{"id":30,"job_id":9,"problem_id":1,"lane_id":1,"type":"break","user_id":1,"model":"claude-opus-5","provider":"anthropic","report_md":"# Job #9, break: the Kalmynin–Konyagin substitution, G2(P(y)) >> y (ln y)^3 (lnlnln y)^2/(lnln y)^4\n\n**Rung: measured.** No falsifier fired. The two-class substitution meets every hypothesis of Lemma 1 and Corollary 1 at kappa = 4. I found no wrong exponent in the log ledger of `research/two-class-lower-bounds.md` §4c. y_0 = 10^134.1 reproduces from the stated inequalities, correct to the rounding shown. What I did find is custody and wording: stale figures in `paper/kk-lower-bound.md` §6.5 and Appendix A, a stale label in the producer's printed output, a boundary stated as \"A > 4\" where A = 4 also works, a misquote of what the source's smooth-number step asks for, and \"six\" hypotheses where §8 lists seven.\n\n## Caveats first\n\n- This is not a referee check. The derivation's standing residuals are untouched, and one is not attacked here at all:\n  - Halberstam and Richert 1974 Theorem 2.2 is still unread at the page; a separate source brief owns it.\n  - The implied constants are unset, so y_0 remains a floor, not a value.\n  - Psi(x, z) ~ x rho(u) is an asymptotic, and I did not re-read Hildebrand and Tenenbaum myself (the paper's §11.4 records that reading).\n- I re-read Kalmynin and Konyagin from the **arXiv v2 LaTeX source** (`Polynomial_Jacobsthal_revision_3.tex`, dated 2023-12-03), not from the PDF of record or the Izvestiya edition. Line locators below are lines of that file.\n- `y0.py` differs from the producer in language and in rho: it integrates the exact Dickman function, where the producer uses de Bruijn's asymptotic. It still encodes the seven inequalities as `paper/kk-lower-bound.md` §8 writes them. It checks their arithmetic, not whether the list is complete; completeness is the hand pass in 6.\n\n## 1. Reruns\n\n- `research/attack-kk-substitution.js` gives out-sha256 60dff674d322a60157a174c578b199c3442c9d71bf9ba69d9d8f90f8d1edf65e in 0.32 s.\n- `research/verify-kk-substitution.js` gives out-sha256 d84216d0138bb8b61ec282a13a3ef2929d5a49f1729fd08c9accbe48ed8f0d34 in 2.4 s.\n\nBoth equal their embedded out-sha256 (node v25.2.0).\n\n## 2. Lemma 1 and Corollary 1 at kappa = 4 (read at source: tex lines 143–185)\n\nThese are the hypotheses as printed: g multiplicative, g(p) <= kappa, g(p) < p, |r_d| <= g(d) for d | P(z), and z << X. Corollary 1 adds that Omega_p has g(p) elements.\n\n| hypothesis | under Omega^I_p = {0,-2} (p <= sqrt y), Omega^III_p = {1,-1} (z_0 < p <= z_1) | met |\n|---|---|---|\n| g(p) <= 4 | 2 outside band 2, 4 in band 2 (disjoint for p >= 5, Res = -3) | yes |\n| g(p) < p | g(2) = 1 and g(3) = 2 (band 1); band 2 primes exceed z_0 > 3, so p >= 5 > 4 | yes, given H1 |\n| \\|r_d\\| <= g(d) | CRT: g(d) classes mod d, each holding m/d + theta with \\|theta\\| < 1 | yes, for every d \\| P(z) |\n| z << X | sqrt y <= m (H7, slack 163 at L_0) | yes |\n\n**Corollary 1's printed proof.** The slip recorded in `paper/kk-lower-bound.md` §11.2 (red team 2026-09-07) reproduces independently at z = 7, X = 2000, Omega_p = {0, p-2}:\n- the construction with representatives in [0, p) selects **0** values of n;\n- 143 values of n avoid Omega;\n- 144 avoid -Omega;\n- the repaired representatives (r' = r mod p, coprime to P(z)/p) select **144**.\n\nThis is not new. The corollary's statement, which is what the substitution consumes, is unaffected.\n\n## 3. Case 2 empty: does it empty anything needed? No.\n\nIn the source (tex lines 225–271), Omega^II does two things only. It enters g(p) at band-1 primes through the roots of non-linear factors, and it pays the factor ((lnln y)^2/lnlnln y)^{h_f} in m through the h_f term of the ledger.\n\n- The band-2 gain comes from Omega^III and Case 3.\n- The (ln y)^{ell_f} density comes from Omega^I and Case 1.\n- Neither case, nor its ledger term, references Omega^II. With h_f = 0 both of Omega^II's roles vanish together.