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Subject and independence

Role. Independent reviewer in a fresh context, commissioned for refutation and given only the assignment. The reviewer took no part in writing the connections page, the reconstruction pages or their reviews, and had no contact with the page author or with the reviewers and grader of the reconstruction pages.

Subject. The source as it stood on 2026-09-28T13:50:48Z (called "the subject commit" below), page wiki/research/erdos_1171/connections.md, section ## Tools, lines 32--123; the next H2 heading is line 125. The section records a definition and transport fact (T1), restriction and weakening (T2), the induction with general targets (T3), the descent lemma (T4) with its proof, witness extraction (T5) with its proof, and a closing equivalence between $\alpha\to(\alpha,3)^2$ and its finite-color form. The lead page is the same page; its Scope paragraph (lines 18--30) was read as the page's standing statement.

Reading. Read in full at the subject commit: the section; the two reconstruction pages Lemma 2.1 and Theorem 3.1, as premises; the Statement paragraphs of Problem 1171 and Problem 118; the provenance paragraph of the library card Gao (2026) and the Statement section of its Lemma 2.1 result page; the held four-page PDF of that source, all pages by text extraction and pages 2 and 4 on page images (Lemma 2.1 with its proof; Remark 3.2); and the guidance pages named in the commission (docs/verification.md, sections "Whole-claim report", "Audit checklist" and "Grading and claim standing"; docs/approach_vetting.md; docs/evidence.md, section "Source fidelity"; docs/math_authoring.md). The other sections of the lead page were consulted only through a keyword scan for the uses of (T1)--(T5), of graph targets and of order types, and one paragraph (lines 458--463, on Problem 118) was read in full as the consumer of the order-type remark in (T1).

Exposure. The folder index is excluded and was not read, so the attribution of (T3) to it is unchecked here; (T3) is verified below as the constant case of (T4). A directory listing of evidence/verify/ showed the file names of the two reconstruction reviews and the grade; none was opened. No Current assessment, Known results, standing or acceptance text was read. Nothing else reached the reviewer.

Restatement

Throughout, a coloring of $[X]^2$ with the colors $0,\ldots,k$ is a function on the two-element subsets of a linearly ordered set $X$; a subset is homogeneous in color $i$ when all its pairs have color $i$; a target is an order type or a graph; and a copy of a target $H$ in $X$ is a subset of $X$ of order type $H$ when $H$ is an order type, and a subgraph of the complete graph on $X$ isomorphic to $H$ when $H$ is a graph. The copy is in color $i$ when all pairs of the subset, respectively all edges of the subgraph, have color $i$. This is the reading of "all of whose pairs have color $i$" adopted here; F2 records the ambiguity. A triangle is a copy of the order type $3$.

