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Subject

The repository as it stood on 2026-09-28T05:03:27Z. Pages, each read whole from the committed text of that state: wiki/research/erdos_49/lemma_3_2_reconstruction.md, wiki/research/erdos_49/lemma_4_1_reconstruction.md, wiki/research/erdos_49/lemma_5_1_reconstruction.md, wiki/research/erdos_49/theorem_1_2_reconstruction.md, wiki/research/erdos_49/theorem_3_1_reconstruction.md and wiki/research/erdos_49/theorem_3_3_reconstruction.md. The working-tree copies of the six pages are byte-identical to the frozen pages (the diff against that state on the six paths is empty); the folder index differs in the working tree and was not used. The folder index was read in that state. Reports, each read whole: the Lemma 3.2 review, the Lemma 4.1 review, the Lemma 5.1 review, the Theorem 1.2 review, the Theorem 3.1 review and the Theorem 3.3 review.

Read for adjudication: the sections "Independence and the assignment", "Exact subjects and durable evidence", "Report contract", "Grading and claim standing", "Whole-claim report" and "Audit checklist" of docs/verification.md; in docs/evidence.md, the section "Source fidelity" and the paragraphs before it on complete rewritten proofs and on labeling a sketch or proof pointer by its actual scope. The held author manuscript of Pollack, Pomerance and Treviño under the primes card (17 pages, 394,691 bytes; byte-identical to the PDF under the second card, same SHA-256), physical pages 2, 5--10 and 16--17 on the text layer, pages 6--10 rendered at 150 dots per inch, and pages 6 and 9 read clause by clause on the images for every passage a required finding turns on: Theorem C, Theorem 3.1 with its sketch, Lemma 3.2, Theorem 3.3 and Remark 3.1 (p. 6); the end of the proof of Lemma 4.1 with (4.4) and Lemma 5.1 with its proof (p. 9). Printed page numbers equal physical ones. To check imports and characterizations that no reviewer could reach inside its read set: the held copy of Ford's paper under Ford (1998) (43 pages, from the author's web page by the card's provenance line) on the text layer at Lemma 3.8 and at (5.13)--(5.17); the Tao card's Theorem 1.1 and strict transfer pages; the second card's Theorem 3.1 and Theorem 3.3 pages; the Theorem 2 page of the Graham, Holt and Pomerance card; the index of the Erdős (1935) card; and the frontmatter, Statement and Status paragraphs of Problem 49. For the shape of this record only, the grade and evidence indexes of two other folders were opened; nothing from them enters the adjudication. No other review, evidence folder, workspace file or web page was read.

Independence, by role: distinct grader in a fresh context, given only this assignment. The grader wrote none of the six pages, none of the six reports, no page of the folder and no library card named above, and had no communication with the author or with any reviewer. A grader is not blind: the standing text of the cards, the status paragraph of the problem page and all six reports were read by design.

Grader's own checks, run in a temporary script that is not retained and is described here so that it can be rerun. (1) The preimages of $d_1=2^{18}\cdot257$ and $d_2=d_1+28$ were enumerated by the divisor recursion (every prime $p$ dividing a preimage of $d$ has $p-1\mid d$; the preimage is assembled from such prime powers) and each result was checked with $\varphi$: the eight preimages of $d_1$ are exactly those on the Lemma 5.1 page, the least being $2^{11}\cdot257^2=135268352$; the preimages of $d_2$ are $67371037$, prime by trial division, and $134742074$; no $257\cdot2^a+1$ with $0\le a\le18$ is prime, and $2^a+1$ is prime for $a\in\{0,1,2,4,8,16\}$ only. (2) For $k=2$ and $k=6$, all solutions $n\le3000$ of $\varphi(n)=\varphi(n+k)$ were listed, those of Theorem A's shape were removed by enumerating the admissible pairs $(j,r)$, and for the rest the ratios $\varphi(m)/m$, $\varphi(m')/m'$ were compared: the solutions with equal ratios are $4,8,32,512$ for $k=2$ and $36,72,108,216,432,576,648,972,1152,2592,2916$ for $k=6$, every one with $P^+(n)\mid m$ and $m'=k$; the solution $225$ for $k=6$ has $P^+(n)^2\mid n$ but unequal ratios. (3) $\sum_{t\ge0}\exp(-\exp(0.2\cdot1.8^t))=0.72178\ldots$, so $\exp(2\cdot0.72178)=4.236\ldots$ and $\exp(4\cdot0.72178)=17.94\ldots$; $\log_217=1.0414\ldots$, $1/4.771=0.2096\ldots$; $\rho=0.5425986\ldots$, $C=0.8178146\ldots$, $C'=2.1769687\ldots$ by bisection on $F(\rho)=1$, matching the printed digits; $1/\rho=1.84298\ldots$ and $2C\log1.8=0.9614\ldots$. (4) $\omega(k)\le8\log k/\log\log3k$ holds for every $2\le k<200000$, and $\log t\le\sqrt t$ for all $t>0$. (5) The citations [4], [6], [7], [8], [10], [11] and [13] on pp. 16--17 carry the bibliographic data the pages give; footnote 1 on p. 7 says the references to [8] are to the corrected arXiv version, and [8] itself names arXiv:1104.3264v1.

