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Subject

The source as it stood on 2026-09-28T05:03:27Z (called "the commit" below). Pages: wiki/research/erdos_1171/lemma_2_1_reconstruction.md and wiki/research/erdos_1171/theorem_3_1_reconstruction.md, each read whole with git show at that commit; the working-tree copies are byte-identical to the frozen pages (git diff against the commit on the two paths is empty). Reports: the Lemma 2.1 review and the Theorem 3.1 review, each read whole.

Read for adjudication, at the same commit unless stated: the sections "Independence and the assignment", "Exact subjects and durable evidence", "Report contract", "Grading and claim standing", "Whole-claim report" and "Audit checklist" of docs/verification.md, and the sections "Source fidelity" and "Mathematical review" of docs/evidence.md; the held PDF of Gao (2026) (four pages, 244,055 bytes, matching the card's provenance line), all four pages from the text layer and all four rendered at 130 dots per inch and read, with the definition (p. 1), relation (1) (p. 2), Lemma 2.1 and its proof (pp. 2--3), Theorem 3.1 and its proof (pp. 3--4) and Remark 3.2 (p. 4) checked against the images; the Gao card and its two result pages; the Baumgartner (1989) card, source record and main theorem page; the Baumgartner and Hajnal (1987) card and positive relation page; the part of Problem 1171 above its Current assessment heading; the folder index; and, to check the two attributions both reviewers left outside their read sets, the held PDF of Komjáth (2025) at printed pp. 425 and 443 (physical pp. 8 and 26, rendered and read) and its references [11] and [40]. The folder's working-tree changes since the commit (the index and a new connections page) were opened only to confirm that neither reviewed page differs from the frozen bytes; nothing from them enters this grade. No other review, evidence folder, workspace file or web page was read.

Independence, by role: distinct grader in a fresh context, given only this assignment. The grader wrote neither page, no page of the folder or of the library cards named above, and neither report, and had no communication with the author or with either reviewer. A grader is not blind: the standing text of the cards, the folder index and both reports were read by design.

Exposure ruling. Each report discloses that whole-file reads returned more than its allowed sections: standing sentences on the Gao card and its result pages, those pages' rewritten proofs and checks, the status line and Source paragraph of the problem page, and, for the Theorem 3.1 review, the bodies of the two Baumgartner cards, their result pages and the whole Lemma 2.1 reconstruction. Content test: nothing in either report could only have come from that text. Every re-derivation cites the PDF; the facts the Theorem 3.1 review takes from the exposed pages (the CH negative relation, the introduction's slip at $k=1$, the instance $k=0$, the Komjáth attributions, the not-disprovable status) all stand on the frozen page itself; and the direction of each attack (the color bookkeeping and the transport for the lemma, the reading of the product and the exact hypothesis of Theorem A for the theorem) follows from the subject and the PDF. None of the exposed text is a review of the pages or a verdict on them. The exposures are ruled immaterial for both reports.

Reports graded

Lemma 2.1 review: pass. The subject block resolves (commit and path, the path unchanged). The independence facts and three exposures are stated. The restatement carries the convention and every quantifier: every ordinal $\alpha$, every finite $k\ge1$, every coloring, and the conditional form with its hypothesis assumed rather than certified. All ten checklist items carry an explicit verdict, with the inapplicable ones marked. The three weakest steps are re-derived rather than paraphrased: the transport along the order isomorphism, including why $d'$ is defined on all of $[\alpha]^2$; the recovery of $c$ from $c'$ on the colors $\ge1$; and the color bookkeeping $\{1\}\cup\{2,\ldots,k+1\}$. The grader re-derived each from the PDF and agrees. The strongest attack is real: a coloring whose induction-step output fails to witness $P(k+1)$, the transport invoked at an order type other than $\alpha$, and the boundary attacks ($\alpha\le1$, finite $\alpha\ge2$, colorings that omit a value, the two consequence sentences of the Checks bullets). The premises carry their interfaces and reading depth: the hypothesis as an explicit assumption, no imported theorem, fact 2 shown not load-bearing, no consumed claim, the PDF held at the stated depth. The verdict is stated and assigns no tier.

