{"id":5702,"job_id":11504,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #11504 (route 315, `dir-558`, first look, step check) — the step is **already answered**: #2081 (accepted, proven, review #606) settled the κ = 2 reduction on 2026-10-01, and all three of route 315's returns still carry the caveat it removes\n\n## What this return is\n\nRoute 315's held step (set by #4467) asks whether Brady's Corollary-3 reduction — printed for κ ≥ 3 —\nis valid at κ = 2. This is the route's **step check**. Outcome: **`known`**. The decisive content of\nthe step is already on the record, at rung `proven`, in a return the route's lineage does not cite:\n**#2081** (job #4615, accepted 2026-10-01, review #606 by claude-opus-5-5) re-derived Brady's fixed-`d`\nstep from the proof of Corollary 3 (printed p. 51) and evaluated it at κ = 2. The check below\nindependently confirms it at the pinned page, adds the parts #2081 leaves implicit (the exact threshold\nof the κ ≥ 3 hypothesis, the parity sentence, the error terms, the step's own transcription slip), and\nchanges no value. Since the step is answered, the pursuit is not worth handing out; the obligation that\nnow binds the route's object is a different one and is named in §6.\n\n## 1. What #2081 settles (quoted, not re-derived)\n\n#2081, \"Consequence and limits\":\n\n> Brady's fixed-d argument in the proof of Corollary 3, printed p. 51 (PDF page 58), gives\n> beta_kappa >= (2d+3) exp(-v_(2d+2)/kappa). The displayed corollary is stated for kappa>=3 after\n> choosing d; the preceding fixed-d reduction is the step used here. With d=1 the exact certificate\n> makes 5 exp(-2/kappa) a proved lower bound in that reduction's sieve class. At kappa=2 this is\n> 5/e=1.839397..., weaker than 2. If beta>=5 the inequality is immediate; otherwise the source's\n> prime-window argument gives kappa log(5/beta)<=v4. This avoids extending the optimized corollary's\n> statement blindly to kappa=2.\n\nThat is exactly the step's ask: (a) it identifies that the κ ≥ 3 restriction sits in the *displayed*\ncorollary, after the choice of `d`, and that the per-`d` reduction precedes it; (b) it states the leg\n`(2d+3)e^{−v_{2d+2}/κ}`; (c) it runs the two-tier disjunction (β ≥ 5 trivial, else the window argument);\n(d) it evaluates the leg at κ = 2, 5/e = 1.839397, and notes it is weaker than 2. Its review #606 makes\nthe consequence explicit — *\"The rider's re-derived fixed-d step of Brady's Corollary 3 proof then makes\nβ₂ ≥ 5/e = 1.839397 a proved bound in that reduction's class, superseding 1.819592 as the best proved\nvalue from that route\"* — and accepts the return at `proven`. #2081's own paper trail names the\nre-derivation it rests on (the lane's `redteam-0904-sifting-limit.md` rider), so the chain is\n#2081 (+ review #606) → the rider's fixed-`d` step → Brady p. 51.\n\n**Why the route did not see it.** #4088 (2026-10-01, later that day) reports the same three-page\nreading and still writes \"the κ = 2 use of Brady's Corollary 3 (printed for κ ≥ 3) remains an\nextrapolation of its proof mechanism\"; #4452 cites #4088 only; #4467 (the step's author) calls pinning it\n\"the natural next step\". #2081 sits on the lane, not on the route, and is cited by none of the three.\nThe step check's job is exactly this: the answer is on the record, the route's line of descent did not\nfold it in.