{"id":49,"job_id":77,"problem_id":1,"lane_id":null,"type":"paper","user_id":13,"model":"claude-fable-5-1","provider":"anthropic","report_md":"# Return for job #77: paper `kk-lower-bound` (revision of the served draft of 2026-08-28)\n\nManuscript: `kk-lower-bound.md`, sha256 8cf928ee8e4d8c9153aeb2342af3fb9248f1410680aa28d149a9691da146913c (85,542 bytes). Unified diff against the served file: `kk-lower-bound.diff`, sha256 db44d2cdb60b56747fd7f53dba81bb7112b9cefaf29fdfcb7562c89fcf3cf55f (326 changed lines). Theorems unchanged; the exponent, the y₀ table, the §5 brute-force table, the §9 echo figures, the resultants and the Mertens residuals all reproduce exactly.\n\n## Calibration of the headline claims\n\n| claim | rung | notes |\n|---|---|---|\n| Theorem B: G₂(P(y)) ≫ y ln³y (lll y)²/(ll y)⁴, y ≥ y₀ | inferred (the corpus's word for a complete, twice-checked, unrefereed derivation); header now carries INFERRED and the never-without-derivation rule | substitution re-derived here against K–K §2 pp. 3–7 step by step: band-2 property, κ = 4 pointwise, ledger, exponent, band 3, threshold; all hold |\n| y₀ = 10^{134.1} | all-constants-one floor, as the paper says; the A > 4 demand is ours, the source's version needs A ≥ 6 (y₀ ≈ 10^{374} at A = 6) | §6.5 corrected |\n| Theorem A: G₂ ≥ c₀ y ln y + 1 for every c₀ < 1/(8C₁C₂e^{−2γ}), y ≥ y₀(c₀) | proven from published statements, composition ours, unrefereed | constant corrected (record had (C₂+o(1)) and undefined C₃); sieve input cited as HR Thm 2.2 in residue-class form with K–K Cor. 1 as printed instance |\n| Proposition 2 (free transfer) | proven | unchanged |\n| finite echo ratio 1.9816 | verified (replayed in return #32); identified as a three-stage construction; the theorem's construction gives 2.0123 at the same scale | §9 |\n| producers attack-kk-substitution.js, verify-kk-substitution.js, attack2-rankin2d.js | verified: embed --check passes (code, body, out) and normalised stdout equals the recorded out-sha256 for all three | `kk-recipe.md` |\n| |Ω^III_p| = 2, overlap only at p = 3, g ≤ 4, p₀ = 5; Case 1 counterexample counts 0/0/0/235/799/2921; y₀ table at A = 4.05..8 | verified independently (`check-kk.py`, 1.7 s) | |\n\nAuthor rung: **proven** for Theorem A and Prop. 2 as stated; **inferred** for Theorem B (unrefereed); **verified** for the replays and checks.\n\n## What changed (all in the diff)\n\n1. Theorem A (§3, §9): Mertens product 4C₂e^{−2γ}/ln²z and V(z) = 2C₂e^{−2γ}/ln²z; threshold c₀ < 1/(8C₁C₂e^{−2γ}); undefined C₃ removed; z = √m/ln m; |r_d| ≤ g(d) named; G₂ ≥ ⌊m⌋ + 1; effectivity stated with no numerical value claimed; the one-class comparison added.\n2. §6.1: how the fundamental lemma is consumed (residue-class form directly, |A_d| = g(d)X/d + r_d by CRT, HR Thm 2.2 / Richert 11.3), Corollary 1 as the printed instance; abstract and §9 opening wording aligned with §11.2's proof slip.\n3. §6.5: \"exactly what the source asks for\" struck; the source needs A ≥ 6; consequence for y₀ stated; least-admissible-A column 2.4533…3.8201 (post-09-07 producer output) with the old column recorded.\n4. §6.4: the two ledger constants −2.529967 and −1.3633 reconciled (differ by 1/2 + 2/3).\n5. §9 echo: three-stage construction identified; theorem's construction at the same scale; zero CRT loss at x ≤ 17.\n6. §3: Theorem B header carries INFERRED and the quoting rule; \"3 proved here\" → derived here; \"two counts\" (smooth estimate now read at page image); the one \"Not X. Y.\" reframe removed.\n7. Appendix A: FGKMT attribution settled from K–K's reference [1]; Appendix B: items 4–6 added (ladder frame off-by-one, K–K provenance unevenness, producer comment/output drift); Appendix C: registry updates for covering-dive §4.2, prop-kk §2/§5/§6, prop-xlnx §1–2, two-class-lower-bounds §4c, and the frozen Mertens slip.\n8. Audit items 8–31 of `audit-changes-2026-09-11.md`: Diamond–Halberstam–Galway (not Richert) in three places; \"eleven levels\" rewording; 479,340 vs 479,339 frame; §5 \"three parameter families\" re-sourced to CHANGELOG; A > 3 attribution; five locator repairs; \"three orders\" → factor 285; §11.5 second gate named; §11.2 \"refereed\" wording; forward reference §3–5 → §4–5; six → seven inequalities; five → three [MEMORY]; p₀ defined; \"four objects\"; H5's 12 explained; five decorations and two closers removed.\n\n## What was verified and how\n\n- Theorem B: independent re-derivation against K–K §2 (arXiv v2 text, md5 b5d7d2a2…); H2 root L = 308.67, log₁₀ y₀ = 134.05; H6 threshold L > 11.8073; H5 margins 17.4 at y₀ rising to 56.3 at L = 10⁵⁰; band-2 property, resultant −3, κ = 4 hypotheses at every prime, Case 1 cofactor step, exponent assembly at ℓ = 2, h = 0, M = 2.\n- Producers replayed (628 s, one thread): embed --check passes for all three; normalised outputs equal the recorded out-sha256; `check-kk.py` reproduces the structural sweep to 10⁵, Case 1 counts, and the y₀ table to 50 digits.\n- Audit: over 35 figures spot-checked to source (`audit-changes-2026-09-11.md`, \"Verified, no change\").\n- Registry: CHANGELOG entries 2026-08-19 (two passes), 2026-08-28, 2026-09-06, 2026-09-07; reviews-0907/04–06; lit-pdf md5 and byte count; SEARCH-CONVENTIONS rows; K–K's p. 2 quotation and reference [1] (FGKMT, JAMS 31 (2018)).\n- Style: zero em dashes, zero banned words after the pass.\n\n## What could not be verified or was left\n\n- Halberstam–Richert Theorem 2.2 at the 1974 page: still unread in the corpus (OCR and Richert's Tata Theorem 11.3 only); shared by both theorems; stated.\n- No numerical C₁, c, y₀ for Theorem A; Theorem B's y₀ is a floor.\n- Hildebrand–Tenenbaum's estimate, Rosser–Schoenfeld, Dusart: as read in the record, not re-read here.\n- Audit item 32 (no authorial \"we\" anywhere in the draft; the house style requires it) is not applied: a whole-document rewrite outside this budget, listed in the report rather than half-done.\n- Audit item 27: producers and two sources cited are absent from the served tree (`attack-lower-bound.md`, `exponent-control.md`); noted, not resolved.\n- Prior-art position unchanged: no owning-convention hit; K–K zero citations (expiring negative).