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The whole-conclusion grade assessed the source deductions needed by the supplied proof. The two examiners read text excerpts; neither inspected a PDF or checked a transcription against its printed edition. Earlier reading described on a library card is a different examination and does not enlarge their scope. The retained source identities fix editions, not the accepted standing of every result in those editions.

OpenAI plane-coloring manuscript

The canonical source card identifies The Euclidean plane is not five-colorable, September 23, 2026 Math Release, external release revision adc7f1241, and records the Apache-2.0 terms. Its PDF and matching transcription are retained at library/discrete_geometry/openai_2026_euclidean_plane_not_five_colorable/, with the common stem openai_2026_euclidean_plane_not_five_colorable. These canonical files supply the lawfully retained proof text; there is no duplicate source transcription among the research assets.

The examined scope is the transcription's original lines 83--545: Sections 2 and 3 and the supplied part of Section 4 through Proposition 4.2, together with selected bibliography. The review derives the relative-singularity, tower, multiple-average, number-field selection and continuous-factor interfaces needed by the native argument. The new finite-function-field selection replaces the step confined to number fields. The grade covers these deductions and their stated specialization, not the manuscript's full five-color theorem or its separate Lean formalization.

Finite triangle forcing

The edition is Jelínek, Kynčl, Stolař and Valla, Monochromatic triangles in two-colored plane, arXiv:math/0701940v1, January 31, 2007. The source card also identifies the later Combinatorica publication; the review used the arXiv edition, not that later typesetting. Its consumed inputs are the direct-motion and degenerate-triangle conventions, Lemma 1.3 and Corollary 1.4, including the finite forcing proof.

The report's historical transcription locators are lines 48--54 and 72--94. That restricted transcription is not distributed in the public copy, and no republication license is inferred from its availability to the examiners. The exact public edition linked above supplies the source; the result card states its interface. The native circle proof and independent review reconstruct the forcing configuration in coordinates and justify its positive-measure use. Their argument depends on those written coordinates, not on an unread figure or a copied source proof.

Empty measure-theoretic boundary

The edition is P. Lahti, Federer's characterization of sets of finite perimeter in metric spaces, arXiv:1804.11216v1, April 27, 2018. The source card fixes that edition; its title page is dated May 1, 2018. The supplied TeX excerpt included the Euclidean introduction on p. 1, Theorem 1.1 on p. 2, the standing metric hypotheses on p. 3, the boundary definition and formulas (2.2)--(2.3) on p. 5, and (2.4) on p. 6, with required macros and bibliography. These statements and their hypotheses were examined; the criterion's and perimeter representation's original proofs were not.

The retained theorem interface states the exact boundary convention, finite-perimeter criterion, representation and relative isoperimetric inequality. In the Euclidean application, an empty measure-theoretic boundary gives finite and then zero perimeter on each ball, so one color is null there; nested balls give a globally null or conull color. This application is checked in the whole review. The cited theorems remain external boundaries. No restricted source transcription or TeX excerpt is copied into these assets.

Other classical boundaries

The review and grade explicitly identify the compactness, real-closed field, spectral, probability, amenability, density and finite-perimeter theorems used, with the hypotheses checked at each application. Selected source bibliography supplies further locators; the original external proofs were not read merely because their titles occur there. The Bessel identities and analytic inequalities used in the argument are derived in the retained text, with the finite rational part independently runnable. These boundaries give no acceptance to omitted source conclusions, unexamined formalizations or future revisions of the native proof.