Proof scope and dependencies in Bernstein 1931

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The copy read, the MathNet scan named on the source card, has 26 physical pages corresponding consecutively to printed pp. 1025--1050. All pages were visually read for this compilation. Visual reading is distinct from complete proof reconstruction: the completed chain here is the historical local argument supporting Problem 1153.

Completed local chain

The logical dependencies are:

1. Interpolation extremum plus the explicit Chebyshev identities gives the local gap test.
2. The midpoint product inequality gives the logarithmic pair estimates.
3. The interpolation formula and pair estimates give the finite one-sided and two-sided telescoping inequalities.
4. The local gap test and the one-sided telescoping inequality give the $1/4$ local-maximum theorem, including boundary maxima.
5. The local gap test, two-sided telescoping, and a uniform separation hypothesis give the qualified $1/2$ version.

No external theorem is needed in this completed local chain. Proofs live on the linked result pages rather than being repeated here.

Corrections and limits that affect the mathematics

The source's finite Chebyshev expression on p. 1037 has an inexact replacement of a reciprocal power. The companion keeps the exact identity and supplies a finite bound depending on the maximum on the prescribed interval. This also avoids using a global small-maximum restriction as a local premise. These are compilation repairs, not a published erratum.

The stronger display (33) is retained as a source claim. Its bare interiority wording does not explicitly supply the uniform outer-distance control used in its displayed deduction. The interior-case page states the exact residual obligation. It is neither silently promoted to a complete proof nor declared false.

The reconstructed all-cases theorem concerns $\max_I F$. In its large-maximum branch it does not identify that value with the value at an arbitrary maximizer of the nodal polynomial's modulus.

The shrinking-interval discussion on pp. 1040--1041 is not used to obtain the fixed-interval theorem. Its additional asymptotic claims are not credited as reconstructed here. The finite bounds retain interval length explicitly; they should not be converted into a uniform shrinking-interval assertion by dropping that dependence.

Other source branches and external interfaces

The sharp global minimax asymptotic (2), p. 1026, is separate from the local coefficient $1/4$. The following branches were visually read and located, but their full rewritten proofs remain a separate queue:

• Sections 2--3, pp. 1027--1036, equations (3)--(26 bis), stated on the perturbed-Chebyshev page: perturbed Chebyshev nodes, product asymptotics, derivative asymptotics, and asymptotically equal gap maxima. In the singular-integral and Fourier steps on pp. 1030--1031, Bernstein refers to his 1930 memoir Polynômes orthogonaux relatifs à un segment fini, chapter II, section 9. The interface used there is uniform convergence and continuity of the relevant singular-integral/conjugate-series expressions under the stated logarithmic modulus of continuity with exponent greater than one. That external proof is not included.
• The trigonometric interpolation Théorème and its algebraic transfer, pp. 1041--1049, equations (35)--(58): circular node gaps, product estimates, averaging, truncation and the final harmonic sum. The derivative step on p. 1048 uses the trigonometric Bernstein inequality $\|S_d'\|_\infty\le d\|S_d\|_\infty$ for a trigonometric polynomial of order at most $d$. Its proof is not imported into the local chain. The matching equidistant circular upper bound is attributed on p. 1041 to Grandjot's 1925 paper cited earlier in the source.
• The unit-circle polynomial Corollaire, pp. 1049--1050, depends on that trigonometric branch.

These entries are dependency interfaces and proof-coverage limits, not assertions that the global chain or its external sources have been independently checked in this compilation.

Relation to other problems and review scope

The global minimax setup also appears in Problem 1129. Its asymptotic value alone does not characterize the exact minimizing node configurations. The fixed-point and almost-everywhere questions in Problem 1132 have different quantifiers from a maximum over a fixed interval; no status transfer is made.

Independent mathematical review on 6 September 2026 covered the selected local proof chain and its separately attributed compilation companion. It did not extend to the uncompiled global branches or external interfaces above. This publication composition adds no formal-verification or additional acceptance evidence and does not change any problem's imported statement or status.