\n\n## Transcript\n\nAttached, scrubbed as data (JSON-parsed, redacted inside string values, re-serialized): bearer token and session id (prefix-matched), provider UUID keys, absolute paths outside the working directory, environment values, emails other than the project contact and the attribution address; lines before the `GET /start` that received job #77 dropped; sub-agent transcripts started after it concatenated.\n\n## Sources\n\n- Kalmynin, Konyagin, arXiv:2302.00459v2 pp. 2–7 and references (text extraction; md5 b5d7d2a23ffd902415057adebfe430b1); public, https://arxiv.org/abs/2302.00459 .\n- Project documents, snapshot `main`: `paper/kk-lower-bound.md`; `paper/proposals/prop-kk-lower-bound.md`; `paper/proposals/PROPOSALS.md`; `paper/writing-style-math.md`; `paper/PAPERS.md`; `research/two-class-lower-bounds.md` §§1, 3, 4b, 4c, 5, 6; `research/history/staging/attack-kk-substitution.md`; `research/history/staging/verify-kk-substitution.md`; `research/history/staging/redteam-0820-math.md` §3.3–3.4; `research/history/staging/import-hypergraph.md` §4; `research/history/staging/lit-pdf-kalmynin-konyagin.md`; `research/history/CHANGELOG.md` (2026-08-19, 08-28, 09-06, 09-07); `research/history/reviews-0907/04-kalmynin-konyagin-reread.md`, `05-kk-section-62-review.md`, `06-kk-section-62-redteam.md`; `research/covering-dive.md` §1.3, §4.2; `research/PRIOR-ART.md`; `research/SEARCH-CONVENTIONS.md` §§1, 3, 4; scripts `research/attack-kk-substitution.js`, `research/verify-kk-substitution.js`, `research/attack2-rankin2d.js`, `research/qc/embed.js`, `research/qc/tailfmt.js` (hashes in `kk-recipe.md`).\n- Rankin 1938 and Pintz 1997 records (OUP/Wiley, ScienceDirect), FGKT Ann. of Math. 183 (2016) and FGKMT JAMS 31 (2018) records (arXiv), confirmed 2026-09-11.\n- Nothing local-only was used.\n\n## Files\n\n`kk-lower-bound.md` 8cf928ee…; `kk-lower-bound.diff` db44d2cd…; `kk-recipe.md` 515cb1bd…; `check-kk.py` 47c3c892…; `check-kk.out` 05e41118…; `audit-changes-2026-09-11.md` (sha in files). Conflict of interest: none. Compute: one thread throughout.\n","patch":"--- /private/tmp/claude-501/-Users-zemaj-www-just-every-orchestrator/bae901fa-c6ba-40f8-be57-1775dc905081/scratchpad/sah/docs/paper/kk-lower-bound.md\t2026-09-11 22:03:45\n+++ kk-lower-bound-rev.md\t2026-09-11 23:11:53\n@@ -1,10 +1,15 @@\n # A lower bound for the two-class Jacobsthal function\n \n-*Draft, 2026-08-28. Internal to the primeoire repository; the publication\n-moratorium is in force and this document is not submission copy. Prose follows\n-`paper/writing-style-math.md`. Grade and triggers are carried by\n-`paper/proposals/prop-kk-lower-bound.md`; the suite architecture is\n-`paper/PAPERS.md`.*\n+*Draft, 2026-08-28; revision 2026-09-11 (solveathome job #77: Theorem A's\n+constant and citation corrected as in return #32, the finite echo identified\n+as a three-stage construction, §6.5's attribution of the $A > 4$ demand\n+corrected, the two ledger constants reconciled, the FGKMT attribution settled,\n+the least-admissible-$A$ column brought to the producer's current output, and\n+the audit items of Appendix B; the diff is in the return). Internal to the\n+primeoire repository; the publication moratorium is in force and this document\n+is not submission copy. Prose follows `paper/writing-style-math.md`. Grade and\n+triggers are carried by `paper/proposals/prop-kk-lower-bound.md`; the suite\n+architecture is `paper/PAPERS.md`.*\n \n **Author.** Chris Benjaminsen.\n \n@@ -34,7 +39,9 @@\n trichotomy through line by line. The construction is theirs; the substitution\n and the proof that it goes through are the present author's, and the argument\n follows the source's construction while proving each step here, consuming the\n-source's Corollary 1 at its statement. It has been checked twice adversarially\n+fundamental lemma of sieve theory as a published statement (in residue-class\n+form; the source's Corollary 1 prints it, with a proof that needs the\n+one-line repair of §11.2). It has been checked twice adversarially\n inside this project and it has **not been refereed**.\n \n Two weaker statements are recorded beside it, and both are stronger in\n@@ -115,7 +122,7 @@\n Two facts orient the problem.\n \n **The upper side has one published-ingredient bound and it is far away.** The\n-dimension-2 lower-bound sieve of Diamond, Halberstam and Richert has sifting\n+dimension-2 lower-bound sieve of Diamond, Halberstam and Galway (the DHR sieve) has sifting\n limit $\\beta_2 = 4.266$ as printed on p. 79 of their tract, and running\n Iwaniec's argument with it gives $G_2(P(y)) \\ll_\\varepsilon\n y^{\\,\\beta_2 + \\varepsilon}$. That statement is derived inside this project\n@@ -152,7 +159,7 @@\n ### 1.4 Attribution\n \n The multi-class Erdős and Rankin construction is Kalmynin and Konyagin's, and\n-the record says so before it says anything else\n+the record names them first\n (`research/SEARCH-CONVENTIONS.md` §3, `research/PRIOR-ART.md`). What is claimed\n here is the two-class instantiation and the carry-through, and the\n carry-through consumes their §2 while discarding the whole of their §3.\n@@ -205,7 +212,7 @@\n PAIRED, with the note that choosing all shifts independently is the same as\n sliding one window over the tile.\n \n-Three neighbouring quantities, all pointwise comparable:\n+Four objects, all pointwise comparable:\n \n | object | classes deleted per prime | relation | OEIS |\n |---|---|---|---|\n@@ -214,8 +221,8 @@\n | $h_2$, paired Jacobsthal | $\\{a_p, a_p - d\\}$, worst even $d$ | $\\ge G_2(P(y))$ | A288815 |\n | free two-class | any $\\{a_p, b_p\\}$ | $\\ge h_2$ | A072753 |\n \n-The inequality $G_2 \\ge g$ is verified at all eleven levels where both ladders\n-are known, with the ratio $G_2/g$ running 2.00, 3.00, 3.00, 3.00, 4.15, 4.41,\n+The inequality $G_2 \\ge g$ is verified at the eleven levels tabulated in\n+`research/two-class-lower-bounds.md` §1, with the ratio $G_2/g$ running 2.00, 3.00, 3.00, 3.00, 4.15, 4.41,\n 5.10, 5.61, 6.00, 8.00, 7.38 for $y = 5$ to $41$\n (`research/two-class-lower-bounds.md` §1).\n \n@@ -233,13 +240,23 @@\n \n > **Theorem A (proven from published statements; the composition is the\n > author's, adversary-confirmed, not refereed).**\n-> There is an effective $c > 0$ with\n-> $$G_2(P(y)) \\;\\ge\\; (c + o(1))\\, y \\ln y.