# Review: does Theorem B of paper/kk-lower-bound.md need section 6.2? Read-only review, 2026-09-07. No repository file was edited. Sources read at source: arXiv:2302.00459v2 (`scratchpad/kk/kk.pdf`, md5 `b5d7d2a23ffd902415057adebfe430b1`, matching the manuscript's recorded artifact, pp. 3 to 4 via `pdftotext -layout`) and Richert, *Lectures on Sieve Methods*, Tata 1976 (`scratchpad/kk/richert.pdf`, md5 `d2915a3eca4436a760dfaf50e1a1c335`, via the existing `richert.txt` extraction, Chapter 11 pp. 125 to 128 and the chapter notes p. 142). Caveat on both: text extraction, not page images. The project's own record (CHANGELOG 3425 to 3428) says a `pdftotext` linearisation once nearly produced a false finding against this same paper. Every quotation below from the two PDFs carries that caveat. Halberstam and Richert 1974 was not read; nothing below claims to know its page. ## Verdict, first **(a). Section 6.2 can be demoted to a remark, and Theorem B consumes Corollary 1 directly at z = sqrt(y).** The theorem consumed has no support parameter and no remainder sum; every hypothesis it prints is discharged for the manuscript's sequence at z = sqrt(y) with no size restriction on d; the step uses only the upper-bound direction; and the manuscript's own ledger (sections 6.3 and 6.4) already computes at z = sqrt(y), not at z = xi, so the "repair" of 6.2 is never carried through the argument as written. The "kappa = 4 load-bearing, margin one sieve dimension wide" claim of 11.2 is a statement about a different theorem (a Selberg-form bound with an explicit remainder sum) that the second adversarial pass introduced without the source PDF, and it is not correct even inside that theorem's frame (section 3 below). Residual that keeps this from being a clean "proven" demotion: the chain is manuscript -> Corollary 1 (printed, refereed) -> Lemma 1 (printed, refereed) -> "[5, Theorem 2.2]" (cited, unread here). The demotion needs only the first two links, which are read. If a reader with the 1974 text found Theorem 2.2 printed in a remainder-sum form, that would be a defect in Kalmynin and Konyagin's citation inside a refereed paper, and by the manuscript's own custody rule at lines 865 to 868 it would land on Theorem A exactly as it lands on Theorem B. The manuscript currently applies that rule to A and not to B with no stated reason, and the sieve step of B consumes the same printed statement in the same way. ## 1. What the manuscript's Theorem B step actually needs Line 446 to 449: > `R \le S(m, \Omega) + O(\sqrt{y}) + 2\Psi(m+2, z_1)`, where > `S(X, \Omega) = \#\{n \le X : n \bmod p \notin \Omega_p \ \forall p \le z\}` with X = m (line 298) and z = sqrt(y) (line 317: "The sieve sets fed to the fundamental lemma at sifting level z = sqrt(y)"). What is needed is an upper bound for S(m, Omega) of size O(m V(sqrt y)), consumed at lines 552 to 553 and 559: > `S(m, \Omega) \ll m \prod_{p \le \sqrt{y}}(1 - g(p)/p) \ll A^4 y/(B \ln y)` > ... "taking B large against A gives S(m, Omega) <= y/(4 ln y)." That is exactly the statement of Corollary 1 with X = m, z = sqrt(y). ## 2. Lemma 1 and Corollary 1 as printed, and each hypothesis for this sequence Lemma 1, arXiv v2 p. 3 to 4, extracted text (`kk-p3-4.txt` lines 64 to 84): > Suppose that kappa > 0, z >= 2. Assume that {a_n} is a sequence of > non-negative real numbers such that for all d | P(z) we have > sum_{n = 0 (mod d)} a_n = g(d) X/d + r_d, where g(d) is a multiplicative > function with g(p) <= kappa, g(p) < p for all primes p, |r_d| <= g(d) and > z << X. Then we have S(a, z) = sum_{(n, P(z)) = 1} a_n <<_kappa X V(z), > where V(z) = prod_{p <= z}(1 - g(p)/p). > Proof. This is a version of the fundamental lemma of sieve theory. See, for > example, [5, Theorem 2.2] Corollary 1, p. 4 (lines 