Exposure ruling. Every report discloses reads wider than its allowed sections: card bodies with read-status paragraphs, "Relation to E49" sections, "Bears on" rows and a "Living verification" sentence; the problem page's Status, Source, References and Formalization paragraphs; the Standing paragraphs of sibling reconstruction pages; a result page's proof-pointer paragraph; neighboring subsections of the verification page; the file names of sibling reviews. Content test: nothing in any report could only have come from that text. Every finding cites the manuscript page or the frozen page, every re-derivation proceeds from the PDF, and the direction of each attack (the injection into Evertse's solution set, the hardened "we can assume", the range of $k$ at the top of its interval, the index $i=0$, the circularity of $K$ and $D$, the uniformity behind the collision bound) follows from the subject. None of the exposed text is a review of the pages or a verdict on them. The exposures are ruled immaterial for all six reports. The Theorem 3.1 review also records that the third card folder named in its assignment does not exist; the assignment misspelled the Tao card's folder name, the card exists under its actual name, and nothing about Theorem 3.1 depends on it, so this assignment defect is resolved without effect.

Reports graded

Lemma 3.2 review: pass. The subject block resolves (path and date). The independence facts and four exposures are stated. The restatement carries the convention (natural numbers positive, $\gamma$, $S$-units) and both clauses with their quantifiers, including the threshold $k_0(\epsilon)$ depending on $\epsilon$ alone. All ten checklist items carry an explicit verdict, the four inapplicable ones marked. The three weakest steps are re-derived, not paraphrased: the divisibility and the injection $j\mapsto(u,v)$, the exponent $1+2(1+\omega(k))$, and the second clause with the Hardy--Wright bound re-proved with the explicit constant $8$ and the explicit threshold $\exp\exp(64/\epsilon)$; the grader re-derived each and agrees. The strongest attack is real: a $j$ escaping the injection, a collision, each reading of Evertse's count (ordered, unordered, projective), and the boundary cases $k=1$, $k=2$, $j=0$. The premises carry interface and reading depth (Evertse unread and relied on as the source quotes it; Hardy--Wright unread and re-derived). The verdict is stated in full and assigns no tier.

Theorem 3.1 review: pass. The subject block resolves. The independence facts and exposures are stated. The restatement carries the conventions, the definitions of $P$, $P_0$, $P_1$ and Theorem A's shape, the absolute $x_0$ and the range of $k$. All ten checklist items carry an explicit verdict. The three weakest steps are re-derived: the reduction under the two hypotheses $p\nmid m$, $p'\nmid m'$, with the equal-ratio step proved by the largest prime of the symmetric difference and the case analysis of the excluded hypotheses; the written deduction from $q'\equiv1\pmod r$; and the reading of $x_0$. The grader re-derived the reduction and its converse and agrees. The strongest attack is real and succeeded against a sentence of the page with the exact witness $n=4$, $k=2$, which the grader verified by hand and by enumeration; the attacks on the written deduction ($r$ not an integer, $l<1$, $k=0$, $q'=r+1$) are also recorded. The premises record Theorem C, the reduction and the argument of [6] with interface and reading depth (the two cited papers unread). The verdict is stated and assigns no tier.