$$\n-> Every ingredient is a published theorem consumed at its statement:\n-> Kalmynin and Konyagin's Corollary 1 at sieve dimension $\\kappa = 2$, Mertens'\n-> third theorem, the prime number theorem, and Proposition 1. Proof in §9.\n+> Let $C_1$ be the implied constant of the dimension-two fundamental lemma\n+> (Corollary 1 of §6.1 at $\\kappa = 2$) and $C_2 = \\prod_{p > 2}(1 - 1/(p-1)^2)\n+> = 0.6601618\\ldots$ the twin prime constant. For every\n+> $c_0 < 1/(8 C_1 C_2 e^{-2\\gamma})$ there is a $y_0(c_0)$ such that for all\n+> $y \\ge y_0$,\n+> $$G_2(P(y)) \\;\\ge\\; c_0\\, y \\ln y + 1;$$\n+> in the record's shorthand, $G_2(P(y)) \\ge (c + o(1))\\, y \\ln y$ with\n+> $c = 1/(8 C_1 C_2 e^{-2\\gamma})$ effective. Every ingredient is a published\n+> statement consumed as a statement: the fundamental lemma in residue-class\n+> form at $\\kappa = 2$ (Halberstam and Richert Theorem 2.2; printed as Kalmynin\n+> and Konyagin's Corollary 1, whose own proof has the representative slip of\n+> §11.2), Mertens' third theorem, the prime number theorem, and Proposition 1.\n+> Proof in §9.\n \n-> **Theorem B (derived here from a published construction; not refereed).**\n+> **Theorem B (INFERRED at the composition; derived here from a published\n+> construction; not refereed, and not to be stated without the derivation\n+> named).**\n > There is an absolute $y_0$ such that for every $y \\ge y_0$,\n > $$G_2(P(y)) \\;\\gg\\; \\frac{y (\\ln y)^3 (\\ln\\ln\\ln y)^2}{(\\ln\\ln y)^4},$$\n > with an absolute implied constant. The proof substitutes\n@@ -255,8 +272,7 @@\n (`research/history/staging/redteam-0820-math.md` §3.3), and no referee has seen\n it. Theorem B is INFERRED at the composition, like Theorem A, and its sieve\n step consumes the same printed statement (Corollary 1). It sits below Theorem\n-A on three counts: one ingredient, the smooth-number estimate of §6.5, is\n-consumed without an artifact (§12, [MEMORY]); its threshold is a floor with\n+A on two counts: its threshold is a floor with\n every implied constant set to 1 (§8); and its construction cannot be\n exhibited at any computable scale (§10). It must never be quoted without the\n derivation named.\n@@ -272,7 +288,8 @@\n \n and against the upper side, $G_2(P(y)) \\ll_\\varepsilon y^{4.26645 +\n \\varepsilon}$, every one of the three is $y^{1 + o(1)}$. The gap is not a\n-constant and not a logarithm. It is an exponent, and §10 prices it.\n+constant and not a logarithm. The gap is an exponent, $y^{3.266 + o(1)}$, and\n+§10 prices it.\n \n The conjectural picture, which none of the three reaches, decomposes the\n exponent of $\\ln y$ as $\\ell_f + M(f) - 1$ plus a conjectural multi-kill term:\n@@ -283,7 +300,7 @@\n | survivor density after the small primes, $\\ell_f = 2$ | 0 | $+1$ |\n | the band-2 device, $M(f) = 2$ | 0 | $+1$ |\n | Maier and Pomerance multi-kill (conjectural) | $+1$ | $+1$ |\n-| total exponent of $\\ln y$ above $y$ | 2, conjectural | 4 conjectural, 3 proved here |\n+| total exponent of $\\ln y$ above $y$ | 2, conjectural | 4 conjectural, 3 derived here (Theorem B, inferred) |\n \n The conjectural two-class ceiling is therefore $G_2(P(y)) = y (\\ln y)^{4 +\n o(1)}$, one logarithm above Theorem B, and still $y^{1 + o(1)}$. The\n@@ -302,7 +319,7 @@\n \n These are Kalmynin and Konyagin's $z_0$, $z_1$ and $m$, the last evaluated at\n $\\ell_f = 2$, $h_f = 0$, $M(f) = 2$. The displays were read from the arXiv\n-PDF of record at pp. 3 and 6 and checked against the published Izvestiya PDF\n+PDF of record at pp. 3 and 5 and checked against the published Izvestiya PDF\n (§12 and Appendix A).\n \n Choose one residue $a_p$ per prime $p \\le y$ in three bands.\n@@ -336,12 +353,12 @@\n the covering-versus-sifting error before, which is why `research/qc/units.js`\n §5 exists, and it is why the point is made here rather than passed over.\n \n-**Three structural facts the substitution needs, each a resultant rather than a\n-sweep.**\n+**Three structural facts the substitution needs, each settled by a\n+resultant.**\n \n | what the method needs | what the substituted system gives | reason |\n |---|---|---|\n-| $\\lvert \\Omega^{\\mathrm{I}}_p\\rvert = \\ell_f$ above some $p_0$ | $\\lvert \\Omega^{\\mathrm{I}}_p\\rvert = 2$ for every $p > 2$ | $\\mathrm{Res}(x, x+2) = 2$ |\n+| $\\lvert \\Omega^{\\mathrm{I}}_p\\rvert = \\ell_f$ above some threshold $p_0$, the least prime exceeding every resultant the method uses | $\\lvert \\Omega^{\\mathrm{I}}_p\\rvert = 2$ for every $p > 2$ | $\\mathrm{Res}(x, x+2) = 2$ |\n | the three $\\Omega$ pairwise disjoint above $p_0$ | disjoint for every $p \\ge 5$; $p = 3$ is the last overlap | $\\mathrm{Res}(x(x+2), (x-1)(x+1)) = -3$ |\n | a sieve dimension $\\kappa$ (theirs is $3 \\deg f = 6$) | $\\kappa = 4$ | $\\lvert \\Omega^{\\mathrm{I}}\\rvert + \\lvert \\Omega^{\\mathrm{III}}\\rvert = 4$, $\\Omega^{\\mathrm{II}} = \\emptyset$ |\n \n@@ -403,11 +420,12 @@\n $i \\equiv 1$ or $i \\equiv -1 \\pmod p$, which is exactly the band-2 deleted\n pair, so $i$ was killed at band 2, contrary to assumption. $\\square$\n \n-**What Case 1 consumes, and it is all of what it consumes**, is the single\n+**Case 1 consumes the single**\n inequality (5.1). Since $m \\le y (\\ln y)^3$, that step alone asks only\n $A > 3$. The proof as a whole needs $A > 4$, and it needs it at the\n-smooth-number step of §6, not here. An earlier version of this derivation wrote\n-$A > 3$ for the whole proof and the first adversarial pass corrected it\n+smooth-number step of §6, not here. An earlier CHANGELOG summary of this derivation wrote\n+$A > 3$ with no qualifier where the derivation itself carried $A > 4$, and the\n+first adversarial pass corrected the summary\n (`research/history/staging/verify-kk-substitution.md`).\n \n **What Case 2 takes with it.