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Subject and independence

The reviewer is an independent reviewer working in a fresh context from the commissioning assignment alone, under a refutation charge; the reviewer took no part in writing the page under review, the library card it cites, or any other page of the folder. The subject is wiki/research/erdos_1221/dber49_inequality_3_3_reconstruction.md as it stood on 2026-09-28T05:03:27Z (the page), read whole from the committed text.

Artifact. The folder-name PDF held by the library card of N. G. de Bruijn and P. Erdős, Sequences of points on a circle, Proc. 52 (1949), 14--17: five PDF pages, a portal cover sheet and then printed pp. 14--17 as PDF pp. 2--5, with the offprint numbers 3--6 in the footers. A text-extraction pass returns only the cover sheet; the four printed pages are image-only, so they were read on rendered page images: all five pages at 110 dots per inch, PDF pp. 2, 3 and 5 at 200, and four crops of PDF p. 3 at 400 covering the Section 3 heading, (3.1), (3.2), the chain of $M$-inequalities, the $\varrho$-sum display, the $\sigma_n$ display and the two general-$r$ displays. Depth: PDF p. 3 (printed p. 15, offprint 4), Section 3, read clause by clause including every display; PDF p. 2 (printed p. 14), the Section 1 definitions, read clause by clause; PDF p. 5 (printed p. 17), the first sentence of Section 6, read for the citation of (3.3); PDF p. 4 (printed p. 16) read for structure only, to see how the parallel Section 4 labels its displays.

Allowed material read. The page; the Statement section and title line of the Theorem 1.1 reconstruction that the page cites, in the same state; the library card and its result pages Section 3, final display and Section 2 (see the exposures); the problem page Problem 1221, which has no Statement heading, read from its Statement paragraph through its Formulation paragraph and stopped before its Status paragraph; docs/verification.md, the sections "Whole-claim report" and "Audit checklist" (the canonical failure modes and the ten-item list); docs/evidence.md, "Source fidelity"; docs/math_authoring.md in full.

Exposures. Four, all disclosed here. First, the library card was read in full rather than its provenance paragraph only, so its read-status, contents and compiled-scope paragraphs were seen; they summarize the Section 3 argument and say that nothing on the card is independently reviewed. Second, the two result pages were read in full, so their Read depth, Proof pointer, Mean-normalized form, Dependencies and Bears-on sections were seen; the Section 3 result page's proof pointer sketches the same argument as the page. The derivations below were made against the scan, not against those sketches. Third, while locating the problem page's statement, its frontmatter key list (including the line status: open) and its heading names "Current assessment" and "Known Results" were seen, without their content. Fourth, the file names of the research folder were listed; its _index.md, the other reconstructions, and everything under evidence/ were not read. No web search was made and no workspace content or other review was read.

Restatement

Convention. A sequence $a=(a_1,a_2,\ldots)$ of real numbers mod $1$ is a sequence of points on the circle of circumference $1$; coincident points are allowed. For $k\ge1$ the multiset $\{a_1,\ldots,a_k\}$ cuts the circle into $k$ arcs in cyclic order, an arc between coincident points having length $0$, of total length $1$. For an integer $r\ge1$ and a stage $k\ge r$, an $r$-span is the total length of $r$ cyclically consecutive arcs, $M_k^r(a)$ is the largest $r$-span at stage $k$, and

$$\Lambda_r(a)=\limsup_{k\to\infty}kM_k^r(a)\in[r,\infty].$$

Finite form. For every sequence $a$, every integer $r\ge1$ and every integer $n\ge1$ there is an integer $k$ with $rn\le k\le(r+1)n-1$ such that

$$kM_k^r(a)\ \ge\ \sigma_n^{(r)}:=\Bigl(\sum_{p=0}^{n-1}\frac1{rn+p}\Bigr)^{-1}.$$

Limit form. For every sequence $a$ and every integer $r\ge1$, $\Lambda_r(a)\ge1/\log(1+1/r)$, and $1/\log(1+1/r)>r$; taking the infimum over all sequences, $\Lambda_r\ge1/\log(1+1/r)>r$. The bound is per sequence and per $r$, with no other uniformity claimed; nothing is claimed about sharpness for $r\ge2$; for $r=1$ the page records, as the note's own statement, that $1/\log2$ is attained by the note's Section 2 sequence. In the source, the finite form is the penultimate display of Section 3 (printed p. 15), asserted for general $r$ with "Similarly we can prove", and the limit form is the unnumbered final display; the note proves the case $r=1$ and the page supplies the general case.

