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Subject

Frozen subject. The page wiki/research/erdos_1221/ko26b_lemma_4_2_reconstruction.md as it stood on 2026-09-28T05:03:27Z (ko26b_lemma_4_2_reconstruction), read whole from the committed text of that state, every deduction of its proof checked line by line by the grader.

Report graded. The second independent focused review, ko26b_lemma_4_2_reconstruction_review_2, read whole.

Relation to the first grade. The first review of this page, ko26b_lemma_4_2_reconstruction_review, was graded void on form in the first grade (its checklist was filed against the shared list, not the ten Erdos items); that grade's correction C6 for this page has since been applied to the working tree. The first review's body was not read here; only the first grade's Subject, Corrections and Graded verdicts sections were, as the assignment directs. The working-tree page was compared with the frozen bytes: the only differences are the two C6 edits (the Source paragraph's locators and the locator "(p. 8)" on the subheading "The source's derivation, as stated") and the updated time. The second review assessed the frozen bytes, so its F1 addresses text that C6 has already replaced.

Rules read. docs/verification.md, the sections "Independence and the assignment", "Exact subjects and durable evidence", "Report contract", "Grading and claim standing", "Whole-claim report" and "Audit checklist"; docs/evidence.md, the section "Source fidelity".

Source artifacts read for adjudication. The PDF held by Korsky 2026, resolution: text layer, pp. 7--8 (the definition of $H_L$, Theorem 4.1, the derivation paragraph, Lemma 4.2 and its proof) read whole and p. 8 also read on a rendered page image, display by display; p. 6 (the statement of Proposition 3.1), p. 9 (Section 5 through Remark 5.1) and the reference entry [11] on p. 16 read from the text layer. The grading assignment named the folder of the other Korsky preprint, Korsky 2026, improved lower bound, as the source; the grader searched its text layer and found no Section 4, Theorem 4.1, Lemma 4.2 or reference to Larcher, so that naming is an assignment defect, resolved as the reviewer resolved it: the page's Source paragraph names the resolution preprint, which holds Section 4, and that PDF is the artifact. The card Schmidt 1972 was opened to settle which Schmidt paper the corpus holds. The papers of Larcher (2015) and Schmidt (part VII, 1972) were not read; the corpus holds neither.

Independence. The grader is a role distinct from the author of the page and from both reviewers: a fresh context given only the grading assignment, taking no part in writing the page or either report, with no contact with their authors. The first review's body was not read. The working-tree status of the folder was seen while resolving the subject. No web search was made.

Ruling on the reviewer's disclosed exposures. The report discloses three: the assignment's naming of the wrong Korsky folder, resolved by reading the preprint the page names and not opening the other; the sight of the first review's file name in a directory listing, without opening it; and the read-status paragraphs and the truncated opening of a standing sentence on the two library cards. Ruling by the content test: immaterial. Nothing in the report could only have come from that text, no attack direction or verdict followed it, and no verdict on the subject page reached the reviewer. The review stands as independent.