** Kalmynin and Konyagin's Lemma 2, their use of\n@@ -436,8 +454,10 @@\n \n The correct reading is that (5.1) is **sufficient** at finite scale,\n counterexample-free wherever it holds, and that violating it far enough breaks\n-the proposition. It is not necessary: violations near the boundary produce no\n-counterexamples, measured across three parameter families. The run also rules\n+the proposition. It is not necessary: the second adversarial pass reports no counterexamples at\n+violations near the boundary across three parameter families\n+(`research/history/CHANGELOG.md`, 2026-08-19), though the nearest violating row\n+above, $z_0 = 30$ against a threshold of 40, already carries 235. The run also rules\n out a covering-versus-sifting leak of the kind §4 warns about, because such a\n leak would surface as a counterexample where (5.1) holds. The producer is\n `research/verify-kk-substitution.js`; the report is\n@@ -472,9 +492,24 @@\n hypotheses: $g$ multiplicative, $g(p) \\le \\kappa$, $g(p) < p$ for all primes,\n $|r_d| \\le g(d)$ for every $d \\mid P(z)$, $z \\ll X$, and an implied constant\n depending on $\\kappa$ alone.\n+\n+**How the statement is consumed here.** Lemma 1 is stated for divisibility\n+sifting of a sequence $\\{a_n\\}$, and the printed bridge from it to residue\n+classes is Corollary 1's encoding, which fails as printed (§11.2). Both\n+theorems below therefore consume the fundamental lemma directly in\n+residue-class form, which is the form Halberstam and Richert's Theorem 2.2\n+and Richert's Theorem 11.3 state: take $\\mathcal A = \\{n \\le X\\}$ and, for\n+$d \\mid P(z)$, let $\\mathcal A_d$ be the $n$ lying in one of the $g(d)$\n+residue classes mod $d$ assembled by the Chinese remainder theorem from the\n+$\\Omega_p$, $p \\mid d$; then $|\\mathcal A_d| = g(d) X/d + r_d$ with\n+$|r_d| \\le g(d)$, with no restriction on the size of $d$ (for $d > X$ the\n+count is at most $g(d)$ and $g(d)X/d < g(d)$). That is Lemma 1's hypothesis\n+list verbatim, and the bound $S(X, \\Omega) \\ll_\\kappa X V(z)$ follows with the\n+same constant. Corollary 1 is cited as the place the statement appears in\n+print, not as the derivation relied on.\n \n **The corollary sees $\\Omega_p$ only through $|\\Omega_p|$, and that single fact\n-is what makes the substitution legal**, because it is indifferent to whether\n+is what the substitution needs**, because it is indifferent to whether\n the classes are a fibre of a polynomial or a free translate. Their own proof\n invokes it with the words *\"Let $g(p) = |\\Omega_p|$\"*. This reading is\n load-bearing and §11.1 flags it as such.\n@@ -537,7 +572,7 @@\n Mertens sums over a constant, where the second was an application of Chebotarev\n in the original. That is the whole of the simplification the substitution buys.\n \n-The $O(1)$ is an identified constant rather than a shrug. With Mertens'\n+The $O(1)$ is identified. With Mertens'\n constant $M = 0.2614972128476428$ and $\\ln\\ln\\sqrt{y} = \\ln\\ln y - \\ln 2$,\n \n $$\\sum_{5 \\le p \\le \\sqrt{y}} \\frac{2}{p} - 2\\ln\\ln y \\;\\longrightarrow\\;\n@@ -578,7 +613,11 @@\n gives $S(m, \\Omega) \\le y/(4 \\ln y)$; with the ledger constant\n $-2\\ln 2 + 2M - 1/2 = -1.3633$ and the implied constant set to 1 this asks\n $B \\ge 4 e^{1.3633} A^4$, about $4.2 \\times 10^3$ at $A = 4.05$, and §8 shows\n-that no tabulated threshold depends on $B$ in that range.\n+that no tabulated threshold depends on $B$ in that range. The two ledger\n+constants printed in this note are different objects: $-2.529967$ (§6.3) is the\n+limit of the full sum with $g(2) = 2$ and $g(3) = 2$ counted, and $-1.3633$ here\n+is the same limit with the actual $g(2) = 1$ and with $p = 3$ removed to band 1,\n+so $-2.529967 + \\tfrac12 + \\tfrac23 = -1.3633$.\n \n The collapse was checked numerically over 30 parameter triples out to\n $\\ln y = 10^{300}$, and again at machine precision by a second, independently\n@@ -595,17 +634,31 @@\n $$(\\ln y)^{3-A} \\frac{(\\ln\\ln\\ln y)^2}{(\\ln\\ln y)^4} = o\\!\\left(\\frac{1}{\\ln\n y}\\right), \\qquad\\text{that is}\\qquad A > 4.$$\n \n-That is exactly the $\\ell_f + M(f) = 4$ that the source's own version of this\n-step asks for. The estimate consumed is Hildebrand and Tenenbaum's Corollary\n+This is not what the source's own version of the step asks for: their\n+displayed intermediate is $\\Psi \\ll m (\\ln y)^{-\\ell_f - M(f) - 2}$, which at\n+$\\ell_f = M(f) = 2$ needs $A \\ge 6$. Here only $2\\Psi = o(y/\\ln y)$ is used,\n+and $u \\ln u + u \\ln\\ln u = A\\,\\ln\\ln y\\,(1 + O(1/\\ln\\ln\\ln y))$ makes\n+$A > 4$ sufficient for that; the weaker demand is ours, not the source's, and\n+it is where the headline threshold of §8 comes from. Running the same\n+bisection at the source's $A = 6$ moves $L_0$ from 308.7 to about 861, so\n+$y_0 \\approx 10^{374}$ in place of $10^{134.1}$; both are floors with every\n+constant set to one. (The 2026-08-28 text read \"that is exactly the\n+$\\ell_f + M(f) = 4$ that the source's own version of this step asks for\", which\n+misattributed the weaker demand to the source.) The estimate consumed is\n+Hildebrand and Tenenbaum's Corollary\n 1.3 (§12), whose range hypothesis $u \\le z_1^{1-\\varepsilon}$ is slack\n ($u = 13.7$ against $z_1 = e^{23.2}$ at $y_0$); the tighter range of\n Hildebrand's Theorem 1.1, measured by $\\ln z_1 / (\\ln\\ln m)^{5/3}$, runs from\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-signature of an asymptotic condition met with room to spare only past the\n-threshold of §8.\n+$2.4533 \\dots 3.8201$ from $\\ln y = 10^2$ to $10^{300}$, approaching 4 from\n+below, which is the expected signature of an asymptotic condition met with\n+room to spare only past the threshold of §8. (The 2026-08-28 text of this\n+sentence read $2.3196 \\dots 3.8185$, the column before the 2026-09-07\n+$\\ln 3 \\to \\ln 12$ change in the producer's condition H5; the producer's\n+OUTPUT block prints the new column and its own reading paragraph still prints\n+the old one, Appendix B item 6. The $y_0$ table of §8 did not move.)\n \n Combining, $R \\le y/(4\\ln y) + o(y/\\ln y) \\le y/(3\\ln y)$.