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Record, attribution and exact subject

PASS. A fresh reviewer rendered printed pp. 193--195 and checked the compilation-supplied finite symmetrization correction: the exact counting repair $L\geq U/4$, the off-diagonal pair estimate, the diagonal case through the Erdős--Turán adjacent-polynomial interface, and the resulting $1/16$ prefactor in (C3). Completed 2026-09-06T07:49:18Z. 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 reviewed bytes of the correction are not retained; the current page carries the reviewed estimates. 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 exact reviewed copies were review-packet candidates and are not retained, and the comparison recorded in this section says how the committed pages relate to them.

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: PASS. No author-package delta is required.

The reviewed author manifest is a working-storage receipt; the correction was read from the review packet's copy of finite_symmetrization_correction.md (exact copy not retained). The five-page primary PDF is the source card's erdos_szabados_1978_integral_lebesgue_function_interpolation.pdf.

Source check

Printed p. 193 / physical p. 3 confirms that the inner triangular sum in (7) literally begins at k=m. Its diagonal brace is A_mm+A_mm, while the preceding rectangular sum contains A_mm once. The printed comparison therefore double-counts the diagonal. Printed p. 194 / physical p. 4 again begins at k=m, although its ratio derivation uses separated positive gaps and does not cover m=k.

The cited interface is exactly the adjacent-polynomial inequality l_k(y)+l_(k+1)(y)>=1 on [x_k,x_(k+1)], located at the foot of p. 193 and continued with “cf. [5, Lemma IV]” on p. 194. Printed p. 195 expands [5] as P. Erdős and P. Turán, On interpolation. III, Annals of Mathematics 41 (1940), 510–552. The companion correctly treats that lemma as an external input and claims no review of its proof.

Mathematical check

The finite counting repair is exact. At each point, the sum over adjacent pairs counts each included fundamental polynomial at most twice, so T<=2L. The triangular sum satisfies U=T+D, and nonnegativity gives D<=T; hence U<=4L and L>=U/4.

For m<k, the affine map carries the middle half of I_m onto the middle half of I_k. The first ratio estimate has coefficient d_k/[4(x_(k+1)-x_m)]; the Jacobian changes this to d_m/[4(x_(k+1)-x_m)]. Interchanging the intervals gives the same coefficient for the reciprocal ratio. Neither nodal polynomial vanishes on the middle halves, and u+u^(-1)>=2 over length d_k/2 yields

A_mk+A_km >= d_m d_k/[4(x_(k+1)-x_m)].

For m=k, the adjacent-polynomial inequality gives A_mm>=d_m, which is more than enough for the same displayed pair bound. Combining the pair bound with L>=U/4 gives the stated 1/16 prefactor in (C3). Restricting the outer sum to a<=x_m<=(a+b)/2 removes only nonnegative terms.

The note does not prove the remaining harmonic-block, threshold, node-free-gap, or endpoint steps. It does not retain the source's downstream numerical constant. This review therefore approves only the diagonal companion, conditional on the stated Erdős–Turán interface. It grants zero full-theorem, external-lemma, formal, status, or new-solution credit.

Visual evidence

The selected PDF pages were independently rendered with Poppler 26.07.0 at 220 dpi. The inspected review renders are:

• physical p. 3 / printed p. 193;
• physical p. 4 / printed p. 194;
• physical p. 5 / printed p. 195, bibliography only.

Advisory source-context notes were received from two other participants. They were not used as review authority; every conclusion above was checked independently against the pinned candidate and rendered primary pages.

This review did not edit the author package, canonical corpus, Git state, or the full-proof compilation.