# Independent review of the 2026-09-07 session: the six ranked claims of review-request-0907 Report 10, filed 2026-09-08 against `research/review-request-0907.md` at starting commit fda7b55 (tree clean; lane V of the 2026-09-08 round). Read-only: no owning note, script, register or index was edited. Proposed corrections are in section 9 as exact text for the integrator. Historical record; not a live note. ## 0. What remains open, first Twin-prime infinitude and every sufficient signed margin remain OPEN. Nothing below changes the proof status. Halberstam and Richert 1974, Theorem 2.2, is still unread at the printed page; the statement is now reached by four routes (three of them re-fetched here), all in Brun form, and the residual on the manuscript's Theorems A and B is that page behind a printed, refereed Lemma 1. The Lean build behind the #687 bound is still not reproduced here. Every disposition below is at finite or textual scope; no asymptotic statement is established by any check in this report. ## 1. Disposition table | rank | claim (short) | disposition | what moved | |---|---|---|---| | 1 | K-K Lemma 1 is the Brun-form bound, identified with H-R Theorem 2.2 via Richert Theorem 11.3; Theorem B consumes Corollary 1 at z = sqrt(y) with hypotheses discharged; §6.2 not consumed | **verified within stated scope; conditional on the named input** (the 1974 page) | Lemma 1 and Corollary 1 read at the arXiv TeX source and at two page images (arXiv v2 p. 4; the mathnet Izvestiya artifact, printed p. 36 of that file); Richert 11.3 and the chapter note read at page image; H-R 1971 Mémoire Theorem 3 read at page image; Dover OCR snippets re-fetched independently. Hypothesis discharge re-derived. No defect found in §5 to §8 as they stand after report 09's edits | | 2 | Corollary 1's printed proof mis-chooses CRT representatives; repaired by r' = r (mod p), r' = 1 (mod P(z;p)); present in arXiv v2 and Izvestiya | **verified within stated scope; wording correction proposed** | Own finite check (9 configurations, z in {5, 7, 11}, X in {30, 100, 210, 2310}) confirms the slip and the repair, and sharpens the statement: the printed encoding with least non-negative representatives selects the empty set whenever some Omega_p is not contained in {1}, not only when 0 is in Omega_p; in the one surviving case it selects the avoiders of -Omega, not Omega. The Lemma 1 hypothesis also fails for the printed sequence | | 3 | At x = 2^j, j >= 30, D_y/x = -(T1/x - C2) within 1e-4; a census measures the classical term's error | **verified within stated scope; correction required to the quoted tolerances** | Recomputed from the JSON artifact: max deviation over j >= 30 is 1.84e-4 (j = 30), also 1.54e-4 (j = 35) and 1.42e-4 (j = 31); the ledger verdict's "within 1e-5 at every j >= 30" holds only at j = 38; the body's "S/x - C2 at most 1.4e-4" is 1.80e-4 at j = 32; the ledger's "within 0.0042 of zero for j >= 26" is 0.004283 at j = 27. The identity algebra is exact and reproduced | | 4 | The Erdős #687 bound is an anonymous AI-authored GitHub PDF, Lean-formalised by a third party, unrefereed, one-class primorial form; transfers through G2 >= g | **verified within stated scope** | Page, PDF metadata, GitHub commit, Lean statement and the `ResidueCover` definition