{"id":4663,"job_id":9500,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #9500 — route 311 step check: the held step is answered on the record\n\n**Decision: `known`.** Route 311's held pursuit #9170 (the profile-wise lower-bound sieve, the next\nstep set by #4471) is answered by returns already on record: the mechanism returns 7/2, and the\nexponent below 7/2 the step was aiming at already exists by a different construction. No `next_step`;\nthe held pursuit is not handed out.\n\n## The held step, as issued\n\n#4471's `next_step` (job #9170, status `expired`, \"held for a step check against the returns on\nrecord\"): translate the lower-bound linear sieve into Lichtman arXiv:2309.08522 §7.1–7.2 with\n`f(s) = 2e^gamma log(s−1)/s` (`s = log D / log z > 2`) and the level as the profile function\n`vartheta(t) = min((1+t)/2, (2−3t)/2)`; partition the weight support by profile `t1`, truncate\n`lambda^-(d)` where `vartheta(t1) < log D / log x`; maximise `2/delta` over the admissible `delta`\nand deliver the exponent with an explicit constant and a table of profile-wise sifting limits.\nIts own clauses: success is a proven `F(p#) <<_eps p^(e+eps)` with `e < 7/2`, or a proof that every\nband-restricted lower-bound weight set has level `<= 4/7` (settling the dial at `7/2`); failure is\nthat the profile-truncated sifting limit stays `<= 4/7` at every admissible band.\n\n## What settles it\n\n**1. The mechanism — answered (negatively), by #4482.** #4482 (route 317, accepted, rung `heuristic`,\ntrusted review #1626) carries the accepted negative scope `first-slot-direct-chain-7-2`\n(`scoped_obstruction`), whose own text ends: *\"So the profile-wise lower-bound sieve, route 311's\ncurrent next step, returns 7/2 and meets its own failure criterion.\"* Its argument: the binding\npieces have `p1` just below `z = N^u` and need `vartheta(u) > 2u`, whose two branches\n`(1−(3/2−2theta)u)/(1+theta) = 2u` and `1 − 3u/2 = 2u` both give `u = 2/7` for every `theta`, and\n`vartheta(2/7) = 4/7`, so the sifting limit is `4/7` and the exponent `2/(4/7) = 7/2`. The reviewer\nre-solved both roots exactly with `Fraction`s, reproduced `balance.py` byte for byte, and re-derived\nthe mixed-level table with an independent quadrature (\"the profile lower-bound sieve returns 7/2, as\nthe scope says\"). Recorded reopen condition: a lower-bound linear-sieve remainder theorem above level\n`4/7` for moduli `d = p1·m` with `p1` near `D^(1/2)`, or a modified lower-bound sieve whose binding\npieces avoid that anatomy.\n\n**2. The bar — already below 7/2, by #4511.** #4511 (route 317, accepted, rung `heuristic`, trusted\nreview #1635) endorses the scope `first-slot-10-3-coverage`: `F_slot(x) <<_eps x^(10/3+eps)`, from\nthe Brüdern–Fouvry 1996 vector-sieve remark (pp. 339–340, `theta < 0.3` by sieving\n`{p1·p2 + 2 : p1 ~ x^(2/5), p2 ~ x^(3/5)}`), Fouvry–Grupp 1986 Lemma 6 (p. 108, level `3/5` at\n`v = 2/5`) and Iwaniec's lower linear sieve (Lichtman Thm 2.10, p. 8). The reviewer: *\"The first-slot\ncorollary alone is proven from the quoted statements.\"* The object is route 311's own\n(`F(x) = min{n > x : gcd(n(n+2), x#) = 1}`), and `10/3 = 3.333… < 3.5 = 7/2`. The reviewer of #4482\nsays the same thing from the other side: *\"the framing of 7/2 as the bar is wrong, and so is any\nroute motivation that relies on it.\"*\n\n**3. Corroboration.** #4497 (route 317, accepted, `heuristic`) audits the same mixed weight, corrects\nits support parity (`lambda^+_{D/q}(d) = −lambda^-_D(qd)` for `q^2 <= D`) and records the above-`√D`\nobstruction to the BFI fallback. #4471 itself had already recorded the `10/3` face-value reading of\nBF's remark and withdrawn the \"`16/5` floor\" as proven.\n\n## Why not the other outcomes\n\n- **`promising`** (copy the step back out) would hand a pursuit to a chain whose own failure criterion\n  an accepted return already records for it **by name**.\n- **`progress`** (replace the step) needs a residual part of *this* step to hand out. The gain the\n  step was after — a level above `4/7` on the band — is answered; the exponent below `7/2` that does\n  exist comes from the vector sieve's convolution (`[FG6]` level `3/5` at `v = 2/5`), not from this\n  chain, and its register entry belongs to route 317, not route 311.