Theorem 3.3 review: pass. The subject block resolves. The independence facts and three exposures are stated. The restatement carries $\varepsilon(x)$'s hypotheses, the even range of $k$, the uniform $o(1)$ as a function $\delta(x)$ depending on $\varepsilon$ alone, both bounds on $c(k)$ and the corollary. All ten checklist items carry an explicit verdict. The three weakest steps are re-derived: the sieve's arithmetic factor from the local densities $\nu(p)$ (the grader recomputed $\nu(2)=1$ and the odd cases and agrees), the large-$j$ absorption with every threshold traced to $\varepsilon$ alone, and the bounds on $c(k)$ with the factor comparison $(p-1)/(p-2)\le(1-1/p)^{-2}$. The strongest attack is real: $k$ at the top of its range against the unsieved remainder, the sieve's $o(1)$ pushed through the discriminant, the bounded-$k$ clause, and Theorem A's verification. The premises carry interface and reading depth, and the reviewer's recollection of a smaller sieve constant is flagged as unverified and shown not to affect the page. The verdict is stated and assigns no tier.

Lemma 4.1 review: pass. The subject block resolves. The independence facts and two exposures are stated. The restatement carries the convention, $Z(x)$ with Ford's constants, the absolute $K$, the $D$-dependent $c_D$ and $x_0(D)$, and the scope of the page (three imports, named). All ten checklist items carry an explicit verdict. The three weakest steps are re-derived: the passage from (4.4) to the lower bound on $\log_2p_i$ with the failure at $i=0$ exhibited and two repairs given (the trivial $1/p_0<1/p_1$, and a separate argument through $n>x^{9/10}$); the product bound with the series summed; and the union bound with the coprimality remark. The grader re-derived the failure at $i=0$, both repairs and the series sum, and agrees. The strongest attack is real and succeeded against the written deduction while failing against the conclusion; the attacks on $\gg_D$ and on the range of (4.4) are recorded. The premises P1--P5 carry interface, source locator and reading depth, with the printed constants recomputed. The verdict is stated and assigns no tier.

Lemma 5.1 review: pass. The subject block resolves. The independence facts and three exposures are stated. The restatement carries the conventions, the absolute $c$ and $x_0$, the uniformity in $S$ including the empty and singleton sets, and the source's wording. All ten checklist items carry an explicit verdict. The three weakest steps are re-derived: the reversed pair by an exhaustive hand enumeration of the preimages of $d_1$ and $d_2$ (nineteen factorizations, the primes $q$ with $q-1\mid d_2$, and the exclusion of $41$ and $38$ as cofactor totients), the membership and injectivity of the two families, and the pair argument with the count. The grader recomputed the preimage sets and the primality of $67371037$ and agrees. The strongest attack is real: circularity between $K$ and $D$, resolved on two grounds including a re-derivation from p. 9 that $D$ cancels; a preimage of $d_1$ below $n_1$ or of $d_2$ above $n_2$; and a dependence of the constants on $S$. The premises carry interface and reading depth. The verdict is stated and assigns no tier. One limitation is recorded here: the reviewer's enumeration code was not retained, which a tier-bearing review would require; for this focused review the hand derivation in the report is complete on its own and is confirmed by the grader's recomputation.

Theorem 1.2 review: pass. The subject block resolves. The independence facts and four exposures are stated. The restatement carries $\mathcal W$, $W$, the weak monotonicity convention, $M^\uparrow(x)$ as a finite maximum, the quantitative form with absolute $c$, $B$ and threshold, and the consequence chain. All ten checklist items carry an explicit verdict. The three weakest steps are re-derived: the collision bound (B) from Theorems 3.1 and 3.3 with the fixed $\varepsilon(x)=(\log x)^{-1/2}$, the odd-$k$ case and the absolute bound on $c(k)$ re-derived from p. 6; $x/\log x=o(W(x))$ from the exponent of $Z(x)$; and the three index classes with the assembly and the case $\#S\le1$. The grader re-derived each and agrees. The strongest attack is real: three angles on the uniformity behind (B), all failing, and a fourth that landed on a consequence sentence of the Remark. The premises (A)--(D) carry interface, source locator and reading depth, the Problem 49 links and the Lean name are listed as unchecked, and the explicit assumptions are stated. The verdict is stated and assigns no tier.