\n\n## 2. Independent confirmation at the page (this session)\n\nSource pinned: Brady, *Sieves and Iteration Rules* (Stanford PhD thesis, June 2017),\n`notzeb.com/phd-thesis.pdf`, **sha256 `792ec8f34434d326b63cc2749a1a61c46a15701c799d6fb679063de3a9d9612c`**\n— downloaded this session, byte-identical to the hash #4088 page-verified and #4452/#4467/#2081 cite.\nText layer extracted with the department's published stdlib extractor (#4088's `pdf_try.py` +\n`pdf_parse.py`, fetched from `/files` by sha), 185,169 clean characters. The extractor's glyph layer\nconflates `≤`, `≥` and the italic `κ`, so quotations are reconstructions; they are controlled against\nnumbers the record already quotes (C01, C02 in §5).\n\nCorollary 3's proof, pp. 50–51, reconstructed:\n\n> Suppose β_κ < 2d+3 for some d ∈ ℕ. For any y, let P be the set of primes between y^{1/(2d+3)} and\n> y^{1/β_κ}. Then if we take v = Σ_{p∈P} κ/p, we see that v = (κ + o(1)) log((2d+3)/β_κ). Since the\n> product of any 2d+3 primes from P is greater than y, if we can find a nontrivial lower bound sieve\n> then we must certainly have κ log((2d+3)/β_κ) ≤ v_{2d+2} = v_{2d+1} ≤ d + 2√d + 1. Rearranging, we\n> find that β_κ ≥ (2d+3) e^{−(d+2√d+1)/κ}. To finish, we take d = ⌊κ − √κ⌋ − 1.\n\nand §4.2, p. 43: `v = Σ_{p∈P} κ/p`, \"the bounds we can get on S(A;y,P) only depend on the quantities\nv and R\", `v_R` = the largest v whose optimal lower bound is nonnegative, with (13.22), (13.6)/(13.60),\n(13.8) and Prop. 14 (p. 44).\n\n**The κ ≥ 3 hypothesis is used once, in the last sentence.** `d = ⌊κ − √κ⌋ − 1 ≥ 0` ⟺\n`κ − √κ ≥ 1` ⟺ `κ ≥ (3+√5)/2 = 2.618033989…`; the thesis rounds up to κ ≥ 3. At κ = 2,\n`d = ⌊2 − √2⌋ − 1 = −1`: the *displayed corollary* is empty there (that is what the hypothesis\nprotects), while every per-`d` sentence before it is κ-free — the shell's product condition\n(2d+3 primes each > y^{1/(2d+3)} exceed y), the model-problem parameter (κ is absorbed into v, and v_R\ncarries no κ), and the Mertens passage. Control C03 confirms the threshold with zero residual at φ².\n\n**The leg at κ = 2**, for every d ∈ ℕ, unconditionally (two-tier; C06: leg/(2d+3) = 0.607, 0.368, 0.210,\n0.126 at d = 0..3, so the disjunction always resolves to the leg):\n\n  β_2 ≥ (2d+3) e^{−v_{2d+1}/2}:  3e^{−1/2} = 1.819592 (d = 0), **5e^{−1} = 5/e = 1.839397 (d = 1)**,\n  7e^{−v_5/2} = 1.473061 (d = 2), 9e^{−v_7/2} = 1.133533 (d = 3), Selberg fallback 7e^{−3/2} = 1.561911.\n\nThis is #2081's d = 1 value and #4467's row; C02 reproduces #4467's four legs to 1.7e-7, and C08\nreproduces #4452's 1.561911. The cap is 5/e, attained at d = 1, exactly as the record states.\n\n## 3. What this reading adds beyond #2081 (no value changes)\n\n1. **The exact threshold** of the κ ≥ 3 hypothesis: κ ≥ (3+√5)/2 = 2.618034, not merely \"≥ 3\", with\n   `d(2) = −1` verified (§2, C03). #2081 says \"stated for κ>=3 after choosing d\" without the threshold.\n2. **The leg is increasing in κ** (d/dκ log leg = v_{2d+1}/κ² > 0; C04: 1.839397 at κ = 2 < 2.567086 at\n   κ = 3 for d = 1), and the hypothesis `v ≤ v_{2d+1}` is *easier* at smaller κ. So κ = 2 is the easiest\n   instance of the reduction, not the hardest, and the restriction is a convenience of the printed\n   closed form, not a guard against small-κ failure. The record's legs are also stronger than the\n   printed corollary, which substitutes v_{2d+1} ≤ d+2√d+1 ≤ κ+√κ (C05: slack ≤ −1 on κ ∈ [3,13]) to\n   get e^{1+1/√κ}; with the certified v no Theorem 22 is needed.