88 to 110), matches the manuscript's line 459 to 462 quotation word for word from "Suppose". Its printed proof encodes "n avoids every Omega_p" into "(m, P(z)) = 1" by a CRT change of variable and ends "Conditions of Lemma 1 also clearly hold." So the manuscript, consuming Corollary 1, has to discharge only: g multiplicative, g(p) = |Omega_p| <= kappa, g(p) < p for all p, z << X. The |r_d| <= g(d) condition is discharged inside the corollary's own proof. It is checked here anyway, as the task asks. There is no support parameter xi, no remainder sum, and no restriction on d beyond d | P(z) anywhere in Lemma 1 or Corollary 1. Confirmed at the extracted text. Hypothesis by hypothesis for the manuscript's Omega_p (lines 319 to 321 and 307 to 311; Omega_p is Omega^I_p union Omega^III_p, Omega^II empty): | hypothesis | manuscript's discharge | checked here | calibration | |---|---|---|---| | g multiplicative | not stated explicitly; g(d) := prod_{p \mid d} g(p) on squarefree d is the definition Lemma 1 intends and Corollary 1's proof uses | trivially satisfiable; worth one clause in 6.1 | proven | | g(p) <= kappa | line 474 to 475: "g(p) = \|Omega_p\| <= 4 everywhere, with 4 attained only on band 2, so kappa = 4" | g = 1 at p = 2 ({0, -2} mod 2 = {0}); g = 2 at p = 3 ({0, 1}); g = 2 on band 1 for p >= 5; g = 4 on band 2 (z_0, z_1], since {0, -2} and {1, -1} are disjoint mod p for p not dividing Res = -3, line 342. kappa = 4 is correct as a sup. **Any kappa >= 4 is also a valid hypothesis**; kappa enters Lemma 1 only through <<_kappa | proven | | g(p) < p for all p | line 475 to 478: fails only at p = 2, 3 if they carried a band-2 count; they sit in band 1 because z_0 > 3 (H1, line 614) | confirmed: g(2) = 1 < 2, g(3) = 2 < 3, g(p) <= 4 < 5 <= p otherwise | proven, conditional on H1 | | \|r_d\| <= g(d) for every d \mid P(z) | discharged inside Corollary 1's proof ("clearly hold") | Direct check: for squarefree d, the n mod d with n mod p in Omega_p for every p \mid d form, by CRT, exactly g(d) = prod g(p) residue classes mod d. Each class a (1 <= a <= d) contains floor((X - a)/d) + 1 integers n in [1, X], which lies in [X/d - 1, X/d + 1]. Summing over g(d) classes: #\{n <= X : n mod p in Omega_p for all p \mid d\} = g(d) X/d + r_d with \|r_d\| <= g(d). **This holds for every squarefree d, with no restriction on the size of d**; d > X is allowed and gives \|r_d\| <= g(d) still. This is the (R) hypothesis of Richert 11.3 in the same form | proven | | z << X | line 478: "Finally z = sqrt(y) << m" | m / sqrt(y) = sqrt(y) (ln y)^3 (lnlnln y)^2 / (B (lnln y)^4) -> infinity for fixed B; so z <= X with constant 1 for all y >= some y_1(B), and at y = 10^{134.1} the ratio is about 10^67 times a log factor over B. Not one of the six H-conditions in section 8's table (lines 612 to 619); it is slack by a wide margin but should be listed if section 8 claims to list "every hypothesis" | proven | So Corollary 1 applies at z = sqrt(y), X = m, giving S(m, Omega) <<_4 m V(sqrt y). Nothing else is required. On the implied constant. Line 466 says it depends "on kappa alone", quoting Kalmynin and Konyagin's <<_kappa. Richert's Theorem 11.3 makes it depend on A (the exponent in z <= X^A), A_1 from (Omega_1), A_2 from (Omega_2(kappa)) and kappa. For this sequence: exponent A = 1 (z = sqrt(y) <= m); (Omega_1) is g(p)/p <= 1 - 1/A_1, met with A_1 = 5 (worst case 4/5 at p >= 5, 1/2 at p = 2, 2/3 at p = 3); (Omega_2(kappa)) is sum_{w <= p < z} g(p) ln p / p <= kappa ln(z/w) + A_2, met at kappa = 4 with an absolute A_2 by Mertens, or replaced by Richert's (Omega_0) g(p) <= A_0 = 4 (line 6772 of `richert.txt`: "(Omega_0) implies (Omega_2(kappa))"). None of these depends on the manuscript's A or B. So the step at line 557 to 559, "since the implied constant depends on kappa alone and not on A or B, taking B large against A", survives under either printed form of the constant. Calibration: derived here from Richert's stated hypotheses; the derivation is three lines and is included above; not adversarially checked. ## 3. Is the 6.2 support restriction needed? What is lost or gained? **Not needed.