Record, attribution and exact subject

Correction required before integration; five components PASS unconditionally at the frozen bytes. A fresh reviewer read physical pages 1--2 and 12--17 of the source (printed pp. 1025--1026 and 1036--1041) and all nine candidate pages. The interpolation extremum, midpoint inequality (27), pair bounds (29)--(31), the local gap companion and the quarter-logarithm theorem (34) pass; the finite telescoping (32) required one correction, a plus sign omitted from display (T1); and the uniformly separated half-logarithm theorem (33) passes conditionally on that fix. Completed 2026-09-06T07:49:26Z. The corrected successor was accepted in the publication review, which confirms the plus sign; the intermediate approval record of the one-byte successor that the publication review cites is not retained in this repository. 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 nine candidate pages were read from the review packet's copies of the pages under library/polynomials/bernstein_1931_limitation_values_polynomial_segment/ (exact copies not retained); the pages the report names are identified as they stood on 2026-09-15, before this record's filing on 2026-09-16. The review was completed before this repository's first commit; the committed equation (32) page as it stood on that date (unchanged since 2026-09-08) already carries the plus sign the review required, so it is a later successor of the reviewed candidate, not the candidate itself. The current equation (32) page carries the corrected (T1) with the plus sign, and the other pages carry the reviewed constants and cases. 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 nine frozen candidate pages each carried an author-recorded Proof scope standing line stating that the reconstruction was complete and awaiting independent review, and local_gap_test_companion.md named the author agent; the review-complete successors of those lines stood on 2026-09-15 at (_index.md lines 88-90, equation_27_midpoint_product.md line 72, equation_32_telescoping.md line 105, equation_33_interior_case.md line 96, equation_34_local_growth.md lines 140-141, equations_29_31_logarithmic_pairs.md line 75, interpolation_extremal_identity.md line 69, local_gap_test_companion.md lines 131-133, source_proof_scope.md lines 115-120), the pre-edit wording is not retained, and no tier, roadmap, research-plan, acceptance or earlier-review text was in the subject; on 2026-09-18 a separately spawned grader (Claude Fable 5.1) ruled this exposure immaterial under the content test, because the text stated only the author's own unreviewed claim that the review was commissioned to test and the report's reasoning does not lean on it.

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

Final verdict: correction required before integration. The intended mathematics of the quarter-logarithm chain, the separately attributed local gap companion, and the half-logarithm theorem with uniform interior separation checks out. The frozen candidate has one blocking source-fidelity and proof display defect: equation (T1) on equation_32_telescoping.md omits the plus sign between its two quarter-logarithms.

The reviewed author manifest is a working-storage receipt. The source PDF is the source card's bernstein_1931_limitation_values_polynomial_segment.pdf (26 physical pages, printed pages 1025–1050). I independently rendered and visually read physical pages 1–2 and 12–17, corresponding to printed pages 1025–1026 and 1036–1041. That is the actual source-reading scope for this review. I read all nine candidate Markdown files in full. I did not reconstruct or award credit for the other branches of the 26-page paper.

Required correction

Frozen equation_32_telescoping.md, line 36, currently has

\frac14\log\frac{\xi-a_p}{\xi-a_h}
\frac14\log\frac{a_q-\xi}{a_{h+1}-\xi}.

It must be

\frac14\log\frac{\xi-a_p}{\xi-a_h}
+\frac14\log\frac{a_q-\xi}{a_{h+1}-\xi}.

Printed equation (32), physical page 16 / printed page 1040, visibly has this plus sign. The proof immediately below also obtains two additive telescoping sums. Without the plus, TeX juxtaposes the factors and states their product; that is neither Bernstein's inequality nor an implication strong enough to deduce (T2).

The correction inserts exactly one ASCII plus at zero-based byte offset 1040 of the frozen candidate. The review's correction record gives the exact span guard and the minimal seven-file manifest cascade.