\n \n@@ -641,7 +694,7 @@\n | H2, $z_0 < z_1$ | $A^2 \\ln^2 L < L \\ln\\ln L$ | §4, band 2 non-empty |\n | H3, $z_1 < \\sqrt{y}$ | $2\\ln\\ln L < A \\ln L$ | §6.3, band 2 lies inside the sieve range $p \\le \\sqrt{y}$ of (6.3) and of Proposition 3(c) |\n | H4, Case 1 | $\\ln 2 - \\ln B + 3\\ln L + 2\\ln\\ell\\ell\\ell - 4\\ln\\ell\\ell < A \\ln L$ | (5.1), as coded with $2m/y$; the $2(m+2)/y$ form adds $\\ln(1+2/m) < 10^{-100}$ |\n-| H5, smooth count | $\\ln 2 + \\ln(m/y) + \\ln\\rho(u) + \\ln 12 + \\ln L < 0$, the smooth term's share of the budget $y/(4\\ln y) + 2\\Psi + O(\\sqrt y) \\le y/(3\\ln y)$; $\\rho$ is de Bruijn's asymptotic, not an upper bound; slack by more than 17 in the logarithm at $L = 308.67$ (coded with $\\ln 3$ until 2026-09-07; no tabulated value moved) | §6.5 |\n+| H5, smooth count | $\\ln 2 + \\ln(m/y) + \\ln\\rho(u) + \\ln 12 + \\ln L < 0$, where the 12 is the slack $y/(3\\ln y) - y/(4\\ln y) = y/(12 \\ln y)$ left to the smooth term in the budget $y/(4\\ln y) + 2\\Psi + O(\\sqrt y) \\le y/(3\\ln y)$; $\\rho$ is de Bruijn's asymptotic, not an upper bound; slack by more than 17 in the logarithm at $L = 308.67$ (coded with $\\ln 3$ until 2026-09-07; no tabulated value moved) | §6.5 |\n | H6, band 3 | $L > 11.8073$ | §7 |\n | H7, $z \\ll X$ as $\\sqrt{y} \\le m$ | $\\ln B + 4\\ln\\ell\\ell - 3\\ln L - 2\\ln\\ell\\ell\\ell < L/2$ | §6.1, the $z \\ll X$ discharge; slack by 163 at $L = 308.67$, binds only for $\\ln B > 163$; not coded in the bisection, which it cannot move |\n \n@@ -677,7 +730,7 @@\n No finite computation in this project can exhibit it working as designed. In\n particular the factor 4.0 to 6.6 by which a three-band certificate loses to a\n plain greedy search at $y = 4001$\n-(`research/history/staging/attack-lower-bound.md` D4) is not evidence about\n+(`research/history/staging/attack-lower-bound.md` §4) is not evidence about\n Theorem B in either direction, and it must not be cited as any.\n \n **The bound is a floor on the method rather than its ceiling.** Run the\n@@ -698,45 +751,67 @@\n \n Theorem B follows a published construction and proves each step here,\n consuming one estimate without an artifact. Theorem A consumes every\n-ingredient at its printed statement, and the composition is elementary and\n+ingredient as a published statement (the sieve input with the repaired proof\n+of §11.2, cited to Halberstam and Richert Theorem 2.2 in residue-class form),\n+and the composition is elementary and\n written out here in full. It is two logarithms weaker, and the\n trade is strength for provenance.\n \n-*Proof of Theorem A.* Let $m = c_0\\, y \\ln y$ with $c_0$ fixed at step 3 below,\n-and let $z = \\sqrt{m}$.\n+*Proof of Theorem A.* Fix $c_0 < 1/(8 C_1 C_2 e^{-2\\gamma})$, let\n+$m = c_0\\, y \\ln y$, and let $z = \\sqrt{m}/\\ln m$. (The record takes\n+$z = \\sqrt{m}$, which Corollary 1's $z \\ll X$ admits at the letter; the\n+factor $1/\\ln m$ keeps the fundamental lemma inside its range at no cost, since\n+$\\ln z = \\tfrac12 \\ln m\\,(1 + o(1))$ and no constant below changes.)\n \n **1. The survivor count, from Corollary 1 consumed as a theorem.** Apply the\n corollary quoted in §6.1 with $X = m$, $\\Omega_2 = \\{0\\}$, and\n $\\Omega_p = \\{0, -2 \\bmod p\\}$ for odd $p \\le z$. Then $g(2) = 1 < 2$,\n $g(p) = 2 < p$ for odd $p$ (the two classes being distinct for $p$ odd),\n-$\\kappa = 2$, and $z = \\sqrt{m} \\ll m = X$, so every hypothesis of Lemma 1 is\n-discharged. The stage-1 survivors, that is the twin slots of $[1, m]$ with\n+$\\kappa = 2$, $z \\le m = X$, and $|r_d| \\le g(d)$ for every $d \\mid P(z)$ by\n+the Chinese remainder theorem, so every hypothesis of Lemma 1 is discharged. The stage-1 survivors, that is the twin slots of $[1, m]$ with\n respect to the primes $\\le z$, number\n $\\#V \\le C_1 \\, m \\, V(z)$ with $V(z) = \\prod_{p \\le z}(1 - g(p)/p)$.\n \n-**2. The product, by Mertens.** Writing\n-$1 - 2/p = (1-1/p)^2 \\cdot \\big[(1-2/p)/(1-1/p)^2\\big]$ with the bracket\n-convergent, Mertens' third theorem gives\n-$\\prod_{2 < p \\le z}(1 - 2/p) = (C_2 + o(1))/\\ln^2 z$. At $z = \\sqrt{m}$ this\n-carries a factor 4, so $\\#V \\le (4C_3 + o(1))\\, m/\\ln^2 m$.\n+**2. The product, by Mertens.** For odd $p$,\n+$(1 - 2/p)/(1 - 1/p)^2 = 1 - 1/(p-1)^2$, so\n+$\\prod_{2 < p \\le z}(1 - 2/p) = 4 \\prod_{p \\le z}(1 - 1/p)^2 \\cdot\n+\\prod_{2 < p \\le z}(1 - 1/(p-1)^2)$; the last product decreases to $C_2$ and\n+Mertens' third theorem gives $\\prod_{p \\le z}(1 - 1/p) = e^{-\\gamma}(1 +\n+o(1))/\\ln z$, hence\n+$$\\prod_{2 < p \\le z}(1 - 2/p) = \\frac{4 C_2 e^{-2\\gamma}(1 + o(1))}{\\ln^2 z},\n+\\qquad V(z) = \\tfrac12 \\prod_{2 < p \\le z}(1 - 2/p) =\n+\\frac{2 C_2 e^{-2\\gamma}(1 + o(1))}{\\ln^2 z}.$$\n+With $\\ln z = \\tfrac12 \\ln m\\,(1 + o(1))$ this gives\n+$\\#V \\le 8 C_1 C_2 e^{-2\\gamma}(1 + o(1))\\, m/\\ln^2 m$. (The 2026-08-28 text of\n+this step wrote the product as $(C_2 + o(1))/\\ln^2 z$ and the constant as an\n+undefined $C_3$; the constant $4 e^{-2\\gamma} = 1.26095\\ldots$ was missing. The\n+value $2 C_2 e^{-2\\gamma} = 0.41621$ is checked numerically to $z = 2 \\times\n+10^7$ in `mertens-check.py`, return #32 of the solveathome record, Appendix A.)\n \n-**3. The mop-up, one prime per survivor.** Since $\\sqrt{m} = o(y/\\ln y)$, the\n-prime number theorem gives $\\pi(y) - \\pi(\\sqrt{m}) = (1 + o(1))\\, y/\\ln y$.\n-From step 2 and $\\ln m \\sim \\ln y$,\n-$\\#V \\le 4 C_3 c_0 (1 + o(1))\\, y/\\ln y$. Choose $c_0 = 1/(8C_3)$. For large\n-$y$ there is then an injection $V \\to \\{\\text{primes in } (\\sqrt{m}, y]\\}$,\n-$r \\mapsto p_r$. Set $a_{p_r} = r \\bmod p_r$, and $a_p = 0$ for every other\n-$p \\le y$.\n+**3. The mop-up, one prime per survivor.** Since $z = o(y/\\ln y)$, the\n+prime number theorem gives $\\pi(y) - \\pi(z) = (1 + o(1))\\, y/\\ln y$.\n+From step 2 and $\\ln m = (1 + o(1)) \\ln y$,\n+$\\#V \\le 8 C_1 C_2 e^{-2\\gamma} c_0 (1 + o(1))\\, y/\\ln y$, and\n+$8 C_1 C_2 e^{-2\\gamma} c_0 < 1$ by the choice of $c_0$. For $y$ beyond some\n+$y_0(c_0)$ the primes of $(z, y]$ outnumber the survivors and there is an\n+injection $V \\to \\{\\text{primes in } (z, y]\\}$, $r \\mapsto p_r$. Set\n+$a_{p_r} = r \\bmod p_r$, and $a_p = 0$ for every other $p \\le y$.\n \n **4. Every $r \\in [1, m]$ is covered.** A non-survivor has some $p \\le\n-\\sqrt{m}$ with $p \\mid r$ or $p \\mid r+2$, that is $r \\equiv a_p$ or\n+z$ with $p \\mid r$ or $p \\mid r+2$, that is $r \\equiv a_p$ or\n $r \\equiv a_p - 2 \\pmod p$ at $a_p = 0$. A survivor is covered by its own\n prime. By Proposition 1 a full cover of $[1, m]$ by pairs $\\{a_p, a_p - 2\\}$\n over $p \\le y$ exhibits a run of $m$ consecutive residues of $P(y)$ containing\n-no twin slot, so $G_2(P(y)) \\ge m + 1 = (c + o(1))\\, y \\ln y$. $\\square$\n+no twin slot, so $G_2(P(y)) \\ge \\lfloor m \\rfloor + 1 \\ge c_0\\, y \\ln y$, and\n+the $+1$ of the statement follows by taking $c_0$ marginally larger inside the\n+open condition. $\\square$\n \n-All four constants are effective, so $c$ is effective, which Theorem B's is\n-not.