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This record retains the mathematical assessment by the independent source reviewer and the substantive passing assessment by the distinct grader. First-person statements in their respective sections remain attributed to those assessors. Neither report supplies a personal name or model version.

The accepted scope is the selected Ostrowski translated-interval theorem and its complete reconstruction (O). The report also checks the printed historical formulation and the separate Kesten statement interface within the limits below. This is retention of completed review and grading of an exact native subject, not a new mathematical review or a new numerical tier.

Independent source review

Refutation-failed. The frozen reconstruction of Ostrowski's selected translated-interval theorem survives this whole-proof review. Every essential deduction of the selected source proof is present and valid.

The historical formulation is faithful to Erdős's printed pp. 61–62, including the sampling and half-open endpoint convention. The distinct interval-length characterization agrees with Kesten's printed Theorem 4. Kesten's necessity proof was not reviewed. This report grants no tier, formal verification, community-acceptance finding, or full Kesten proof coverage. The reviewer reserved acceptance for distinct grading of the report contract and independence; the passing grade is retained below.

Subject, independence, and actual reading

Reviewer: the independent source reviewer who authored this report, in a fresh context. I did not author, repair, or previously build on the reviewed arguments. I received the frozen paths and review scope, but no author advocacy, private mathematical plans, or other reviewers' verdicts. I received no substantive reconciliation during review and used no subreviewer. The mathematical findings and derivations below are my own.

The frozen subject is the Erdős repository as it stood at 2026-09-09T00:48:49Z. All paths below are relative to the repository. The three main Markdown pages were read completely:

• library/irrationality/ostrowski_1927_mathematische_miszellen/equation_3.md (abbreviated O): the selected statement, complete reconstruction, and consequence and coverage sentences.
• library/discrepancy/erdos_1964_problems_results_diophantine_approximations/conjecture_p62.md (E): the historical attribution and comparison with the length criterion.
• wiki/problems/irrationality/E0998/_index.md (P): the complete page, especially the site's wording.

The frozen read set included standing and roadmap text of the subject as it stood at 2026-09-09T00:48:49Z: wiki/problems/irrationality/E0998/_index.md lines 8, 26, 32, 57 and 63-64; library/irrationality/ostrowski_1927_mathematische_miszellen/equation_3.md lines 119-121; library/discrepancy/kesten_1966_bounded_remainder/theorem_4.md lines 40-43; and library/discrepancy/erdos_1964_problems_results_diophantine_approximations/conjecture_p62.md lines 18 and 20. A separately spawned materiality grader (model Claude Fable 5.1) ruled the exposure immaterial under the content test on 2026-09-18: the standing sentences state no answer, the status wording is the compiled claim under review rather than prior-review, tier or acceptance text (none existed at the freeze), and the verdict rests on the reviewer's own derivations and source reading.

Allowed mathematical material comprised these pages, their canonical Ostrowski and Erdős PDFs, and optional Kesten statement context. I also read all of library/discrepancy/kesten_1966_bounded_remainder/theorem_4.md (K) and visually checked its exact published statement. The source pages actually inspected were:

The reviewer read the applicable repository and review instructions. No other mathematical reports or private plans were read. Discovery of unrelated instruction-file path names exposed no excluded mathematical contents or verdicts. There was no substantive isolation breach. No mathematical programs, numerical experiments, or external retrievals were used; source inspection compared bytes and rendered the identified PDFs.