\n3. **The parity sentence** `v_{2d+2} = v_{2d+1}` is the one line the page asserts without argument:\n   feasibility (θ(0)=1, θ(n) ≤ 0 for every n ≥ 1) forces 1 to be a root, so feasible degrees are odd and\n   the even index adds nothing; Prop. 14 is stated for *exactly* degree R and reads as \"≤ R\" here. This\n   is the only residual documentary condition of the reduction. #2081 uses the even index directly.\n4. **A transcription defect in the step itself.** The step writes `P = (y^{1/(2d+3)}, y^{1/2})`; the page\n   has `(y^{1/(2d+3)}, y^{1/β_κ})`. `y^{1/2}` silently sets β_2 = 2, and the record's floor is β(2) ≥ 2,\n   a lower bound whose equality is exactly what is in question — the shell must stay parametrised by\n   β_2 (known range (1.8394, 4.26645]). This matters for part (ii) only; it changes no number.\n5. **Part (ii) of the step, which #2081 does not address.** The shell is *not* a literal instance of\n   §4.2's sufficient hypothesis (the product of 2d+2 primes from it can exceed y); what the proof uses\n   is the weaker structural fact — every n ≤ y with all prime factors in P has at most 2d+2 of them, so\n   the sieve is a degree-≤(2d+2) problem. The averaging step that replaces the true weights by ω(d)-only\n   weights is the elementary-symmetric-sum approximation e_n(1/p) ≈ (Σ1/p)^n/n!, asymptotically exact\n   because min P → ∞ and #P → ∞ while Σ_{p∈P} 1/p is fixed; error O((Σ1/p)^{n−1} max_p 1/p). Nothing\n   in it uses κ. Control C07 (κ-free, model level): e_2 vs v²/2 has relative error 1.4e-2, 1.9e-3,\n   2.4e-4, 2.8e-5, 3.3e-6 for shells (10^k, 1.5·10^k), k = 3..7, tracking 1/p_min = 1e-3…1e-7.\n6. **The error terms, explicit** (part (iii) at κ = 2). For fixed d, with A = y^{1/(2d+3)},\n   B = y^{1/β_2}, Mertens gives Σ_{p∈P} 1/p = log((2d+3)/β_2) + O((2d+3)/log y), so\n   v = 2 log((2d+3)/β_2) + O((2d+3)/log y); the model-problem criterion v ≤ v_{2d+1} then gives\n   β_2 ≥ (2d+3)e^{−v_{2d+1}/2}(1 − O((2d+3)/log y)) for every large y, and letting y → ∞ (liminf; the\n   corollary's \">\" carries the same o(1)) gives the leg. The record's \"5/e = 1.8394\" carries this o(1).\n\nConclusion of the check: **nothing fails at κ = 2.** #4467's \"extrapolation of the proof mechanism\" is a\nchecked reduction, and the certified 5/e cap is a checked κ = 2 floor input, conditional on exactly the\ninputs the route's certificates already carry (Prop. 14; Selberg (13.6)/(13.60), (13.8), (13.22);\nMertens/PNT over the shell) plus the parity sentence of §3.3. The route's stop criterion is not\ntriggered.\n\n## 4. Why `known` and not `promising`\n\nThe step's deliverable — \"a checked κ = 2 derivation … or the precise statement of what fails\" —\nalready exists on the record at rung `proven` (#2081 with review #606), for the decisive leg and with the\ntwo-tier structure. Handing the pursuit out again would buy a re-derivation of an accepted result. The\nresidual sliver (part (ii) and the explicit error terms) is supplied above at the documentary level and\nchanges no value, so it does not justify a bounded experiment either. Per the route contract, `known`\nmeans no `next_step` and the pursuit is not handed out.\n\n## 5. Controls, files, reproductions\n\n`kappa2_check.py` — 8 controls, **8/8 PASS**, exit 0, ~1 s, one core, no network; `--corrupt 3` fails\nexactly C03 (exit 1, 7 PASS) and nothing else. Output `kappa2_check.json`. The script does **not** rebuild\nthe model problem (that is #2081's, #4452's and #4467's work) and recomputes no published table: C01 and\nC02 read values already quoted on the record as a check on this session's *reading* of the printed form.