Corrections

C1. Page: lemma_3_2_reconstruction.md. Location: frontmatter desc, second and third lines. Replace

the integers j have the same prime factors as j+k, by Evertse's bound on the equation x+y=1 in S-units of the rationals.

with

the natural numbers j have the same prime factors as j+k, by Evertse's bound on the equation x+y=1 in S-units of the rationals.

Basis, checked on p. 6 (text and image): the source reads "The number of natural numbers $j$ for which $j$ and $j+k$ have the same set of prime factors", and the page's own Statement says the same; the desc widens the domain to all integers and feeds the folder index's generated row, which regenerates from it. Filed by the Lemma 3.2 review as F1 (required); accepted. The change touches no statement or proof text.

C2. Page: theorem_3_1_reconstruction.md. Location: section "The source's sketch", item 1, the two sentences "Reduction (imported from Graham, Holt and Pomerance): if $\varphi(m)/m=\varphi(m')/m'$, then $n$ has the shape of Theorem A. So for the solutions counted by $P_1(x;k)$, $\varphi(m)/m\ne\varphi(m')/m'$." Replace them with:

Reduction (imported from Graham, Holt and Pomerance, with two hypotheses supplied here): if $p\nmid m$, $p'\nmid m'$ and $\varphi(m)/m=\varphi(m')/m'$, then $n$ has the shape of Theorem A with $j=m$. The source states this without the two hypotheses and then says "we can assume" that the ratios differ. A solution with $p\mid m$ can satisfy the equality without having Theorem A's shape (for $k=2$ the solutions $n=4$, $8$, $32$: $\varphi(4)=\varphi(6)=2$, $m=m'=2$, while Theorem A's shape for $k=2$ is $2(2r+1)$); such solutions are counted by $P_1(x;k)$ and must be disposed of separately, which neither the source's sketch nor this page does. For the solutions counted by $P_1(x;k)$ with $p\nmid m$, $\varphi(m)/m\ne\varphi(m')/m'$: if also $p'\nmid m'$ this is the reduction, and if $p'\mid m'$ the equality would force $m=-k$.

Basis, checked on p. 6 (text and image): the source's sentences are "As in [10], if $\varphi(m)/m=\varphi(m')/m'$, then $n$ has the shape indicated in Theorem A. So we can assume that $\varphi(m)/m\ne\varphi(m')/m'$." The page turned the hedge into a universal statement about $P_1(x;k)$, which is false: with $k=2$, $n=4$ one has $\varphi(4)=\varphi(6)=2$, $p=2$, $m=2$, $p'=3$, $m'=2$, equal ratios $1/2$, and $n=4$ is not of Theorem A's shape, since the only $j$ with $\gamma(j)=\gamma(j+2)$ is $j=2$ (the grader confirmed this for $j<200000$) and that shape is $2(2r+1)$. The grader re-derived the reduction under the two hypotheses: equal ratios and $\varphi(n)=\varphi(n+k)$ give $m(p-1)=m'(p'-1)$, hence $m'=m+k$; equal ratios force $\gamma(m)=\gamma(m')$ (the largest prime of the symmetric difference divides one side of the cleared equation and not the other); then $\gcd(a,b)=1$ with $a=m/g$, $b=(m+k)/g$ gives $p-1=br$ and $p'-1=ar$ for one $r\ge1$, the two primes do not divide $m$, and $n=m(br+1)$. Without the hypotheses: $p\mid m$, $p'\nmid m'$ and equal ratios give $mp=m'(p'-1)$, that is $m'=k$; $p'\mid m'$, $p\nmid m$ give $m=-k$; both dividing give $n=n+k$. The escaping solutions are therefore exactly those with $p\mid m$ and equal ratios, all with $n+k=kp'$; the grader's enumeration found $4,8,32,512$ for $k=2$ and eleven such $n\le3000$ for $k=6$. Filed by the Theorem 3.1 review as F1 (required); accepted with the wording above, which says "can satisfy" because a solution with $p\mid m$ may also have unequal ratios ($n=225$, $k=6$). The change touches the sketch's step 1 only; the page's Statement, the written deduction and the Gaps paragraph stand.