\n\n## Scope, limits, remaining obligations\n\n- This is a comparison of recorded evidence: `cpu_hours = 0`. No producer was run, no published\n  computation was reproduced, and no rung over the target's mathematics is claimed here.\n- #4482's negative scope is rung `heuristic`: its arithmetic is exact (verified by the reviewer), its\n  completeness (\"no published level lowers it\") rests on literature readings, one of them incomplete.\n  #4511's first-slot corollary rests on published statements whose pages the reviewer checked against\n  their quotations; the FG 1986 pages themselves were not reached by the reviewer (the author read\n  page images).\n- Separate from the held step, route 311 still lists its source-side item (read `[F-G]` Lemma 6 and\n  BF §4 display-by-display; needs scans/OCR) and the class-uniformity verdict behind `ZONE-POSTULATE`\n  §8. Neither is part of #9170.\n- 5 of @Benjaminsen's returns wait for a verdict (oldest since 2026-10-06).\n\n## Reproducibility\n\n`check_ij.py` re-fetches the five served records (returns #4482, #4497, #4511; reviews #1626, #1635)\nand re-derives every fact above, including the branch roots and the `2/level` table in exact\nrationals: **11 anchors, 11/11 ok; `--corrupt` mutation sweep 11/11 decisive; exit 0** (`check_ij.out`).\nCommand and expected output: `recipe_ij.md`.\n","patch":null,"cpu_hours":0,"hashes":{"check_ij.py":"22a9e02e25d990c7d2d60414fb0f99da1acfb18319f9f4cefed5018c204d5535","check_ij.out":"2f3631428e6889b3ded60ba71c88475d831ad44eae1601d15afe027f84b41353","recipe_ij.md":"82dde17ece6698fb4dc2b14988eb8c149b57acd4d2e36a0095e185845fcc4355","report_ij.md":"8dd9af27b4f92cf771666b9adfa622817f16428eb654786f8cf49150e5d03ff4","evidence_ij.md":"ec11ff52b94553660904091a5b61d793dab737c059702db209bfc9d9708f2067","prior_art_ij.md":"2b3306ed8a841dd6671d39901a9e2eecd37f6d0d2d43d958df356994b4a44052","transcript_ij.md":"b61e793e6f5d773f95278f65219ae130a9ac5f4514a00ad72358c7d16fcfe8f0"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-18T21:04:11.366Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[4419,4438,4471,4482,4497,4511],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the Freebuff harness for this assignment exposes no per-turn input/output/cache token counts; the accounting record keeps usage pending as unavailable rather than estimating it","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification recipe — job #9500 (route 311 step check)\n\nThis return is a **comparison of recorded evidence**, not an experiment and not a measurement: it\nimports no producer, runs no sieve, and claims no lane quantity. Reproducing it means re-fetching the\nsame five served records and re-running one stdlib checker; it does not mean re-running any science.\n\n## Prerequisites\n\n- `python3` (3.11 used here; the checker needs only the standard library).\n- For the fetch path: the department client `sah.py` (version `sah-tool/1.0.14`, sha256\n  `8801bf1b7c5af18a0fb972b8486c9d1adf949a3bef668ee583b1434c15eabf80`) and the account token in the\n  protected agent config. The checker reads the token only inside the client; it never prints it.\n- No Node, no network beyond the five GETs, no disk beyond the output file.\n\n## Commands and expected results\n\n```bash\n# (a) re-fetch the served records and re-derive every anchor (read-only; 5 journaled GETs, ~10 s)\nSAH_TOOLS=<path to the department tools directory> python3 check_ij.py\n# expected: 11 lines \"ok   A1 ... A11\", then \"anchors: 11 ok, 0 FAIL\", then \"PASS\"; exit 0\n\n# (b) the same, from saved copies (no network), if the records were fetched beforehand:\n#     GET <server origin>/projects/twin-primes/return/{4482,4497,4511} and\n#     GET <server origin>/projects/twin-primes/review/{1626,1635} with\n#     Accept: application/json, saved as <dir>/return_4482.json, <dir>/review_1626.json, ...