---

Subject and independence

Role and independence. The reviewer is an independent reviewer working in a fresh context from the commissioning assignment alone, took no part in writing the page under review or any page in its folder, and received the assignment from the commissioning process, identified here by role. The charge is refutation; agreement was not the goal.

Frozen subject. wiki/research/erdos_1221/dber49_inequality_5_7_reconstruction.md as it stood on 2026-09-28T05:03:27Z, read from the committed text.

Artifact and reading depth. The held PDF under the library card (five physical pages: a portal cover sheet, then printed pp. 14--17). The file is image-only; its text layer holds the cover sheet only, so every statement was read on rendered page images. Physical pages 4 and 5 (printed pp. 16--17, offprint pp. 5--6, Section 5 with displays (5.1)--(5.7) and footnote 3) were rendered whole at 150 dpi and again at 260 dpi in three crops (the p. 16 Section 5 opening with footnote 3; the p. 17 text from "Clearly" through the $r=1$ line; the p. 17 text from (5.5) through (5.7)), and read clause by clause. Physical page 2 (printed p. 14, Section 1) was rendered at 150 dpi and read for the definitions and the display $nM_n^r(a)\ge r\ge nm_n^r(a)$, which the page's Definitions and Source notes consume. No canonical conversion sits beside the PDF.

Allowed material actually read. The page; the Definitions and Statement sections of the folder's Section 3, 2026-note Theorem 1.1 and 2026-preprint Theorem 1.1 reconstruction pages in the same state; the library card and the result pages (5.1) and (5.7) and Section 2 in the same state; the Statement paragraph of Problem 1221; docs/verification.md "Whole-claim report" and "Audit checklist"; docs/evidence.md "Source fidelity"; docs/math_authoring.md in full. The other three library folders named by the assignment were not opened.

Exposures. Four pieces of excluded or out-of-set text reached the reviewer through over-wide prints and are disclosed here; none was used for any verdict below. (1) The problem page's Status paragraph, which sits in the body between the Formulation and Source paragraphs and was printed with the statement region; its Source, References and Formalization paragraphs came with it. (2) The library card beyond its provenance paragraph: the generated rows, the read-status paragraph, the contents list and the bears-on and results lists. (3) The two result pages beyond their Statement sections: the proof pointer, dependencies and bears-on lists; the (5.1)-and-(5.7) proof pointer summarizes the same argument as the page, so it is the one exposure that could have biased the reading, and every step below was re-derived from the page images rather than from it. (4) The general "Audit checklist" section of docs/verification.md, which precedes the repository-specific section of the same name, and the Source and Standing paragraphs that precede the Statement sections of the three folder pages. No standing, acceptance or assessment text other than these reached the reviewer.

Restatement

Convention. A sequence $a=(a_1,a_2,\ldots)$ of numbers mod $1$ is a sequence of points on the circle of circumference $1$; the source does not exclude coincident points. At stage $n$ the points $a_1,\ldots,a_n$ cut the circle into $n$ intervals (an interval has length $0$ at a coincidence). An $r$-span at stage $n$ is the sum of $r$ cyclically consecutive intervals of stage $n$; $M_n^r(a)$ and $m_n^r(a)$ are the largest and the smallest $r$-span. The $n$ spans together count each interval exactly $r$ times, so they sum to $r$ and $m_n^r(a)\le r/n\le M_n^r(a)$. The ratio constant of a sequence is $\mu_r(a)=\limsup_{n\to\infty}M_n^r(a)/m_n^r(a)$, with $M/0$ read as $+\infty$ (the page's reading; the source is silent), and $\mu_r=\inf_a\mu_r(a)$ over all sequences.

Claim (5.1), as the page states it. For every sequence $a$, every integer $r\ge1$ and every integer $n\ge2r-1$,

$$M_n^r(a)\ \ge\ \Bigl(1+\frac1r\Bigr)m_{n+1}^r(a).$$

The source asserts this for every $n\ge1$; the page omits the range $1\le n<2r-1$ (nonempty only for $r\ge2$) and says so.

Claim (5.7). For every sequence $a$ and every integer $r\ge1$, $\mu_r(a)\ge1+1/r$; hence $\mu_r\ge1+1/r$, that is, $r(\mu_r-1)\ge1$. The page's route passes through the intermediate statement: for every integer $n\ge2$ there is a $k$ with $rn\le k\le(r+1)n$ such that either $m_k^r(a)=0$ or $M_k^r(a)/m_k^r(a)\ge(1+1/r)/(1+1/k)^2$.