** Lemma 1 as printed is the Brun-form bound: pointwise |r_d| <= g(d) for every d | P(z), conclusion S << X V(z), no xi. Richert's Theorem 11.3 (`richert.txt` lines 6755 to 6768), which his chapter notes (line 7414) identify as "cf. l.c. Theorem 2.2", has exactly that shape: > (R) |R_d| <= omega(d) if mu(d) != 0 and (d, p-bar) = 1. (11.9) > Theorem 11.3. (Omega_1), (Omega_2(kappa)), (R): For any A > 0 > S(A, p, z) << X prod_{p < z}(1 - omega(p)/p) if z <= X^A, (11.10) > where the <<-constant depends at most on A, A_1, A_2 and kappa. > In the literature, usually, the phrase 'by Brun's sieve...' refers to the > statement (11.10). and the chapter note adds (line 7414 to 7416): "Actually, Theorem 11.3 holds for z >= X^A also when the restriction in the product, p < z, is replaced by p < X." Richert derives 11.3 "from (the second inequality of) Theorem 9.1 and (11.5)" (line 6757 to 6758), that is, from the Selberg sieve with the remainder handled internally under (R). In other words, the manuscript's 6.2 manoeuvre (shrink the support to a power of log below sqrt(X), then note the product only changes by a bounded factor) is what the proof of the theorem consumed does on the reader's behalf, and the resulting factor is absorbed into the constant. Doing it again outside the theorem is not wrong; it is redundant. The remainder that 6.2 analyses, `sum_{d <= xi^2} 3^{omega(d)} |r_d|`, is the remainder of Richert's Theorem 11.1 (Halberstam and Richert Theorem 4.1, the Selberg upper bound; `richert.txt` lines 6735 to 6740): > Theorem 11.1. (Omega_1), (Omega_2(kappa)): S(A, p, z) << X W(z) + > sum_{d <= z^2, (d, p-bar) = 1} mu^2(d) 3^{nu(d)} |R_d|. (11.7) That is a different theorem, with a different hypothesis set (no (R)), and the manuscript neither cites nor needs it. Under |R_d| <= g(d) <= kappa^{nu(d)} the sum is << z^2 (ln z)^{3 kappa - 1}, which is where 6.2's exponent 11 comes from. So 6.2's arithmetic is internally correct for Theorem 11.1 and irrelevant to Lemma 1. **Loss or gain at z = sqrt(y) versus z = xi.** None that matters, in either direction. V(sqrt y) / V(xi) = prod_{xi < p <= sqrt y}(1 - g(p)/p), and on (xi, sqrt y] every g(p) = 2 (band 2 ends at z_1 << xi by H3), so the ratio is (ln xi / ln sqrt(y))^2 (1 + o(1)) = (1 - 14 ln ln y / ln y)^2 (1 + o(1)) -> 1. The manuscript's line 495 says the same with exponent kappa = 4; the exact exponent on this interval is 2, but either way the factor is 1 + o(1) and it is the direction that helps (V(sqrt y) < V(xi)), so sieving at sqrt(y) gives a marginally smaller bound. The threshold table of section 8 does not depend on xi at all (H1 to H6, lines 614 to 619, contain no xi), so nothing in section 8 moves. **The internal inconsistency this exposes.** Line 501 to 502 says the repair is to "sieve at z = xi", but lines 512 to 515 (the ledger, sum over p <= sqrt y), line 517 ("Omega^I charged over all p <= sqrt y"), and lines 550 to 553 (product over p <= sqrt y) all compute at z = sqrt(y). The manuscript already runs Theorem B exactly as it would under (a); 6.2 is a paragraph that is never consumed downstream. That is harmless numerically and it is a defect in the exposition: as written, the proof claims to use one sifting level and computes with another. **Is "kappa = 4 load-bearing, margin one dimension wide" about the theorem consumed?