• S0 (provenance sentence). "Five facts are used below. The first two are on the reconstruction pages, the third is stated on the folder index, and the last two are recorded here with their proofs." Not a mathematical claim; assessed under Source and verdict fidelity.
• S1 (T1, definition and transport). For an order type $\gamma$, an order type $\beta$, targets $H_1,\ldots,H_k$ and $k\ge1$, the relation $\gamma\to(\beta,H_1,\ldots,H_k)^2_{k+1}$ means: every coloring of $[\gamma]^2$ with the colors $0,\ldots,k$ has a subset of order type $\beta$ homogeneous in color $0$, or, for some $i\in\{1,\ldots,k\}$, a copy of $H_i$ in color $i$. If $Y$ is a linearly ordered set (in the section, a set of ordinals) with $\operatorname{otp}(Y)=\gamma$, the same conclusion holds for every coloring of $[Y]^2$, with subsets of $Y$ and copies inside $Y$; the proof of fact 1 of the lemma page carries over. Since only order isomorphisms are used, the facts (T2)--(T5) hold with ordinals replaced by arbitrary linear order types, the relation for an order type being read on any representative.
• S2 (T2, restriction). If $\gamma\to(\beta,H_1,\ldots,H_k)^2_{k+1}$ and $\gamma'$ has a subset of order type $\gamma$ (for ordinals, $\gamma'\ge\gamma$), then $\gamma'\to(\beta,H_1,\ldots,H_k)^2_{k+1}$.
• S3 (T2, weakening). Three monotonicity clauses, each from a fixed relation $\gamma\to(\beta,H_1,\ldots,H_k)^2_{k+1}$: (a) if a set of order type $\beta$ has a subset of order type $\beta'$ (for ordinals, $\beta'\le\beta$), then $\gamma\to(\beta',H_1,\ldots,H_k)^2_{k+1}$; (b) a target $H_i$ may be replaced by any $H_i'$ that embeds in $H_i$; (c) a color may be left unused, so the relation with $k$ triangle targets implies the relation with $k-1$ triangle targets, and in particular $\alpha\to(\alpha,3,\ldots,3)^2_{k+1}$ implies $\alpha\to(\alpha,3)^2$.
• S4 (T3). If $\alpha\to(\alpha,H_i)^2$ for every $i=1,\ldots,k$, then $\alpha\to(\alpha,H_1,\ldots,H_k)^2_{k+1}$, with color $i$ carrying $H_i$.
• S5 (T4, descent lemma). Let $k\ge1$, let $\gamma_0,\ldots,\gamma_k$ be ordinals (or order types, by S1) and $H_1,\ldots,H_k$ targets, and suppose $\gamma_{j-1}\to(\gamma_j,H_j)^2$ for every $j=1,\ldots,k$, each a two-color relation with the large target in color $0$ and $H_j$ in color $1$. Then $\gamma_0\to(\gamma_k,H_1,\ldots,H_k)^2_{k+1}$, with color $j$ carrying $H_j$.
• S6 (T5, witness extraction). Let $k\ge1$ and let $c:[\alpha]^2\to\{0,\ldots,k\}$ have no subset of order type $\alpha$ homogeneous in color $0$ and no triangle homogeneous in any color $j\ge1$. Then there are $Y\subseteq\alpha$ with $\operatorname{otp}(Y)=\alpha$ and $j\in\{1,\ldots,k\}$ such that the coloring $d_j$ of $[Y]^2$ with $d_j(p)=1$ when $c(p)=j$ and $d_j(p)=0$ otherwise has no subset of $Y$ of order type $\alpha$ homogeneous in color $0$ and no triangle of color $1$; pulled back along an order isomorphism from $\alpha$ onto $Y$, $d_j$ witnesses $\alpha\not\to(\alpha,3)^2$.
• S7 (equivalence). For every order type $\alpha$ and every finite $k\ge1$: $\alpha\to(\alpha,3)^2$ if and only if $\alpha\to(\alpha,3,\ldots,3)^2_{k+1}$ with $k$ triangle targets. The section derives this from Lemma 2.1, (T2) and (T5).
• S8 (consequence). A witness for the negative relation with $k$ triangle targets contains, on a subset of order type $\alpha$, a witness for $\alpha\not\to(\alpha,3)^2$ (this is S6), and adding finitely many triangle colors changes neither the positive nor the negative relation (this is S7 together with its negation).

Checklist

• Quantifiers and scope. No defect. Every statement is for each finite $k\ge1$ and each order type separately; none claims the relation with infinitely many colors, a uniform bound, or an "almost all" reading. The existential quantifiers of S6 ($Y$ and $j$) are constructed by the proof, and S5's conclusion names which color carries which target. The scope of "everything below" in (T1) is verified for (T2)--(T5) only (F3).
• Circularity. No defect. S4 is derived from S5; S5 and S6 are proved by finite induction on $k$ whose base case is the hypothesis itself; S7 uses Lemma 2.1, a premise from another page, only for the direction that S6 also proves; nothing assumes S7 or an equivalent of it.
• Model and convention changes. No defect in substance; one wording finding. The passage from ordinals to order types (S1) is stated with its bridge (order isomorphisms only) and is checked below for each of (T2)--(T5); graph targets are defined in the section, and the definition agrees with the lemma page's relation when every target is an order type. The word "pairs" for the edges of a graph copy admits a stronger misreading (F2).
• Finite and statistical overreach. Inapplicable: no finite case, sample or heuristic is used as evidence.
• Uniformity. No defect. No constants or bounds occur; the only parameter is $k$, every statement is proved for each $k$ by an induction that yields no limit statement, and the section claims none.
• Extremal conclusions. Inapplicable: no infimum, supremum, attained value or sharpness sentence occurs.
• Consequences and composition. No defect in substance; findings on attribution. The "so" and "gives" sentences (S3(c), S4 as the constant case of S5, S7 with S8) were checked separately and each follows. The consumed interfaces (fact 1 and fact 2 of the lemma page, Steps 2 and 3 of the theorem page, the statement of Lemma 2.1) were confirmed on their pages at the subject commit. Lemma 2.1 as filed is for ordinals, so the forward direction of S7 over non-ordinal order types rests on (T3) or (T5), not on the citation as written (F3).
• Computation. Inapplicable: the section has no computation.
• Reproduction. Inapplicable: the section carries no rerun command or coverage claim.
• Source and verdict fidelity. No defect in substance; findings on provenance. "Fact 1 of the lemma page is stated for ordinal targets" and "The lemma page merges the colors $0$ and $1$ and renames the rest" match the lemma page; "Steps 2 and 3 of the theorem page" are the restriction and reading-back steps as described; the held source states Lemma 2.1 for an ordinal $\alpha$ (p. 2) and Remark 3.2 (p. 4) as the reconstruction page reports. The sentence placing (T1) and (T2) "on the reconstruction pages" overstates: those pages carry fact 1 and the restriction principle for ordinal targets only (F1). The attribution of (T3) to the folder index could not be checked (F5).

