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Subject and independence

The reviewer is an independent reviewer working in a fresh context from the review assignment alone. The reviewer took no part in writing the page under review, had no contact with its author, and saw no draft, note or discussion from its preparation. The charge was refutation.

Frozen subject: path wiki/research/erdos_501/glazer_lemma_4_1_reconstruction.md as it stood at 2026-09-28T05:03:27Z (the reconstruction page), read in full as of that time.

Artifact: the eight-page PDF glazer_2026_erdos_problem_501_after_adding_random_reals.pdf under the library card Glazer (2026), draft rev10; its printed page numbers coincide with the physical ones. Physical pp. 4 and 5 were read clause by clause, in the text layer and on page images rendered at 130 dots per inch: the opening paragraph of Section 4 and the statement of Lemma 4.1 (p. 4), its proof, Lemma 4.2 and the statement and the start of the proof of Proposition 4.4 (p. 5), the source's own first consumer of the lemma. Pages 1, 6, 7 and 8 were read in the text layer for the notation the page attributes to the source: display (1.1) on p. 1, the end of the proof of Proposition 4.4, the proof of Lemma 4.5 and the proof of Theorem 5.1 on p. 6, the application to $\mathcal O^{\mathbb Z}$ on p. 7, and the reference list on p. 8. Pages 2 and 3 were skimmed in the text layer for the coding space $\mathcal O$ (p. 3). No canonical conversion sits beside the PDF.

Allowed material read: the provenance paragraph of the library card; the sections "Whole-claim report" and "Audit checklist" of docs/verification.md, together with the shared "Audit checklist" of the same file; the section "Source fidelity" of docs/evidence.md; and docs/math_authoring.md in full. The page cites no reconstruction page as an input; the two pages it names as consumers were confirmed to exist as of that time by a tree listing and were not read. No problem page, no folder index, no evidence folder and no other review was read.

Exposure: the library card was read in full rather than only its provenance paragraph, so its "Read status" paragraph, the acceptance paragraph of its "Relation to E501" section and its Overview summary of Section 4 reached the reviewer. None of that text was used; every finding below rests on the PDF and the page alone.

Restatement

Fix any set $\Theta$ of coordinates. Let $\mu_\Theta$ be the completion of the product of the fair-coin measures on $2^\Theta$, and let $\mathbb B(\Theta)$ be the measure algebra of $\mu_\Theta$: measurable sets modulo null sets, a complete Boolean algebra. Forcing is Boolean valued, $\|\varphi\|\in\mathbb B(\Theta)$, and $\dot G$ names the generic point $u_G\in2^\Theta$, with $u_G(\theta)=1$ exactly when the class of $\{u:u(\theta)=1\}$ lies in the generic filter. Borel sets, standard Borel spaces and Borel maps of the ground model are reinterpreted in the extension from their codes.

The lemma, provable in ZFC: for every standard Borel space $X$ and every $\mathbb B(\Theta)$-name $\dot z$ with $\Vdash\dot z\in X$, there exist a countable $S\subseteq\Theta$ and a Borel map $F\colon2^S\to X$ of the ground model such that

$$\Vdash_{\mathbb B(\Theta)}\ \dot z=F(\dot G\restriction S).$$

The quantifiers are: for all $\Theta$, all $X$ and all such $\dot z$, there exist $S$ and $F$. $S$ is countable, possibly finite or empty; $F$ is total on $2^S$ and Borel for the product Borel structure; the equation is forced by the top condition. Nothing is claimed about uniqueness of $S$ or $F$, and no cardinal hypothesis on $\Theta$ is made. The page's added convention "$S$ reads $\dot z$ through $F$" names exactly this conclusion, and its remark that a countable $S'\supseteq S$ reads $\dot z$ through $u\mapsto F(u\restriction S)$ is a consequence, since restriction to $S$ is continuous and $(\dot G\restriction S')\restriction S$ is $\dot G\restriction S$.

Checklist

• Quantifiers and scope. Pass. The page's statement carries the source's quantifiers unchanged (p. 4): every standard Borel $X$, every name for an element of $X$, some countable $S$, some Borel $F$, the equation forced outright. Boundary cases checked: $S=\emptyset$ (then $2^S$ is a point and $F$ is constant), $N=\emptyset$ in the general case, $X$ countable or finite, and $X=\emptyset$, where the hypothesis fails and the lemma is vacuous; the proof's "fix $x_0\in X$" is available because $\|\dot z\in X\|=1\ne0$ (F6).
• Circularity. Pass. The proof consumes (R1), (R2), (R4) and (R5), none of which speaks of arbitrary names: (R1) and (R2) concern single measurable sets and single Borel sets, (R4) and (R5) concern codes and isomorphisms. No statement equivalent to the lemma is assumed.
• Model and convention changes. Pass with notes. The page specifies the product measure as the fair-coin product, identifies the generic filter with the point $u_G$, and reads the source's $\mathbb B_{\omega_2}$ as $\mathbb B(\omega_2\times\omega)$ (F4); each is the only reading consistent with the source (the opening paragraph of Section 4 on p. 4, and (5.3) and the proof of Theorem 5.1 on p. 6) and none changes the objects. The completion of the product measure on the cylinder $\sigma$-algebra and the Haar measure on the compact group $2^\Theta$ give the same measure algebra, so (R1) holds on either reading of "product measure" (see Strongest attack).
• Finite and statistical overreach. Inapplicable: no finite check, sample or heuristic appears.
• Uniformity. Inapplicable in substance: the only family is $n\mapsto(S_n,W_n)$, whose countable union is the support; no constant or error term depends on a parameter.
• Extremal conclusions. Inapplicable: no infimum, supremum or sharpness is claimed.
• Consequences and composition. Pass with one precision finding. The "hence" from coordinatewise agreement to $\Vdash\dot z=F(\dot G\restriction S)$ (Weakest step 2), the "so $N$ is $\mu_S$-null" (Weakest step 1) and the enlargement remark were each re-derived. The interfaces (R1), (R2) and (R5) are supplied at the strength used; (R4) is worded below the strength the general case draws on (F1).
• Computation. Inapplicable: the page runs no computation.
• Reproduction. Inapplicable: the page states no rerun command or coverage claim.
• Source and verdict fidelity. Pass with notes. The statement matches the source's Lemma 4.1 (p. 4) word for word in its mathematics; the locators (Section 4 opening paragraph, Lemma 4.1, physical pp. 4–5, the eight-page PDF, draft rev10) are right; the Kunen entry matches [4] on p. 8, which the source lists and never cites in its body, so the page's "lists" is exact. Two characterizations are inexact: the proof's line count (F3) and the attribution of the $\mathbb B_{\omega_2}$ convention to the source's writing (F4). The standing paragraph claims only an author-recorded reconstruction, no review and no tier.