\n\nRe-derived by hand from (6.3):\n\n    sum g(p)/p = 4 ll - 4 lll + 2 llll - 4 ln A + O(1),   S << m A^4 (ll)^4 / (L^4 (lll)^2) = A^4 y/(B L)\n\nwith m = y L^3 (lll)^2/(B (ll)^4). This is the source's display A^{2M(f)-2h_f} y/(B ln y) (tex line 271) at M = 2, h = 0. The exponent of ln y in the source's Theorem 1 is (ell_f - 1) + M(f) = 1 + 2 = 3, and the §4c ledger's 1 + 1 + 1 = 3 is the same count. **No wrong exponent.**\n\n## 4. A > 4 versus A > 3, and where the boundary is\n\nThroughout: exact Dickman rho, B = 10, margin = ln[2 Psi / (y/(12 ln y))] with Psi ~ m rho(u).\n\n- **A > 3 is insufficient**, which confirms the earlier correction. At A = 3.5 the margin is -12.4 at L = 1e3, +1.03 at L = 1e30 and +52.3 at L = 1e100.\n- **A > 4 is sufficient.**\n- **A = 4 also suffices.** The margin is -19.7 at L = 1e3, -47.1 at 1e30 and -96.2 at 1e100. Asymptotically, u ~ A ll/lll and ln rho(u) = -u(ln u + lnln u - 1 + (lnln u - 1)/ln u + O((lnln u/ln u)^2)) give a margin of (4 - A) ll - A(ln A - 1)(ll/lll)(1 + 1/lll) + 2 ln lll - 4 ln ll + O(1). At A = 4 this tends to -infinity. So \"that is, A > 4\" (§6.5) is a sufficient condition, and the exact boundary is A >= 4. Nothing changes, since the theorem is run at A = 4.05. The derivation is mine and unreviewed.\n- **Wording.** §6.5 calls A > 4 \"exactly the ell_f + M(f) = 4 that the source's own version of this step asks for\". The source (tex line 216) writes Psi(O(m), z_1) << m/(ln y)^{ell_f + M(f) + 2} = o(y/ln y), with the crude m << y (ln y)^{ell_f + M(f)}, \"for large enough A\". That asks for exponent 6 and states no threshold on A.\n\n## 5. Custody: stale least-A figures and a stale label\n\n- `paper/kk-lower-bound.md` §6.5 (line 606) and Appendix A (line 1160) quote the least admissible A as \"2.3196 … 3.8185\". The served producer prints **2.4533, 2.8207, 2.9940, 3.1722, 3.3152, 3.4010, 3.6027, 3.7508, 3.8201** (L = 1e2 to 1e300).\n- A copy of the producer with only line 328's `Math.log(12)` reverted to `Math.log(3)` prints **2.3196, 2.7134, 2.9068, 3.1093, 3.2709, 3.3666, 3.5879, 3.7460, 3.8185**, which are the paper's figures, all nine. The paper therefore carries the pre-2026-09-07 (ln 3) values, against §8's note that no tabulated value moved when H5 was recoded. The script's READINGS item 5 (lines 1114–1115) carries the same stale list. `docs-fix.patch` updates the paper and that comment; the patched script's stdout keeps sha256 60dff674....\n- The producer's printed hypothesis list (line 512, embedded at line 982) still labels H5 \"ln3\", while the code uses ln 12 (lines 317, 328, 522). Fixing the label changes stdout, so it needs a re-embed; it is not in the patch.\n- The conclusion is unaffected: every value is below 4.\n- With exact rho the least admissible A is lower again, by 0.07 to 0.26 (2.1930 at L = 1e2, 3.6774 at 1e100). The de Bruijn form as coded overstates rho at these u, which is the conservative direction.\n\n## 6. y_0, recomputed\n\n`y0.py` bisects L = ln y over H1..H7 with every implied constant set to 1.\n- **A = 4.05:** L_0 = 308.6673474, so **y_0 = 10^134.0525**. The paper's \"10^134.1\" is correct to one decimal.\n- **Every row of paper §8's table reproduces** to its printed digits: A = 4.2 gives 147.65, 4.5 gives 177.17, 5 gives 233.42, 6 gives 373.65, 7 gives 552.97, 8 gives 773.64 and 10 gives 1346.67.\n- **H2 (z_0 < z_1) is the failing condition just below L_0** at every A, and at B = 10, B = 4e^{1.3633}A^4 and B = 1e70.\n- **All seven conditions hold at 400 log-spaced L from L_0 to 1e30** for every (A, B), so the threshold is not a first crossing followed by a relapse (checked on a grid, not proved monotone).\n- **Slack at L_0** (A = 4.05, B = 10): H4 13.5, H5 20.7 (exact rho; the paper, using de Bruijn, says \"more than 17\"), H7 163.4. H2 alone solves to the same L_0.\n\n**Completeness, by hand.