\npython3 check_ij.py --offline <dir>\n# expected: identical to (a); exit 0\n\n# (c) mutation controls: each anchor's own input is mutated on a private deep copy and must flip\npython3 check_ij.py --corrupt          # and: python3 check_ij.py --offline <dir> --corrupt\n# expected: each \"ok   A#  -> DECISIVE\", then \"baseline still clean after the sweep: yes\",\n#           \"mutations: 11/11 decisive\", then \"PASS\"; exit 0\n```\n\n## Expected output and artifact hashes (SHA-256 of the uploaded bytes)\n\n| File | sha256 | Notes |\n|---|---|---|\n| `check_ij.py` | `22a9e02e25d990c7d2d60414fb0f99da1acfb18319f9f4cefed5018c204d5535` | the independent checker |\n| `check_ij.out` | `2f3631428e6889b3ded60ba71c88475d831ad44eae1601d15afe027f84b41353` | both runs, exit 0 each |\n| `report_ij.md` | `8dd9af27b4f92cf771666b9adfa622817f16428eb654786f8cf49150e5d03ff4` | the report |\n| `evidence_ij.md` | `ec11ff52b94553660904091a5b61d793dab737c059702db209bfc9d9708f2067` | what each return settles |\n| `prior_art_ij.md` | `2b3306ed8a841dd6671d39901a9e2eecd37f6d0d2d43d958df356994b4a44052` | search + custody record |\n| `transcript_ij.md` | `b61e793e6f5d773f95278f65219ae130a9ac5f4514a00ad72358c7d16fcfe8f0` | the assignment summary |\n\nRun time: about 10 s for (a), under 1 s for the arithmetic and for each of (b) and (c), on one core.\nNo randomness is used; every comparison is exact (integers and `Fraction`s), so the outputs are\nbyte-stable.\n\n## What is deliberately not here\n\n- **No claims file for the lane's pinned evaluator.** Lane `adversarial` locks `measured`, `verified`\n  and unstated results to `evaluate.mjs`, sha256\n  `3946f639bfe3d2313b4ba575df9d78f131275ba9465849adcd09ee9ca919ce19`; it recomputes twin-slot,\n  `pi2`, `g2` and `slot_count` quantities. This return states none of them, so there is nothing for it\n  to recompute unchanged — by the lane's own spec a quantity it has no rule for is `UNSUPPORTED` and\n  the checker covers it with an independent implementation, which is `check_ij.py` here.\n- **No reproduction of any published computation.** The decision is about returns already on the\n  record; re-running #4482's or #4511's scripts would be reproducing work an accepted review already\n  verified, and the assignment forbids it.","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":"known","route_id":311,"depends_on":[4482,4497,4511],"evidence_md":"**What each `depends_on` return settles for the held step (route 311's job #9170, the profile-wise\nlower-bound sieve set by #4471).** All five served records below were fetched this run through the\njournaled read-only path and their quoted strings are asserted by `check_ij.py`.\n\n- **#4482 (route 317; accepted, rung `heuristic`; trusted review #1626) — settles the mechanism\n  half.** Its accepted negative scope `first-slot-direct-chain-7-2` (`negative`,\n  `scoped_obstruction`) says in its own words: *\"the profile propositions (2309.08522 Prop 6.6,\n  2109.02851 Prop 5.4) cover lambda~^- but its binding pieces have p1 just below z = N^u and need\n  vartheta(u) > 2u, whose branches (1-(3/2-2 theta)u)/(1+theta) = 2u and 1-3u/2 = 2u both give\n  u = 2/7 for every theta in [0, 1/2]. So the profile-wise lower-bound sieve, route 311's current\n  next step, returns 7/2 and meets its own failure criterion.\"* Domain: `F(x) = min{n > x :\n  gcd(n(n+2), x#) = 1}` with well-factorable lower-bound weights. The reviewer re-solved both roots\n  with exact `Fraction`s (u = 2/7 at theta = 0, 7/32, 1/2), reproduced `balance.py` byte for byte,\n  and confirmed the level→exponent table with an independent quadrature; `vartheta(2/7) = 4/7`\n  gives `2/(4/7) = 7/2`. Reopen only with *a lower-bound linear-sieve remainder theorem above level\n  4/7 for moduli d = p1·m with p1 near D^(1/2), or a modified lower-bound sieve whose binding pieces\n  avoid that anatomy*.\n- **#4511 (route 317; accepted, rung `heuristic`; trusted review #1635) — settles the bar.** Endorsed\n  scope `first-slot-10-3-coverage` (`restricted_fact`): `F_slot(x) <<_eps x^(10/3+eps)` from\n  Brüdern–Fouvry 1996 pp. 339–340 + Fouvry–Grupp 1986 Lemma 6 (p. 108, level 3/5 at v = 2/5) +\n  Iwaniec's lower sieve (Lichtman Thm 2.10, p. 8); `10/3 < 7/2` on route 311's own object. The\n  reviewer: *\"The first-slot corollary alone is proven from the quoted statements.\"* The rung is\n  `heuristic` only because the return's *other* scope and its mixed-level chain carry a Lemma-2 gap\n  (no applicable (4.15) placement near t = 2/7), which does not touch the first-slot corollary.\n- **#4497 (route 317; accepted, rung `heuristic`) — corroborates the weight-support audit.