Report graded

ko26b_lemma_4_2_reconstruction_review_2: pass. The subject block names the path and the date of the frozen text, read from the committed text. The independence section states the reviewer's role, the material allowed and read, and three exposures, ruled immaterial above. The restatement carries every quantifier: the sequence of distinct points, $B\ge1$, $S\ge2$, the threshold $n_0$ for all $n\ge n_0$, all $x\in\mathbb T$ and all real $D\in[0,S]$, the condition $\lfloor S\rfloor\ge L_0$, and the absolute $L_0$ of Theorem 4.1, with the sequence-dependence of $n_0$ made explicit. The checklist gives an explicit verdict on all ten Erdos items by name (Quantifiers and scope, Circularity, Model and convention changes, Finite and statistical overreach, Uniformity, Extremal conclusions, Consequences and composition, Computation, Reproduction, Source and verdict fidelity), with a reason for each inapplicable one. The three weakest steps are actually re-derived: the pigeonhole on the $N$ cyclic $L$-spans with the translation window $0<\varepsilon<\min(g_1,g_2)$ and the bound $\ell<\delta\le\frac12$; the prefix-to-arc identity at the insertion time with the forced $j=1$ for early prefixes; and the convex combination $(1-u)f(u)-u(f(1)-f(u))$ with the length budgets $D,D'\le n\ell\le L\le S$ and the passage from the $\le$ to the $<$ convention. The strongest attack is real: it tests the two applications of (4.1) at one time with different base points, the possibility of $j=2$ below $n_0$, and the convention change at $u=0$ and $u=1$. The premises name the interface of Theorem 4.1, its reading depth, the unheld Larcher paper, the consumer-side reading of Proposition 3.1 and the unheld Schmidt paper with the interface the authored remark uses. The verdict is explicit on fidelity and argument with its limitations. The grader's own checks agree: the arithmetic $(a-2)(8a+3)/(16(1-2a)^2)$ at $a=7/2$ is $(3/2)\cdot31/576=31/384$, and $31/(384\log3.5)=0.0644\ldots>0.064>1/16$; the Section 5 comparison on p. 9 reads $B=\frac3{100}\log r+O(1)$ against $\frac1{16}\log\lfloor S\rfloor=\frac1{32}\log r-O(\log\log r)$; and Proposition 3.1 on p. 6 sets $S=\sqrt{Ar}/\Lambda^2$.

Corrections

None. The report's one suggested finding is the correction C6 already applied; its two notes are optional, below.

Rejected and downgraded findings

• F1 (suggested): already handled by C6 of the first grade, applied to the working tree. Verified again on the p. 8 image: the paragraph headed "Derivation from Larcher's proof" opens p. 8 and Theorem 4.1 closes p. 7. The current page reads "Theorem 4.1 (p. 7), its derivation from Larcher's proof (p. 8) and Lemma 4.2 (p. 8)" and the subheading carries "(p. 8)"; the reviewer's proposed wording differs from C6 only in word order. No further change.
• F2 (note): downgraded to optional. The reviewer is right that Schmidt's part VII is not held: the corpus card for Schmidt 1972 is Irregularities of distribution VI (Compositio Math. 24, 63--74), a different paper from the VII (Acta Arith. 21, 45--50) the remark cites, so the optional pointer in the first grade's entry "ko26b L4.2 F2" does not apply to this remark. The remark is page-authored, marked "checked here", cited in full, and used only for the qualitative alternative $H_L\ge c\log L-1$, not by the lemma's proof. The reviewer's own re-derivation shows the conclusion survives either box convention (the point $(z_L,1)$ on the top edge costs at most one unit of count, and the closed-in-$v$ count is the one-sided limit of the half-open count), which the grader confirms. Saying that the paper is not held and naming the box convention would be tidy; no result or attribution changes.
• F3 (note): rejected as wording. The source says "pp. 12--13 of the preprint" and its reference [11] on p. 16 identifies that preprint as "arXiv:1407.2094" while listing the journal version at pp. 474--485; the page's "pp. 12--13 of the arXiv preprint, per the source" is a faithful reading of the source together with its reference list, not an inference beyond it.

Graded verdict

No tier is assigned and no status changes. The page remains an author-recorded reconstruction; the verdict below is that of the graded focused review, read with the first grade's correction C6 applied.

• ko26b_lemma_4_2_reconstruction. Fidelity: faithful with correction (C6, applied); the definition of $H_L$, Theorem 4.1 as used, the derivation paragraph with its constant $c_a$ and the value at $a=7/2$, the statement of Lemma 4.2 with (4.1) and its quantifiers, the proof's steps and the Role paragraph all match pp. 6--9 of the resolution preprint, and the current page's locators are right. Argument: sound relative to the imported Theorem 4.1; every supplied detail (the pigeonhole on $L$-spans, the translation by $\varepsilon$, the prefix-to-arc identity, the early-prefix case, the convex combination, the convention change) is correct and expands the source's own steps. The composition inherits the unchecked premise the page discloses: the finite-list form of Theorem 4.1 with an absolute threshold rests on the source's derivation from Larcher's unheld paper.

No tier is assigned and no status changes.