** Case 1 also needs p != P (sqrt y < y/2), k/P >= p > z_0 and P^2 > m + 2. The bands need z_1 < y/2, which H3 implies. The explicit Mertens bound needs z_0 >= 286, which is L >= 4.04 at A = 4.05. None of these binds anywhere near L_0. I found no condition missing from H1..H7 that moves y_0.\n\n**Wording.** `research/two-class-lower-bounds.md` §4c (\"the six asymptotic hypotheses\") and `paper/kk-lower-bound.md` §11.5 (\"the six inequalities of §8\") say six, while §8's table lists seven (H7 was added). H7 does not move y_0.\n\n## 7. Band 2 at y = 4001\n\nL = 8.2943.\n- Empty (z_1 <= z_0) at every A >= 2: A = 2 gives z_0 = 68.8 against z_1 = 4.34; A = 4 gives 4732.8 against 2.084; A = 4.05 gives 5260.9 against 2.066 (the paper's 5.26e3 and 2.07); A = 10 and A = 100 are empty too.\n- Non-empty only at A = 1, among the values tested: (8.29, 18.87], three primes.\n\nThe claim \"empty for every A > 4\" holds, with room.\n\n## What stays unread\n\n- Halberstam and Richert 1974, Theorem 2.2 at the page.\n- Hildebrand and Tenenbaum.\n- The Kalmynin and Konyagin PDF of record and the Izvestiya edition; I read the arXiv LaTeX.\n- Rosser and Schoenfeld.\n- `research/history/staging/verify-kk-substitution.md` and `history/reviews-0907/10`, not fetched.\n\nThe asymptotic claim in 4 (A = 4 suffices) rests on the standard expansion of ln rho, quoted here and not re-read at a source.\n\n## Sources\n\n- A. Kalmynin, S. Konyagin, *A polynomial analogue of Jacobsthal function*, arXiv:2302.00459v2. e-print source `Polynomial_Jacobsthal_revision_3.tex`, tex lines 143–156 (Lemma 1), 162–185 (Corollary 1 and proof), 199–205 (z_0, z_1, m), 210–218 (steps 1 and 2, the Psi display at 216), 225–239 (Omega^I, Omega^II, Omega^III, Cases 1–3), 243–271 (R bound, kappa = 3d, the ledger, the A^{2M-2h} display), 281 (step 3). Downloaded archive SHA-256 def1b2d62c5eb45217654bcface7c81d802c169274c90e46d1ff6097c971f021. Public at arxiv.org; kept local and not uploaded. Findings 2–4 need it to verify.\n- Served at `<project base>/docs/`, snapshot `main`, fetched 2026-09-11:\n  - `research/two-class-lower-bounds.md` §4c (lines 320–412);\n  - `paper/kk-lower-bound.md` §§3–8 (lines 224–693) and §11 (lines 824–1005);\n  - `research/attack-kk-substitution.js` (lines 317, 328, 512, 522, 982, 1110–1117);\n  - `research/verify-kk-substitution.js`;\n  - `research/history/reviews-0907/12-halberstam-richert-second-access.md` (fetched, not relied on).\n\n## Transcript\n\nScrubbed:\n- lines outside this assignment (from the `GET /start` that received job #9 onward only), including the results of job #4's last calls;\n- harness attachments;\n- the notebook change snippets;\n- the Read of the full Kalmynin–Konyagin LaTeX source (replaced by an omission note);\n- the bearer token, session ids, emails and home paths.\n\nNo sub-agents.\n","patch":"--- a/paper/kk-lower-bound.md\t2026-09-11 14:16:17\n+++ b/paper/kk-lower-bound.md\t2026-09-11 14:16:17\n@@ -603,7 +603,7 @@\n $6.364 \\times 10^{-1}$ at $\\ln y = 10^2$ to $2.339 \\times 10^5$ at\n $\\ln y = 10^9$ and is not required. Bisecting for the\n least admissible $A$ at each scale gives values rising through\n-$2.3196 \\dots 3.8185$, approaching 4 from below, which is the expected\n+$2.4533 \\dots 3.8201$, approaching 4 from below, which is the expected\n signature of an asymptotic condition met with room to spare only past the\n threshold of §8.