Checklist

• Quantifiers and scope: pass. Both claims are stated for every sequence and every $r\ge1$, matching the source's "Let $\{a\}$ be a sequence" and "$r\ge1$" (p. 16). The one scope change, $n\ge2r-1$ in (5.1), is declared in the Statement section and in the Source notes, and the (5.7) argument stays inside it. The limit superior is handled by an explicit sequence $k_n\to\infty$; the boundary case of a vanishing span is covered by the declared $+\infty$ reading and the multiplicative form. The restriction $n\ge2$ in the (5.7) proof is visible and reasoned but not attributed (F1).
• Circularity: pass. (5.7) consumes (5.1) at stages $k\ge rn\ge2r$, which the page proves first; nothing equivalent to either claim is assumed.
• Model and convention changes: pass. The page's multiplicative form of (5.1) is equivalent to the source's ratio form whenever $m_{n+1}^r(a)>0$ and is trivially true otherwise; the $+\infty$ reading of $M/0$ only adds sequences with $\mu_r(a)=+\infty$, which cannot lower the infimum. Both are labeled as the page's conventions.
• Finite and statistical overreach: inapplicable. No finite case, sample or heuristic average carries any step; the reviewer's own random spot-check of (5.1), mentioned under "Strongest attack", is not evidence and is not relied on.
• Uniformity: pass. The constant $1+1/r$ is explicit, and (5.1) holds with it for every $n\ge2r-1$; no hidden dependence on $n$ or $a$ enters, and the $o(1)$ in the closing sentence is the explicit factor $(1+1/k_n)^{-2}$.
• Extremal conclusions: pass. $\mu_r\ge1+1/r$ follows from the per-sequence bound by taking the greatest lower bound. The sharpness sentence for $r=1$ imports $\mu_1(a)=2$ for the Section 2 sequence from its result page; it is attributed, not re-derived here.
• Consequences and composition: pass. "Hence $\mu_r\ge1+1/r$", "$r(\mu_r-1)\ge1$" and "in particular $m_k^r(a)>0$" were each checked separately. (5.1) is consumed exactly at the strength proved (stages $k\ge2r-1$), and the mean identity is consumed at stages $rn$ and $(r+1)n$, where it holds.
• Computation: inapplicable. The page carries no code; every arithmetic identity was re-derived by hand in "Weakest steps".
• Reproduction: inapplicable. The page states no rerun commands and holds no evidence folder.
• Source and verdict fidelity: pass. Displays (5.1)--(5.7), footnote 3 and the surrounding sentences were compared clause by clause with the page images; the page's characterization of the printed denominator $rn+n-1$ as a slip is correct (witness in "Weakest steps", step 3), is not presented as an erratum, and the page's replacement is weaker than the printed line and suffices. No source statement is strengthened; three routine justifications are supplied without a label (F2).

Weakest steps

Step 1: the long neighbor and (5.3). Fix $r\ge2$ and $n\ge2r-1$, and number the $2r-1$ consecutive stage-$n$ intervals around the one that receives $a_{n+1}$ by $-r+1,\ldots,r-1$ with lengths $\beta_i$; they are distinct because $n\ge2r-1$. A block of $r$ consecutive members has index set $\{i,\ldots,i+r-1\}$ with $-r+1\le i\le0$, so it contains index $0$; there are $r$ blocks, each an $r$-span of stage $n$, so their maximum $M_1$ satisfies $M_1\le M_n^r(a)$. In a block realizing $M_1$ the $r-1$ members other than the central one sum to $M_1-\beta_0$, so some $\beta_j$ with $j\ne0$ is at least $(M_1-\beta_0)/(r-1)$; reversing the orientation of the circle swaps $i$ with $-i$ and the two pieces of the central interval and changes none of $M$, $m$, $M_1$, $\beta_0$, so $1\le j\le r-1$ may be assumed. The block $\{j-r+1,\ldots,j\}$ lies in the window (it needs $j\ge0$ and $j\le r-1$) and contains $0$, so its length is at most $M_1$. At stage $n+1$ the members with indices $j-r+1,\ldots,-1$ ($r-1-j$ of them), the two pieces of the central interval, and the members with indices $1,\ldots,j-1$ ($j-1$ of them) are $r$ consecutive intervals of total length equal to that block's length minus $\beta_j$. Hence