Weakest steps

W1: the induction step of (T4), where the last relation is applied on the merged set. Assume the lemma for chains of length $k$, let the $k+1$ relations hold, and fix $c:[\gamma_0]^2\to\{0,\ldots,k+1\}$. Put $c'(p)=0$ when $c(p)\in\{0,k+1\}$ and $c'(p)=c(p)$ otherwise. The first $k$ relations form a chain of length $k$ from $\gamma_0$ to $\gamma_k$, so the lemma gives, for $c'$, either a copy of some $H_j$ ($1\le j\le k$) in $c'$-color $j$, which is in $c$-color $j$ because $c'=c$ off $\{0,k+1\}$, or a set $Y\subseteq\gamma_0$ with $\operatorname{otp}(Y)=\gamma_k$ in the order of $\gamma_0$ and $c[\,[Y]^2\,]\subseteq\{0,k+1\}$. In the second case let $d$ be the coloring of $[Y]^2$ with $d(p)=0$ when $c(p)=0$ and $d(p)=1$ when $c(p)=k+1$, a bijective relabeling (fact 2 of the lemma page). Fix an order isomorphism $\pi$ from $\gamma_k$ onto $Y$ with the order inherited from $\gamma_0$, and pull $d$ back to $d'$ on $[\gamma_k]^2$. The relation $\gamma_k\to(\gamma_{k+1},H_{k+1})^2$ gives $X\subseteq\gamma_k$ of order type $\gamma_{k+1}$ with $d'\equiv0$ on $[X]^2$, or a copy of $H_{k+1}$ in $d'$-color $1$. Then $\pi[X]\subseteq Y\subseteq\gamma_0$ has order type $\gamma_{k+1}$ in the order of $\gamma_0$, because $\pi$ is an order isomorphism onto $Y$ in that order, and $c\equiv0$ on $[\pi[X]]^2$; and the image of a copy of $H_{k+1}$ is a copy of $H_{k+1}$ (an order isomorphism is a bijection) with every edge of $c$-color $k+1$. Both cases give the conclusion for $k+1$ with color $j$ carrying $H_j$ for every $j\le k+1$. The base case $k=1$ is the hypothesis itself. The step is sound; the point that carries it is that $Y$'s order type and the isomorphism $\pi$ are both taken in the order of $\gamma_0$.

W2: the inductive case of (T5), where the two extracted colorings must coincide. Let $k\ge2$ and let $c$ be a witness. Put $c'(p)=0$ when $c(p)\in\{0,k\}$ and $c'(p)=c(p)$ otherwise. Case A: $c'$ has $Y\subseteq\alpha$ of order type $\alpha$ with $c'\equiv0$ on $[Y]^2$. Then $c[\,[Y]^2\,]\subseteq\{0,k\}$, so $d_k$ on $[Y]^2$ has color $0$ exactly on the pairs of $c$-color $0$ and color $1$ exactly on those of $c$-color $k$. A subset of $Y$ of order type $\alpha$ homogeneous in $d_k$-color $0$ would be homogeneous in $c$-color $0$, and a $d_k$-color-$1$ triangle would be a $c$-color-$k$ triangle; $c$ has neither. Case B: otherwise $c'$ has no subset of order type $\alpha$ homogeneous in color $0$, and a $c'$-triangle in a color $j\in\{1,\ldots,k-1\}$ would be a $c$-triangle in color $j$, so $c'$ is a witness with $k-1\ge1$ triangle targets. The induction hypothesis gives $Y$ and $j\le k-1$ such that the coloring $d'_j$ built from $c'$ witnesses. For $1\le j\le k-1$ and any pair $p$, $c'(p)=j$ if and only if $c(p)=j$, so $d'_j=d_j$ on $[Y]^2$. The base case $k=1$ takes $Y=\alpha$, $j=1$ and $d_1=c$. The pull-back clause: if $\pi$ is an order isomorphism from $\alpha$ onto $Y$ and $e(\{\xi,\eta\})=d_j(\{\pi\xi,\pi\eta\})$, then a subset of $\alpha$ of order type $\alpha$ homogeneous in $e$-color $0$ maps onto a subset of $Y$ of order type $\alpha$ homogeneous in $d_j$-color $0$, and an $e$-color-$1$ triangle maps onto a $d_j$-color-$1$ triangle; neither exists. The step is sound.