** No. Under Lemma 1, kappa enters only <<_kappa, and kappa = 6 (the source's own value) is a weaker hypothesis than kappa = 4 that the sequence satisfies trivially. The claim at lines 498 to 500 and 832 to 835 is a claim about Theorem 11.1's remainder. And even in that frame it is overstated: at general kappa with xi = sqrt(y)/(ln y)^C the remainder is y (ln y)^{3 kappa - 1 - 2C}, which sits below the main term y/ln y as soon as C > 3 kappa / 2, so kappa = 6 is repaired by C = 9 (remainder y (ln y)^{-1} against y/(B ln y) needs C > 9; take C = 10) with the main term still losing a factor -> 1. The specific support (6.2) fails at kappa = 6; the method does not. "One sieve dimension wide" is a property of the constant 7 chosen in (6.2), not of any theorem. Calibration: arithmetic checked here by hand; not by script. ## 4. Direction: only the upper bound is used Every consumer of the sieve bound uses it as an upper bound on a count that must be shown small: - line 446, (6.1): `R <= S(m, Omega) + O(sqrt y) + 2 Psi(m+2, z_1)`, an upper bound on R (Proposition 3 says "at least one of the following holds", which is a containment of the unkilled set in the union of three sets); - line 552 to 559: `S(m, Omega) << ... <= y/(4 ln y)`, upper bound; - line 584: `R <= y/(4 ln y) + o(y/ln y) <= y/(3 ln y)`, upper bound; - line 597 to 600: band 3 needs `pi(y) - pi(y/2) > y/(3 ln y) >= R`, so the survivors can be injected into the primes of (y/2, y]. Upper bound on R against a lower bound on the prime count (Rosser and Schoenfeld), which is a separate published statement. No step of Theorem B needs a lower bound on S(m, Omega) or an asymptotic for it. The same is true of Theorem A (line 686: `#V <= C_1 m V(z)`). Lemma 1's one-sided conclusion is exactly the direction consumed. Confirmed by reading sections 5 to 7 and 9 in full. ## 5. The exact lines that would change under (a) In `paper/kk-lower-bound.md`: - **Lines 478 to 481**, "A smaller kappa than the source's own kappa = 6 is a tightening rather than a weakening, since kappa enters only the implied constant, and §6.2 shows it does more than tighten." Delete the final clause; keep "kappa enters only the implied constant". Add one clause defining g(d) = prod_{p | d} g(p) on squarefree d, and one sentence noting that the CRT remainder |r_d| <= g(d) holds for every d | P(z) (section 2 table above), which is what Corollary 1's proof calls "clearly". - **Lines 483 to 508** (all of §6.2). Demote to a remark of roughly this content: "Lemma 1 is the Brun-form bound under the pointwise hypothesis |r_d| <= g(d) (Richert, Tata 1976, Theorem 11.3, cross-referenced by its author to Halberstam and Richert Theorem 2.2), and carries no support parameter. A Selberg-form bound (Richert Theorem 11.1) would instead carry the remainder sum_{d <= xi^2} 3^{omega(d)} |r_d| << xi^2 (ln xi)^{3 kappa - 1} and would need xi = sqrt(y)/(ln y)^C with C > 3 kappa / 2, sieving at z = xi, at a cost of 1 + o(1) in V; that route is available and is not the one consumed." The riders at 497 to 502 go, or survive inside the remark with "load-bearing" removed. - **Line 317**, "at sifting level z = sqrt(y)": unchanged, and now consistent with 512 to 553. - **Lines 824 to 835** (§11.2 first paragraph). Replace "The Selberg support correction of §6.2 uses ... The exponent 3 kappa - 1 = 11 and the choice (6.2) are load-bearing, and a different hypothesis on xi against z in the printed theorem would need a different repair. The direction of the risk is known: at kappa = 6 that support fails, so the margin is one sieve dimension wide." with a statement that the residual is only the unread 1974 page behind a printed, refereed Lemma 1, and that it is shared with Theorem A. The 2026-09-07 rider at 837 to 863 becomes the body of §11.2; its last two sentences ("§6.2 has not been rewritten on its strength. A reader with the 1974 