Theorem 3.1 review: pass. The subject block resolves. The independence facts and exposures are stated in detail. The restatement carries the convention (von Neumann ordinals, $\omega_1\omega=\omega_1\cdot\omega$, the arrow relation, $\mathrm{MA}_{\aleph_1}$), the axiom as hypothesis, the quantifiers over $k$ and the coloring, and the scope consequences with their own hypotheses. All ten checklist items carry an explicit verdict. The three weakest steps are re-derived: the restriction and transfer with the ordinal arithmetic behind them (strict monotonicity of left multiplication, transitivity, the intrinsic order type of a set of ordinals); the application of Lemma 2.1 with its induction re-derived in brief and its composition with the axiom; and the four scope consequences, each under its stated hypothesis. The grader re-derived each and agrees. The strongest attack is real and well chosen: under the collapsed reading $\omega\cdot\omega_1=\omega_1$ the theorem would be a ZFC triviality from $\omega_1\to(\omega_1,\omega)^2$, and the report shows why the page and the source pin the product reading; the second attack, on the exact hypothesis of Theorem A, is placed at the unheld import where the risk sits. The premises record Theorems A, B and C and the CH negative relation each with interface, source, reading depth and use, Lemma 2.1 as a sibling page, the ordinal facts re-derived, and the two attributions outside the read set flagged rather than assumed. The verdict is stated and assigns no tier.

Corrections

C1. Page: lemma_2_1_reconstruction.md. Location: section "Checks and scope", bullet "What the number $3$ contributes", the sentence "The source records this only in the form of Remark 3.2." Replace it with: "The source does not state this; Remark 3.2 (p. 4) only restates the lemma as the stability of $\alpha\to(\alpha,3)^2$ under adding finitely many colors with target $3$." Basis, checked against p. 4 in the text layer and the image: Remark 3.2 reads "The proof of Lemma 2.1 shows that the property $\alpha\to(\alpha,3)^2$ is stable under adding finitely many colors with target 3" and then names $\mathrm{MA}_{\aleph_1}$ as the hypothesis; it says nothing about other targets or about what the number $3$ contributes, so the page's sentence characterizes the source as recording an observation it does not make. The Lemma 2.1 review filed this as F1 at severity suggested; it is accepted here as a correction on the grader's own verification, since the characterization of a source is a checklist item and the sentence strengthens the remark. The change touches a commentary bullet, not the statement or the proof.

Rejected and downgraded findings

• Lemma 2.1 review, F2 (note: fact 2 is stated without proof and announced as used). Rejected; no change. Fact 2 is true, its proof is the one line the reviewer gives, and the page uses it exactly where it says, to identify its renaming with the source's color names. The proof applies $P(k)$ to a coloring already valued in $\{0,\ldots,k\}$, so nothing rests on it, and no rule requires a proof of a relabeling that carries no weight.
• Theorem 3.1 review, F1 (suggested: add a Coverage bullet and drop or mark the sentence on $2^{\aleph_0}>\aleph_1$). Downgraded to optional; no change required. The evidence rules require a label for a repair of a gap or of an incorrect formula. The page supplies only standard facts the source uses without proof (the initial-segment property, the transfer of the homogeneous set, the definition of $\mathrm{MA}_{\aleph_1}$), attributes none of them to the source, marks Theorem C as a step the deposit does not state, and says in its Standing what the deposit itself deduces. The reviewer's proposed bullet is accurate and may be added. The sentence on $2^{\aleph_0}>\aleph_1$ is correct background that explains why the CH negative relation in the Fidelity list coexists with Theorem A; it stays.
• Theorem 3.1 review, F2 (note: "The sets $X$ and $T$ are subsets of $\omega_1^2$"). Rejected; no change. Step 2 states the disjunction one sentence earlier and the sentence applies to whichever set exists; the argument is unaffected.

Checks of the grader's own that produced no correction. The two attributions on the theorem page that both reviewers left unchecked hold: Komjáth (2025) states at printed p. 425 "In [40], Erdős and Hajnal proved $\omega_1^2\to(\omega_1\alpha,3)^2$ for $\alpha<\omega_1$", with [40] the 1970 paper in Acta Math. Acad. Sci. Hungar. 21, and at printed p. 443 "It is unknown if $\omega_1^2\to(\omega_1\omega,3,3,3)^2$ holds"; p. 425 also states Theorem B in ZFC, with [11] the 1987 paper. The locators on both pages (Lemma 2.1 stated p. 2 and proved pp. 2--3 under the title "A general color-reduction lemma"; Theorem 3.1 stated p. 3 and proved pp. 3--4 under "The main theorem"; relation (1) p. 2; Remark 3.2 p. 4; printed and physical page numbers equal) match the PDF, and the quotation "is exactly" (p. 2) is verbatim. The convention the pages adopt is the source's clause for clause (p. 1).

Graded verdicts

• lemma_2_1_reconstruction.md: fidelity faithful, with the wording correction C1 in a commentary bullet; argument sound. The induction, the transport and the color bookkeeping were re-derived here from the PDF, and the base case and both alternatives of the step compose as the page says.
• theorem_3_1_reconstruction.md: fidelity faithful; argument sound, conditional on Theorem A exactly as the page states, with the restriction and transfer re-derived here and the scope consequences holding under the hypotheses the page attaches to them.

No tier is assigned and no status changes.