W3: the equivalence over order types that are not ordinals. The lemma page states Lemma 2.1 for an ordinal $\alpha$, and the held source does the same (p. 2). For a linear order type $\alpha$ that is not well ordered, the forward direction of S7 is obtained inside the section twice over: as the contrapositive of S6, since a witness for the $k$-color negative relation yields, after the pull-back of W2, a witness for $\alpha\not\to(\alpha,3)^2$; and as (T3) at $H_1=\cdots=H_k=3$, the constant case of W1. Neither argument uses that $\alpha$ is well ordered: the inductions are on $k$, the only sets compared are subsets of one representative of $\alpha$ in its own order, and order types of such subsets are transported by order isomorphisms. The backward direction is S3(c): a coloring with the colors $0,1$ is a coloring with the colors $0,\ldots,k$ in which $2,\ldots,k$ are unused, and a triangle in an unused color cannot occur. Sanity checks at the small types: for $\alpha=0$ and $\alpha=1$ both sides hold vacuously (the empty set and a singleton are homogeneous); for $\alpha=2$ both sides fail (color the single pair $1$); for $k=1$ the two sides coincide. The equivalence holds as stated; only its attribution is imprecise (F3).

Strongest attack

• S1. Attacked the transport of a graph target by asking whether an order isomorphism is needed at all: it is not, since a copy of a graph is carried by any bijection, and the order matters only for the order-type targets; the fact holds a fortiori. Attacked the order-type extension by looking for a hidden use of well-foundedness in (T2)--(T5): none (no induction on $\alpha$, no least element, no ordinal arithmetic). Attacked the definition by reading "all of whose pairs have color $i$" as all pairs of the copy's vertex set; under that reading a graph target is equivalent to the clique on its vertex set, and the two readings give different relations (witness in F2), so the wording matters, but the phrase "a subgraph isomorphic to $H_i$" fixes the edge reading. Refutation failed.
• S2. Attacked with a subset $Y$ of $\gamma'$ of order type $\gamma$ that is not an initial segment, and with a graph copy found inside $Y$: the argument uses only the order inherited from $\gamma'$, which $Y$ carries by definition, and a subgraph of the complete graph on $Y$ is a subgraph of the complete graph on $\gamma'$. For ordinals, $\gamma'$ has a subset of order type $\gamma$ if and only if $\gamma'\ge\gamma$, since a subset of an ordinal has order type at most that ordinal. Refutation failed.
• S3. (a) Attacked with $\beta'$ embeddable in $\beta$ but not an initial segment: the clause only needs a subset of order type $\beta'$. (b) Attacked with $H_i'$ a non-induced subgraph of $H_i$: the edges of $H_i'$ map to edges of $H_i$, so a copy of $H_i$ in color $i$ contains a copy of $H_i'$ in color $i$. (c) Attacked the unused-color clause with an edgeless target: for $E$ the one-vertex graph, $2\to(2,3,E)^2_3$ holds, because a single vertex is a copy of $E$ in color $2$ under every coloring of the one pair, while $2\not\to(2,3)^2$; so "a color may be left unused" is false for targets without a pair. The clause's stated conclusion concerns triangle targets, for which a copy in an unused color cannot exist; the section applies the clause to triangles only (F4). Refutation failed as stated.
• S4. Attacked whether the constant chain meets the hypotheses of S5: with $\gamma_0=\cdots=\gamma_k=\alpha$ the $j$-th relation is $\alpha\to(\alpha,H_j)^2$, exactly the hypothesis of S4, and the conclusion of S5 is that of S4. Refutation failed.
• S5. Attacked with a non-constant chain, checking which relation the step applies where: the induction hypothesis on the first $k$ relations returns a color-$0$ set of order type $\gamma_k$, and the $(k+1)$-st relation has $\gamma_k$ on its left, so the types match (W1). Attacked the merge order: merging $0$ with color $1$ instead would leave a color-$0$ set of order type $\gamma_{k+1}$ on which the relation with $H_1$ would be needed from $\gamma_{k+1}$, which the chain does not supply; the proof merges $0$ with the last color, which the chain does supply, so the proof is unaffected and only the closing remark needs its scope (F6). Attacked the relabeling of $\{0,k+1\}$ to $\{0,1\}$: fact 2. Refutation failed.
• S6. Attacked with $k=2$ and a witness $c$ whose merged coloring $c'$ has no color-$0$ set of order type $\alpha$: the proof returns $Y=\alpha$, $j=1$, and $d_1$ has color $0$ on the pairs of $c$-colors $0$ and $2$ together; $d_1=c'$, and $c'$ was shown to be a witness, so $d_1$ is one. Attacked case A by asking whether $d_k$'s color-$0$ class could contain pairs of a color other than $0$: not on $[Y]^2$, where $c$ takes only the values $0$ and $k$. Attacked the pull-back clause: W2. Refutation failed.
• S7. Attacked the forward direction for a non-ordinal order type through the citation of Lemma 2.1, which the lemma page and the source state for ordinals only; the direction is supplied inside the section by (T5) and by (T3) (W3), so the equivalence holds as claimed and the attribution is the only defect (F3). Attacked the quantifier "for every order type" by asking whether the relation depends on the representative of the type: it does not, by transport. Checked the small types and $k=1$ (W3). Refutation failed.
• S8. The first clause is S6 and the second is S7 read together with its negation; nothing is added. Refutation failed.