$$m_{n+1}^r(a)\ \le\ M_1-\beta_j\ \le\ M_1-\frac{M_1-\beta_0}{r-1} =\frac{r-2}{r-1}M_1+\frac{\beta_0}{r-1},$$

which is the source's (5.3) (p. 17). This composes with (5.4), which needs only the two outer blocks $\{-r+1,\ldots,0\}$ and $\{0,\ldots,r-1\}$ and the pieces $\gamma_1+\gamma_2=\beta_0$: $m\le M_1-\gamma_1$ and $m\le M_1-\gamma_2$, whose average is $m\le M_1-\beta_0/2$. The edge values $j=1$ (no members on the positive side) and $j=r-1$ (none on the negative side) give counts $(r-2)+2$ and $2+(r-2)$, both $r$.

Step 2: the case split. If $\beta_0\le2M_1/(r+1)$, the coefficient of $\beta_0$ in (5.3) is positive, so

$$m\ \le\ \frac{r-2}{r-1}M_1+\frac{2M_1}{(r-1)(r+1)} =\frac{(r-2)(r+1)+2}{(r-1)(r+1)}M_1=\frac{r^2-r}{(r-1)(r+1)}M_1 =\frac r{r+1}M_1 ,$$

since $(r-2)(r+1)+2=r^2-r-2+2$. If $\beta_0\ge2M_1/(r+1)$, the coefficient of $\beta_0$ in (5.4) is negative, so $m\le M_1-M_1/(r+1)=\frac r{r+1}M_1$. The two cases cover every $\beta_0$, and $M_1\le M$ gives $m\le\frac r{r+1}M$, which is (5.1). For $r=2$ this reads $M\ge\tfrac32m$: after the reduction the block realizing $M_1$ is $\{0,1\}$, so $M_1=\beta_0+\beta_1$, (5.3) is $m\le\beta_0$ and (5.4) is $m\le M_1-\beta_0/2$, and the split at $\beta_0=2M_1/3$ gives $m\le2M_1/3$ on both sides. For $r=1$ the page's separate line $m\le\min(\gamma_1,\gamma_2)\le\beta_0/2\le M/2$ is the source's (p. 17) and needs no window.

Step 3: the telescoping and the contradiction, with the printed slip. Fix $r\ge1$ and $n\ge2$; then $rn\ge2r>2r-1$, so (5.1) is available at every stage $k\ge rn$. Suppose (5.5) holds for all $k$ with $rn\le k\le(r+1)n$; then $m_k^r(a)>0$ there. For $rn\le k<(r+1)n$, (5.1) gives $m_{k+1}^r(a)\le M_k^r(a)/(1+1/r)$ and (5.5) gives $M_k^r(a)<(1+1/r)\,m_k^r(a)/(1+1/k)^2$, so $m_{k+1}^r(a)<m_k^r(a)\,k^2/(k+1)^2$. The $n$ factors from $k=rn$ to $k=(r+1)n-1$ are positive and multiply to $(rn)^2/((r+1)n)^2=r^2/(r+1)^2$, so

$$m_{(r+1)n}^r(a)\ <\ \frac{r^2}{(r+1)^2}\,m_{rn}^r(a)\ \le\ \frac{r^2}{(r+1)^2}\cdot\frac1n ,$$

the last by the mean identity at stage $rn$. At $k=(r+1)n$, (5.5) and $(1+1/k)^2\ge1$ give $m_k^r(a)>\frac r{r+1}M_k^r(a)$, and the mean identity gives $M_k^r(a)\ge r/k=r/((r+1)n)$, so $m_{(r+1)n}^r(a)>\frac{r^2}{(r+1)^2}\cdot\frac1n$, a contradiction. The source's chain (p. 17) writes the middle bound as $M_{rn+n}^r(a)\ge r/(rn+n-1)$. That inequality is not true in general: if $a_1,\ldots,a_k$ are the $k$ equally spaced points $i/k$ with $k=rn+n$, every $r$-span at stage $k$ equals $r/k$, so $M_k^r(a)=r/k<r/(k-1)$; for $r=2$, $n=2$ this is $1/3<2/5$. The page's weaker bound $r/(rn+n)$ is what the mean identity gives, and the contradiction survives because (5.6) is strict. The closing step is routine: for every $n\ge2$ some $k_n\in[rn,(r+1)n]$ has $m_{k_n}^r(a)=0$ or a ratio at least $(1+1/r)/(1+1/k_n)^2$; since $k_n\ge rn\to\infty$ and $(1+1/k_n)^{-2}\to1$, the limit superior over all $k$ is at least $1+1/r$, and the greatest lower bound over $a$ preserves the bound.