text should check ...") become "§6.2 is retained as a remark". - **Lines 865 to 868**: the paragraph granting Theorem A "published theorem consumed as a theorem" status applies verbatim to Theorem B's sieve step; say so, and confine Theorem B's INFERRED grade (line 256) to the trichotomy and the ledger, which are the parts that re-derive the source's proof. - **Line 915** (§11.4 table): "Halberstam and Richert Thm 2.2's hypotheses | a xi-versus-z condition incompatible with (6.2) | no." Replace the falsifier with "Theorem 2.2 printed in 1974 in a form other than the Brun-type bound under (R)" and the status with "Richert Tata Theorem 11.3 read at extracted text 2026-09-07; 1974 page unread; shared with Theorem A". - **Line 932**: "or at §11.2, which costs the support repair and needs the sieve step rebuilt" becomes "or at §11.2, which would be a defect in a refereed citation and would reach Theorem A equally". - **Line 641** ("the fundamental lemma's constant at kappa = 4"): unchanged. - **Section 8 table, lines 612 to 619**: optionally add the row "z << X, sqrt(y) <= m" with the inequality `ln B + 4 ln lnln L - 3 ln L - 2 ln lnlnL < L/2`, marked slack; it is currently the one hypothesis of §6.1 the table omits. - **Line 1069** (Appendix A row): the provenance "second adversarial pass" stays, with the note that the pass "had no K–K PDF" (CHANGELOG 2948) and priced a Selberg-form remainder that Lemma 1 does not carry. Outside the manuscript, the same text lives at `research/two-class-lower-bounds.md` lines 362 to 365, `paper/proposals/prop-kk-lower-bound.md` lines 146 and 161, and `paper/proposals/draft-kk-lower-bound.md` lines 267 to 275 and 482 to 485. `research/history/staging/verify-kk-substitution.md` line 195 and CHANGELOG 2934 and 3416 are history and stay as written. ## 6. Provenance of the 6.2 claim, for the record CHANGELOG 2927 to 2950: the second adversarial pass "found kappa = 4 load-bearing in the Selberg remainder", and "the second checker had no K–K PDF ... and could not check Halberstam–Richert Thm 2.2's xi-vs-z hypothesis at source". CHANGELOG 3415 to 3417: "The §3a check tests z << X but not the Selberg support, and a naive xi = sqrt(y) leaves a remainder (ln y)^12 above the main term." The "xi-vs-z hypothesis" was hypothesised by a checker working from the general shape of upper-bound sieves without the paper in hand; it was never read from either source. That is the same failure mode the project's campaign lessons name (a claimed hypothesis must be read, not inferred from the theorem's family), running in the conservative direction: it added a constraint the theorem does not impose. The manuscript's line 505 to 507 already says as much ("from the standard theory of upper-bound sieves rather than from the printed hypotheses"). ## 7. What would falsify (a), and whether it has run - Kalmynin and Konyagin's Lemma 1 printed with a remainder sum or a support parameter: read at extracted text of the PDF of record, none present. Not read at page image in this pass. - Corollary 1 needing a hypothesis beyond Lemma 1's: read; it needs only "Omega_p contains g(p) elements" for p <= z. - The CRT remainder failing |r_d| <= g(d) for large d: checked by the two-line count in section 2; it cannot fail. - A Theorem B step using the lower-bound direction or an asymptotic: read sections 5 to 7 in full; none. - Halberstam and Richert 1974 Theorem 2.2 printed with a remainder sum and without (R): not checked, book unreachable. If true it is a defect in a refereed citation and reaches Theorem A equally; it does not reinstate 6.2 as the manuscript's responsibility rather than the source's, and Richert's own Theorem 11.3 (a proved statement in a book, proof not re-derived here) is available as the citation in its place.