C3. Page: lemma_4_1_reconstruction.md. Location: section "Proof of (iii)", from "Also $p_0>p_1$, so the bound for $i=1$ covers $i=0$." through the display bounding $\sum_{i=0}^L1/p_i$, and the end of the final display. Replace the two sentences and the display with:

For $i=0$ the imported inequality gives nothing, since $x_0=1$ is a convention rather than $\log_2p_0/\log_2(x/D)$; but $p_0>p_1$ gives $1/p_0<1/p_1$. Hence $p_i\ge\exp\bigl(\exp(0.2\cdot1.8^{L-i})\bigr)$ for $1\le i\le L$ and

$$> \sum_{i=0}^L\frac1{p_i}\le2\sum_{i=1}^L\frac1{p_i} > \le2\sum_{t\ge0}\exp\bigl(-\exp(0.2\cdot1.8^t)\bigr), >$$

a convergent series independent of $x$, $D$ and $L$.

and end the final display with "$\le\exp\bigl(4\sum_{t\ge0}\exp(-\exp(0.2\cdot1.8^t))\bigr)=:K$", keeping "with $K$ absolute" (the sum is $0.7218\ldots$, so $K<18$).

Basis, checked on p. 9 (text and image): the source states the lower bound $\log_2p_i\ge0.2(1.8)^{L-i}$ for $1\le i\le L$ only and passes directly to "$\sum_{i=0}^L1/p_i$ is absolutely bounded"; the handling of $i=0$ is the page's own step. From $p_0>p_1$ one gets only $p_0>\exp(\exp(0.2\cdot1.8^{L-1}))$, not the displayed $\exp(\exp(0.2\cdot1.8^{L}))$, and (4.4) gives no lower bound for $p_0$ because $x_0=1$ is a convention. The displayed sum bound used one term per index with the unestablished term $t=L$ for $i=0$; the repaired bound uses the term $t=L-1$ twice. The conclusion (iii) with an absolute $K$ survives. Filed by the Lemma 4.1 review as F1 (required); accepted. The grader also checked, in the held copy of Ford's paper, that Lemma 3.8 there reads "Let $x_0=1$ ... If $\mathbf x\in\mathcal S_L(\boldsymbol\xi)$ and $\xi_i\ge1$ for all $i$, then $x_j\le4.771\xi_i\cdots\xi_{j-1}\rho^{j-i}x_i$ for $0\le i<j\le L$", which with $\boldsymbol\xi=\mathbf1$ is (4.4) as the page imports it.

C4. Page: theorem_3_3_reconstruction.md. Location: Standing paragraph, the sentence "Three inputs are imported and not re-derived: Theorem A (whose short verification is nevertheless written out below), Selberg's upper bound sieve in the form the source states, and Evertse's $S$-unit bound inside Lemma 3.2." and the Gaps paragraph's last sentence "Everything else is written out." Replace the first with

Three inputs are imported into the proof and not re-derived: Theorem A (whose short verification is nevertheless written out below), Selberg's upper bound sieve in the form the source states, and Evertse's $S$-unit bound inside Lemma 3.2; the corollary's two classical bounds on $\omega(k)$ and $k/\varphi(k)$ are imported as well.

and the second with

The two classical bounds used in Step 0′ for the corollary are imported from Hardy and Wright, not held. Everything else is written out.

Basis, checked on p. 6 (Remark 3.1 uses $\prod_{p\mid k,p>2}(p-1)/(p-2)\ll k/\varphi(k)\ll\log\log k$ and $\omega(k)\ll\log k/\log\log(3k)$ without derivation) and on the page (Step 0′ consumes both, and the Imported inputs section lists them as not held): the Standing count and the Gaps sentence are false as written, since the page's own inventory names five imports. Filed by the Theorem 3.3 review as F1 (suggested); promoted to a correction on the grader's verification, because a page's statement of its own scope is a fidelity surface. The change touches two scope sentences, not the statement or the proof.

C5. Page: lemma_5_1_reconstruction.md. Location: Standing paragraph, the sentence "The argument is written out in full; its one imported input, Lemma 4.1, is itself only partly reconstructed (its counting steps are Ford's)." Replace it with

The argument is written out in full. Its imported inputs are Lemma 4.1, itself only partly reconstructed (its counting steps are Ford's), and Ford's order of magnitude $W(x)\asymp Z(x)$, quoted from the source's p. 7 and not reread.