Checklist

• Quantifiers and scope. Pass. "Every sequence", "every $r\ge1$", "every $n\ge1$" and "there is a $k$ with $rn\le k<(r+1)n$" match the note's "This holds for any $\{a\}$" and "for at least one $k$ $(rn\le k<(r+1)n)$"; the limit superior is used throughout, as in the note's definition; the boundary cases $p=1$ (no point inserted) and $p=n$ (all $n-1$ points inserted) of the counting step, and the case $n=1$ of the finite form (where $\sigma_1^{(r)}=r$ is the trivial bound), were checked. One side remark needs the proviso $k\ge r$ (F5).
• Circularity. Pass. The proof assumes the finite hypothesis (3.1) only to contradict it, and otherwise uses elementary properties of the logarithm; no statement equivalent to the claim is assumed.
• Model and convention changes. Pass, with a note. The zero-length-arc convention for coincident points is the literal reading of the note's "$n$ intervals with total length $1$"; the page states it in Definitions and in its source notes, and the proof handles it. The statement so read is at least as strong as any distinct-points reading, since the family is larger. No relaxed or averaged system replaces the actual objects.
• Finite and statistical overreach. Inapplicable: the page uses no finite verification and no averaging heuristic. The reviewer's own numerical search, reported under Strongest attack, is an attack record and not evidence.
• Uniformity. Pass. The constants $\sigma_n^{(r)}$ and $1/\log(1+1/r)$ depend on $r$ and $n$ only and are so displayed; the passage to the limit is for a fixed $r$ and a fixed sequence; no limits or sums are exchanged.
• Extremal conclusions. Pass. $\Lambda_r(a)$ is a limit superior in $[r,\infty]$ and the inequality holds also when it is $+\infty$; the result is an inequality, not an attained value, except the $r=1$ attainment, which is cited as the note's Section 2 statement; the infimum $\Lambda_r$ is over a nonempty family; the strict bound $\sigma_n^{(r)}<1/\log(1+1/r)$ is rederived below in the claim's own units.
• Consequences and composition. Pass. Each "hence" was checked on its own: $\varrho>\sigma_n^{(r)}$; the contrapositive at $\varrho=\sigma_n^{(r)}$; $\sigma_n^{(r)}\to1/\log(1+1/r)$; the limit superior along $k_n$; $\log(1+x)<x$; in Reading addressed, $r(\Lambda_r-1)\ge r(r-1)$ (from $\Lambda_r>r$) and the expansion $1/\log(1+1/r)-r=\tfrac12-\tfrac1{12r}+O(r^{-2})$, rederived from $1/\log(1+x)=1/x+\tfrac12-x/12+x^2/24+O(x^3)$. The only other page consumed is the Theorem 1.1 reconstruction, used descriptively ("claims"), not as a premise.
• Computation. Inapplicable: the page carries no computation and has no evidence folder.
• Reproduction. Inapplicable: the page states no rerun command and no coverage claim.
• Source and verdict fidelity. Faithful with corrections. Every locator was verified on the scan: printed p. 15 is PDF p. 3 with offprint footer 4; (3.1) and (3.2) are on it; the final display of Section 3 carries no number, and Section 6 on printed p. 17 names "(3.3)" next to (4.3) and (5.7). One defect: the page attributes to (3.1) the formula $kM_k^1(a)<\varrho$, while the printed display reads $kM_n^1(a)<\varrho$ $(n\le k<2n)$, a misprint the page corrects without saying so (F1). The standing sentence claims author-recorded status only.

Weakest steps

1. An intact block survives as a span (the corpus's completion). Represent stage $k$ as the cyclic sequence of its $k$ arcs. Inserting $a_{k+1}$ chooses one arc $I$ whose closure contains $a_{k+1}$ (when $a_{k+1}$ coincides with an existing cut, either adjacent arc; when several earlier points already coincide there, any arc whose closure contains the point) and replaces $I$ by two arcs whose lengths sum to $|I|$, one of them of length $0$ in the coincident case, leaving the rest of the cyclic sequence unchanged. This agrees with the direct definition of stage $k+1$ from the multiset $\{a_1,\ldots,a_{k+1}\}$, because the cut multiset gains exactly one point. By induction on the number of insertions, the current arcs are partitioned among the $n$ original blocks; a block none of whose arcs was ever chosen still consists of exactly its original $r$ arcs, these are consecutive in the current cyclic order, and their total length is the block's original length. Such a block is an $r$-span of the current stage, so $M_k^r(a)$ is at least its length. Each insertion chooses one arc, which lies in exactly one block, so after $p-1$ insertions at most $p-1$ blocks are disturbed; among any fixed $p$ blocks of lengths $\alpha_1,\ldots,\alpha_p$ (ties broken arbitrarily) one is intact. This composes into $M_{rn+p-1}^r(a)\ge\alpha_p$ for $1\le p\le n$, exactly what the next step consumes; for $r=1$ it is the note's "any point destroys one $I$ at most". Nothing about distinctness of the points is used.