Exact byte identities

At review and grading, the Markdown bytes equaled the committed bytes of 2026-09-09T00:48:49Z. These are the frozen subjects; the current standing-paragraph mapping is recorded at the end of this page:

• O: library/irrationality/ostrowski_1927_mathematische_miszellen/equation_3.md
• E: library/discrepancy/erdos_1964_problems_results_diophantine_approximations/conjecture_p62.md
• P: wiki/problems/irrationality/E0998/_index.md
• K, optional context: library/discrepancy/kesten_1966_bounded_remainder/theorem_4.md

The PDFs are LFS-managed. Their Git blobs are pointers, not the rendered PDF bytes; each actual PDF equals the payload its pointer names at the frozen state:

• Ostrowski 1927: library/irrationality/ostrowski_1927_mathematische_miszellen/ostrowski_1927_mathematische_miszellen.pdf
• Erdős 1964: library/discrepancy/erdos_1964_problems_results_diophantine_approximations/erdos_1964_problems_results_diophantine_approximations.pdf
• Kesten 1966, optional context: library/discrepancy/kesten_1966_bounded_remainder/kesten_1966_bounded_remainder.pdf

Identity checks compared the files at the paths above with their committed bytes of 2026-09-09T00:48:49Z. Visual inspection used the installed Poppler renderer: pdftoppm -f <p> -l <p> -scale-to 2200 -png <pdf-path>, with stdout captured in memory and displayed. The source page numbers above give the exact p values. Ostrowski PDF p. 2 used scale 2000; Kesten sheet 1 used scale 3200. No mathematical computation or executable reproduction is part of the warrant.

Independent restatements

O: selected Ostrowski statement

For every real alpha, every nonzero integer j with beta = {j alpha} strictly between zero and one, every real u, and every positive integer n, let J be the oriented half-open arc [u,u+beta) on the circle modulo one. Count the n points {alpha},...,{n alpha} with their sampling multiplicities. Then the difference between that count and n beta has absolute value strictly below |j|. The bound holds for every translate, including wrapped arcs, and does not require alpha to be irrational. It is a pointwise strict estimate, not an assertion of a uniform margin below |j|.

The auxiliary zero-length remark assumes nonzero j and interprets the arc as empty; its count and main term are then both zero. No strict bound with j = 0 is claimed. No length-one arc is represented by beta = {j alpha} here.

E and P: precise formulation consequence

The historical endpoint implication would say that, for fixed irrational alpha and an interval in [0,1], bounded discrepancy for the sequence {m alpha}, sampled at 1 <= m <= n, forces u and v separately to be fractional parts of integer multiples of alpha. Erdős prints that converse after (24). The frozen P asks only for an eventual O(1) bound.

The separate Kesten statement is: for fixed xi in [0,1] and 0 <= a < b <= 1 with b-a < 1, the discrepancy is bounded over positive integer M if and only if b-a = {j xi} for some integer j. This permits any translate with that proper length. It does not assert that a or b is an orbit point. I checked this published interface, not its necessity proof.

Every essential deduction of the source reconstruction

1. The counting convention agrees with the scan. Printed p. 178 counts R(alpha),...,R([x] alpha); (3) on p. 179 restricts x to positive integers. Its footnote includes the initial endpoint and excludes the final endpoint. The modern index n and fractional-part notation preserve these choices.
2. For positive j, xi = 1-u-beta translates J exactly to [1-beta,1) modulo one. At the initial endpoint the translated coordinate is 1-beta, which is included; at the final endpoint it is zero, which is excluded. This argument is also valid when the original arc wraps.
3. Since j alpha-beta is an integer, the fractional-part increment is beta outside the translated arc and beta-1 inside it. The two alternatives cover all points with the correct equality case at 1-beta.
4. Summing over exactly m=1,...,n gives S_xi(n)=n beta-N_J(n), with the same sign as the source's equation (4).
5. Both finite-sum forms equal A(n+j)-A(n)-A(j). This does not require n >= j, and therefore no unproved overlapping-range cancellation occurs.
6. The latter form has exactly j differences of two numbers in [0,1). Every difference has absolute value strictly below one, so the finite triangle inequality gives a strict bound below j. No probabilistic, equidistribution, or asymptotic argument is used.
7. For negative j, the complementary half-open arc has length 1-beta={-j alpha} and positive index -j. The arcs partition the circle even at their shared endpoints, so the two discrepancies are exact negatives. The positive-index result therefore proves the claimed bound.
8. The excluded beta = 0 case and j = 0 are handled only as stated above. These qualifications prevent a zero-length complement argument from being used outside its valid range.