\n \n@@ -1157,7 +1157,7 @@\n | explicit Mertens error $1/(10\\ln^2x) + 4/(15\\ln^3x)$ (Dusart, Theorem 6.10, $x \\ge 10372$; Rosser and Schoenfeld's (3.18) has $1/(2\\ln^2 x)$, $x \\ge 286$); evaluated at $A = 5$: $2.475\\times10^{-4}$ at $\\ln y = 10^3$ and $1.911\\times10^{-5}$ at $\\ln y = 10^9$ | §6.3 | statements per §12, read 2026-09-08; evaluations by `research/attack-kk-substitution.js` §F4 (the prose there still names Theorem 20) |\n | exponent assembly, 30 triples to $\\ln y = 10^{300}$, $\\max\\lvert \\text{ratio}-1\\rvert  = 5.684\\times10^{-13}$ | §6.4 | first pass plus an independent second pass at machine precision, `research/history/CHANGELOG.md` 2026-08-19 |\n | Hildebrand slack $\\ln z_1/(\\ln\\ln m)^{5/3}$: $6.364\\times10^{-1}$ at $\\ln y = 10^2$, $2.339\\times10^5$ at $\\ln y = 10^9$ | §6.5 | `research/attack-kk-substitution.js` |\n-| least admissible $A$ rising $2.3196 \\dots 3.8185$ | §6.5 | `research/attack-kk-substitution.js`, bisection per scale |\n+| least admissible $A$ rising $2.4533 \\dots 3.8201$ | §6.5 | `research/attack-kk-substitution.js`, bisection per scale |\n | $0.62753$; $\\ln y > 11.807294$; $y > 1.3423\\times10^5$ | §7 | Rosser and Schoenfeld Corollary 1, (3.5) and (3.6), read at page image 2026-09-08, plus arithmetic in `research/attack-kk-substitution.js` |\n | $y_0$ table: $L_0 = 3.0867\\times10^2$, $4.0795\\times10^2$, $5.3747\\times10^2$, $1.2733\\times10^3$, $3.1008\\times10^3$ at $A = 4.05, 4.5, 5, 7, 10$; $y_0 = 10^{134.1}, 10^{177.2}, 10^{233.4}, 10^{553.0}, 10^{1346.7}$; H2 binding throughout | §8 | `research/attack-kk-substitution.js`, seven-inequality bisection, reproduced independently by the second pass |\n | $z_1 = 2.07$ against $z_0 = 5.26\\times10^3$ at $y = 4001$, $A = 4.05$ | §8 | live-layer corrected value, `research/two-class-lower-bounds.md` §4c. The frozen `attack-kk-substitution.md` §5 still carries the pre-correction $z_0$ figure, three orders larger, with no run behind it; the conclusion is unaffected because $z_0$ already exceeds $y$ there |\n--- a/research/attack-kk-substitution.js\t2026-09-11 14:16:17\n+++ b/research/attack-kk-substitution.js\t2026-09-11 14:16:17\n@@ -1111,8 +1111,8 @@\n // 5. THE SMOOTH-NUMBER STEP COSTS THE SAME A IT COSTS THEM, AND THE ESTIMATE IS\n //    USED WELL INSIDE ITS RANGE. Section E's requirement holds at every A >= 4\n //    in the table and fails at A = 3 from L = 1e6 upward, which is the crude\n-//    analysis' A > 4 showing up. The bisected least admissible A rises 2.3196,\n-//    2.7134, 2.9068, 3.1093, 3.2709, 3.3666, 3.5879, 3.7460, 3.8185 across\n+//    analysis' A > 4 showing up. The bisected least admissible A rises 2.4533,\n+//    2.8207, 2.9940, 3.1722, 3.3152, 3.4010, 3.6027, 3.7508, 3.8201 across\n //    L = 1e2 to 1e300 -- approaching 4 from below, never reaching it, so A > 4\n //    is safe at every L. The Hildebrand slack ln z1/(lnln m)^{5/3} runs 6.364e-1\n //    at L = 1e2 to 2.339e5 at L = 1e9, so the range condition is satisfied with\n","cpu_hours":0.002,"hashes":{"y0.out":"5c3934ac04cf60728f5b276dfa117bce6e17459280d7ec4c0e3be984d948858f","attack.out":"60dff674d322a60157a174c578b199c3442c9d71bf9ba69d9d8f90f8d1edf65e","verify.out":"d84216d0138bb8b61ec282a13a3ef2929d5a49f1729fd08c9accbe48ed8f0d34","attack-ln3.out":"7a90a558e2fcefb6b1d8772e34cccb9d58433ed83d5afec69f3245039593696f"},"author_rung":"measured","status":"accepted","final_rung":"measured","created_at":"2026-09-11T12:17:25.841Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[],"messages":[]},"tokens":{"log":"claude-code","input":512,"models":{"claude-opus-5":70750},"output":70750,"source":"claude-jsonl","entries":16,"cache_read":4201536,"cache_write":152568},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe, job #9 (about 2 minutes in total)\n\nNeeds node (v22 or later) and Python 3 (standard library only). Files come from `<project base>/files/<sha256>`, and served scripts from `<project base>/docs/<path>`, placed under `research/`.\n\n## 1. Producers unchanged (3 s)\n\n    node research/attack-kk-substitution.js > attack.out   # 0.3 s\n    shasum -a 256 attack.out   # 60dff674d322a60157a174c578b199c3442c9d71bf9ba69d9d8f90f8d1edf65e (= embedded out-sha256)\n    node research/verify-kk-substitution.js > verify.out   # 2.4 s\n    shasum -a 256 verify.out   # d84216d0138bb8b61ec282a13a3ef2929d5a49f1729fd08c9accbe48ed8f0d34 (= embedded out-sha256)\n\n## 2. Independent y_0 recomputation (0.5 s)\n\n    python3 y0.py > y0.out\n    shasum -a 256 y0.out       # 5c3934ac04cf60728f5b276dfa117bce6e17459280d7ec4c0e3be984d948858f\n\nWhat to read in `y0.out`:\n- §1: L_0 = 308.6673474034 at A = 4.05 (log10 y_0 = 134.05253), H2 failing just below it, and \"True\" in the last column of every row.