Strongest attack

The strongest attempt was to break (5.1) inside the page's own range at its boundary, where the argument has the least room: $n=2r-1$, so that the window (5.2) is the whole circle, and $r=2$, where (5.3) degenerates to $m\le\beta_0$. At $n=2r-1$ the $r$ non-wrapping blocks of the window are still genuine $r$-spans of stage $n$, the stage-$(n+1)$ spans used in (5.3) and (5.4) are $r$ consecutive of $2r$ distinct intervals, and no step uses more than that; at $r=2$ the two bounds $m\le\beta_0$ and $m\le M_1-\beta_0/2$ meet exactly at $\beta_0=2M_1/3$ with common value $2M_1/3=\frac r{r+1}M_1$, so the split is tight but closed. The attempt failed. A second attack looked for a configuration in which the long neighbor $\beta_j$ found in a block realizing $M_1$ is not in the block $\{j-r+1,\ldots,j\}$ that (5.3) actually uses; that can happen, but the deduction never needs it, since the (5.3) block is bounded by $M_1$ on its own. A third attack was the printed chain: the intermediate bound $r/(rn+n-1)$ is false for equally spaced prefixes (step 3), so a page that had transcribed it would have carried a false line; this page does not, and its replacement suffices. A fourth attack tested whether the page's per-sequence form of (5.7) and its $+\infty$ reading claim more than the source: the source's own argument is run for a fixed sequence and ends with the bound for that sequence, and the reading only adds sequences with an infinite constant. A fifth attack, on the restriction to $n\ge2$ in the (5.7) proof, found no mathematical gap but an unattributed departure from the source's "natural number $n$" (F1). As a non-load-bearing sanity check, the reviewer also evaluated (5.1) on several thousand random rational configurations with occasional coincident points, $r\le6$ and $n\ge2r-1$, without finding a violation; this is recorded as a check performed, not as evidence.

Premises

• Section 1 of the source (printed p. 14, physical page 2): the definitions of the intervals, of $M_n^r(a)$, $m_n^r(a)$, $\mu_r(a)$ and $\mu_r$ (greatest lower bound over all sequences), and the display $nM_n^r(a)\ge r\ge nm_n^r(a)$. Held; read on the page image, clause by clause for these items. Interface used: the mean identity at stages $rn$ and $(r+1)n$, in the form $m\le r/k\le M$, re-derived above from the count of spans containing each interval.
• Section 5 of the source (printed pp. 16--17, physical pages 4--5): displays (5.1)--(5.7), footnote 3, and the connecting sentences. Held; read clause by clause on the page images at two resolutions. Interface used: the statements of (5.1) and (5.7) and the printed argument, which the page follows.
• The Section 2 result page (statement section, in that state): the value $\mu_1(a)=2$ for the sequence $\log_2(2k-1)$ mod $1$, cited by the page for sharpness at $r=1$. Held on the card; the Section 2 computation was not re-derived in this review, and the sharpness sentence stands on the result page's statement.
• The two 2026 theorem reconstruction pages (statement sections, in that state): the fixed-$r$ bound $1+r/(r^2-1)$ for $r\ge2$ over distinct points, and the claimed bound $\mu_r-1\ge\log r/(100r)$ for large $r$. Only the page's descriptive sentences in "Reading addressed" depend on them, and those sentences were checked against the statements alone.
• Explicit assumptions of the page: the $+\infty$ reading of $M/0$; the restriction of (5.1) to $n\ge2r-1$; the choice $n\ge2$ in the (5.7) proof. Each is stated on the page, the third with its reason but without attribution.