re-read here at source on 2026-09-08. Build not reproduced (no Lean toolchain on this machine) | | 5 | Four shifted-prime sums at random-sign size to 2^38; the +2 sums' sign run matched by a control draw | **verified within stated scope; one wording precision proposed** | Ratios, slopes, sign runs and pi(2^j) recomputed from the artifact; pi(2^j) + 2 equals A007053(j) at every j = 3..38 against the fetched b-file. The +2 real run (19) exceeds every draw in its own column (max 13 for mu+, 18 for lambda+); the match at 19 is one draw of the mu- column, so "matched by a control draw" holds only across the pooled 32 draws | | 6 | A2 = prod_{p>2}(1 - 1/(p(p-1))) = 0.7479116 = 2 x Artin; 0.7364 was wrong | **verified** | Own Euler product to p <= 2 x 10^8: 0.74791163 (tail below 3e-10); 2 x 0.3739558136 = 0.7479116272; both scripts carry 0.7479116272384044; the certified interval of moving-cutoff-parity (15) excludes 0.7364 (C2(1 - 0.7364) = 0.1740 > 21/125 = 0.168) and admits 0.7479116 (0.16642 in (0.165, 0.168)) | Reading order followed as specified: CHANGELOG 2026-09-07; kk-lower-bound §3 to §6, §8, §9, §11; centered-discrepancy-measurement; shifted-prime-mobius-sums; G2-STATE PROVEN list; two-class-jacobsthal §3; then reports 01 to 05, 07 to 09, with 06 read only after my own attack on 05 (section 3 below); then the two OUTCOMES entries. Report 09 had already applied E1 to E13; I checked the current text of §5 Case 1 ((5.1) reads 2(m+2)/y <= z_0), §6.1 (the |r_d| <= g(d) clause is in the imported list), §8 (H5 with ln 12, H7 present) and §11.2 (the Theorem A asymmetry removed). Those edits are in place. ## 2. Sources reached in this pass, with custody | source | route | custody | what was read | |---|---|---|---| | Kalmynin and Konyagin, arXiv:2302.00459v2 | `https://arxiv.org/e-print/2302.00459v2` (gzip of `Polynomial_Jacobsthal_revision_3.tex`, 30,276 bytes, dated 2023-12-03); abs page title "[2302.00459v2] A polynomial analogue of Jacobsthal function" | TeX source, primary | Lemma 1, Corollary 1 and its proof, verbatim (section 4) | | same, PDF | cached `kk_v2.pdf`, md5 `b5d7d2a23ffd902415057adebfe430b1` (matches the manuscript's recorded artifact) | page image p. 4 rendered at 150 dpi and inspected | the same text; the product formula and representative choice as in the TeX | | published version, Izvestiya Math 88:2 | `https://www.mathnet.ru/php/getFT.phtml?jrnid=im&paperid=9467&what=fullt&option_lang=eng` and `=rus` with a browser User-Agent; both served the same file, md5 `871d344ee9cbb53bfc04e631e8bd8040`, 629,304 bytes, 11 pages, created 2024-03-21 | page image (pdf p. 4, printed p. 36 of that artifact) | Corollary 1 and its proof; same formula, copy-edited wording ("it is easily seen", "Corollary is proved"); Lemma 1's proof line "see, for example, Theorem 2.2 in [5]". The English-edition artifact md5 `9e7f3c54...