Premises

• Lemma 2.1 reconstruction page at the subject commit, read in full. Interface used: the statement (an ordinal $\alpha$ with $\alpha\to(\alpha,3)^2$ satisfies $\alpha\to(\alpha,3,\ldots,3)^2_{k+1}$ for every finite $k\ge1$); fact 1 (transport along an order isomorphism, ordinal targets); fact 2 (bijective relabeling of colors carries the list of targets). Standing: treated as established at the standing the commission states, an independent review as it stood on 2026-09-28T05:03:27Z with corrections applied since; the review itself was not read. The section uses the statement within its hypotheses for ordinals; for other order types see F3.
• Theorem 3.1 reconstruction page at the subject commit, read in full. Interface used: Steps 2 and 3 (restriction of a coloring of $[\omega_1^2]^2$ to $[\omega_1\omega]^2$ and reading the homogeneous set back in $\omega_1^2$) and the closing paragraph stating restriction for any $\alpha$ with a subset of order type $\beta$, ordinal targets. Same standing. S2 is re-proved here rather than imported.
• Gao (2026), the held deposit, cited through the lemma page. Held; read by text extraction of all four pages and on the page images of pp. 2 and 4. Lemma 2.1 (p. 2) is stated for an ordinal $\alpha$ with the two-color hypothesis, and Remark 3.2 (p. 4) restates the lemma as stability of $\alpha\to(\alpha,3)^2$ under adding finitely many colors with target $3$. Standing per its library card: unrefereed deposit, proofs followed, no independent review.
• Folder index, cited for (T3). Excluded from the read set and not read; (T3) is verified as the constant case of (T4) (F5).
• Problem statements. Problem 1171 asks $\omega_1^2\to(\omega_1\omega,3,\ldots,3)^2_{k+1}$ for all finite $k$; Problem 118 concerns a cardinal, ordinal or order type $\alpha$ with the two-color triangle relation. Used only to confirm that "the setting of Problem 118" is the order-type setting; for a cardinal read as its initial ordinal, a subset of full cardinality has full order type and conversely, so the readings agree there.