Findings

F1. Severity: suggested. Location: "Fix $r\ge1$ and an integer $n\ge2$, so that every $k\ge rn$ satisfies $k\ge2r-1$" and, in the Source notes, "The proof of (5.7) uses (5.1) only at stages $k\ge rn\ge2r-1$". Defect: the source runs the argument for every natural number $n$ (p. 17, physical page 5: "Now suppose that $n$ is a natural number and that for $nr\le k\le n(r+1)$ we have (5.5)"); for $n=1$ and $r\ge2$ its chain invokes (5.1) at stage $k=r<2r-1$, inside the range the page leaves unreconstructed. The page's $n\ge2$ is therefore a departure from the source made to stay inside the reconstructed range, and it is what makes the Source-note sentence true; the page states the reason but not that the restriction is its own, so a reader comparing with the note cannot tell which of the two took it. Proposed replacement: in the proof, "Fix $r\ge1$ and an integer $n\ge2$ (the note takes every natural number $n$; the restriction is the reconstruction's, so that every $k\ge rn$ satisfies $k\ge2r-1$ and the reconstructed (5.1) applies at stage $k$; nothing is lost, since only $n\to\infty$ matters)"; in the Source notes, "The reconstruction's proof of (5.7), which takes $n\ge2$, uses (5.1) only at stages $k\ge rn\ge2r$; the note's own run with $n=1$ would use it at stage $r$."

F2. Severity: note. Location: "Take a block realizing $M_1$", "Reflecting the circle exchanges $j$ with $-j$ and $\gamma_1$ with $\gamma_2$", and "Since $k_n\to\infty$, the ratio exceeds". Defect: these three justifications are the page's; the source writes "Clearly at least one of the numbers ... is $\ge(M_1-\beta_0)/(r-1)$; we may suppose that $j>0$" and "It follows that" (p. 17), giving no reason. Each supplied reason is correct (steps 1 and 3 above), and the Standing sentence announces only that the block counts are made explicit. Proposed replacement, in the Standing paragraph: "makes the block counts explicit, supplies the reasons behind the note's 'clearly', its 'we may suppose that $j>0$' and its closing 'it follows', restricts (5.1) ...".

F3. Severity: note. Location: Source notes, "The note's Section 1 allows coincident points". Defect: Section 1 (p. 14) defines a sequence as points on the circle, "in other words numbers mod 1", and says nothing either way about coincidences; "allows" reads as a positive statement of the source, while the folder's Section 3 page says "does not exclude". Proposed replacement: "The note's Section 1 does not exclude coincident points, so a span can vanish".

F4. Severity: note. Location: Reading addressed, "The fixed-$r$ improvement $\mu_r\ge1+r/(r^2-1)$ over sequences of distinct points". Defect: the cited statement holds for $r\ge2$ (its Statement section says so, and the expression is undefined at $r=1$); the sentence gives no range. Proposed replacement: "The fixed-$r$ improvement $\mu_r\ge1+r/(r^2-1)$ for $r\ge2$ over sequences of distinct points".

F5. Severity: note. Location: "the ratio exceeds $1+1/r-o(1)$ along the subsequence $k_n$". Defect: the violation of (5.5) gives a ratio at least $(1+1/r)/(1+1/k_n)^2$, with equality possible, so "exceeds" overstates a non-strict bound; the conclusion is unaffected. Proposed replacement: "the ratio is at least $(1+1/r)/(1+1/k_n)^2$, which tends to $1+1/r$, along the sequence $k_n$".

Verdict

Source fidelity: faithful. The statements of (5.1) and (5.7), their hypotheses and quantifiers, the locators (printed pp. 16--17, physical pages 4--5, offprint pp. 5--6, displays (5.1)--(5.7), footnote 3) and the characterization of the printed denominator all match the page images; the one scope restriction is declared, and no source statement is strengthened. No finding is required; F1 asks for an attribution, and F2--F5 are wording.

The argument as reconstructed: sound. Every deduction on the page was re-derived above; the case split closes, the block counts are exact at both edge values of $j$, the telescoping product is exact, and the contradiction holds with the corrected denominator.

Limitations. The omitted range $1\le n<2r-1$ of (5.1) was not examined, as the page does not claim it. The sharpness of $1+1/r$ at $r=1$ rests on the Section 2 result page's statement and was not re-derived. The random spot-check of (5.1) is not retained and warrants nothing. The review read only the material listed above, plus the disclosed exposures.

This focused review assigns no tier and changes no status.