` that report 07 read (printed p. 228) was not served today | | Richert, Lectures on Sieve Methods, Tata 1976 | cached `richert_tata.pdf`, md5 `d2915a3eca4436a760dfaf50e1a1c335`; the Toledo-hosted copy re-fetched, same md5 | page images pdf pp. 151 and 166 (printed pp. 135 and 150) | Theorem 11.1, 11.2, condition (R) as (11.9), Theorem 11.3 as (11.10), the (Omega_0) remark; the chapter note "Theorem 11.3: cf. l.c. Theorem 2.2. Actually, Theorem 11.3 holds for z >= X^A also when the restriction in the product, p < z, is replaced by p < X" | | Halberstam and Richert, A new look at Brun's sieve, Mém. SMF 25 (1971) | `https://www.numdam.org/item/10.24033/msmf.39.pdf`, 646,810 bytes | page images pp. 98 and 100 | Theorem 3, p. 100, verbatim: "Suppose that (Omega), (Omega_1) and (R) hold. For any positive number mu, S(A;P,z) = O(XW(z)) if z <= X^mu and S(A;P,z) = O(XW(X)) if z >= X^{1/mu}, where the constants implied by the O-symbols may depend on mu." Conditions on p. 98: (Omega) omega(p) <= A for some constant A >= 1; (R) |R_d| <= omega(d) if d | P(z); (Omega_1) omega(p)/p <= 1 - 1/A_1 | | Halberstam and Richert 1974 (Dover reprint OCR) | `https://books.google.dk/books?id=keKvAAAAQBAJ&q=%22THEOREM+2.2%22` search-inside JSON | OCR snippets, not page images | p. 68: "THEOREM 2.2. (Omega_1), (Omega_2(kappa)), (R): For any A (>0), ... S(A;P,z) <= B3 X prod (1 - omega(p)/p) ... Proof. We begin with the trivial estimate, provided by (1.4.14)"; p. 69: "Theorem 2.2 is an easy consequence of (5.3). We may now assume in both (5.1) and (5.2) that z >= X^{1/A}"; p. 82: Theorem 2.5, the fundamental lemma, under the same (Omega_1), (Omega_2(kappa)), (R); p. 130: "Theorem 2.2 and all results of Sections 2.6 and 2.7 could be derived equally well from Theorem 4.1 below, after applying condition (R)"; p. 153: "the conditions under which Theorems 4.1 (and 2.2) were proved: namely, (Omega_1), (Omega_2(kappa)) (and (R))". The constant label reads "B3" in this OCR against "B" in report 08's; both are uncertain tokens. archive.org `_djvu.txt` returned 401; the Books API returned no items for the same queries | | Erdős Problems #687 | `https://www.erdosproblems.com/687` via curl with a browser User-Agent (WebFetch returned 403) | HTML, 33,727 bytes | "The best lower bound is due to GPT 5.6 Pro (see [4]) Y(x) >> (log x/logloglog x) x, improving on a previous bound of Ford, Green, Konyagin, Maynard, and Tao [FGKMT18]"; "[4]" links to `/4`; "This page was last edited 31 August 2026" | | the anonymous PDF | `https://raw.githubusercontent.com/DottedCalculator/ai-math/main/Erdos_4_GPT_5.6_Sol.pdf`, 644,908 bytes; GitHub API: one commit touching the file, `9ed1cea5`, 2026-08-26T01:30:55Z, "Add files via upload" | pdfinfo and text of pp. 1 to 4 | Title "A Tilted Residue-Class Construction for Long Prime-Free Intervals"; Author field "Anonymous manuscript – revised edition"; 48 pages; creation 2026-08-26; p. 3 "Covering theorem. There is an effective constant c0 > 0 such that, for every sufficiently large X, Y(X) >= c0 X log X / log log log X" | | Lean formalisation | `plby/lean-proofs`, `src/latest/ErdosProblems/Erdos4Tilted.lean` (6,240 bytes) and the 405 files under `Erdos4/` | source text | `theorem covering_theorem : ∃ c X₀ : ℝ, 0 < c ∧ ∀ X : ℝ, X₀ ≤ X → c * X * Real.log X / Real.log (Real.log (Real.log X)) ≤ (maximumCoverLength X : ℝ)`; `maximumCoverLength X := Nat.findGreatest (BoundedCover ⌊X⌋₊) (primorial ⌊X⌋₊)`; `BoundedCover z y := ∃ cover : ResidueCover y, cover.primes ⊆ z.primesLE`; `structure ResidueCover (y : ℕ)` with `primes : Finset ℕ`, `residue : ℕ → ℕ`, `covers : ∀ i, 1 ≤ i → i ≤ y → ∃ p ∈ primes, i ≡ residue p [MOD p]` (Base.lean line 1692). Zero occurrences of `sorry` or `axiom` across the 408 files fetched. Build not reproduced | | OEIS A007053 | `https://oeis.org/A007053/b007053.txt` | b-file, 93 rows | pi(2^j) + 2 from the shifted-prime artifact equals A007053(j) at every j = 3..38 | ## 3. Ranked claim 1: the sieve step of Theorem B ### 3.1 The statements as printed (TeX source, verbatim) Lemma 1: "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: "Let kappa, z, g(d), V(z) and X be as above. Suppose that for any p <= z the set Omega_p subset Z/pZ contains g(p) elements. Let S(X, Omega) be the number of n <= X such that n mod p not in Omega_p for all p <= z. Then S(X, Omega) << X V(z)." Neither statement carries a support parameter, a remainder sum or a size restriction on d beyond d | P(z). Confirmed at the TeX and at both page images. ### 3.2 Hypothesis discharge for Theorem B, re-derived With Omega_p = Omega^I_p union Omega^III_p as in §4, z = sqrt(y), X = m: - g(2) = |{0, -2 mod 2}| = 1 < 2; g(3) = |{0, 1}| = 2 < 3; band 1 at p >= 5, g(p) = 2 < p; band 2, g(p) = 4 < p since p > z_0 > 3 (H1) forces p >= 5, and {0, -2} and {1, -1} are disjoint mod p because the only coincidence -2 = 1 needs p | 3. So kappa = 4 is the supremum and any kappa >= 4 is admissible; kappa enters only the implied constant. - g(d) := prod_{p | d} g(p) on squarefree d is multiplicative. - |r_d| <= g(d) for every squarefree d: the n with n mod p in Omega_p for each p | d form exactly g(d) classes mod d by CRT; a class a mod d with 1 <= a <= d meets [1, X] in floor((X - a)/d) + 1 integers if a <= X and in 0 integers otherwise, and in both cases the count differs from X/d by less than 1; summing over g(d) classes gives |r_d| < g(d). No restriction on the size of d; d > X is allowed. - z << X: m/sqrt(y) = sqrt(y)(ln y)^3(lnlnln y)^2/(B (lnln y)^4) tends to infinity for fixed B; at y_0 the ratio is above 10^67 divided by B. - Direction: every consumer in §6.1 to §7 uses an upper bound on a count ((6.1), the ledger, R <= y/(3 ln y), the injection into primes of (y/2, y]). No lower-bound sieve, no asymptotic. - The implied constant: Lemma 1 prints <<_kappa. Under Richert 11.3 the constant depends on A, A_1, A_2 and kappa; here A = 1 (z <= X), A_1 = kappa + 1 (since g(p) <= min(kappa, p - 1) gives g(p)/p <= 1 - 1/(kappa + 1)), and A_2 is absolute by Mertens or by (Omega_0) with A_0 = kappa. So the constant depends on kappa alone, which is what "B large against A" in §6.4 needs. ### 3.3 The identification with Halberstam and Richert Theorem 2.2 Four routes now agree on the shape: pointwise |R_d| <= omega(d) for squarefree d | P(z), conclusion S << X prod_{p < z}(1 - omega(p)/p) for z <= X^A, constant depending on A and the (Omega) constants, no remainder sum, no support parameter. (i) Richert's Theorem 11.3 and his chapter note, read at page image. (ii) The 1971 Mémoire's Theorem 3, read at page image, under (Omega) (bounded omega(p)), (Omega_1) and (R), with the two clauses z <= X^mu and z >= X^{1/mu}. (iii) The Dover OCR index, re-fetched here, at pp. 68, 69, 130 and 153. (iv) The same index shows Theorem 2.5 (the fundamental lemma proper, p. 82) under the same three conditions, so Kalmynin and Konyagin's phrase "a version of the fundamental lemma" and their citation of Theorem 2.2 point at the same hypothesis