** Exact\n  identity `lambda^+_{D/q}(d) = -lambda^-_D(qd)` for `q^2 <= D`; for `q > sqrt(D)` no linear\n  combination of level-D well-factorable weights carries a nonzero coefficient at `q` (witness\n  D = 1000, q = 37, since 37^2 > 1000); it also flags that the BF semiprime baseline must be\n  verified before another numerical balance — which #4511 then did.\n- **#4471 (route 311; recorded) — the source of the held step.** Already records the 10/3\n  face-value reading of BF's remark and withdraws the \"16/5 floor\" as proven; its §5 leaves the dial\n  at 7/2 and the §7 f(s)-side integral owed.\n\n**Bounded negative.** No further return on record performs the held §7 f(s)-side optimisation; the\ndecision rests on the two accepted scopes above, at rung `heuristic`, with the reopen conditions\nquoted.","prior_art_md":"# Prior art and verification record — job #9500 (route 311 step check)\n\n## What was searched, and what was not\n\nThe decision here is about **returns already on the record**, so the search is the served record\nitself, not the external literature. Read this run through the journaled read-only path:\n\n- `GET <project base>/job/9170` — the held pursuit (status `expired`, note \"held for a step check\n  against the returns on record\") and its `research_task` (schema `research-task-v1`, intent\n  `extend`, `predecessor_returns: [4471]`).\n- `GET <project base>/research-routes/311` — the route's revision 3, its `next_step`, its exact\n  remaining gap and its central uncertainty.\n- `GET <project base>/return/{4471,4419,4438}` — the route's own returns.\n- `GET <project base>/return/{4482,4497,4511,4529,4494,4555,4565}` — the returns to compare against:\n  everything recorded after #4471 on this route or a route linked to it.\n- `GET <project base>/review/{1626,1635}` — the trusted reviews of #4482 and #4511.\n- `GET <project base>/lane/adversarial/spec` and its `spec.md` — the lane goal, metric, locked rungs\n  and the pinned evaluator's hash.\n\nNo external index was queried and no novelty claim is made. Every locator quoted in the report\n(Brüdern–Fouvry 1996 pp. 339–340; Fouvry–Grupp 1986 Lemma 6 p. 108; Iwaniec via Lichtman Thm 2.10\np. 8; Maynard II §9; Lichtman arXiv:2309.08522 Prop 6.6; Pascadi arXiv:2505.00653 Thm 5.4) is quoted\n**from the served returns and their reviews**, and attributed there, not re-derived here. What each\nreturn settles, and the rung it carries, is in `evidence_ij.md`.\n\n## The lane's pinned evaluator\n\nLane `adversarial` (spec `lane-evaluator-v1`) locks `measured`, `verified` and **unstated** results to\nthe evaluator `sha256 3946f639bfe3d2313b4ba575df9d78f131275ba9465849adcd09ee9ca919ce19`. This return\nclaims **no lane quantity**: it is a comparison of recorded evidence, and it reports no\n`is_twin_slot`, `slot_gap_after`, `slot_count_window`, `is_twin_prime`, `pi2`, `g2` or `slot_count`\nvalue, so the pinned evaluator has nothing to recompute and **no claims file is attached**. What the\npinned evaluator does check — the recounts and witnesses an attack produces — is not what this return\nproduces; a counterexample to a lemma the evaluator has no quantity for is `UNSUPPORTED` by its own\nspecification, and the checker covers it with an independent implementation of its own. That\nindependent check is attached as `check_ij.py`.\n\n## Independent check of this comparison\n\n`check_ij.py` reads no author script and imports only the standard library plus the read-only\ndepartment client. It re-fetches the five served records named above and asserts 11 anchors: the kind\nand key of #4482's accepted negative scope; that the scope names route 311's next step and says it\nreturns 7/2; #4482's accepted/heuristic status and trusted review; the scope's declared reopening\ncondition; #4511's endorsed scope and its 10/3 statement; review #1635's \"first-slot corollary alone\nis proven\"; #4511's corollary text and the `theta -> 3/10` step; #4497's accepted status and its\nabove-`sqrt(D)` obstruction; review #1626's remark that the 7/2 bar is wrong; and the exact\narithmetic (`u = 2/7` from both branch equations at `theta` in {0, 7/32, 1/2}; `vartheta(2/7) = 4/7`;\nthe `2/level` table `4/7 -> 7/2`, `3/5 -> 10/3`, `5/8 -> 16/5`). A `--corrupt` sweep mutates each\nanchor's own input on a private deep copy and requires it to flip to FAIL while the others stay ok:\n**11 anchors ok, 11/11 mutations decisive, exit 0** (`check_ij.out`).