Findings

• F1 (suggested). Location: "Five facts are used below. The first two are on the reconstruction pages, the third is stated on the folder index, and the last two are recorded here with their proofs." Defect: (T1) in its mixed-target form and three of the four clauses of (T2) are not on the reconstruction pages, which carry fact 1 and the restriction principle for ordinal targets only; the sentence invites a reader to extend the reviewed standing of those pages to extensions that are this page's own. Witness: fact 1 of the lemma page quantifies over ordinals $\beta_0,\ldots,\beta_{n-1}$, and the theorem page's closing paragraph states restriction for ordinal targets; neither page states weakening of the large target, replacement of a target by one that embeds in it, or the unused color. Replacement: "The first two extend facts on the reconstruction pages (fact 1 of the lemma page and the closing restriction paragraph of the theorem page, both for ordinal targets), the third is stated on the folder index, and the last two are recorded here with their proofs."
• F2 (suggested). Location: (T1), "a copy of $H_i$ all of whose pairs have color $i$, a copy being a subset of type $H_i$ when $H_i$ is an order type and a subgraph isomorphic to $H_i$ when $H_i$ is a graph". Defect: for a graph target, "pairs" can be read as all pairs of the copy's vertex set, under which a graph target is equivalent to the clique on its vertices; the two readings define different relations. Witness: with $P_3$ the path on three vertices, $5\to(3,P_3)^2$ holds under the edge reading, since if the color-$1$ edges contain no two adjacent edges they form a matching of at most two edges and the color-$0$ graph, a $K_5$ minus a matching, contains a triangle; under the clique reading the relation is $5\to(3,3)^2$, which the two-coloring of $K_5$ by a pentagon and its complement refutes. Replacement: "a copy of $H_i$ in color $i$, a copy being a subset of type $H_i$ all of whose pairs have color $i$ when $H_i$ is an order type, and a subgraph isomorphic to $H_i$ all of whose edges have color $i$ when $H_i$ is a graph".
• F3 (suggested). Location: (T1), "Only order isomorphisms are used, so everything below also holds with the ordinals replaced by arbitrary order types", and the closing sentence "Together with Lemma 2.1 and (T2), (T5) gives, for every order type $\alpha$ and every finite $k\ge1$". Defect: "everything below" is unbounded and is verified here for (T2)--(T5) only; and Lemma 2.1 as filed on the lemma page, and in the source, is for ordinals, so for a non-ordinal order type the forward direction of the equivalence rests on (T3) or on (T5) after transport, not on the citation as written. The equivalence itself holds (W3). Witness: the lemma page's Statement begins "Let $\alpha$ be an ordinal". Replacement: scope the first sentence to "(T2)--(T5) below", and write the closing sentence as "By (T2) in one direction and (T5) after transport in the other (Lemma 2.1 when $\alpha$ is an ordinal), for every order type $\alpha$ and every finite $k\ge1$".
• F4 (note). Location: (T2), "a color may be left unused, so a relation with fewer triangle colors is weaker". The general clause needs the dropped target to have a pair (an order type of at least two elements, or a graph with an edge); with an edgeless target the copy exists vacuously and the clause fails, as the witness under S3 shows. As stated the conclusion concerns triangle targets and holds. If the clause is ever applied to a general target, add "whose target has a pair".
• F5 (note). Location: (T3), "This is the observation recorded on the folder index". The folder index is outside the read set, so the match between the index's wording and (T3) is unchecked here; (T3) is verified as the constant case of (T4), and the lemma page's Checks and scope confirm that the index states a generalization to distinct fixed targets.
• F6 (note). Location: end of the proof of (T4), "merging $0$ with the last color is the same induction with the colors relabeled". This holds for the constant chain of the lemma page. For a non-constant chain the merge order is forced: with $k=2$, relations $\gamma_0\to(\gamma_1,H_1)^2$ and $\gamma_1\to(\gamma_2,H_2)^2$, merging $0$ with color $1$ would produce a color-$0$ set of order type $\gamma_2$ on which a relation from $\gamma_2$ with target $H_1$ would be needed, and the chain supplies none. The proof merges with the last color and is unaffected; the remark should say it concerns the constant case.

Verdict

• S0: not a mathematical claim; provenance sentence corrected by F1.
• S1 (T1, definition and transport): refutation-failed.
• S2 (T2, restriction): refutation-failed.
• S3 (T2, weakening, clauses (a)--(c)): refutation-failed.
• S4 (T3, the induction with general targets): refutation-failed.
• S5 (T4, descent lemma): refutation-failed.
• S6 (T5, witness extraction): refutation-failed.
• S7 (the equivalence for every order type and every finite $k\ge1$): refutation-failed.
• S8 (the closing consequence sentence): refutation-failed.

Limitations. The folder index was not read, so (T3)'s attribution to it is verified only through the mathematics. The reconstruction pages were taken at the standing the commission states, without reading their reviews. The uses of (T1)--(T5) in the other sections of the page were not reviewed here beyond the keyword scan that located them; in particular the graph-target definition was checked for internal consistency and for agreement with the lemma page's relation, not against the literature's convention for the problems that later sections target. No required correction was found; the three suggested corrections concern provenance and wording, and the three notes record scope provisos. This focused review assigns no tier and changes no status.