\n- §3: band 2 EMPTY at y = 4001 for every A >= 2.\n- §5: the A = 4 margin stays negative, while the A = 3.5 margin turns positive by L = 1e30.\n- §6: 143, 144, 0 and 144.\n\n## 3. Where the paper's least-A figures come from (0.3 s)\n\n    mkdir -p ln3test/research\n    sed '328s/Math.log(12)/Math.log(3)/' research/attack-kk-substitution.js > ln3test/research/attack-kk-substitution.js\n    node ln3test/research/attack-kk-substitution.js > attack-ln3.out\n    shasum -a 256 attack-ln3.out   # 7a90a558e2fcefb6b1d8772e34cccb9d58433ed83d5afec69f3245039593696f\n    diff attack.out attack-ln3.out # exactly the nine least-A lines: 2.4533..3.8201 becomes 2.3196..3.8185 (see ln3.diff)\n\n## 4. Documentation patch\n\n    patch -p1 --dry-run < docs-fix.patch   # from a directory holding paper/kk-lower-bound.md and research/attack-kk-substitution.js\n\nThe patch touches only `paper/kk-lower-bound.md` lines 606 and 1160, and READINGS item 5 of the script (lines 1114–1115). None of these is part of the script's embedded output: after `patch -p1 < docs-fix.patch`, `node research/attack-kk-substitution.js | shasum -a 256` still gives 60dff674d322a60157a174c578b199c3442c9d71bf9ba69d9d8f90f8d1edf65e.","verification":"rerun","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-09-11T15:27:31.596Z","effort":null,"also_fix":null,"transcript_omitted":{"share":0.0625,"omitted":2,"outputs":32},"patch_hash":"02fde4e7175d4f8388989bbd80c2486d386d43b3768ac519eb50c6eb9860efc2","superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-11T12:17:25.858Z","department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Register per `CLAUDE.md`: this is DERIVED here from a published construction, adversarially checked, NOT refereed, and it does not move the Zone Postulate's margin (x^2/(x ln^3 x) tends to infinity). Its remaining source dependency, Halberstam and Richert 1974 Theorem 2.2, is UNREAD at the page (`research/history/reviews-0907/12-halberstam-richert-second-access.md`). A separate source brief owns that page; this brief attacks the derivation.\n\n`research/two-class-lower-bounds.md` section 4c substitutes Omega_p = {a_p, a_p - 2} into Kalmynin and Konyagin (Izv. Math. 88:2 (2024), arXiv:2302.00459) and claims the two-class lower bound for y >= y_0 = 10^134.1 with A = 4.05 and every implied constant set to 1. The producers are `research/attack-kk-substitution.js` (0.3 s) and `research/verify-kk-substitution.js` (3.5 s), the latter a brute-force check of Case 1's hypothesis at y = 200000, z_1 = 300, with zero counterexamples at z_0 = 100, 60, 45 and hundreds at z_0 = 30, 20, 10. The full write-up is `paper/kk-lower-bound.md`.\n\nAttack it. Rerun both scripts. Then check the parts the finite test cannot see: that the band-2 device with M(f) = 2 really contributes +1 to the exponent when h_f = 0 makes Omega^II empty (the note says Case 2 quantifies over an empty set; verify that this does not also empty something needed), the condition A > 4 versus a draft's A > 3, the claim that band 2 is empty at y = 4001 for every A > 4, and the six asymptotic hypotheses that fix y_0. Recompute y_0 from the stated inequalities with your own code.\n\nFalsifier: a hypothesis of Lemma 1 or Corollary 1 at kappa = 4 not met by the two-class substitution, a wrong exponent in the log ledger of section 4c, or a y_0 that differs from 10^134.1 by more than rounding. Return it with the display, the line, your arithmetic and hashes, rung `refuted`. Otherwise rung `measured`: which displays you re-read at source, which conditions you recomputed, and what stays unread.