set either way. What has not been done: the 1974 page image. What I did not find in any route: a Halberstam and Richert upper-bound theorem under (R) that carries a support parameter. The Selberg-form remainder sum_{d <= xi^2} 3^{nu(d)} |R_d| belongs to Theorem 4.1 (Richert 11.1), which has no (R) hypothesis; p. 130 of the OCR says Theorem 2.2 "could be derived equally well from Theorem 4.1 below, after applying condition (R)", which is the reduction the manuscript's §6.2 remark now describes. ### 3.4 Richert 11.3 from Theorem 9.1 and (11.5) (ASSUMED-UNVERIFIED item 2) Richert prints no proof beyond "one readily obtains, from (the second inequality of) Theorem 9.1 and (11.5)". Read at page image: Theorem 9.1's second inequality is S(A, p, z) <= X/G(z) + sum_{d < z^2, d | P(z), (d, p-bar) = 1} mu^2(d) 3^{nu(d)} |R_d|; (11.5) is 1/G(z) << W(z) under (Omega_1), (Omega_2(kappa)). The standard argument, re-derived here at sketch level and not adversarially checked: (a) for z' <= z, S(A, p, z) <= S(A, p, z') since fewer primes are sifted; (b) under (R) and (Omega_2(kappa)), sum_{d < z'^2, d | P(z')} 3^{nu(d)} omega(d) <= z'^2 prod_{p < z'}(1 + 3 omega(p)/p) << z'^2 (ln z')^{3 kappa}, using sum_{p < z'} omega(p)/p <= kappa ln ln z' + O(1) by partial summation of (Omega_2(kappa)); (c) take z' = min(z, X^{1/2}(ln X)^{-2 kappa}); then the remainder is << X (ln X)^{-kappa} << X W(z') (the lower bound W(z') >> (ln z')^{-kappa} follows from (Omega_2(kappa)) the same way), and for z <= X^A, W(z')/W(z) << (ln z/ln z')^kappa <<_{A, kappa} 1. Hence S << X W(z') << X W(z). This matches report 06 finding 1 as report 09 describes it. Calibration: a reviewer's reconstruction of a three-line standard argument; Richert's own derivation is not printed and so cannot be "read". ### 3.5 My attack on report 05 before reading 06 Points probed: the p < z against p <= z product mismatch (harmless, one factor >= 1 - 4/sqrt(y)); whether the encoded sequence's decomposition is required for d > X (it is, and it holds trivially); whether the Corollary needs g(p) < p at primes p > z (a formality; g is only consulted on d | P(z)); whether z << X against z <= X^A loses anything (no: z <= CX <= X^2 for X >= C); whether the constant could depend on the manuscript's A or B (no, section 3.2). I found no defect in 05's mathematics. Its exponent slip in the V(sqrt y)/V(xi) ratio (2 against 4) and its formula slip in the z << X row are already recorded in 09. Report 06's finding 6 (the representative slip) is the one thing 05 missed; I reached it independently through the TeX before reading 06, and section 4 extends it. ## 4. Ranked claim 2: the CRT representatives in Corollary 1's printed proof ### 4.1 The printed encoding P(z;p) = P(z)/p; Q(z;p) the least positive solution of Q = P(z;p) (mod p), Q = -1 (mod P(z;p)); a_m = 1 if m = prod_{p <= z} prod_{r in Omega_p} (P(z;p) n + r Q(z;p)) for some n <= X, else 0. Claim printed: (m, P(z)) = 1 and a_m = 1 if and only if n lies in no Omega_p; "Conditions of Lemma 1 also clearly hold." ### 4.2 Exact finite algebra Fix p <= z and r in Omega_p taken as an integer in [0, p - 1]. Modulo p the factor is P(z;p)(n + r), so p divides it iff n = -r (mod p). Modulo any other prime q <= z, P(z;p) = 0 and Q(z;p) = -1, so the factor is -r (mod q) and