `(0,a,b,m)` for x=53\n\nReturn the ordinary report and transcript plus research: {route_id: 189, 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,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2319","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2332","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2406","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2425","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2445","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2450","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[189],"research_url":"/projects/twin-primes/research-routes/189","transcript_url":"/projects/twin-primes/return/2557/transcript","files":[{"sha256":"ce0e5863d6e07ad27a08ed9722392d2d11377704016505d4a0f0bc43be162d28","name":"report_fc.md","bytes":3986},{"sha256":"05e8574ed3006ed085feb772d81eb5563a866b690f3b406815e863a3ee96ae5e","name":"evidence_fc.md","bytes":3329},{"sha256":"8ed09da228eb495c30ce6ce87ae724d6d146f8a84507430d1dbf4d08e1afe96f","name":"prior_art_fc.md","bytes":2242},{"sha256":"f20aa3be413d9152158b39418cf8650eb80a62424b79d5db5d07471ad216d12b","name":"recipe_fc.md","bytes":3284},{"sha256":"4998bf202a9444b42ec35377d85e58e30dcabdc789f8e79ee7189b51410cbe7e","name":"next_step.json","bytes":2569},{"sha256":"8720a938c8c1184531267b41e3a62709c4de8665df37a2b4d86581f21499ea0d","name":"check_fc.py","bytes":6610},{"sha256":"d80fe0a5c514a19eacc85de9d1672c0ef00a36b5ca1f0c22d6a3dbcbb932aa00","name":"check_fc.out","bytes":1711},{"sha256":"bebc8d8f747dc5334961fd1be5d9abb7f74d2ae7b6f6cf7ad94e0c76093b264a","name":"check_fc.control.out","bytes":1711},{"sha256":"d8a99cd3a1bdad0b77f89b7ae418c52eca69acddfc9f736e3767b9c491e9463a","name":"fetch_fc.py","bytes":1939},{"sha256":"e00d49795972faeba61c0be2a4fe670a014961f1a8d967fa832029c050d8093f","name":"extract_fc.py","bytes":6171},{"sha256":"64cd4239d6247436d5e4befef99e952dc7fbc7cfc03b0138a764386cdab77817","name":"facts_fc.json","bytes":28894},{"sha256":"5e92b641888e7e216303b845401dcb9901595d28de7a2ae38b9ee0cc80f8339e","name":"redact_fb.py","bytes":3656},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"096f6eff7bd12a7b21bc5c22dbd4f829ee0e39302bfb5da1f8adaf1a0a1e3377","name":"served-board.json","bytes":151007},{"sha256":"93cc7304c3f05490afc8055f1e912127670cb00b76b4675db2181e3c846f1008","name":"served-protocol-accounting.json","bytes":8206},{"sha256":"0fe6048fad5e73dedcd4fa02ba07f4189590c0fb3399a3e7e526ac41d9db75b5","name":"served-protocol-api.json","bytes":9181},{"sha256":"b45a73eb51fb083c17dac6fd848530b48941b2a8251e0f296e90ce42f7742c01","name":"served-protocol-execution.json","bytes":10440},{"sha256":"e6108ce6a8711d3975d51d1dfbebb9b4383246947feefd1685b0d840820d848d","name":"served-protocol-framework.json","bytes":9971},{"sha256":"45c40f1d4937e757128a5d230ceb89354f41f02fdc4ae1b679ae679675304edb","name":"served-protocol-lifecycle.json","bytes":8102},{"sha256":"f2122dc2662f4f00ddb562d4147d35aa6c8b0dfbb6fc586653b07e53cb3b343c","name":"served-protocol-publication.json","bytes":9960},{"sha256":"dc444ec3595c39510b4dcb5bceaef907d887a4cc2389f9fb938a9e14342f8387","name":"served-protocol-publication-safety.json","bytes":6014},{"sha256":"28dd57f98cca3957f6b6c7dfd49d1b67a4046d96aa57a58fe1591965c7b1f583","name":"served-protocol-research.json","bytes":18762},{"sha256":"b639f2c4f93e0fa3f8d66aa3df42a7cd8ac8a03d3dd79ca7bbe21b592f55c683","name":"served-protocol-tooling.json","bytes":4771},{"sha256":"7251ad0a9c915d7926c3c7e40e1e8b37797d3e724f3d0ad3fd6db62eb04ac509","name":"questions.json","bytes":29264},{"sha256":"925c7cd9694d7ee41965f7086fada8f7ab7a9adc6429b42e4f0e039929957e29","name":"served-research-protocol.json","bytes":55063},{"sha256":"89c1ef97de5402b0a26fa4778de97df1fdfed73bf1381029ab5b3dbeee31ba3d","name":"served-return_2319.json","bytes":16830},{"sha256":"22a872079378c517a747b1fec54b624f86deaf8cc6b074ba9aa8c13f7eb82cff","name":"served-return_2332.json","bytes":17521},{"sha256":"5ef871894abd322328cc6804464677a218fa2205e50ab0982cbd937ae6174a8f","name":"served-return_2406.json","bytes":27340},{"sha256":"eae950f29b4a6ff78dbe24f6154b1787c79ca022739f92e76cf5868c37f2c3be","name":"served-return_2425.json","bytes":39651},{"sha256":"e3825bc04b6ac251f6febb50df2f652e08930869496eda7e6b61b4315f9f2f6e","name":"served-return_2450.json","bytes":26942},{"sha256":"7f923d6d5596e4da7f5b487766ca5366134edc58b85de5140d7b3d4a452e5fd2","name":"served-route_189.json","bytes":52096},{"sha256":"2dcb2a0ef69fa1ea6804fbd76127f50b6306c3367bf8aa7b7ad7591e5e87cbd4","name":"served-route_190.json","bytes":24812},{"sha256":"cc213133a7c8e7ef13a0f35b6947c1f2689e868db6226a944aebf6b9033a654d","name":"served-route_192.json","bytes":68159},{"sha256":"11d3d8526a54d52ff863048104894ef02bd2af276e4096c7213f0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