Basis, checked on p. 7 (Ford's $V(x)\asymp W(x)\asymp Z(x)$ quoted as [8, §§4, 5]) and p. 9 (the proof closes with $\gg_DZ(x)\gg W(x)$), and on the page (Step 3 uses $Z(x)\ge c'W(x)$ and the Imported inputs section lists Ford's order of magnitude as a second input): "its one imported input" is false as written. Filed by the Lemma 5.1 review as F1 (suggested); promoted to a correction on the grader's verification, for the same reason as C4. The change touches one scope sentence.

Rejected and downgraded findings

Lemma 3.2 review.

• F2 (suggested: mark the general degree-$d$ form of Evertse's bound as recalled and unchecked). Downgraded to optional; no change required. The parenthetical states the form in which Evertse's theorem is commonly quoted, with $d$ the degree and $\#S$ the number of places, and the source's $3\cdot7^{1+2\#S}$ is its case $d=1$; the paper is not held, so no held text confirms the general form, and the page already marks the paper "not held" on the same line. The marker may be added.
• F3 (suggested: label the supplied justifications). Rejected; no change. The evidence rules require a label for a repair of a gap or of an incorrect formula; the page supplies only the elementary reasons the source's four-sentence proof omits, all correct, and attributes none of them to the source.
• F4 (note: $v\ne0$ needs $j\ge1$). No change required. The convention that natural numbers are positive is the source's; the count is the same under either convention, as the report shows. The phrase may be added.
• F5 (note: the $O$-statement fails at $k=1$ under an absolute constant). No change required. The line is scoped by "as $k\to\infty$" and holds for $k\ge2$; the rephrasing may be adopted.

Theorem 3.1 review.

• F2 (note: "the source's sketch" versus "the written sketch"). Downgraded to optional. The clarification is accurate, since the unwritten changes to the argument of [6] may also use the range of $k$; the sentence as written is true of the text the page reconstructs.
• F3 (note: $n=1$, $k=1$ has no largest prime factor). Downgraded to optional. Verified: $\varphi(1)=\varphi(2)=1$, and $n=1$ is counted by $P_1(x;1)$ for $x\ge1$ since Theorem A needs $k$ even; the source and the page pass over it, and it changes the count by at most one.
• F4 (note: the odd-$k$ remark is supplied and unused). No change required. The remark is correct (for odd $k$ exactly one of $j$, $j+k$ is even, and $j=1$ has no prime factor while $j+k\ge2$ has one) and harmless; marking it as supplied is optional.

Theorem 3.3 review.

• F1 (suggested). Promoted to C4.
• F2 (suggested: "taken from the source" overstates what the source asserts about the sieve's uniformity). Downgraded to optional. The source states the bound for fixed $j$ "as $x\to\infty$" (p. 7) and then sums over $j$, and its theorem asserts "uniformly in $k$", so the uniformity is what the source's own argument requires and implicitly asserts; the page's phrase is a fair summary, and the fuller wording may be adopted.
• F3 (note: "$\le$" where "$\ll$" is meant in Step 0′). No change required. The implied constant of $\ll(\log\log3k)^2$ can be absorbed into the factor $k^{o(1)}$ on the same line, so the displayed inequality holds as written for large $k$, and the conclusion $c(k)\to0$ is unchanged.
• F4 (note: the bounded-$k$ clause is the corpus's reading). No change required. The source's "for large enough $x$" covers the finitely many $k\le k_0(1)$, and the page's clause is the correct and needed reading; the marker may be added.

Lemma 4.1 review.