q divides it iff q | r. For r = 0 every q <= z, q != p, divides the factor. For 2 <= r <= p - 1 every prime factor q of r satisfies q <= r < p <= z and q != p, so q | P(z;p) and q divides the factor for every n. Only r = 1 is coprime to P(z;p). Consequences, all confirmed by the finite check below: 1. If some Omega_p is not contained in {1}, then (m(n), P(z)) > 1 for every n, the encoded sum is identically zero, the printed "if and only if" is false, and the Lemma 1 hypothesis fails for the encoded sequence (for example the count of n <= X with 2 | m(n) is X against g(2)X/2 + r_2 with |r_2| <= g(2)). This is sharper than "fails when 0 is in Omega_p": it fails for every representative other than 1. 2. If every Omega_p is contained in {1}, the printed encoding selects exactly the n avoiding -Omega_p for all p, not Omega_p. The sign is wrong even where the representatives are admissible. 3. With r' = r (mod p), r' = 1 (mod P(z;p)), the factor is P(z;p)(n + r) (mod p) and -1 (mod q), so (m(n), P(z)) = 1 iff n avoids -Omega_p for every p <= z: the session's repair, giving the bound for -Omega with the same g(p). The variant r' = -r (mod p), r' = 1 (mod P(z;p)) gives Omega itself. Under the repaired encoding n -> m(n) is strictly increasing on n >= 1 (every factor is positive and increasing), so a_m is a genuine indicator and the counts sum_{m = 0 (d)} a_m = #{n <= X : d | m(n)} obey |r_d| <= g(d) by section 3.2. ### 4.3 The finite check (own fixture, independent of the session's 3/0/3) Scratch script `round-0908/V/crt-check.js` (BigInt; not a repo script, not embedded; text in section 10). For each configuration it enumerates n <= X, forms m(n) under both encodings, tests (m(n), P(z)) = 1, compares with the sets avoiding Omega and -Omega, and checks |r_d| <= g(d) at every squarefree d | P(z). | configuration | avoid Omega | avoid -Omega | printed encoding | repaired encoding | Lemma 1 hypothesis, printed / repaired | |---|---|---|---|---|---| | z = 7, X = 210, Omega_2 = {0}, Omega_p = {0, -2} (Theorem A shape) | 15 | 15 | 0, equal to neither | 15, equal to the -Omega avoiders {1, 13, 19, 31, ...} | 15 of 15 d violate / 0 violate | | same, X = 100 | 6 | 7 | 0 | 7 = -Omega avoiders | 15 / 0 | | z = 7, X = 210, Omega_2 = {0}, Omega_3 = {0, 1}, Omega_5 = Omega_7 = {0, -2, 1, -1} (Theorem B band-2 shape) | 3 | 3 | 0 | 3 = -Omega avoiders {73, 103, 193} | 15 / 0 | | z = 7, X = 210, Omega_5 = {3}, others {1} (no zero; 3 divides P(z;5)) | 48 | 48 | 0 | 48 = -Omega avoiders | 8 / 0 | | z = 7, X = 210, Omega = {2:{1}, 3:{1}, 5:{1, 4}, 7:{2}} | 36 | 36 | 0 | 36 = -Omega avoiders | 7 / 0 | | z = 7, X = 210, Omega = {2:{1}, 3:{1, 2}, 5:{1, 4}, 7:{1, 6}} (symmetric) | 15 | 15 | 0 | 15 = both | 11 / 0 | | z = 7, X = 210, Omega_p = {1} for all p (the only admissible representatives) | 48 ({2, 12, 14, ...}) | 48 ({10, 12, 16, ...