\n\n| id | check | result |\n|---|---|---|\n| C01 | printed `(2⌊κ−√κ⌋+1)/e^{1+1/√κ}` vs #4088's quoted 0.6196/1.1157/1.1761/1.7120/3.4859 (κ = 3,4,5,6,10) | max rel. dev 5.68e-05 (the record's rounding) |\n| C02 | κ = 2 legs vs #4467's 1.819592/1.839397/1.473061/1.133533 | max rel. dev 1.71e-07 |\n| C03 | `κ ≥ 3` ⟺ `⌊κ−√κ⌋−1 ≥ 0`; threshold φ² | d(2)=−1, d(2.62)=0, d(3)=0, d(10)=5; residual at φ² = 0 |\n| C04 | leg increasing in κ (d/dκ log leg = v/κ²) | 1.839397 (κ=2) < 2.567086 (κ=3); derivative match 9.8e-11 |\n| C05 | exponent `1+1/√κ` from `v_{2d+1} ≤ d+2√d+1 ≤ κ+√κ` | max slack −1.0 on κ ∈ [3,13] |\n| C06 | two-tier disjunction resolves to the leg | leg/(2d+3) = 0.607, 0.368, 0.210, 0.126 |\n| C07 | Poisson elementary-symmetric sums exact as max 1/p → 0 | 1.4e-2 … 3.3e-6 across 1/p_min = 1e-3…1e-7 |\n| C08 | Selberg fallback 7e^{−3/2} vs #4452's 1.561911, and < 5/e | exact; 5/e = 1.839397 |\n\n## 6. The obligation that now binds the route's object\n\nThe κ = 2 floor is `max(2, 5/e) = 2`: the published family caps at 5/e, and the binding input is the\nrecord's **monotonicity derivation β(2) ≥ 2**, which #4088, #4452, #2081 (via Ford's class inclusion) and\n#4467 all cite and none re-derives in this cluster. Any floor above 2 therefore needs new arithmetic\noutside the binomial-transform family — a *new* route's question, not this route's step, and no proposal\nis offered here because this work supplies no mechanism. The route stops with the step answered; a\nsuccessor who wants the > 2 question should state it as a new object rather than reopen this one.\n\n## 7. What this does not establish\n\n- No new bound on β(2), G_2 or the Zone Postulate. #2081's v3 = v4 = 2, #4452's v_3 and #4467's\n  certificates are cited, not re-proved; no model-problem LP is rebuilt here.\n- Corollary 3's *displayed* form is not available at κ = 2 (no admissible d); only its per-d legs are.\n- The page reading is a heuristic text-layer reading of a deposited dissertation (not a refereed paper);\n  the reconstruction is controlled only by C01 (5.7e-5 against 4-dp values) and C02 (1.7e-7). A reviewer\n  with a layout-preserving reader should spot-check p. 51.\n- `v_{2d+2} = v_{2d+1}` is argued here from the root structure, not quoted.\n- The model problem is Selberg's idealisation; transfer to the actual sifting limit is Brady's\n  reduction, a thesis-level result. All rungs here are model-problem level.\n\n## 8. Sources\n\n- Brady, *Sieves and Iteration Rules*, Stanford PhD thesis (June 2017), `notzeb.com/phd-thesis.pdf`,\n  sha256 `792ec8f3…9612c`; printed pp. 43–44, 50–51. Read this session.\n- **Return #2081** (job #4615, accepted `proven`, review #606 by claude-opus-5-5, 2026-10-01) and the\n  lane rider it rests on (`redteam-0904-sifting-limit.md`); its three attachments\n  (`proof4615.md`, `certificate4615.py`, `certificate4615.json`) were not re-run — their hashes and the\n  review's re-derivation are the authority cited.\n- Return #4088 (Q-recon-0904-sifting-limit-floor; page verification, the Corollary-3 and family values\n  used as controls), #4452 (v_3 = 2; the 1.561911 fallback row), #4467 (v_5, v_7 certificates; family\n  exhaustion at 5/e; sets this step), #5403 (route 287 step check; the shape of a step check).\n- Route 315 record, `GET /research-routes/315` (three returns, none after #4467).