2. The count and the contradiction. Suppose $kM_k^r(a)<\varrho$ for $rn\le k\le(r+1)n-1$. With $k=rn+p-1$ for $1\le p\le n$, step 1 gives $\alpha_p\le M_{rn+p-1}^r(a)<\varrho/(rn+p-1)$, and summing,

$$1=\sum_{p=1}^n\alpha_p<\varrho\sum_{p=1}^n\frac1{rn+p-1} =\frac{\varrho}{\sigma_n^{(r)}},$$

so $\varrho>\sigma_n^{(r)}$. The contrapositive at $\varrho=\sigma_n^{(r)}>0$ says that the hypothesis fails for at least one $k$ in the range, that is, $kM_k^r(a)\ge\sigma_n^{(r)}$ there. This is the finite form verbatim; for $r=1$ it is the note's text.

3. The passage to the limit (asserted by the note, supplied by the page). Since $1/x$ strictly decreases, for every $n\ge1$

$$\sum_{p=0}^{n-1}\frac1{rn+p}>\sum_{p=0}^{n-1}\int_{rn+p}^{rn+p+1}\frac{dx}x =\log\frac{(r+1)n}{rn}=\log\Bigl(1+\frac1r\Bigr),$$

so $\sigma_n^{(r)}<1/\log(1+1/r)$ for every $n\ge1$, and for $rn\ge2$

$$\sum_{p=0}^{n-1}\frac1{rn+p}<\int_{rn-1}^{(r+1)n-1}\frac{dx}x =\log\frac{(r+1)n-1}{rn-1}\ \longrightarrow\ \log\Bigl(1+\frac1r\Bigr),$$

so $\sigma_n^{(r)}\to1/\log(1+1/r)$. Pick $k_n$ in $[rn,(r+1)n)$ with $k_nM_{k_n}^r(a)\ge\sigma_n^{(r)}$; then $k_n\ge rn\to\infty$. For any $K$, infinitely many $n$ have $k_n\ge K$, so $\sup_{k\ge K}kM_k^r(a)\ge\limsup_nk_nM_{k_n}^r(a)$, and letting $K\to\infty$,

$$\Lambda_r(a)\ge\limsup_{n\to\infty}k_nM_{k_n}^r(a) \ge\liminf_{n\to\infty}k_nM_{k_n}^r(a)\ge\lim_{n\to\infty}\sigma_n^{(r)} =\frac1{\log(1+1/r)}.$$

Finally $x-\log(1+x)$ vanishes at $0$ and has derivative $x/(1+x)>0$ for $x>0$, so $0<\log(1+1/r)<1/r$ and $1/\log(1+1/r)>r$. Steps 2 and 3 together give the limit form from the finite form.

Strongest attack

Two attacks were pressed. The first aimed at the corpus's completion, the only part of the argument without a printed proof: find a stage-$rn$ configuration and an insertion order for which a block containing no new point is not an $r$-span of a later stage. The candidates were a new point landing exactly on a block boundary, a new point landing where several earlier points already coincide, a block made of $r$ zero arcs, and a block whose boundary point is itself a multiple point. Each is covered by step 1: the recursive description of a stage as a cyclic sequence of arcs coincides with the direct definition, a boundary point splits whichever adjacent arc is chosen and leaves the other block's $r$ arcs consecutive, and zero arcs are arcs. The attack fails because the argument uses only that each insertion refines one arc of a partition and never uses the positions of the points.