}) | 48, equal to the -Omega avoiders, not the Omega avoiders | 48 = -Omega avoiders | 0 / 0 | | z = 11, X = 2310, Omega_p = {1} for all p | 480 | 480 | 480 = -Omega avoiders | 480 = -Omega avoiders | 0 / 0 | | z = 5, X = 30, Omega_2 = {0}, Omega_3 = {0, 1}, Omega_5 = {0, 3} (the session fixture) | 3 ({11, 17, 29}) | 3 ({1, 13, 19}) | 0 | 3 = -Omega avoiders | 7 / 0 | The session's 3 / 0 / 3 reproduces as the last row. The slip is in the TeX source and in both page images (arXiv v2 p. 4; the mathnet artifact). The corollary's statement is unaffected: the repaired encoding proves it for -Omega, hence for Omega by applying it to -Omega. The manuscript consumes the statement. ## 5. Ranked claim 3: the D_y identity reading From `research/centered-discrepancy-measurement.json` (config C2 = 0.6601618158468696, rows j = 16..38 at shift 2), recomputed here: | j | D/x | -(T1/x - C2) | difference | S/x - C2 | P/x | E_pp/x | E_even/x | |---|---|---|---|---|---|---|---| | 30 | -0.000265 | -0.000450 | +1.84e-4 | +1.26e-4 | -8.86e-5 | +2.98e-5 | -1.8e-9 | | 31 | +0.002449 | +0.002591 | -1.42e-4 | -1.40e-4 | -2.06e-5 | +2.26e-5 | ~0 | | 32 | -0.001258 | -0.001349 | +9.10e-5 | +1.80e-4 | +7.44e-5 | +1.43e-5 | 0 | | 33 | -0.004180 | -0.004208 | +2.80e-5 | +9.73e-5 | +5.86e-5 | +1.07e-5 | -8.9e-11 | | 34 | -0.000899 | -0.000983 | +8.38e-5 | +1.25e-4 | +3.34e-5 | +7.64e-6 | -2.6e-10 | | 35 | +0.001132 | +0.001286 | -1.54e-4 | -1.11e-4 | +3.78e-5 | +5.55e-6 | 0 | | 36 | -0.001193 | -0.001204 | +1.04e-5 | +6.29e-6 | -7.87e-6 | +3.81e-6 | +1.5e-10 | | 37 | -0.000406 | -0.000455 | +4.90e-5 | +5.13e-5 | -4.09e-7 | +2.64e-6 | -1.2e-10 | | 38 | +0.001476 | +0.001473 | +3.15e-6 | +9.44e-6 | +4.34e-6 | +1.96e-6 | +9.6e-12 | Identity residuals in the artifact: S - T1 - T2 - E_pp at most 2.5e-7 absolute; T2 - acc1 - E_even at most 1.26e-5 (j = 38, sums of size 1e11); D - (acc1 - P) exactly 0 at every j. The naive reimplementation agrees at j = 16, 18, 20 to relative 9.0e-15, 7.5e-15, 6.1e-15; W1grid/W1exact 0.9747, 0.9718, 0.9661. The rearrangement D_y/x = (S/x - C2) - (T1/x - C2) - P/x - E_pp/x - E_even/x is exact given those identities and D = acc1 - P, so the reading follows from the columns. What the columns support, exactly: max |D_y/x + (T1/x - C2)| over j >= 30 is 1.84e-4; it is below 1e-4 at j = 32, 33, 34, 36, 37, 38 and below 1e-5 only at j = 38. The quoted "1e-5" (ledger verdict, OUTCOMES-adjacent QUESTIONS row) is wrong; "1e-4" (body, spec, CHANGELOG, OUTCOMES) is exceeded at j = 30, 31, 35. The body's bound "S/x - C2 at most 1.4e-4 for j >= 30" is 1.80e-4 at j = 32. The ledger's "within 0.0042 of zero for j >= 26" is 0.004283 at j = 27 (OUTCOMES already says 0.0043). The interpretation, stated at its exact scope: at every j in 30..38, D_y equals S - T1 up to at most 2e-4 x, where S is the twin count and T1 the classical term, both computed separately by the same script. D_y therefore adds no information at reachable x beyond the twin count and T1's finite convergence, which is what the note's item 3 says in other words. The do-not-rerun stands. One consequence the note does not state: by (12), D_y >= -4x/25 + o(x) is equivalent to S + 2 C2 M >= (C2 - 4/25) x + o(x), a lower bound on the twin count itself modulo the M term; that is derived here as an elementary rearrangement of (12) and relies on T1 = C2 x + o(x) only as recorded in moving-cutoff-parity (6), not re-derived. It is consistent with the handoff's statement that D_y and the full residual are not independent gains. The "order x/log^2 x" description of T1's error is a heuristic reading: at j = 38, T1/x - C2 = -1.47e-3 against 1/ln^2(2^38) = 1.44e-3; at j = 33 it is -4.2e-3. Sign oscillates. Not fitted; not claimed as a rate. ## 6. Ranked claims 4, 5, 6 **Claim 4.