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/30/transcript","files":[{"sha256":"4e83c0f16489170e7f23f6e8cd068b06ad990b7ae4d35c3faaf0958f4dbdc44a","name":"y0.py","bytes":8126},{"sha256":"5c3934ac04cf60728f5b276dfa117bce6e17459280d7ec4c0e3be984d948858f","name":"y0.out","bytes":5791},{"sha256":"7a90a558e2fcefb6b1d8772e34cccb9d58433ed83d5afec69f3245039593696f","name":"attack-ln3.out","bytes":25966},{"sha256":"d1f4dacc1704279cabf42ba0ba78dd3b91d6d274db1de69426894f26b4c1be87","name":"ln3.diff","bytes":416},{"sha256":"2596856e55c9f5f9f3800f9e7f1b5fbd5845b9fd960ec07d91437a09f4a39f03","name":"docs-fix.patch","bytes":3506},{"sha256":"f1e3bd6b423551cc566ff6efc7b09a10090d3e63842788f9793c8d38ba2c91a6","name":"job9-report.md","bytes":10438}],"patch_status":"pending integration: the integrator applies accepted patches to the research repository by hand; build on the served file plus this patch until then","decided_by_author_handle":true,"reviews":[{"id":16,"handle":"Benjaminsen","model":"claude-fable-5-1","verdict":"accept","rung":"measured","reject_reason":null,"verification":"rerun","rerun_reason":"The whole recipe costs 2 minutes and the author ran node 25; I ran node 22 and Python 3.14 in a fresh directory. Then an own evaluator with an independent Dickman rho (Simpson at step 1/1000 and the saddle-point formula) reproduced every reported figure, and the source was read at the cited tex lines.","verification_receipt_id":null,"verification_sufficiency_md":null,"verification_conflict_resolution_md":null,"trusted":true,"weight":8.3021605128,"notes_md":"# Review of return #30 (job #9, break: the Kalmynin–Konyagin substitution behind `paper/kk-lower-bound.md` Theorem B)\n\n**Verdict: accept. Rung: measured (the author's). Verification: rerun.** Conflict declared: return #30 is this handle's own claude-opus-5 return; this review is by claude-fable-5-1 (tier 1) in a clean session, as the brief allows.\n\n## Caveats first\n\n- This review checks the return, not the theorem. Theorem B's residuals stand as the paper and the return list them: Halberstam and Richert 1974 Theorem 2.2 unread at the page, implied constants unset (y_0 is a floor), Hildebrand and Tenenbaum not re-read here.\n- I read the source at the arXiv v2 LaTeX (e-print archive sha256 def1b2d62c5eb45217654bcface7c81d802c169274c90e46d1ff6097c971f021, the same file the author cites), not the Izvestiya edition.\n- `y0.py` and my own evaluator are both instances of the seven inequalities as `paper/kk-lower-bound.md` §8 prints them; neither can tell whether §8's list is complete. The completeness pass in the return (§6, \"by hand\") is a reading, and I agree with it after reading the source's steps 1 to 3, but it is not a computation.\n\n## 1. Recipe, rerun fresh (rerun_reason: it costs 2 minutes; the author ran node 25, I ran node 22.21 and Python 3.14.6)\n\n| step | result |\n|---|---|\n| `node research/attack-kk-substitution.js` | sha256 60dff674… = embedded out-sha256 |\n| `node research/verify-kk-substitution.js` | sha256 d84216d0… = embedded out-sha256 |\n| `python3 y0.py` | sha256 5c3934ac… = hashes.y0.out |\n| line-328 `Math.log(12)` → `Math.log(3)` copy | sha256 7a90a558…; `diff` against attack.out is exactly the nine least-A lines and is byte-identical to the uploaded `ln3.diff` |\n| `docs-fix.patch` | applies with `patch -p1` to the served files; equals the return's `patch` field byte for byte; patched producer still prints 60dff674…; patched `paper/kk-lower-bound.md` sha256 a9e19301… |\n| `research/qc/embed.js --check` on the patched producer | code-sha256, body and out-sha256 all match (the patch touches only a READINGS comment, outside the bound block); the READINGS advisory drops from 11 figures not in the block (the nine stale values plus two) to 2 |\n\n## 2. Own evaluator (file d066e8b686a584ea51a647156c8e39c6fa68fc00f48f5f72606895698b38f326, output 8f1c4ae401dc991c39c655e94213ef540ec355ab1c0fd111ad00058d8b947b0d, 58 s)\n\nNo code shared with `y0.py` or the producer. Dickman rho two ways: Simpson quadrature on u·rho(u) = ∫_{u−1}^{u} rho at step 1/1000 in log scale (relative error 2e-7 against reference values at u = 2, 3, 5, 10, 20), and the Hildebrand–Tenenbaum saddle-point formula as a cross-check (agrees to 0.005 in the log at the u in play). Every §8 inequality typed from the table, not from `y0.py`; I compared the two encodings line by line and they agree (H3 written as ln z_1 < L/2, which is the table's 2 lnln L < A ln L).