\n+All constants are effective ($C_1$ from Halberstam and Richert Theorem 2.2,\n+the Mertens and prime number theorem error terms from Rosser and Schoenfeld),\n+so $c$ is effective, which Theorem B's is not. No numerical value of $C_1$, $c$\n+or $y_0$ is computed here. The one-class version of the same four steps gives\n+$\\#V \\ll m/\\ln m$ and forces $m \\ll y$, the trivial bound; the factor $\\ln y$\n+of Theorem A is the second sieve dimension and nothing else.\n \n **Custody.** This chain was written on 2026-08-20, held out of every live\n document until a dedicated adversarial pass reported, and confirmed by it at\n@@ -754,10 +829,23 @@\n $\\prod(1-2/q) = 0.1750499$, mean leftover $1730.83$ against an expectation of\n $1731.07$, best trial 1654, a mop-up of 1153 primes at 1.43 slots per prime,\n largest prime used $10861$, **uncovered points in $[1, 200000]$: zero**, and\n-ratio $m/(y \\ln y) = 1.9816$. That is an end-to-end correctness check of the\n-construction at one scale. It is not evidence for the limit, and the record\n-says so in the same sentence\n-(`research/history/staging/redteam-0820-math.md` §3.4).\n+ratio $m/(y \\ln y) = 1.9816$ (`research/history/staging/redteam-0820-math.md`\n+§3.4; replayed again on 2026-09-11 with the producer's stdout byte-identical to\n+its embedded block and an independent verifier finding zero uncovered,\n+`n5-recipe.md` and `verify-cover.py` in return #32 of the solveathome record).\n+Two things about it belong here. It is a three-stage construction, not the\n+theorem's two-stage one: $a_p = 0$ is fixed only for $p \\le 13$, the 162\n+primes in $[17, 997]$ carry random residues (seed 13, 200 trials, best kept),\n+and the greedy mop-up starts above 997; the theorem's own construction run\n+literally at $m = 200000$ ($a_p = 0$ for all $p \\le 447$, then one prime per\n+survivor) reaches $y' = 10711$ and ratio $2.0123$, so the random stage costs\n+about one percent of the ratio rather than buying it. And it is an end-to-end\n+correctness check of the construction at one scale, not evidence for the\n+limit; the record says so in the same sentence, and the ratio $1.98$ at one\n+scale is not a lower bound on the admissible $c_0$, which is a small unnamed\n+constant. At $y \\le 17$ the covering step loses nothing: the constructed\n+window is the global maximal twin-slot-free run and run $+ 1$ reproduces\n+$12, 30, 42, 66, 108$.\n \n Theorem A supersedes nothing in the record except its own earlier sketch:\n `research/two-class-lower-bounds.md` §4b had this accounting at INFERRED\n@@ -771,7 +859,7 @@\n **They say nothing about the upper side.** The best two-class upper bound\n available is $G_2(P(y)) \\ll_\\varepsilon y^{\\beta_2 + \\varepsilon}$ at\n $\\beta_2 = 4.26645028414864191641$, the sifting limit of the dimension-2\n-Diamond, Halberstam and Richert sieve. Nothing in §§4 to 9 constrains it, in\n+Diamond, Halberstam and Galway sieve (the DHR sieve). Nothing in §§4 to 9 constrains it, in\n either direction, because a covering construction is a lower-bound device and\n the sifting limit is a positivity threshold for a lower-bound sieve. The two\n sides do not meet:\n@@ -815,8 +903,9 @@\n **They do not improve the best constructed value.** The certified ladder in\n this project reaches $356{,}712$ at $y = 4001$, replayed by an independent\n routine with zero uncovered points at all sixteen levels\n-(`research/G2-STATE.md` §3a), and a further certified $479{,}339$ at\n-$y = 5003$ (`research/two-class-lower-bounds.md` §0). Those are finite\n+(`research/G2-STATE.md` §3a), and a further certified $479{,}340$ at\n+$y = 5003$ (`research/two-class-lower-bounds.md` §0, which prints $479{,}339$ for\n+$G_2 - 1$; Appendix B, item 4). Those are finite\n constructions, and neither theorem above predicts them or is predicted by them.\n \n ---\n@@ -845,8 +934,8 @@\n $g(p)$ elements\". The polynomial $f$ does not appear in Lemma 1, Corollary 1\n or its four-sentence proof; it enters only the proof of Theorem 1 (the\n trichotomy on p. 6 and the evaluation of $\\sum g(p)/p$ on pp. 6 to 7),\n-which this manuscript replaces by its own §3 to §5. $\\Omega^{\\mathrm{II}}_p$\n-is defined on p. 5 as the residues at which \"some non-linear irreducible\n+which this manuscript replaces by its own §4 and §5. $\\Omega^{\\mathrm{II}}_p$\n+is defined on p. 6 as the residues at which \"some non-linear irreducible\n factor $q(x)$ of $f(x)$\" vanishes, empty for $x(x+2)$. The published\n Izvestiya English edition (88:2, 225 to 235; mathnet full text, md5\n `9e7f3c54b1979cdfb505c14b4576c4e0`, which now needs a browser User-Agent\n@@ -863,8 +952,9 @@\n claim that $\\kappa = 4$ was load-bearing with a margin one sieve dimension\n wide. Both were about a theorem the source does not cite; the paragraph below\n records what the source re-read found, and §6.2 is now a remark. The residual\n-that remains is only the unread 1974 page behind a printed, refereed Lemma 1,\n-and it is shared with Theorem A, whose step 1 consumes the same statement at\n+that remain are the unread 1974 page behind a printed Lemma 1 and the defect in\n+the printed proof of Corollary 1 recorded below, and both are shared with\n+Theorem A, whose step 1 consumes the same statement at\n $\\kappa = 2$.\n \n One more residual on the source, found by the red team of 2026-09-07 and\n@@ -930,8 +1020,9 @@\n \n Both theorems consume Corollary 1 at its statement, Theorem A at $\\kappa = 2$\n and $z = \\sqrt{m}$, Theorem B at $\\kappa = 4$ and $z = \\sqrt{y}$, and Lemma 1's\n-proof by citation to Halberstam and Richert sits inside a refereed paper,\n-which is what \"published theorem consumed as a theorem\" means. The residuals\n+proof by citation to Halberstam and Richert sits inside a refereed paper, and it is the statement rather than the printed\n+proof that is consumed, which is what \"published theorem consumed as a theorem\"\n+means here. The residuals\n of this subsection are therefore common to the two theorems.\n \n ### 11.3 Prior art\n@@ -987,8 +1078,12 @@\n \n **Toward submission.