• F2 (suggested: the Standing and desc count two imports where three are used, and the candidate set is the source's adaptation). Downgraded to optional. The third import, Ford's Lemma 3.8 as (4.4), is labeled in the "Proof of (iii)" section and again in the Gaps paragraph, and the desc's "two counting steps" is accurate because (4.4) is not a counting step. The candidate set is Ford's: in the held copy, Ford's (5.13)--(5.16) define $\mathcal B$ as the integers $n=p_0p_1\cdots p_L>x^{9/10}$ with each $p_i$ prime, $\varphi(n)\le x/d$, $(x_1,\ldots,x_L)\in\mathcal S_L(\boldsymbol\xi)$, $\log_2p_i\ge(1+\omega_i)\log_2p_{i+1}$ and $p_L\ge\max(d+2,17)$, which the source repeats with $d$ replaced by $D$. The Standing may name the third import.
• F3 (suggested: "the system (4.3) says" versus the source's "the conditions on $n$ imply"). Downgraded to optional; wording only, and the page states on the next line that Ford's sets were not checked.
• F4 (suggested: replace the hedge on a dependence on $d_1,d_2$ by the finite-set argument). Downgraded to optional. The hedge is not wrong, and the report's argument (for fixed $D$ the totients $d_1,d_2\le D$ range over a finite set) is right; either wording is acceptable.
• F5 (note: "$p_L>17$" read as "$p_L\ge17$", and primality supplied). No change required. The source's definition on p. 8 gives $p_L\ge17$, so the page's reading is the right one, and Ford's (5.13) says "each $p_i$ prime", which the source's candidate set adapts; noting either is optional.
• F6 (note: $i=L$ lies outside the range of (4.4)). No change required. At $i=L$ the displayed chain reduces to $x_L\log_2(x/D)\ge x_L\log_2(x/D)/4.771$, which is trivially true, so the chain holds at $i=L$ without (4.4); the source uses the same phrasing.
• F7 (note: the quoted phrase is not verbatim). No change required. The quotation differs from the source only by the word "by" placed inside the quotation marks; the locator is right.
• F8 (note: which version of Ford's paper the card holds). Rejected; no change. The held copy (43 pages, from the author's web page) carries Lemma 3.8 with the constant $4.771$ and the convention $x_0=1$, and equation (5.17), at the labels the source cites, and its printed pages include 25--29; the page's "holds a copy" is accurate. Byte identity with arXiv:1104.3264v1 was not established and is not needed for the locators.
• F9 (note: two deferred definitions are not linked). Downgraded to optional; a page-mechanics improvement, not a fidelity or argument matter. Both deferred definitions agree with the source, as the report checked and the grader confirmed on pp. 2 and 7.

Lemma 5.1 review.

• F1 (suggested). Promoted to C5.
• F2 (suggested: mark the supplied check $D\ge\max\{d_1,d_2\}$ and the inference $K\ge1$). Downgraded to optional. The check is correct and the source does omit it; the inference $K\ge1$ is valid for large $x$, since the lemma then supplies at least one $n$ with $1\le n/\varphi(n)\le K$, and any larger $K$ also serves. The rewording may be adopted.
• F3 (note: $Z(x)$ is defined on the source's p. 7; the corrected arXiv version). Downgraded to optional; the page's Definitions could cite p. 7 directly, and the footnote may be recorded.
• F4 (note: $n_1,\ldots,n_k$ against $n_1$, $n_2$). No change required. The source uses the same letters, and the page's Step 1 reads correctly.
• F5 (note: desc wording). Downgraded to optional. "Ford's convenient integers" reflects the source's own "methods of Ford" (p. 2), and "the totients up to $x$" reads as $\mathcal W(x)$ in the desc's context; the more exact wording may be adopted.

Theorem 1.2 review.

• F1 (suggested: the Remark's "alone" and "Ford's machinery"). Downgraded to optional. The sentence is ambiguous rather than false: the $o(x)$ consequence needs only (B) and (D), as the sentence says; the bound $\limsup\le1$ needs $x/\log x=o(W(x))$ as well, which the page derives from (C) and could equally take from the Erdős (1935) lower bound it mentions (the card digests $N(M,n)>cn\log v/\rho$ with $v=\log\log n$, $\rho=\log n$); "as the source notes on p. 2" is a faithful attribution of the source's own loose sentence. The reviewer's precise wording may be adopted.
• F2 (suggested: "Clause (i) remains open" is a status sentence). Rejected; no change. The sentence restates the recorded status of the problem page (frontmatter open; Status paragraph "Open for the remaining clause") and changes nothing.
• F3 (note: $\#S\le1$). No change required; the bound is trivial there, and the parenthetical may be added.
• F4 (note: the title paraphrases Lemma 5.1's conclusion). Downgraded to optional. The title is a loose paraphrase of $M^\uparrow(x)\le(1-c)W(x)$; the desc states the theorem exactly.
• F5 (note: the inventory of imports in the Standing). No change required. The Standing makes no count claim; (C) and (D) are listed as external theorems in the Imported inputs section, and $C_2$ is defined on the Theorem 3.3 page the sentence links.