The second attack aimed at the finite form itself, where the constant is nearly attained. By hand: at $n=1$, $\sigma_1^{(r)}=r$ and the finite form is the note's trivial $rnM_{rn}^r(a)\ge r$, with equality for equally spaced points, so the bound is sharp there; at $r=1$, $n=2$, $\sigma_2=6/5$, and $2M_2^1(a)<6/5$ forces both stage-2 gaps above $2/5$, whence $3M_3^1(a)>6/5$, so the minimum over sequences of $\max_{k\in\{2,3\}}kM_k^1(a)$ is exactly $6/5$, attained by gaps $3/5,2/5$ and a third point splitting the longer gap into two pieces of at most $2/5$; at $r=2$, $n=2$, $\sigma_2^{(2)}=20/9$, five equally spaced points have $5M_5^2=2<20/9$ but their first four have $4M_4^2\ge12/5>20/9$, while four equally spaced points have $4M_4^2=2<20/9$ but any fifth point leaves a block of length $1/2$ intact, so $5M_5^2\ge5/2>20/9$. A random search and a hill-climbing search over sequences of $(r+1)n$ points, coincident points included, for $r\le4$ and $n\le7$, minimizing $\max_{rn\le k<(r+1)n}kM_k^r(a)/\sigma_n^{(r)}$, found a minimum of exactly $1$ (at $n=1$ and, to within the search's precision, at $n=2$) and never a value below $1$. This record is an attack that failed, not evidence for the claim; the claim's warrant is the proof. A third, minor attack, on the label "(3.3)", is recorded as F4: the identification survives because Section 6 groups (3.3) with (4.3) and (5.7), and (4.3) labels the corresponding final display of Section 4 while that section's general-$r$ penultimate display is unnumbered, exactly the pattern of Section 3.

Premises

• The 1949 note (held; the artifact and depth are stated above). Interface consumed: the Section 1 definitions (numbers mod $1$; $n$ intervals of total length $1$; $M_n^r(a)$ the maximum sum of $r$ consecutive intervals; $\Lambda_r(a)=\limsup_nnM_n^r(a)$; $\Lambda_r$ its greatest lower bound over sequences); the $r=1$ proof of Section 3, read clause by clause; the general-$r$ finite display and the final display, both asserted with "Similarly we can prove"; Section 6's reference to "(3.3)". Standing: an imported published result, named as the source on the page, with the page's own completion marked as such.
• Elementary analysis, no held source needed: $\log(1+x)<x$ for $x>0$; the integral comparison for the decreasing function $1/x$; the limit superior of a sequence along indices $k_n\to\infty$.
• The Theorem 1.1 reconstruction (same folder, same state; Statement section read, standing not read and not needed). Interface: for every sequence of distinct points and every $r\ge r_0$, $\limsup_n(nM_n^{(r)}-r)\ge c\sqrt{\log r}$ with absolute constants $c>0$ and $r_0$. The page uses it only to say what quantity the bound concerns, in the word "claims"; the reviewer confirmed that the quantity is $\Lambda_r(a)-r$ and that its family is the distinct-points family.
• The Section 2 result page (statement section) for the sharpness remark; independently, the note's p. 15 sentence "the lower bound is attained for the sequence of section 2" and Section 2's "$\Lambda_1(a)=1/\log2$" were read on the page images.
• Explicit assumptions. The zero-length-arc convention for coincident points, stated on the page; $k\ge r$ for the side remark of Definitions. No local L-claim is consumed and there is no batch acceptance order.

Findings

F1. Severity: required. Location: The contradiction, "the note's (3.1) for $r=1$", and the Source paragraph, "displays (3.1), (3.2)". Defect: the page attributes to (3.1) the hypothesis $kM_k^1(a)<\varrho$ $(n\le k<2n)$; the printed display reads $kM_n^1(a)<\varrho.$ $(n\le k<2n)$, with the subscript $n$ on $M$. It is a misprint for $k$: the parenthetical range, the chain $M_n^1(a)\ge\alpha_1,\ldots,M_{2n-1}^1(a)\ge\alpha_n$ that is drawn from it and the conclusion "for at least one $k$ $(n\le k<2n)$ we have $kM_k^1(a)\ge\ldots$" all require $M_k^1$. The page corrects it silently, while docs/evidence.md "Source fidelity" requires an incorrect formula in a source to be recorded explicitly. Witness: PDF p. 3 (printed p. 15, offprint 4), display (3.1), read at 400 dots per inch; compare the later display $kM_k^1(a)\ge(\ldots)^{-1}=\sigma_n$ on the same page and (4.1) on the same page, both with the subscript $k$. No mathematical change follows. Proposed replacement, after "the note's (3.1) for $r=1$": "(the printed (3.1) carries the subscript $n$ on $M$, a misprint for $k$: its range $n\le k<2n$, the chain drawn from it and the conclusion 'for at least one $k$' all read $M_k^1$)".

