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Record, attribution and exact subject

PASS for the selected lemma, with three exact publication edits. A fresh reviewer inspected printed pp. 510 and 528--530 of the source and checked the complete reconstruction of Lemma IV, $l_k(x)+l_{k+1}(x)\geq1$ on every adjacent-node interval for $n\geq2$, including the source's literal $\phi(\xi_0)=1$ sentence, the bounded strict-inequality completion and the increasing-node form. Reviewed 2026-09-06T12:02:58Z. The three required edits were non-substantive: naming the fundamental theorem of calculus among the dependencies and two review-state wordings. Reviewer: a fresh review context distinct from the author of the reconstruction and from the compilation-supplied corrections; it did not build on the subject before reviewing it. No distinct grader is recorded, so no numerical claim tier is assigned.

The body of the source _index.md as it stood on 2026-09-15 is byte-identical to the reviewed body (checked at filing on 2026-09-16; the body is unchanged since). The frozen bytes of the lemma page are not retained; the current page carries the reviewed argument and names the fundamental theorem of calculus. The exact reviewed copies are not retained in this repository. On 2026-09-16 the current pages were compared with the report's description of the reviewed statements, constants and proof steps and agree with it; the retained version history since the earliest corpus snapshot shows only attribution and standing wording changes on these pages. A match of description is not a byte match, and any substantive change to the mathematics requires a new assessment. The pages the report names are identified as they stood on 2026-09-15, before this record's filing on 2026-09-16; the pages named byte-identical above carry the reviewed bodies as of that date, the other reviewed copies were review-packet candidates that are not retained, and the comparison recorded in this section says how the committed pages relate to them.

The frozen lemma candidate and the frozen reciprocal page carried the lemma's own review-state sentence, "pending independent mathematical review" (removed by edits 2 and 3, quoted in the retained report below; the pre-edit bytes are not retained, and the committed pages as they stood on 2026-09-15 — lemma_iv_adjacent_fundamental_polynomials.md lines 209-210 and endpoint_harmonic_completion.md line 89 — already read the post-edit form), and the reciprocal page as of that date carried prior-review text about the 1978 endpoint/harmonic component at lines 19-25; a separately spawned materiality grader (Claude Fable 5.1) ruled on 2026-09-18 that this exposure is immaterial, because the text asserts no verdict on Lemma IV and the review's reasoning rests on its own reading of printed pp. 510 and 528-530 and its own root count, and the unretained author FINAL remains a possible but unverifiable exposure.

This record was filed on 2026-09-16 from a retained report, the review text and its machine-readable companion. The report text is retained below in full. The filing changed only the wrapper, participant identifiers, private paths and operating-history material; it records no new verdict, and the first-person readings and judgments below belong to the historical reviewer, not to the filing author.

Retained report

Verdict: The mathematics, primary-source fidelity, reciprocal overlay, and bounded credit scope pass. The exact frozen proof correctly establishes $l_k(x)+l_{k+1}(x)\ge 1$ on every adjacent-node interval for every $n\ge2$, and its increasing-node/nonnegative form is valid.

I independently inspected the title page (physical 1 / printed 510), the complete Lemma IV page (physical 20 / printed 529), and the neighboring pages 528 and 530. The source visibly prints $\phi(\xi_0)=1$ in its final contradiction sentence. The candidate reports that literal symbol honestly and supplies a correct, bounded mean-value/root-count argument excluding $\phi(\xi)<1$.

The root counts are complete. For interior indices there are $n-4$ exterior derivative zeros; the endpoint-sign contradictions and a hypothetical dip below one add enough distinct central or neighboring zeros to exceed degree $n-2$. The $k=1$ proof works when $n=3$, reflection gives $k=n-1$, the inclusive interior argument covers $k=2,n-2$, and the separate $n=2$ polynomial argument is correct. Reversing the node order exchanges the two adjacent fundamental polynomials, and their factor signs make both nonnegative on the increasing-node interval.

The retained 44-page PDF is accurately qualified as a Rényi Institute archive scan rather than asserted to be separately authenticated publisher bytes. The source home limits its read scope to printed pp. 510 and 528–530. The reciprocal change replaces exactly one obsolete no-acquisition span and preserves every other byte.

Three exact, non-substantive publication edits remain:

1. Add “the fundamental theorem of calculus” to the dependency sentence. The valid endpoint-sign argument twice uses that the integral of $\phi'$ equals the endpoint difference.
2. Replace the lemma page’s “pending independent mathematical review” wording with passed-review wording.
3. Make the same review-state replacement in the reciprocal Erdős–Szabados page.

No substantive proof repair is required. After those three edits, this review supports one complete selected proof component only: Erdős–Turán Lemma IV together with its increasing-node/nonnegative interface. It supplies no review credit for the rest of the 1940 article, the 1978 printed constant $1/40$, the sharp $2/\pi$ bound, a complete E1153 proof, formal verification, acceptance, or problem status.

Author freeze, read from the review packet:

• proposal manifest and author FINAL (working storage; not retained)
• candidate lemma (packet copy; not retained)
• source index (packet copy; its body below *** is the body of the committed _index.md as it stood on 2026-09-15, named in the record section above)
• reciprocal page (packet copy; not retained)
• primary PDF: erdos_turan_1940_on_interpolation_iii.pdf in the source home