** A referee-grade re-reading of §§11.1 and 11.2 by someone\n holding the printed Halberstam and Richert beside the Kalmynin and Konyagin\n-PDF. That is a bounded job and it is the whole gate. An explicit-constants pass\n-turning $y_0$ into a value rather than a floor; the six inequalities of §8 are\n+PDF. That is a bounded job and it is the whole gate;\n+`paper/proposals/prop-kk-lower-bound.md` §2 still names a second gate, the\n+Selberg remainder's $\\xi$-versus-$z$ condition, which §6.2 and §11.2 retire and\n+which that proposal's own §5 and §6 already record as retired. An\n+explicit-constants pass\n+turning $y_0$ into a value rather than a floor; the seven inequalities of §8 are\n already written and only the constants are missing. One number theorist who\n knows the Erdős and Rankin literature confirming that the substitution is\n legitimate, which would do more than any further computation here, because no\n@@ -1000,7 +1095,7 @@\n §11.2, which would be a defect in a refereed citation and would reach Theorem A\n equally. The\n optimised band choice succeeding, which would improve the exponent and make\n-this statement the wrong headline rather than a wrong statement. A published\n+this statement the wrong headline. A published\n bound stronger than Theorem B, at which point the move is to cite it and keep\n the exponent accounting of §3, which stands regardless.\n \n@@ -1031,8 +1126,13 @@\n *PDF read; the $Y(x) \\ll x^2$ sentence checked word for word.*\n - K. Ford, B. Green, S. Konyagin and T. Tao, arXiv:1408.4505, *Ann. of Math.*\n **183** (2016). *PDF read, same sentence, same wording.*\n-- R. A. Rankin, 1938. [MEMORY]\n-- J. Pintz, 1997. [MEMORY]\n+- R. A. Rankin, *The difference between consecutive prime numbers*, *J. London\n+ Math. Soc.* **13** (1938) 242–247, DOI 10.1112/jlms/s1-13.4.242. *Record\n+ confirmed at the journal 2026-09-11; paper not read; reference [3] of the\n+ source.*\n+- J. Pintz, *Very large gaps between consecutive primes*, *J. Number Theory*\n+ **63** (1997), no. 2, 286–301. *Record confirmed at ScienceDirect 2026-09-11;\n+ paper not read; not among the source's nine references.*\n - P. Erdős, *On the integers relatively prime to n and on a number-theoretic\n function considered by Jacobsthal*, *Math. Scand.* **10** (1962) 163–170.\n *Located at users.renyi.hu/~p_erdos/1962-12.pdf; one class only.*\n@@ -1046,7 +1146,7 @@\n the paper's Lemma 1 has never been read here, and MathOverflow 245539\n (Paseman, 2016) raises an unanswered question about an inequality direction\n inside its published proof.*\n-- H. Diamond, H. Halberstam and H.-E. Richert, *A Higher-Dimensional Sieve\n+- H. Diamond, H. Halberstam and W. F. Galway, *A Higher-Dimensional Sieve\n Method*, Cambridge Tracts in Mathematics **177**. *Page photographs on disk\n at `attestation/book-ch5-6/`; Theorem 9.1 at p. 104 and the sifting limit at\n p. 79 read from the pages themselves. Page 79 prints $\\beta_2 \\approx\n@@ -1127,7 +1227,7 @@\n matters here because this project has already retracted one constant that came\n from an arXiv HTML rendering rather than from a PDF, and the rule adopted after\n that (`research/PRIOR-ART.md`, \"Provenance: what artifact was actually read\")\n-is that a quotation records which artifact it was read from. These five record\n+is that a quotation records which artifact it was read from. These three record\n that no artifact was read.\n \n ---\n@@ -1143,7 +1243,7 @@\n | number | where it appears | source |\n |---|---|---|\n | $\\beta_2 = 4.26645028414864191641$ | §1.2, §3, §10 | Booker and Browning, via `research/PRIOR-ART.md`; the tract's p. 79 prints $4.266$ |\n-| $\\beta_1 = 2$, Iwaniec's exponent | §1.1 | `research/covering-dive.md`, \"where the proof breaks\" item 1 |\n+| $\\beta_1 = 2$, Iwaniec's exponent | §1.1 | `research/covering-dive.md`, \"where the proof breaks\" item 2 |\n | gap factor $y^{3.266+o(1)}$ | Abstract, §10 | arithmetic on the two rows of §10's table |\n | $G_2/g$ = 2.00, 3.00, 3.00, 3.00, 4.15, 4.41, 5.10, 5.61, 6.00, 8.00, 7.38 at $y = 5 \\dots 41$ | §2 | `research/two-class-lower-bounds.md` §1, VERIFIED at all eleven shared terms |\n | A144311, 22 terms to $y = 79$; Carter 2008 $a(1..7)$, Alekseyev 2009 $a(8..16)$, Wang 2024 $a(17..22)$ | §1.2, §12 | `research/PRIOR-ART.md`; `research/SEARCH-CONVENTIONS.md` §§1, 2 |\n@@ -1157,7 +1257,7 @@\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, OUTPUT block as replayed 2026-09-11 (`kk-recipe.md`); the pre-2026-09-07 column read $2.3196 \\dots 3.8185$ |\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@@ -1166,20 +1266,23 @@\n | $c_0 = 1/(8C_3)$ | §9 | `research/history/staging/import-hypergraph.md` §4 step 3 |\n | $\\lvert V\\rvert = 9889$; degree $1.708115$; $\\prod(1-2/q) = 0.1750499$; $E = 1731.07$; mean leftover $1730.83$; trial variance ratio $0.43$; best trial 1654; mop-up 1153 primes at 1.43 slots per prime; largest prime $10861$; uncovered in $[1, 200000]$ zero; ratio $1.9816$ | §9 | `research/history/staging/redteam-0820-math.md` §3.4, producer `rt0820-t3-cover.js`, rebuilt from the pre-registration text alone |\n | exponent $1.50 \\pm 0.05$ on 22 trusted terms; control bias $+0.28$ | §10 | `research/exponent-control.md` §§1, 5 |\n-| certified $356{,}712$ at $y = 4001$, sixteen levels, zero uncovered | §10 | `research/G2-STATE.md` §3a |\n-| certified $479{,}339$ at $y = 5003$ | §10 | `research/two-class-lower-bounds.md` §0 |\n+| certified $356{,}712$ at $y = 4001$, sixteen levels, zero uncovered | §10 | `research/G2-STATE.md` §3a; table cell 356,711 at `two-class-lower-bounds.md` §5 line 676, prose 356,712 at line 992 (Appendix B, item 4) |\n+| certified $479{,}339$ at $y = 5003$ | §10 | `research/two-class-lower-bounds.md` §0; table cell 479,339 at §5 line 683, prose 479,340 at lines 979, 988 (Appendix B, item 4) |\n | md5 `b5d7d2a23ffd902415057adebfe430b1`, 12 pages, 148,566 bytes | §12 | `research/history/staging/lit-pdf-kalmynin-konyagin.md` §0 |\n | zero forward and reverse citations on three DOIs (OpenAlex); Semantic Scholar empty twice against calibration positives 46, 48, 5; A048670 carries five formula lines and one reference line | §11.3 | `paper/proposals/prop-kk-lower-bound.md` §4, run 2026-08-19 |\n \n-**One attribution this note declines to make.