Mathematical review

• Interpolation extremum: PASS. Lagrange interpolation gives the upper bound, real sign data attain it even when complex coefficients are allowed, and the node, endpoint, continuity, and equality cases are correct.

• Midpoint inequality (27): PASS. Squaring and factoring at consecutive real roots leaves the exact coefficient (b-a)/4. The identity (t+delta/2)^2=t(t+delta)+delta^2/4 handles every remaining root. The positive-gap degree-two equality case, repeated-root derivative-zero case, and degenerate a=b case are all correctly separated.

• Pair bounds (29), (31), and (31 bis): PASS. The derivative-ratio normalization and AM–GM give delta/(2 sqrt((a-x)(b-x))). Setting z=delta/(a-x) and t=(1/2)log(1+z) gives z/sqrt(1+z)=2sinh(t)>2t=log(1+z). Reflection gives the right-side case, and strictness and excluded nodes are handled correctly.

• Finite telescoping (32): CORRECTION REQUIRED. Apart from the missing plus in (T1), the proof is correct. Each selected node weight is counted at most twice, positive outer weights make the inequality strict, empty sums are covered, and the one-sided exterior formulas (T3) are valid. T3 remains sufficient for the quarter-logarithm theorem. The two-sided T1/T2 chain must await the one-byte fix.

• Local gap companion: PASS. For a node-free (c-r,c+r), every node has r <= |a_j-c| <= R and r<R. The polynomial T_m((2(x-c)^2-(R^2+r^2))/(R^2-r^2)) has degree at most d, is bounded by one at every node, and has exact center value cosh(m log((R+r)/(R-r))). The elementary logarithmic estimate and R<=2 yield the strict gap bound with the stated factor 2. Odd degrees, truncated interval-end gaps, and the whole-segment edge case are correct. The page accurately attributes this finite local formulation to the compilation and correctly distinguishes the source's exact reciprocal term from its inexact next finite display; it does not call the repair a published erratum.

• Quarter-logarithm theorem (34): PASS at its stated scope. Condition (L0) is exactly D0<L/4. The large-M branch dominates the finite right side; the small-M branch finds the necessary endpoint-near node and a second node on the chosen side, then applies the valid one-sided T3 bound. Reflection includes both interval endpoints. The constants Lm/(8 log(2 log d)), m>=d/3, and L/24 all check. The conclusion is honestly about max_I F, including the large-M branch, and does not claim the same value at an arbitrary selected maximizer of |A|.

• Uniformly separated half-logarithm theorem: mathematically PASS after the T1 byte fix. With xi separated from both interval ends by fixed eta, the local gap bound produces outer distances exceeding eta/2 and a surrounding gap shorter than D; the product in T2 then gives exactly (1/2)log(eta/D). The large-M case, thresholds, and uniformity in the nodes are correct. The page does not promote bare degree-by-degree interiority to uniform separation and does not claim a nodal-polynomial counterexample.

The degree/node conversion is consistently Bernstein degree d with d+1 nodes versus E1153 node count N=d+1. The local 1/4 result is not presented as the sharp 2/pi answer. E1129 and E1132 links retain their different quantifiers and receive no status transfer.

Byte and scope checks

All 102 frozen artifact pins, ten candidate pins, nine body pins and saved authored-body copies, ten preserved input pairs, 26 author render pins, and the three identical PDF copies match. I independently reproduced all 43 wiki-link occurrences and resolved all nine relative PDF links. The candidate pages each have exactly one body delimiter and contain no unexpected C0 bytes.

Five components pass unconditionally at the frozen bytes: interpolation, midpoint, pairs, the local gap companion, and the quarter-logarithm theorem. The half-logarithm theorem is gated only by the T1 correction. No complete selected-chain, full-paper, external recursive-proof, formal, acceptance, or problem-status credit is awarded to this frozen package. After the author freezes the guarded one-byte successor with refreshed pins, a bounded exact delta review is sufficient; no renewed mathematical reconstruction is needed if every other byte stays fixed.