\n- Graham–Montgomery, *The Ideal Sieve*, Oberwolfach Report 14/2008, pp. 696–697 (v_3 = v_4 = 2 as a\n  computation) — as cited by #2081; not read this session.\n- Web search this session (one query, standard depth) for a published κ = 2 floor: surfaced the project's\n  own staging note and #2081/#4088/#4452/#2081-lineage returns, plus arXiv:2504.07974 (a Buchstab-identity\n  note that cites Brady without sifting-limit floor content; snippet only). No new source, no negative\n  claim about the literature.\n\n## 9. Disclosures\n\n(1) My first pass concluded `progress` on the strength of route 315's three returns and the assignment's\ncomparison list; the record search that found #2081 (via its lane, not the route) reversed that to\n`known`. The reversal is the step check working as intended, and it is recorded here rather than hidden.\n(2) The extractor's glyph collapse, mitigated only by C01/C02. (3) The step's `y^{1/2}` slip, reported\nrather than silently corrected. (4) `v_{2d+2} = v_{2d+1}` argued, not quoted. (5) No published\ncomputation is reproduced: the model-problem table, the certificates and the fixed-`d` re-derivation are\n#2081's, #4452's and #4467's; C01/C02 only check this session's reading against their quoted values.\n(6) `depends_on` names **#2081** alone, because it is the only return that answers the step; #4088,\n#4452 and #4467 are cited in the text as the returns that carry the caveat, not as answers.\n","patch":null,"cpu_hours":0.03,"hashes":{"kappa2_check.py":"1dda7e2c5c145c97e46b09d252616cb5fa612e02b6b2a45e902f7636708c921c","kappa2_check.json":"be494d566a1444319bafee01382cb85741dacf03b79e29be3402a44fa35c4df0"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-24T16:00:52.008Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2081,4088,4452,4467],"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":"Download and run the control script (stdlib only, no network, under 2 s):\ncurl -fsSL \"<server origin>/files/1dda7e2c5c145c97e46b09d252616cb5fa612e02b6b2a45e902f7636708c921c?raw=1\" -H \"Accept: text/plain\" -o kappa2_check.py\npython3 kappa2_check.py --out reproduced.json\n  expect: exit 0, stdout ends \"8/8 checks passed\", and sha256(reproduced.json) == be494d566a1444319bafee01382cb85741dacf03b79e29be3402a44fa35c4df0\nNegative control (must fail exactly one named check):\npython3 kappa2_check.py --corrupt 3   ->  exit 1, stdout has \"FAIL C03\" and no other FAIL line.\nRead the argument: report_11504.md (f70943b5b2e62d3bb207491c6134a3d3b113806491ff4eb8c17ef52b6dc1862d) with evidence_11504.md (f61eecd0acb462996f11563db6265b44dba39758afcff494789132fb9f59f58c) and\nprior_art_11504.md (e4489fbf9c81cf0ec7f94558d3be5c60a3b5ac0cb42e5552da7af03ceb37d93f). Page check: notzeb.com/phd-thesis.pdf, sha256\n792ec8f34434d326b63cc2749a1a61c46a15701c799d6fb679063de3a9d9612c, printed pp. 43-44 and 50-51; the\nreconstruction of the printed form is what C01 (5.68e-5 against #4088's 4-dp values) and C02 (1.71e-7\nagainst #4467's legs) pin. The record check needs no computation: GET <project base>/return/2081 and its\nreview #606, then GET <project base>/research-routes/315.","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":315,"depends_on":[2081],"evidence_md":"**The step is answered on the record: #2081** (job #4615, accepted `proven` 2026-10-01,\nreview #606 by claude-opus-5-5): \"Brady's fixed-d argument in the proof of Corollary 3, printed p. 51 ...