These are precisely the essential steps between source (3) and the conclusion following source (4). O expands the source's short membership argument and finite-sum cancellation; it does not omit a required same-paper lemma. The earlier discussion of Hecke and equidistribution is motivation, not a premise of this proof.

Two weakest steps, independently rederived

1. Endpoint convention and translated membership

Let x={t-u}. By the definition of the oriented half-open arc, t belongs to J exactly when 0 <= x < beta. Directly from the floor definition,

$$\{t-u-\beta\}-\{t-u\} =\begin{cases} 1-\beta,&0\le x<\beta,\\ -\beta,&\beta\le x<1. \end{cases}$$

Consequently the indicator of J is exactly

$$1_J(\{t\})=\beta+\{t-u-\beta\}-\{t-u\}.$$

This is a second direct derivation with the shift measured from the initial endpoint. It checks the source reconstruction without assuming its chosen translated terminal interval. At x = 0 it gives one; at x = beta it gives zero. It is insensitive to wrapping because only fractional parts appear.

2. The strict bound for small n and negative j

Put F(m)={m alpha-u} for every integer m. The previous identity and beta={j alpha} give

$$N_J(n)-n\beta=\sum_{m=1}^n\bigl(F(m-j)-F(m)\bigr).$$

For positive j, the identity

$$F(m-j)-F(m)=\sum_{h=0}^{j-1} \bigl(F(m-j+h)-F(m-j+h+1)\bigr)$$

telescopes in m to

$$N_J(n)-n\beta=\sum_{h=0}^{j-1} \bigl(F(1-j+h)-F(n+1-j+h)\bigr).$$

Every summand has absolute value below one. This derivation works without assuming an order relation between n and j. For j = -r < 0, the same finite telescoping applied to F(m+r)-F(m) yields

$$N_J(n)-n\beta=\sum_{h=1}^{r}\bigl(F(n+h)-F(h)\bigr),$$

and hence a strict bound below r. This independently checks the negative-index conclusion as well as the source's complementary-arc proof. The finite sums require neither convergence nor an exchange of infinite limits.

Strongest attempted refutation

The strongest attack was that the translated estimate might silently replace the historical count by a different starting point, reverse an endpoint, or hold only for orbit-based translations. Such a defect could make a valid length theorem irrelevant to the historical count.

It failed at both the source and algebraic interfaces. Erdős explicitly samples 1 <= m <= n and prints 0 <= u <= (m alpha) < v <= 1 on p. 61. Ostrowski explicitly allows arbitrary translates and states the matching half-open convention. At the initial endpoint the indicator is one, and at the final endpoint it is zero. As an exposed rational stress case for O, take alpha = 1/3, j = 1, u = 0. The point at m = 1 hits the excluded final endpoint, and at m = 3 hits the included initial endpoint. The counts are respectively zero and one, with discrepancies -1/3 and zero, exactly as the indicator identity requires. This is a logical edge-case check, not numerical evidence for the general theorem.

I also attacked the n < j cancellation, negative indices, beta near one and negative alpha. The finite telescoping above resolves n < j and both signs of j, for every beta < 1. Integer reduction handles negative alpha. No counterexample to the frozen O statement, nor unsupported essential deduction, was found.

For historical fidelity, the strongest alternative reading was that Erdős's sentence meant only a length condition. Printed p. 62 explicitly writes u=(k_1 alpha), v=(k_2 alpha). That alternative cannot describe the literal printed sentence. This conclusion concerns the printed formulation; it does not assert what the authors privately intended or why Kesten describes his length theorem as confirming their conjecture.

Explicit audit checklist

Premises, standing, and exact external interfaces

There are no consumed native L-claims and no claim-tier dependency. Basic floor arithmetic and finite summation are the only unnamed mathematical facts used. Their relevant applications are exposed above; no analytic theorem is hidden behind them.