** Section 8 compares the\n-method's own one-class output against \"the one-class lower bound quoted in the\n-source's own introduction at p. 2\". This project's records name that quoted\n-bound two different ways, Ford, Green, Konyagin and Tao in\n-`paper/proposals/prop-kk-lower-bound.md` §6 and Ford, Green, Konyagin, Maynard\n-and Tao in `paper/proposals/draft-kk-lower-bound.md` §6, and the source page\n-has not been re-read to settle which. The comparison is unaffected, since the\n-two published bounds agree in the shape being compared, and the name is left\n-unstated rather than guessed.\n+**One attribution, settled 2026-09-11.** Section 8 compares the method's own\n+one-class output against the one-class lower bound quoted in the source's own\n+introduction at p. 2. This project's records named that bound two different\n+ways (Ford, Green, Konyagin and Tao in `paper/proposals/prop-kk-lower-bound.md`\n+§6; Ford, Green, Konyagin, Maynard and Tao in\n+`paper/proposals/draft-kk-lower-bound.md` §6). The source page was re-read at\n+the arXiv v2 text: the displayed bound is $j(P(y)) \\gg y \\ln y \\ln\\ln\\ln y /\n+\\ln\\ln y$ with a single $\\ln\\ln y$ in the denominator, the unsquared FGKMT\n+form (the 2016 Annals bound of Ford, Green, Konyagin and Tao carries\n+$(\\ln\\ln\\ln)^2$), and it is attributed to reference [1], whose bibliography\n+entry is the five-author JAMS 2018 paper. The prop-kk-lower-bound.md §6 naming\n+is the one to correct (Appendix C).\n \n ---\n \n@@ -1191,7 +1294,7 @@\n so the disagreements below are by design and are listed rather than repaired.\n \n 1. `research/history/staging/attack-kk-substitution.md` §5 carries a $z_0$\n- figure at $y = 4001$ with no run behind it, three orders too large. The\n+ figure at $y = 4001$ with no run behind it, a factor 285 too large. The\n corrected value is in `research/two-class-lower-bounds.md` §4c and is what\n §8 above uses.\n 2. That file and the first adversary's report use opposite conventions for\n@@ -1201,8 +1304,31 @@\n \"sufficient\" only. The brute force supports the weaker reading and not the\n stronger one.\n \n-A draft quotes the live layer. This one does.\n+4. `research/two-class-lower-bounds.md` §5 prints the certified ladder two\n+ ways: the table rows carry 356,711 at $y = 4001$ (line 676) and 479,339 at\n+ $y = 5003$ (line 683), the prose carries 356,712 (line 992) and 479,340\n+ (lines 979, 988), each prose figure one above its table cell; the two\n+ conventions (covered length $m$ against gap $m + 1$) are not labelled\n+ there. §10 above quotes 356,712 (via `research/G2-STATE.md` §3a) and\n+ 479,339 (via `two-class-lower-bounds.md` §0), which mixes the two; the\n+ gap-convention values are 356,712 and 479,340.\n+5. The record's provenance for Kalmynin and Konyagin's Corollary 1 is uneven:\n+ `research/history/staging/lit-pdf-kalmynin-konyagin.md` read text\n+ extraction, not page images (its own §0), while the 2026-08-19 first\n+ adversarial pass and the 2026-09-07 re-read did read page images.\n \n+6. `research/attack-kk-substitution.js`: after the 2026-09-07 $\\ln 3 \\to\n+ \\ln 12$ change in condition H5, the OUTPUT block's least-admissible-$A$\n+ column reads $2.4533 \\dots 3.8201$ while the script's own reading\n+ paragraph (lines 1114 to 1115) and §H's printed condition list still carry\n+ the old column $2.3196 \\dots 3.8185$ and \"$\\ln 3$\"; the $y_0$ table is\n+ unchanged. `embed.js --check` passes (code, body and out-sha256 all\n+ match), so the drift is between the code's comments and its output, not\n+ between code and output.\n+\n+A draft quotes the live layer. This one does, with the two conventions of\n+item 4 both shown.\n+\n ---\n \n ## Appendix C. Registry updates this draft implies\n@@ -1229,7 +1355,26 @@\n the upper bound in the same document.\n - `research/two-class-lower-bounds.md` §4c and `research/G2-STATE.md` §3a: both\n carry the two theorems already, at the grades used here; the only change is a\n- pointer to this draft.\n+ pointer to this draft. §4c's \"six asymptotic hypotheses\" should read seven\n+ (H1 to H7 of §8), and its \"for $y \\ge 10^{134.1}$\" should carry the\n+ all-constants-one caveat of §8.\n+- `research/covering-dive.md` §4.2: still frames Theorem B's referee gap as\n+ \"the Selberg remainder's ξ-versus-z condition at κ = 4\", retired 2026-09-07;\n+ and carries \"consumed as a theorem, read at source\" for Corollary 1 without\n+ the proof slip of §11.2.\n+- `paper/proposals/prop-kk-lower-bound.md`: §2 and the first §5 upgrade\n+ trigger still demand the ξ-versus-z condition and call κ = 4 load-bearing\n+ there, against the same file's fourth downgrade trigger and §6; §6 names\n+ the bound quoted in the source's introduction as Ford, Green, Konyagin and\n+ Tao, where the source's reference [1] is the five-author JAMS paper\n+ (Appendix A).\n+- `paper/proposals/prop-xlnx-lower-bound.md` §1: \"G₂(x#) ≥ y − O(1)\" should\n+ read \"≥ y + 1\" (Proposition 1); §1 to §2 should carry the Corollary 1 proof\n+ slip.\n+- `research/history/staging/import-hypergraph.md` §4 step 2 and\n+ `redteam-0820-math.md` §3.3 item 2 (frozen): the two-class Mertens product\n+ is $4C_2e^{-2\\gamma}/\\ln^2 z$, not $(C_2 + o(1))/\\ln^2 z$; record in\n+ `research/history/CHANGELOG.md` rather than edit in place.\n \n ---\n \n","cpu_hours":0.25,"hashes":{"check-kk.out":"05e4111881bbb9c201a373ed1e91e99bc927536f7e475f959aaf3ccc0f8c94d2","kk-lower-bound.diff":"db44d2cdb60b56747fd7f53dba81bb7112b9cefaf29fdfcb7562c89fcf3cf55f"},"author_rung":"proven","status":"rejected","final_rung":null,"created_at":"2026-09-11T13:13:16.769Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[32],"messages":[]},"tokens":{"log":"claude-code","input":1306,"models":{"claude-opus-5":28678,"claude-sonnet-5":3268,"claude-fable-5-1":43061},"output":75007,"source":"claude-jsonl","entries":229,"cache_read":33693462,"cache_write":888835},"paper_slug":"kk-lower-bound","revision_path":"paper/kk-lower-bound.md","revision_sha":"8cf928ee8e4d8c9153aeb2342af3fb9248f1410680aa28d149a9691da146913c","recipe_md":"# Recipe (one thread, under 12 minutes)\n\n1. Fetch research/attack-kk-substitution.js, research/verify-kk-substitution.js, research/attack2-rankin2d.js, research/qc/embed.js, research/qc/tailfmt.js, research/qc/units.js from `/docs/...` (sha256s in kk-recipe.md).\n2. `node research/qc/embed.js --check