---

This record retains the source-reading scopes and documentary mapping for the anchored irrational necessity proof. It does not perform another proof review. The independent reviewer returned refutation-failed and the distinct grader passed the report contract and independence. Those findings discharge literature-compilation proof-coverage review for this proper subdirection only.

The theorem is: for irrational $0<\xi<1$ and fixed $0<b<1$, boundedness of $N(M,\xi,0,b)-Mb$ for every positive integer $M$ implies $b=\{j\xi\}$ for some $j\in\mathbb Z$. The interval is $[0,b)$ and the count starts at index 1. This review supplies no Bohl/arbitrary-translate, rational-rotation or sufficiency coverage. The separately accepted Ostrowski sufficiency route is unchanged. It does not change E0998 status or award a native claim tier.

Exact historical subjects

Paths in this record are repository-relative. The live source home is library/discrepancy/kesten_1966_bounded_remainder/. The immutable reviewed bytes are opaque assets, not additional live source pages:

• Historical theorem page. Its original live path was library/discrepancy/kesten_1966_bounded_remainder/theorem_4.md.
• Historical source digest. Its original live path was library/discrepancy/kesten_1966_bounded_remainder/_index.md.

The historical two-path reconstruction patch (working storage; not retained) had as its baseline the repository as it stood on 2026-09-10T06:09:00Z; its two preimages are library/discrepancy/kesten_1966_bounded_remainder/_index.md and library/discrepancy/kesten_1966_bounded_remainder/theorem_4.md as they stood then. Both original preimages and the patch were identity-checked during review; the grader independently confirmed the same two preimages. The patch changed only these two pages. The baseline date and the retained snapshots identify the historical scope; later live-page edits are not silently substituted for the subject that was reviewed.

The canonical primary source remains Kesten's 1966 journal PDF, 3476268 bytes (its SHA-256 is recorded by Git LFS, not restated here). A Git LFS pointer is not the PDF payload. The pointer identifies this exact payload; source reading used hydrated bytes matching that identity.

The full original review and grade wrappers are in working storage, not retained here. The distinct grade assessed the original reviewer text. The native review rendition and distinct grade rendition retain their full substantive mathematics and findings, with the explicitly mapped documentary corrections below. They are edited renditions, not byte-identical transcripts or fresh assessments of their own later text. Unverified process intervals and private locators are not part of the native warrant.

Roles, permitted material and exposure

Original reconstruction author: source reconstruction author, not blind to preceding selection context. The author had read native E0998/E0183 pages, their existing scope and acceptance records, and the living docket during triage. Earlier triage visually routed all 11 source sheets without reconstructing the mathematics. The subsequent original authoring read the selected proof pages afresh. No helper supplied mathematical reasoning to that authoring. Prior accepted verdicts were not premises of the new reconstruction.

Independent reviewer: independent whole-claim reviewer, fresh context. The supplied assignment wrapper permitted only the two frozen payload pages, their full patch, both preimage copies, canonical PDF page images and docs/verification.md, docs/evidence.md, docs/anatomy.md. It excluded author handoff/receipts/renderings, sibling verdicts, material from other repositories, conversation history, plans, triage, research pages and URLs. The report itself omitted the full permitted-material list; this list is the disclosed filing addition from the assignment wrapper, not an invented retrospective statement by the reviewer.

The reviewer actually read the full two payload pages, full patch, both preimages, all three rules pages, and PDF sheets 1–4 and 7–11. The report records no prior authorship or exposure to the argument and no excluded material read. It used hand derivations and a hand numerical illustration, not mathematical code.

Distinct grader: distinct grader, fresh context. The retained grading wrapper names the original review, frozen subject packet and canonical PDF; it does not supply a separate exhaustive allowed-material list. Its wrapper excluded author receipts/renderings, sibling verdicts, other repositories or private material, and conversation history. The grader actually read the full report, full two payload pages, full theorem preimage, patch headers only, all three rules pages, and PDF sheets 1, 3 and 7–11. The source-digest preimage and matching baseline pages were hashed only. PDF sheets 2, 4, 5 and 6 were not read by the grader. The grader's exact disclosure, including one floating-point illustration check, is retained in the grade; no repository program or proof evidence was executed.

This filing author is the original reconstruction author and has read both full reports and the transformation agreement. The present source-prose check is therefore not independent mathematical review. No mathematical computation or new proof work was performed in this filing.

A separate documentary assistant compared the rendition with the original reports, with both the author's text and verdicts exposed. It identified a link-label spacing defect and a wrapped inequality that Markdown would treat as a blockquote; both were corrected. It did not inspect the PDF or review the mathematics. This is transformation assistance, not independent proof acceptance.

Actual primary-source reading

The PDF has two leaves per sheet. Sheet 1 carries printed 192 | 193; sheets 2–10 continue consecutively; sheet 11 carries 212 | back matter. The Kesten article begins on the right leaf of sheet 1. Adjacent article text and the closing cover are not premises.