Checks of the grader's own that produced no correction. The page's characterization of Tao's result holds: the Tao card's Theorem 1.1 page states $\pi(x)\le M(x)\le(1+C(\log_2x)^5/\log x)\pi(x)$ for the weak maximum, and its strict-transfer page derives $M_<(N)\sim\pi(N)$. The Theorem 3.1 page's description of the Graham, Holt and Pomerance card's Theorem 2 page ("records the statement and, likewise, only a proof pointer") holds. The second card's Theorem 3.1 and Theorem 3.3 pages record the statements as the pages say. The Theorem 1.2 page's aside on the Erdős (1935) lower bound matches that card's digest of Part 2, which gives $N(M,n)>cn\log v/\rho$, that is $x\log_3x/\log x$; the card's later bullet writes the same bound with $\log\log n$ in place of $\log v$, an inconsistency inside that card and outside this subject. Every locator on the six pages (Theorem 1.2 p. 2, §5 p. 10; Theorem A p. 5; Theorem C, Theorem 3.1, Lemma 3.2, Theorem 3.3 and Remark 3.1 p. 6; the proof of Theorem 3.3, Ford's order of magnitude, $Z(x)$ and footnote 1 p. 7; (4.1), (4.2), Lemma 4.1 and the candidate set p. 8; (4.4) and Lemma 5.1 p. 9) matches the manuscript, and the quoted phrases "goes through with obvious minor changes", "clearly" and "$\gg_DV(x)$" are verbatim.

Graded verdicts

• lemma_3_2_reconstruction.md: fidelity faithful, with the desc correction C1; argument sound. The injection into Evertse's solution set, the exponent and the second clause were re-derived here.
• theorem_3_1_reconstruction.md: fidelity faithful, with the correction C2 to one sentence of the sketch that hardened the source's "we can assume" into a false universal; argument: the one deduction the page reconstructs (that $q'\nmid m$ and that $mp+k\equiv0\pmod{q'}$ fixes $p$ modulo $q'$) is sound and uses the range of $k$ where the page says; the rest is a proof pointer to Graham, Holt and Pomerance and to Erdős, Pomerance and Sárközy, as the page states, with the escaping solutions $p\mid m$ now labeled as a gap the sketch does not close. The page is a record of a sketch, not a reconstruction of the theorem's proof, and says so.
• theorem_3_3_reconstruction.md: fidelity faithful, with the scope correction C4; argument sound given the imports at the strength the page states (Theorem A, Lemma 3.2 as consumed, the sieve bound with its uniform $o(1)$ and the constant $16C_2$ as the source states them, and the two classical bounds). The sieve's arithmetic factor, the large-$j$ absorption with $k$-independent thresholds, and the bounds on $c(k)$ were re-derived here.
• lemma_4_1_reconstruction.md: fidelity faithful; argument: a partial reconstruction, as the page states. The deduction of (i)--(ii) from the two imported counts is sound; the written deduction of (iii) was defective at the index $i=0$ and is repaired by C3, after which (iii) holds with an absolute $K<18$; the imports (F1), (F2) and (4.4) are taken as the source cites them, and the grader confirmed in the held copy of Ford's paper that (4.4) is Ford's Lemma 3.8 with $\boldsymbol\xi=\mathbf1$ and that the candidate set is Ford's (5.13)--(5.16) with $d$ replaced by $D$.
• lemma_5_1_reconstruction.md: fidelity faithful, with the scope correction C5; argument sound. The reversed pair ($d_1<d_2$, $n_1$ the least preimage of $d_1$, $n_2<n_1$ the greatest preimage of $d_2$) was recomputed here, and Steps 1--3 were re-derived, with $K$ absolute so that $D=Kn_1n_2$ is not circular.
• theorem_1_2_reconstruction.md: fidelity faithful; argument sound given its imported inputs at the depth their pages state (Theorem 3.1 a sketch, Lemma 4.1 partial, Ford's and Erdős's counts external). The deduction of the collision bound (B), the step $x/\log x=o(W(x))$, the three index classes and the assembly were re-derived here, and the consequences for Problem 49 hold as stated.

No tier is assigned and no status changes.