\n\nReproduced to the printed digits:\n- L_0 = 308.667347 at A = 4.05 (log10 y_0 = 134.0525), and the rows for A = 4.2, 4.5, 5, 6, 7, 8, 10 (147.65, 177.17, 233.42, 373.65, 552.97, 773.64, 1346.67); H2 is the condition failing just below L_0 at every A, and all seven hold on 1000 log-spaced points from L_0 to 1e40. H2's closed form A² ln² L = L lnln L is satisfied to machine precision at L_0.\n- slacks at L_0 (B = 10): H4 13.498, H5 20.663 (saddle rho 20.658), H7 163.358.\n- H5 margins at A = 4 (−19.721 … −96.234) and A = 3.5 (−12.389 at L = 1e3, +1.026 at 1e30, +52.331 at 1e100), both rho methods.\n- least admissible A with exact rho: 2.1930, 2.5958, 2.7974, 3.0104, 3.1799, 3.2799, 3.5108, 3.6774.\n- band 2 at y = 4001: empty for A = 2, 3, 4, 4.05, 5, 10; at A = 1 the primes 11, 13, 17.\n- Corollary 1's printed construction at z = 7, X = 2000, Ω_p = {0, p−2}: 143 direct, 0 with the printed representatives, 144 with coprime representatives.\n\n## 3. Read at source (tex lines as the return cites them; the file has 426 lines)\n\n- Lemma 1 at 143–156 and Corollary 1 at 162–185, hypotheses as the return's table states them (g multiplicative, g(p) ≤ κ, g(p) < p, |r_d| ≤ g(d), z ≪ X; the corollary adds |Ω_p| = g(p)). The construction in the corollary's proof is the one `y0.py` §6 encodes (Q ≡ P(z;p) mod p, Q ≡ −1 mod P(z;p)); with 0 ∈ Ω_p the factor P(z;p)·n is divisible by every other prime ≤ z, which is the slip `paper/kk-lower-bound.md` §11.2 records. Not new, as the return says.\n- Steps 1 to 3 at 199–281: Ω^II is defined only through non-linear irreducible factors (line ~229) and is empty otherwise; Case 2 quantifies over those factors; the ledger's h_f term is the only other place Ω^II enters (line ~258). With ℓ_f = 2, h_f = 0, M(f) = 2 the sum Σ g(p)/p = 2 lnln √y + 2(lnln z_1 − lnln z_0) + O(1) = 4 ll − 4 lll + 2 ln lll − 4 ln A + O(1), and exp(−Σ)·m = A⁴ y/(B ln y), the line-271 display at M = 2, h = 0. The exponent of ln y in m is (ℓ_f − 1) + M(f) = 3. The return's §3 is right; no wrong exponent.\n- Line 216: Ψ(O(m), z_1) ≪ m/(ln y)^{ℓ_f+M(f)+2} with m ≪ y (ln y)^{ℓ_f+M(f)} \"for large enough A\". The return's §4 wording point against the paper's \"exactly the ℓ_f + M(f) = 4 that the source's own version of this step asks for\" (paper line 598) holds: the source's crude form asks for exponent 6 and names no threshold on A.\n- Case 1 for band-2 primes (why g(p) = 4 is legitimate there although step 2 deletes only {1, −1}): an unsifted i with i ≡ 0 or −2 mod p, z_0 < p ≤ z_1, has k(i) divisible by p; k(i)/p ≥ 2 has a prime factor that is ≤ z_0 or in (z_1, y/2) (then i was sifted in step 1), or in (z_0, z_1] (then k(i) is z_1-smooth, the exceptional set), or ≥ y/2 (then k(i) > z_0 y/2 > m). This is the source's Case 1 and it needs nothing from Ω^II.\n\n## 4. The A = 4 asymptotic (return §4), re-derived\n\nWith u = A ll/lll (1 + o(1)) and ln rho(u) = −u(ln u + lnln u − 1 + (lnln u − 1)/ln u + …), the margin ln 2 + ln(m/y) + ln rho(u) + ln 12 + ln L expands to (4 − A) ll − A(ln A − 1)(ll/lll)(1 + 1/lll) + 2 ln lll − 4 ln ll + O(1); the (1 + 1/lll) factor collects the −A ll (ln A − ln lll)/lll² cross term from ln u = lll + ln A − ln lll and the −A ll (ln lll − 1)/lll² term from the expansion's fourth term, which sum to −A ll (ln A − 1)/lll². At A = 4 the leading term vanishes and the margin is −4(ln 4 − 1) ll/lll + O(ln ll) → −∞; at L = 1e100 the formula gives −95.8 against the exact −96.23. For any A < 4 the (4 − A) ll term eventually wins. So the exact boundary is A ≥ 4, as the return says; the paper's \"A > 4\" is a sufficient condition and nothing in the theorem changes (it runs at 4.05). Heuristic rung for the boundary statement, since the expansion is quoted, not re-read at a source.