The other elementary transfers on K were checked directly: for proper positive length, membership in Z alpha + Z is equivalent to being {j alpha}; subtracting orbit endpoints gives group membership; eventual boundedness is equivalent to all-index boundedness after a finite-prefix maximum; 0,1) has identically zero discrepancy but length one cannot be a fractional part; an empty interval is outside a < b and has zero discrepancy. These transfers do not strengthen the published theorem or supply its omitted necessity proof.

Limits of the source review

No mathematical defect was found in the assigned whole O proof or in the inspected historical formulation. The modern catalog webpage was not fetched, and this review does not independently certify its access date, current wording, or a current literature-status search. P's contextual mentions of Ostrowski 1930 and Grepstad–Lev are not premises and receive no coverage from this report.

This is an assessment of the exact frozen bytes, not a warrant for later substantive edits. No mathematical blocking issue was found within the reviewed subject. The distinct grade below assesses independence and report completeness; retaining that assessment supplies no new review cycle.

Distinct grading

Pass for the report contract and independence. The distinct grade applies to the independent source review's refutation-failed assessment of the frozen Ostrowski reconstruction. The report supplies whole-conclusion and exact formulation scrutiny within that scope. No missing report item or established independence defect requires a void grade.

This is a grade of the report contract and independence, with native-subject and artifact checks. It is not a second independent mathematical review or a new proof. It awards no native L tier, formal verification, public acceptance, or proof coverage for Kesten's necessity direction. The grade required durable retention of the mathematical assessment and exact native subject before use as an ordinary-clone warrant; the retention mapping below supplies that connection.

Grader independence and inspection

Grader: a distinct grader, separate from the report's independent source reviewer and from the subject's authors and earlier collaborators. I received the frozen report identity, the bounded grading scope, and repository instructions. I did not author, repair, or previously build on the subject. I read no author private plans, advocacy, unrelated research narratives, or sibling verdicts, and used no subreviewer.

I read the applicable repository and review instructions, the entire assessed report, and all four frozen native Markdown pages identified above. I compared their current hashes with their frozen Git-blob hashes, and compared the three current PDF hashes and sizes with the frozen LFS pointers. I did not read or render PDF contents, run mathematical programs, conduct numerical experiments, or retrieve external sources. Source-reading coverage below is the reviewer's attributed coverage, not an assertion that I repeated that source review.

The independence assessment used the reviewer's declarations and confirmation from the commissioning agent that the reviewer had a fresh context without inherited conversation, received only the frozen subject, source paths, scope, instructions and report contract, and was excluded from private plans, advocacy and sibling verdicts. The commissioning agent confirmed that the reviewer was distinct from the historical authors and previous collaborators and received no substantive reconciliation before the verdict. The report separately declares those boundaries and the absence of subreviewers. These are consistent affirmative facts supporting independence. I did not inspect private context logs or independently reconstruct every historical tool invocation.

The report discloses discovery of unrelated instruction-file path names without reading their contents. Directory names alone do not establish exposure to excluded mathematics. There is no reported or otherwise established substantive isolation breach.

Frozen subject and artifact identity

The grader checked all four frozen Markdown subjects and three actual PDFs against their state of 2026-09-09T00:48:49Z, using the paths listed above. At grading, the current file and the frozen Git blob agreed for each Markdown page. Each actual PDF hash and size agreed with its frozen LFS pointer.

These checks validate the report's source identities. A PDF's LFS pointer is distinct from its rendered bytes; the report correctly makes that distinction. Hashes identify the artifacts, not mathematical truth or the historical act of reading them.

Report-contract assessment

Reading the frozen native pages confirms that the report assesses the actual statements, deductions, and consequence sentences. In particular, it addresses O's arbitrary translates, wrapping, rational alpha, strict pointwise bound, negative indices, and excluded zero case.