The original author and independent reviewer each visually read sheets 1–4 and 7–11 for the selected reconstruction/review. Their attributed coverage is:

• Sheet 1, printed p. 193: title and source identity; definitions of $N,R$, fractional parts, half-open interval and positive count index; Theorem 4 and rational-case footnote.
• Sheet 2, printed p. 194: continued-fraction recurrences, complete quotients and error denominator, (1.3)–(1.6). The integer expansion (1.7) was read but is not a premise. Neighboring p. 195 results were statement context, not consumed proof.
• Sheets 3–4, printed pp. 196–199: the needed Theorem 1 geometry, (2.1)–(2.10), including residue labels, gap lengths and refinement. The source leaves odd parity analogous; the local proof supplies it. General-$N$ results and the three-distance corollary are not premises.
• Sheets 7–8, printed pp. 204–207: Section 4 scope, excluded sufficiency and Bohl interfaces, endpoint digit definitions, parity conventions, count identities, discrepancy bounds and finite-prefix stability.
• Sheet 9, printed pp. 208–209: separated-block addition and reverse-order telescoping, digit exclusions and the nonterminal transition.
• Sheet 10, printed pp. 210–211: terminal transitions, later source exclusions and case analysis, and eventual terminality.
• Sheet 11, printed p. 212: final telescoping, orbit integer and source bibliography. Cited Bohl, Hecke and Ostrowski papers were not read for this proof and are not consumed as premises.

The original author and reviewer did not freshly proof-read sheets 5–6, printed pp. 200–203. Those metric/Farey discussions are outside the selected dependency closure. Reading the source's (4.27)–(4.31) and cases (i)/(ii)/(iii) does not mean that they were reconstructed: the local direct exclusion bypasses them. The review checks the complete selected local proof.

The distinct grader visually read sheets 1, 3, 7, 8, 9, 10 and 11, not sheets 2, 4, 5 or 6. Its spot-checks do not inherit the reviewer's fuller PDF scope. The full review has a sheet-by-sheet account; the full grade states its narrower actual reading.

For the later prose and standing edits, the author freshly viewed full retained renderings of sheets 3, 8, 10 and 11, corresponding to printed pp. 196–197, 206–207 and 210–212, on 2026-09-10 UTC. Before using those reading aids, the hydrated PDF payload was freshly hash-checked against the exact LFS identity above. The images were produced from that same source during the original authoring at 160 dpi. The present focus was only footnote 4's norm definition, p. 207's threshold symbol and tail notation, and the compensated final limit on pp. 211–212. No fresh whole-proof reading is claimed from these four views.

Dependency closure and standing boundary

The local proof derives continued-fraction identities (A)–(C), consumed partition geometry (D)–(H), endpoint transitions (I)–(K), counting (L), block accumulation (M), exclusions (N)–(Q), and the final integer invariant. No native L-claim or executable artifact is a mathematical premise. The source's general Ostrowski expansion theorem is unnecessary.

The surrounding links to the E0998 endpoint counterexample, Ostrowski equation (3), and the earlier translated-interval review concern other scopes. The independent reviewer did not read them, and this selected proof does not depend on their verdicts. The prior accepted sufficiency and endpoint routes remain separately accepted, not reopened or erased.

The current theorem page retains the full translated Theorem 4, but only its irrational anchored necessity proof has the review coverage recorded here. No page-wide full-proof label, rational case, Bohl reduction, new problem status, native claim tier or formalization is supplied.

Source-prose correction mapping

These three changes affect characterization and notation, not the reconstruction's mathematical deductions:

1. Final paragraph: replace the historical claim that the invariant avoids an “unjustified ordinary real limit” with the accurate statement that Kesten's alternating half-unit terms justify the printed limit. The invariant avoids the unstated final representative check.
2. Notation dictionary: identify the local signed convergent error $\varepsilon_n=q_n\xi-p_n$ separately from Kesten's p. 207 $\varepsilon_n$, the stability threshold called $\eta$ locally.
3. Block accumulation paragraph: replace the oblique fractional-part allusion with an explicit record of the p. 207 braces-versus-norm slip. The intended distance-to-nearest-integer notation is defined in footnote 4, p. 196. The local signed-error and exact-tail proof does not import the literal fractional-part inequality.

The historical proof's displayed mathematics and deduction paragraphs are preserved, apart from these explanatory prose additions and replacements. The opening, selected-scope introduction, current verification and source digest now distinguish independently reviewed historical anchored proof coverage from the separately assessed documentary transformation of later prose. The new, unvalidated review_status scalar is removed; no replacement enum is invented. The earlier Ostrowski acceptance paragraph remains qualified to its original statement/transfer/sufficiency scope.