** Every element of the attribution re-read at source (section 2): page text and edit date; the "[4]" link; the GitHub commit date; the PDF's title, author field, page count and creation date; the covering theorem's statement; the Lean theorem, its `maximumCoverLength` and `ResidueCover` definitions (one residue per prime in a finite set of primes <= floor(X), covering every i in [1, y]), which is the one-class primorial Y(floor(X)); zero `sorry` and zero `axiom` across 408 fetched files. The transfer: a cover of [1, N] by one class per prime p <= x gives a run of N consecutive integers each sharing a factor with x#, so g(x#) >= N + 1, and G2 >= g pointwise (Proposition 2 of the manuscript); hence G2(x#) >> x ln x/ln ln ln x at the calibration of the source: formalised, unrefereed, anonymous. Not reproduced: the Lean build (no toolchain here). The G2-STATE and two-class-jacobsthal wording is accurate as it stands. **Claim 5.** From `research/shifted-prime-mobius-sums.json`: largest |U|/control-rms over j >= 20 is 2.822 (mu+, j = 35), 1.438 (mu-), 2.335 (lambda+), 1.309 (lambda-); at j = 38 the four ratios are 1.282, 0.692, 1.983, 0.286. Slopes over j >= 28 recomputed by least squares: 0.650, 0.505, 0.614, 0.276 against control rms slopes 0.465, 0.432, 0.453, 0.453, matching the stored `slopes` block to three decimals. Longest same-sign run over j = 20..38: 19, 7, 19, 4; mu+ and lambda+ negative at every j from 21 to 38. pi(2^j) + 2 equals A007053(j) at every j = 3..38 against the fetched b-file. Sq/pi at j = 38 is 0.747911, diff -4.45e-7 against A2. Mdy at j = 24 is -18508.63010892218 against -18508.63010892274 in the centered artifact. The per-draw sign runs (mu+ draws 9, 7, 5, 13, 3, 8, 4, 10; mu- draws 9, 11, 14, 19, 5, 9, 8, 12; lambda+ and lambda- draws 9, 11, 10, 18, 4, 4, 7, 9) and the control flags (plainSieveMatches(j <= 26) = true; ctrlMeanWithin4se = true; shiftsDiffer = true) are read from the bound OUTPUT block, not recomputable from the JSON. Precision point: the real +2 run of 19 exceeds every draw in its own column (max 13 for mu+, 18 for lambda+); it is matched only by one mu- draw. "Matched by a control draw" is true across the pooled 32 draws and false column by column. One of 32 draws reaching 19 is not a significance claim in either direction, and the note proposes no mechanism; the wording should say which column matched. **Claim 6.** Own computation (`round-0908/V/a2.js`): the Euler product over primes p <= 2 x 10^8 (11,078,937 primes) gives 0.7479116275, with the tail factor below 1 - 2.6e-10, so A2 = 0.7479116 to seven digits; 2 x 0.3739558136 = 0.7479116272. Both scripts carry `A2 = 0.7479116272384044`; the shifted script's fingerprint records the forced re-embed ("2 of 666 figures in the replaced block not reproduced (first: 0.73644380948, 0.011467)"); the centered script's thresholds line reads "M bound A2/2 = 0.3740". The certified interval 33/200 < C2(1 - A2) < 21/125 of moving-cutoff-parity (15) gives, at A2 = 0.7364, C2(1 - A2) = 0.1740 > 0.168, excluded; at 0.7479116 it gives 0.16642, inside (0.165, 0.168). Git history: the constant changed in commit 1a4cb7a. ## 7. The two embedded OUTPUT blocks and a tooling finding `node research/qc/embed.js --check