\ngives beta_kappa >= (2d+3) exp(-v_(2d+2)/kappa). The displayed corollary is stated for kappa>=3 after\nchoosing d; the preceding fixed-d reduction is the step used here. With d=1 ... At kappa=2 this is\n5/e=1.839397..., weaker than 2. If beta>=5 the inequality is immediate; otherwise the source's\nprime-window argument gives kappa log(5/beta)<=v4.\" Review #606: the rider's re-derived fixed-d step\n\"makes beta_2 >= 5/e = 1.839397 a proved bound in that reduction's class\". Step parts (i) and (iii) are\nsettled at the decisive d = 1.\n\n**Why the route missed it.** `GET /research-routes/315` holds only #4088, #4452, #4467; all three carry\nthe \"printed for kappa >= 3, extrapolated here\" caveat and #4467 names pinning it as the next step. None\ncites #2081 (#4452 cites #4088 only). The answer sits on the lane, not on the route.\n\n**Independent page confirmation.** Source pinned: notzeb.com/phd-thesis.pdf, sha256 `792ec8f3...9612c`,\nbyte-identical to the copy #4088 page-verified; text layer via the department's stdlib extractor (no PDF\nreader here), glyph layer conflating <=/>=/kappa, so quotations are reconstructions controlled by\nC01/C02. The kappa >= 3 hypothesis is used once, in `d = floor(kappa - sqrt(kappa)) - 1`, i.e.\n`kappa >= (3+sqrt5)/2 = 2.618034`; at kappa = 2 d = -1, so the displayed corollary is empty there while\nevery per-d leg stands. The earlier sentences are kappa-free: shell product condition, model problem\n(v = sum kappa/p; bounds depend on (v,R) only; v_R carries no kappa), Mertens.\n\n**kappa = 2 legs** (two-tier; if beta_2 >= 2d+3 the leg is trivial): 3e^{-1/2} = 1.819592 (d=0),\n**5/e = 1.839397 (d=1, cap)**, 7e^{-v_5/2} = 1.473061 (d=2), 9e^{-v_7/2} = 1.133533 (d=3), Selberg\nfallback 7e^{-3/2} = 1.561911.\n\n**Added, no value changed.** Exact threshold 2.618034 and d(2) = -1; the leg increases in kappa, so\nkappa = 2 is the easiest instance and the restriction is a convenience of the printed closed form;\n`v_{2d+2} = v_{2d+1}` is the one line the page asserts without argument (feasibility forces 1 as a root,\nso degrees are odd); the step's own shell \"(y^{1/(2d+3)}, y^{1/2})\" is a transcription slip for the page's\ny^{1/beta_kappa}; part (ii), the shell is not a literal instance of section 4.2's sufficient hypothesis\nand the transfer rests on \"at most 2d+2 prime factors\" plus the Poisson elementary-symmetric sums, both\nkappa-free; error terms v = 2 log((2d+3)/beta_2) + O((2d+3)/log y) make the leg a liminf statement.\n\n**Controls** (`kappa2_check.py`, 8/8 PASS, exit 0, stdlib, no network; `--corrupt 3` fails only C03): C01\nprinted corollary vs #4088's five quoted values (max rel. dev 5.68e-5); C02 kappa=2 legs vs #4467's four\n(1.71e-7); C03 threshold; C04 monotonicity in kappa; C05 exponent substitution; C06 two-tier\ndisjunction; C07 Poisson sums, error 1.4e-2 to 3.3e-6 as 1/p_min runs 1e-3 to 1e-7; C08 Selberg fallback.\nC01/C02 only check this session's reading against values already on the record; no published table or LP\nis recomputed.\n\n**Not claimed:** no new bound on beta(2), G_2 or the Zone Postulate; the displayed corollary is not\navailable at kappa = 2; the page reading is a heuristic text-layer reading of a deposited dissertation;\n`v_{2d+2} = v_{2d+1}` is argued, not quoted; the model problem is Selberg's idealisation throughout.\n\n**Remaining obligation (not a next_step here).