Report and grade transformation mapping

The following records the ten requested report corrections:

1. Add the exact permitted-material list from the supplied assignment wrapper beside actual reading; preserve the grader's finding that it was absent from the original report body.
2. Attribute the first-person review and grade to distinct fresh-context roles. This filing author does not adopt either assessor's independence.
3. Replace private paths with native relative locators and complete subject hashes. Both reviewed pages are immutable opaque byte assets.
4. Remove the failed preimage-check process transcript while preserving the two baseline digests and exact two-path patch scope.
5. Remove the conversational preamble; retain all substantive sections.
6. Correct the W2 decimal illustration: Y4 0.4222048 → 0.4222051; L3 0.1194292 → 0.1194295; delta3 0.0084038 → 0.0084040; y3(21) 0.6417124 → 0.6417116; direct h3 0.0082876 → 0.0082884. The transition h3 is also corrected from 0.0082882 to 0.0082884. These changes adopt the grader's disclosed cross-check without rerunning it. Exact counts N(20)=8 and N(33)=11 and the member list are unchanged.
7. Reconcile Section 4 coverage: (4.27)–(4.31) and cases (i)/(ii)/(iii) were read and bypassed, not reconstructed.
8. Wrap prose and convert wide tables to labeled lists, preserving their mathematical contents and attributed findings.
9. Add native frontmatter and file review, grade and this source-reading record in the source-owned evidence/verify home. New report links are source-internal; no additional incoming-problem rollout is introduced.
10. State the anchored-only literature-compilation coverage discharged by the review and distinct grade, with no native claim tier.

Additional corrections required by the transformation agreement:

• Review §5.5: at a=2, (a−1)/(a+2)=1/4, not 1/3. The 1/4 bound is sharp there; the 5/32 deduction is unchanged. The grader's historical “Nothing else” finding is preserved with an annotation recording this later correction, rather than rewritten as if the grader found it.
• Both displayed h3 values receive the correction above, not only the direct one.
• The erroneous abbreviated theorem-hash suffix was replaced by the full already-correct digest wherever the subject was identified; the subject is now identified by its path, and that digest is not retained.
• Observations j≤0 and j≠0 are conditional on the contradiction's eventual terminal regime. Review restatement, W1, checklist and grading discussion do not strengthen the theorem to a one-sided orbit.

The report rendition also includes these explicit documentary changes:

• Review §11 adds the distance-to-nearest-integer definition in Kesten's footnote 4, p. 196. This clause is the distinct grader's additional source observation, attributed in-line to that grader; it is not recast as a discovery by the original reviewer.
• Review §11 replaces “The prose is, if anything, more conservative than the mathematics warrants” with “The accepted coverage remains only the anchored irrational necessity direction.” This is the filing's scope qualification corresponding to the Section 4 coverage reconciliation, not a new mathematical verdict.
• The review's, grade's and reading record's exclusion statements use generic descriptions of other repositories or private material in place of project-specific provenance wording. The permitted-material lists, actual reading and named exclusions above retain the same boundaries; this rephrasing neither conceals exposure nor expands permitted reading.
• Review W3 and the grade's quotation of that step use “the sum of the preceding blocks” for the same high-index tail. This is a terminology change only, not an alteration of the block ordering or argument.
• Pending-standing reconciliation distinguishes the historical mathematical review from the separately assessed documentary transformation of later prose. It does not extend the mathematical verdict to later edits.

Original grader findings about source fidelity, all ten audit items, independence, the missing permitted list, numerical corrections and exact scope remain visible. Its floating-point arithmetic is disclosed as a reviewer cross-check, not mathematical evidence or a reproduction command.

Rules reconciliation and remaining filing obligation

The review and grade initially read the three rules pages as they stood on 2026-09-09T21:17:39Z. Later anatomy guidance changed on 2026-09-10T05:48:43Z. The source, frozen subjects and preimages were unchanged. The grader subsequently reconciled its cited rules as they stood on 2026-09-10T07:12:00Z and reported no changed finding: the cited verification and evidence passages were identical, while the three anatomy citations shifted. This concrete reconciliation was re-verified as the pages stood on 2026-09-10T07:43:28Z:

• docs/verification.md: lines 73, 97–98, 115–133, 119, 122–123, 135–138, 187 and 189 remain identical to the cited passages.
• docs/evidence.md: lines 32–42, 39–40, 51–57 and 72 remain identical to the cited passages.
• docs/anatomy.md: the historical 282–302 tier citation is now 311–323; the historical 337–341 formatting citation is now 366; the historical 204–206 source-link-generator citation is now 231–232.

The grade retains each historical citation with its current locator beside it. The native rendition keeps the version caveat without unverified process times or stale rule-line numbers as the present authority.

For this filing, the author fully read current anatomy, evidence and verification guidance (docs/anatomy.md, docs/evidence.md and docs/verification.md as they stood on 2026-09-10T07:43:28Z), along with governing AGENTS, math-authoring guidance and the PDF skill.

The relevant current requirements remain: explicit allowed and actual reading, exact native review subjects, full substantive review retention, distinct grading, source fidelity and scoped literature-proof coverage. No later documentary edit receives a new independent mathematical verdict merely because this record is filed.

The transformation and source-delta review of the later prose and standing edits was completed and accepted before filing; it found no mathematical drift outside the mapped corrections. The historical anchored proof review and distinct passing grade stand at their exact subject and scope; the later edits do not claim broader Theorem 4 coverage or reopen the separately accepted sufficiency route.