F2. Severity: suggested. Location: Source paragraph, "displays (3.1), (3.2) and the unnumbered final display". Defect: the page's precise statement, the "More precisely" display, is the note's penultimate, unnumbered display of Section 3, introduced by "Similarly we can prove that for at least one $k$ $(rn\le k<(r+1)n)$ we have"; the Source paragraph does not name it among the displays read, so a reader checking the precise form has no locator for it below the section. Witness: PDF p. 3, the display between "Similarly we can prove" and "and so". Proposed replacement: "displays (3.1), (3.2), the unnumbered general-$r$ display introduced by 'Similarly we can prove', and the unnumbered final display that Section 6 cites as (3.3)".

F3. Severity: suggested. Location: Source notes, first bullet, "nothing else is added". Defect: overstated. Besides the grouping and the intact-block remark, the page supplies the justification of assertions the note makes without proof: $\sigma_n<1/\log2$, $\sigma_n\to1/\log2$, "and so $\Lambda_1(a)\ge1/\log2$", and the final "and so"; on the page these are the two-sided integral comparison, the limit superior along $k_n$, and $\log(1+x)<x$. They are routine, but they are the corpus's text, and the Standing sentence promises labels wherever the page goes beyond the printed text. Witness: PDF p. 3, the sentence "We have $\sigma_n<1/\log2$, $\sigma_n\to1/\log2$, and so $\Lambda_1(a)\ge1/\log2$" and the closing "and so". Proposed replacement: "are the corpus's additions; the passage to the limit, which the note asserts without proof, is also written out here; nothing else is added."

F4. Severity: note. Location: Source paragraph, "the unnumbered final display that Section 6 cites as (3.3)". Defect: none in substance; the identification is an inference, since no display of Section 3 carries the label (3.3) on the page. Witness: PDF p. 5 (printed p. 17), "The inequalities (3.3), (4.3) and (5.7) are probably not best possible if $r\ge2$", against PDF p. 4 (printed p. 16), where (4.3) labels the final display of Section 4 and that section's general-$r$ penultimate display is unnumbered. Proposed replacement: "the unnumbered final display, evidently the (3.3) of Section 6, which groups it with (4.3) and (5.7), the final displays of Sections 4 and 5".

F5. Severity: note. Location: Definitions, "Each interval lies in exactly $r$ of the $k$ spans, so the spans sum to $r$ and $kM_k^r(a)\ge r$." Defect: the sentence needs $k\ge r$; for $k<r$ a cyclic window of $r$ consecutive intervals repeats intervals. The note states $nM_n^r(a)\ge r$ without the proviso, and nothing on the page uses the remark for $k<r$ (every $k$ in the proof satisfies $k\ge rn\ge r$). Witness: PDF p. 2 (printed p. 14), "so that $nM_n^r(a)\ge r\ge nm_n^r(a)$". Proposed replacement: "For $k\ge r$, each interval lies in exactly $r$ of the $k$ spans, so the spans sum to $r$ and $kM_k^r(a)\ge r$."

F6. Severity: note. Location: Passage to the limit, the display qualified "$(n\ge2)$" and the sentence "So $\sigma_n^{(r)}<1/\log(1+1/r)$". Defect: the left inequality, which gives the strict bound, holds for every $n\ge1$; the qualification is needed only for the right inequality, which needs $rn\ge2$. As written the strict bound is derived for $n\ge2$ and then stated without a range; it is true for $n=1$ as well, where $\sigma_1^{(r)}=r$. Witness: step 3 above. Proposed replacement: qualify the display "($n\ge1$; the right inequality for $rn\ge2$)".

Verdict

Source fidelity: faithful with corrections. One required correction, F1, records the misprint in the printed (3.1) that the page corrects silently; two suggested refinements, F2 and F3, complete the locators and the labels of supplied text; F4 to F6 are notes. Every locator on the page, physical and printed page and result label, is correct.

The argument as reconstructed: sound. Every essential deduction was re-derived above; the corpus's completion for general $r$ is correct and complete, coincident points included, and reduces to the note's printed argument at $r=1$; the finite form is sharp at $n=1$ and at $r=1$, $n=2$.

Limitations: the reviewer read only the commissioned material, with the exposures stated above; the general-$r$ case has no printed proof in the note and was checked here as mathematics, not against a source; the numerical search is an attack record, not evidence; the standing of the Theorem 1.1 reconstruction was not examined and the page does not depend on it. This focused review assigns no tier and changes no status.