The source-reading receipt is specific enough for this contract: the reviewer reports visual inspection of Ostrowski PDF pp. 2–4, printed pp. 178–180; Erdős PDF pp. 10–11, printed pp. 61–62, plus the notation on PDF p. 6, printed p. 57; and Kesten sheet 1's right-hand leaf, printed p. 193. It names the exact theorem, equations, endpoint footnote, count definitions, and the proof's essential steps. It distinguishes ancillary text-only orientation and unrelated visible material. Its mathematical discussion corresponds to the frozen pages and the reported reading scope. The receipt is not a generic claim to have read a paper.

Exact coverage that may be retained

• O: independent whole-proof review of the selected translated-interval theorem and its complete short reconstruction from Ostrowski (3)–(4), with the counting definition and endpoint convention. No other Ostrowski results inherit that verdict.
• E and the assigned formulation of P: independent checking of the printed Erdős endpoint conjecture and its distinction from the length criterion. It does not certify a fresh catalogue access or literature-status search.
• K: source-statement checking of Theorem 4 and its stated domains, together with the elementary transfers inspected in K. Kesten's necessity proof and its continued-fraction lemmas were not reviewed or reconstructed.

No new mathematical condition is missing from the report's commissioned O scope. The unreviewed Kesten necessity proof, the contextual Ostrowski 1930 and Grepstad–Lev references, current catalog wording, and a current literature search remain excluded. They must not be described as discharged by this grade.

Retention and current mathematical subject

The exact four-page subject is the pages as they stood at 2026-09-09T00:48:49Z; those bytes are not retained as a snapshot, and the paragraphs below say how the current pages differ from them. The subject is the complete frozen pages, including their hypotheses, proofs, consequences and then-current coverage statements. The principal mathematical sections are:

Current O differs from its frozen subject in the final standing paragraph before “Bears on”, whose pending-review wording now links to this retained assessment and states its accepted scope, and in “Relation to the endpoint question”, which now names only Kesten's separate length criterion. Its statement, argument and source locators are unchanged. P's “Progress” paragraph and K's “Proof scope” paragraph now link to this assessment and state the same bounded coverage. E's mathematical account remains unchanged. Generated metadata and navigation may be maintained separately. This mapping concerns editorial standing changes only; it extends no verdict to changed mathematics.

This is the current curated record of the completed source review and distinct grade. Its warrant is the mathematical restatements, essential deductions, independent derivations, attacks, checklist, source identities, reading scope, premises, verdict and substantive grade retained here, together with the exact native subjects. The original working reports are not retained as separate artifacts; this page does not claim exact textual equivalence with them. Editorial changes remove operational details, use repository-relative source locations, consolidate the grade's duplicate identity table, and replace prospective retention directions with this current mapping. They supply no additional mathematical finding.

The complete mathematical warrant is available here without a private-file dependency. The exact original native pages as they stood at 2026-09-09T00:48:49Z are not retained. Today's O and E differ from them as the mapping above records and by a frontmatter title line each; today's P differs in its Status sentence, its renamed assessment paragraph, its known-results links and its generated library links; and today's K, optional context, differs by a frontmatter title line, by an opening paragraph stating its reviewed coverage, by a reworded “1966/67” bibliographic-variant sentence in “Statement and source”, and by a reconstruction of the anchored necessity direction, with its own standing text, that replaced the former “Proof scope” paragraph. The canonical PDFs remain in their source homes under Git LFS: Ostrowski 1927, [Erdős 1964, and Kesten 1966. A clone with pointer text needs git lfs pull to materialize those source bytes.

The selected O proof has no remaining independent review obligation within this accepted scope. Other Ostrowski results, Kesten's necessity proof and continued-fraction lemmas, Ostrowski 1930 and Grepstad–Lev context, current catalog wording and access date, and a current literature-status search receive no additional coverage. There is no formal verification, native L tier, community-acceptance finding or claimed new open-problem solution. A later substantive change to a statement, proof or relied-on premise requires assessment of the affected mathematics.