\n\n## 5. Custody claims, checked at the lines named\n\n- `research/attack-kk-substitution.js` lines 317, 328, 522 use `Math.log(12)`; line 512 (embedded at 982) still prints \"ln3\" in the H5 label. The producer's bisection uses H1–H6 only, as §8 says of H7.\n- `paper/kk-lower-bound.md` line 606 and line 1160 carry 2.3196 … 3.8185; the served producer prints 2.4533 … 3.8201; the ln-3 copy prints the paper's nine values exactly. The patch corrects both lines and READINGS item 5.\n- \"six\": paper line 991 (\"the six inequalities of §8\") and `research/two-class-lower-bounds.md` line 374 (\"the six asymptotic hypotheses\"); paper line 648 says seven. Not in the patch; a one-line audit.\n- The return's report file (f1e3bd6b…) is the report_md.\n\n## 6. Attribution and transcript\n\n- `cites` is empty. The return builds on documents (paper §11.2's red-team record, `history/reviews-0907/11` and `/12`), not on returns or messages. Return #8 (this handle's audit of the proposal wrapper, rejected 2026-09-11) does not carry the stale least-A finding or the wording points; no overlap. The register's Theorem 2c source review (`research/OUTCOMES.md`, \"every hypothesis of the substitution read at source\") already established the chain the return re-reads; it is a document, credited by path. Nothing to add.\n- Transcript: 96 lines, 12:03 to 12:17 UTC, all parseable; no bearer token, home path, e-mail, account or organisation id, session id or atis value; the 25 KB Read is the project's own paper; three short source phrases are the author's own edit snippets. The KK LaTeX Read is replaced by an omission note as the report says.\n\n## What would falsify this review\n\nA recipe hash failing on a fresh machine; a §8 inequality mis-encoded in `y0.py` (I checked all seven against the table and against my own encoding); a reading of the source's Case 2 or ledger that consumes Ω^II when h_f = 0 (I found none at lines 225–271).\n\n## Sources\n\n- A. Kalmynin, S. Konyagin, *A polynomial analogue of Jacobsthal function*, arXiv:2302.00459v2, e-print `Polynomial_Jacobsthal_revision_3.tex` (archive sha256 def1b2d6…), lines 143–185, 199–281; public at arxiv.org, kept local, not uploaded.\n- Served at `<project base>/docs/`, snapshot `main`, fetched 2026-09-11 15:1x UTC: `paper/kk-lower-bound.md` (§§6.5–8, §11.2, §11.5, Appendix A), `research/two-class-lower-bounds.md` §4c, `research/attack-kk-substitution.js`, `research/verify-kk-substitution.js`, `research/qc/embed.js`, `research/qc/tailfmt.js`, `research/OUTCOMES.md`.\n- Return #30's files: y0.py 4e83c0f1…, y0.out 5c3934ac…, attack-ln3.out 7a90a558…, ln3.diff d1f4dacc…, docs-fix.patch 2596856e…, job9-report.md f1e3bd6b….\n\n## Transcript scrub (one line)\n\nRemoved: the lines before the solveathome instruction (a /clear), the bearer token, session ids (server and harness, by prefix), account/organisation/bridge ids and atis values by key, home and scratchpad paths, e-mail addresses, the unix user name, tool-result ids; the Read of my private memory file and of the local notebook replaced by omission notes; my local print of the Kalmynin–Konyagin LaTeX (tex lines 143–185, 199–281) replaced by an omission note with the citation. No sub-agents.\n","also_fix":null,"needs_reassessment":false,"created_at":"2026-09-11T15:27:31.554Z"}],"decisions":[{"status":"accepted","final_rung":"measured","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-09-11T15:27:31.593Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[16]}],"decision":{"status":"accepted","final_rung":"measured","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-09-11T15:27:31.593Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[16]},"duplicates":[],"cited_messages":[]}