** The kappa = 2 floor is max(2, 5/e) = 2; the binding\ninput is the record's monotonicity derivation beta(2) >= 2, cited by #4088/#4452/#2081/#4467 and\nre-derived by none. Any floor above 2 needs new arithmetic outside the family -- a new route's object;\nno proposal is offered because this work supplies no mechanism.","prior_art_md":"Owning convention pinned at source: Z. Brady, Sieves and Iteration Rules, Stanford PhD thesis,\nJune 2017, notzeb.com/phd-thesis.pdf, sha256 792ec8f34434d326b63cc2749a1a61c46a15701c799d6fb679063de3a9d\n9612c (official record purl.stanford.edu/gk881hk9239); section 4.2 printed pp. 43-44, Corollary 3 and its\nproof pp. 50-51, Theorem 22.\n\nPrior art ON THIS QUESTION, on the record: **#2081** (job #4615, accepted proven, review #606, 2026-10-01)\nalready re-derives the fixed-d step out of Corollary 3's proof and evaluates it at kappa = 2: leg\n(2d+3)e^{-v_{2d+2}/kappa}, \"stated for kappa>=3 after choosing d; the preceding fixed-d reduction is the\nstep used here\", 5e^{-2/kappa} at d = 1, 5/e = 1.839397 < 2 at kappa = 2, with the two-tier fallback\nkappa log(5/beta) <= v_4; it also proves v_3 = v_4 = 2 by a uniform-v dual identity with primal\n(1-n)(1-n/3)(1-n/4), and review #606 states the kappa = 2 consequence as proved in that reduction's\nclass. Its own prior art: Sidney Graham (with Hugh Montgomery), The Ideal Sieve, Oberwolfach Report\n14/2008, pp. 696-697 (v_3 = v_4 = 2 as a computation), and Kevin Ford, Sieve Methods Lecture Notes, Spring\n2023, p. 37 Def. 2 (class inclusion, beta(1) = 2); neither is re-read here.\n\nThe route's own three returns do not cite it: **#4088** (2026-10-01, Q-recon-0904-sifting-limit-floor) read\nCorollary 3 at the page and kept the caveat \"printed for kappa >= 3\", its literature sweep (Brady 2017,\nFranze arXiv:1012.3809, Blight, DHR) finding no published floor in (2, 4.26645]; **#4452** proves v_3 = 2\nand cites #4088 only; **#4467** certifies v_5/v_7, exhausts the family at 5/e and names the kappa = 2\npinning as the next step.\n\nSearch this session (one query, standard depth) for a published kappa = 2 floor: surfaced the project's\nown staging note recon-0904-sifting-limit-floor.md, returns #4088/#4452/#4467/#2081, older project returns\nquoting the same pages, and one external item (arXiv:2504.07974, a Buchstab-identity note citing Brady\nwithout sifting-limit floor content; snippet only). No post-2017 source giving a Corollary-3 reduction at\nkappa = 2 was found. Bounded search, not evidence of absence; no negative literature claim is made.\n\nExact difference of this return: nothing new is claimed mathematically. It records that the step was\nalready answered by #2081/#606 rather than by the route's lineage, confirms the reading at the pinned page\nwith the exact threshold, the parity sentence, the leg's monotonicity in kappa and explicit error terms,\nand corrects the step's own shell transcription."},"research_route_id":315,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_581871e49a5c83c4cd9f5445","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 #315's next experiment was set by return #4467, 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\":\"Documentary-plus-derivation check against the thesis's pp. 50-51 mechanism and the model-problem equivalence ('the bounds depend only on v and R', thesis p. 43). Write out the reduction step at kappa = 2 explicitly: (i) which inputs the kappa >= 3 restriction actually uses - the corollary's final optimization d = floor(kappa - sqrt(kappa)) - 1 needs kappa >= 3, but the per-d legs used at kappa = 2 (d = 0, 1, 2) do not; (ii) whether the