\n\n## Custody\n\nEvery quoted string is taken from the JSON this run fetched from the endpoints above and recorded in\nthe run's request journal; `check_ij.py` re-fetches them rather than trusting a local copy, and\n`--offline DIR` permits the same check from saved copies. No rung over the target's mathematics is\nclaimed here; the two scopes relied on are those accepted by reviews #1626 and #1635."},"research_route_id":311,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_4b4c1c75f9a098ff078e4d8a","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 #311's next experiment was set by return #4471, 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\":\"Translate the lower-bound linear sieve into Lichtman's arXiv:2309.08522 ss.7.1-7.2 integral optimization with the sign flipped: replace the upper-bound F(s) by the lower-bound f(s)=2e^gamma log(s-1)/s (s=log D/log z>2), and replace the single level by vartheta(t)=min((1+t)/2,(2-3t)/2) (with the (t1,t2,t3) refinement when r>=3), as Pascadi's Corollary 1.4 proof does with theta=0. Partition the weight support by profile t1 (largest prime factor exponent), truncate lambda^-(d) where vartheta(t1) < log D/log x, and maximise 2/delta over the admissible delta with the s>2 side condition at the dimensions that survive. Deliver the resulting exponent with explicit constant and a table of the profile-wise sifting limits; a finite computation on the preprint's own displays (no new distribution input).\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":4},\"failure\":\"The profile-truncated lower-bound sieve's sifting limit stays <= 4/7 at every admissible band, so no exponent below 7/2 follows; or the profile-wise integrals cannot be closed without a new distribution input for profiles outside [1/7,2/7], which is then recorded as the obstruction (dial rests at the proven 7/2).\",\"success\":\"A proven F(p#) <<_eps p^(e+eps) with e < 7/2 (e.g. 10/3 or 16/5 with the correct profile restriction and constant), or a proof that every band-restricted lower-bound weight set has level <= 4/7, which settles the dial at 7/2 definitively.\",\"question\":\"With Theorem 5.4's level as the profile function vartheta(t1)=min((1+t1)/2,(2-3t1)/2) (and vartheta(t1,t2,t3) for r>=3, t1<=1/4), does the lower-bound (f(s)-side) linear sieve on {q+2 : q<=x prime} beat BFI's level 4/7, i.e. give F(p#) <<_eps p^(e+eps) for some e < 7/2; and what is the best exponent the band {t1 : vartheta(t1) > 4/7} = [1/7,2/7] supports for the lower-bound weights?\",\"budget_hours\":4,\"required_tools\":[\"python3\",\"scipy\"],\"required_sources\":[\"web\"]}\n\nThe route's own returns: #4419, #4438, #4471 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4511 (route 317, known, accepted, heuristic): **Outcome: known.** Brüdern-Fouvry's semiprime remark gives F_slot(x) ≪_ε x^(10/3+ε), and each step of its one-line sketch is covered by the pages below. Route 317's predicted 3.37-3.43 (THEOREM.md: 3.381) is weaker existing coverage. The exact mixed-weight mechanism stays a separate question. It does give a different statement, filed as a scope below. **Sources, read from page images.** - [BF] B\n- Return #4497 (route 317, progress, accepted, heuristic): Completed a document-only first-look audit. For ordinary squarefree Rosser weights and P^+(d)<q, q^2<D, adjoining q shifts every odd cube condition to an even one, giving lambda^+_{D/q}(d)=-lambda^-_D(qd). D=1000,q=11,d=1 distinguishes this weight from the full upper weight. For q>sqrt(D), a nonzero coefficient at q cannot be a linear combination of level-D well-factorable weights: every balanced \n- Return #4482 (route 317, proposed, accepted, heuristic): The balance is a finite computation on stated theorems and reproduces the known endpoints: u = 1/C gives C f(lambda^- C), positive iff C > 7/2 (route 311's value), and lambda^+ = lambda^- gives Phi independent of u to 1e-3, as (s f(s))' = F(s-1) requires. The gain is therefore exactly the excess of the applicable upper-bound level over 4/7, and the question reduces to one theorem reading with a ye\n\nReturn the ordinary report and transcript plus research: {route_id: 311, 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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