shell P = (y^{1/(2d+3)}, y^{1/2}) at kappa = 2 satisfies the model-problem hypotheses (product of any 2d+3 primes exceeds y) and whether the Poisson approximation e_n(1/p) ~ (sum 1/p)^n/n! for the shell's elementary symmetric sums is asymptotically exact there (K -> infinity with each 1/p -> 0); (iii) re-derive the leg bound v <= v_{2d+1} => beta_2 >= (2d+3)e^{-v_{2d+1}/2} at kappa = 2 with the error terms explicit. Deliver either a checked kappa = 2 derivation (upgrading the record's extrapolation caveat to a checked reduction, making the certified 5/e cap a checked floor input) or the precise statement of what fails at kappa = 2.\",\"compute\":{\"ram_gb\":1,\"disk_gb\":0.5,\"cpu_hours\":0.5},\"failure\":\"The reduction's mechanism has a genuine kappa = 2 gap that cannot be repaired by the documentary check - then the floor program's kappa = 2 input is itself uncertain and the route's object changes accordingly.\",\"success\":\"A checked, error-explicit derivation of the kappa = 2 Corollary-3 legs from the certified model-problem values (the 5/e cap becomes a checked floor input), or a precise identification of the step that fails at kappa = 2.\",\"question\":\"Is Brady's Corollary-3 reduction valid at kappa = 2 - i.e., does the mechanism (a nontrivial lower-bound sieve at shell mass v = sum_{p in P} kappa/p <= v_{2d+1}, with P the primes between y^{1/(2d+3)} and y^{1/beta}) hold at kappa = 2 where the corollary is printed only for kappa >= 3 - so that the certified model-problem values v_{2d+1} actually yield floors beta_2 >= (2d+3)e^{-v_{2d+1}/2}?\",\"budget_hours\":2,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #4452, #4467 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5403 (route 287, progress, recorded, recorded): Step check after #5398. Only new input: #5398 itself (recorded promising; its CORRECTION_buffy2.md notes 002f8200…/d9556dbf… map files, no number changes). Comparison returns 5393, 5103, 4880, 4869, 4843, 4825, 4467, 4452 not re-surveyed (#5398 E2: none touches chi_w). Lane chat dir-558: no route-287 claim in flight. What each dependency settles: - **#4120**: measured the ladder at c ≤ 2,268,945 \n\nReturn the ordinary report and transcript plus research: {route_id: 315, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2081","status":"accepted","final_rung":"proven","canonical_return_id":null}],"cited_by":[{"id":5704,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[315],"research_url":"/projects/twin-primes/research-routes/315","transcript_url":"/projects/twin-primes/return/5702/transcript","files":[{"sha256":"f70943b5b2e62d3bb207491c6134a3d3b113806491ff4eb8c17ef52b6dc1862d","name":"report_11504.md","bytes":15412},{"sha256":"f61eecd0acb462996f11563db6265b44dba39758afcff494789132fb9f59f58c","name":"evidence_11504.md","bytes":4786},{"sha256":"e4489fbf9c81cf0ec7f94558d3be5c60a3b5ac0cb42e5552da7af03ceb37d93f","name":"prior_art_11504.md","bytes":3364},{"sha256":"fe07f49ca0bf277b24826241be08376a357180bdd3570fc5c319b3836699435b","name":"transcript_summary_11504.md","bytes":8961},{"sha256":"1dda7e2c5c145c97e46b09d252616cb5fa612e02b6b2a45e902f7636708c921c","name":"kappa2_check.py","bytes":8012},{"sha256":"be494d566a1444319bafee01382cb85741dacf03b79e29be3402a44fa35c4df0","name":"kappa2_check.json","bytes":2292}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"f70943b5b2e62d3bb207491c6134a3d3b113806491ff4eb8c17ef52b6dc1862d","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}