Weakest steps

Weakest step 1: the null-set repair in the general case. Let $\iota\colon X\to X'\subseteq2^\omega$ be the Borel isomorphism of (R5), and let $S$ and $F_0$ come from the first case, so that $\Vdash\iota(\dot z)=F_0(\dot G\restriction S)$. Put $N=F_0^{-1}(2^\omega\setminus X')$, a Borel subset of $2^S$ of the ground model. Two computations of one Boolean value: by (R2), $\|\dot G\restriction S\in N\|=[N\times2^{\Theta\setminus S}]$; and in the extension $\dot G\restriction S\in N$ holds exactly when $F_0(\dot G\restriction S)\notin X'$, because the reinterpretation of $N$ is the preimage of the reinterpreted complement of $X'$ under the reinterpreted $F_0$, so that

$$\|\dot G\restriction S\in N\|=\|\iota(\dot z)\notin X'\|=0,$$

the last because $\Vdash\dot z\in X$ and the range of the reinterpreted $\iota$ lies in the reinterpreted $X'$. Hence $\mu_\Theta(N\times2^{\Theta\setminus S})=0$, that is $\mu_S(N)=0$, and by (R2) again $\Vdash\dot G\restriction S\notin N$. The map $F$, defined as $\iota^{-1}\circ F_0$ off $N$ and as $x_0$ on $N$, is Borel: for Borel $A\subseteq X$, $F^{-1}(A)$ is $F_0^{-1}(\iota[A])\setminus N$ joined with $N$ or with nothing, and $\iota[A]=(\iota^{-1})^{-1}(A)$ is Borel. In the extension

$$F(\dot G\restriction S)=\iota^{-1}(F_0(\dot G\restriction S)) =\iota^{-1}(\iota(\dot z))=\dot z.$$

The three facts about reinterpretation used here, the preimage identity, the range inclusion and $\iota^{-1}\circ\iota=\mathrm{id}_X$, are universal statements about all points of the extension; they hold because each is a $\Pi^1_1$ statement about Borel codes true in the ground model (F1). The step composes with the first case by consuming its $S$ and $F_0$ and returns the lemma for $X$.

Weakest step 2: coordinatewise reading in the case $X=2^\omega$. For each $n$, $b_n=\|\dot z(n)=1\|$ has, by (R1), a representative $W_n\times2^{\Theta\setminus S_n}$ with $S_n$ countable and $W_n\subseteq2^{S_n}$ Borel. With $S=\bigcup_nS_n$ and $F(u)(n)=1$ if and only if $u\restriction S_n\in W_n$, each coordinate of $F$ is the indicator of the Borel set $W_n\times2^{S\setminus S_n}$, and a map into $2^\omega$ is Borel when its coordinates are, since the coordinate cylinders generate the Borel sets of $2^\omega$. In the extension $F(\dot G\restriction S)(n)=1$ holds exactly when $\dot G\restriction S_n\in W_n$, so by (R2)

$$\|F(\dot G\restriction S)(n)=1\|=[W_n\times2^{\Theta\setminus S_n}]=b_n, \qquad \|\dot z(n)=F(\dot G\restriction S)(n)\| =(b_n\wedge b_n)\vee(-b_n\wedge-b_n)=1.$$

Since $\|\forall n\in\check\omega\,\varphi(n)\|$ is $\bigwedge_n\|\varphi(\check n)\|$, the top condition forces agreement at every $n$, and two points of $2^\omega$ agreeing at every $n$ are equal. This is the whole first case; the general case consumes it for the name $\iota(\dot z)$.

Weakest step 3: the import (R2) itself. Fix countable $S$ and define $\Phi(W)=\|\dot G\restriction S\in W\|$ and $\Psi(W)=[W\times2^{\Theta\setminus S}]$ on Borel $W\subseteq2^S$. For a cylinder $C=\{u:u\restriction E=s\}$ with $E\subseteq S$ finite, $\Phi(C)=\bigwedge_{\theta\in E}\|\dot G(\theta)=s(\theta)\|$; by the definition of $\dot G$ and the identity $\|\check a\in\dot\Gamma\|=a$ for the canonical name $\dot\Gamma$ of the generic filter, $\|\dot G(\theta)=1\|$ is the class of $\{u:u(\theta)=1\}$ and $\|\dot G(\theta)=0\|$ its complement, so $\Phi(C)=\Psi(C)$. Both maps send complements to complements and countable unions to suprema: $\Psi$ because $W\mapsto W\times2^{\Theta\setminus S}$ is a $\sigma$-homomorphism into the measurable sets, $\Phi$ because the reinterpretation of a coded complement or countable union is the complement or union of the reinterpretations and $\|\exists k\in\check\omega\,\varphi(k)\|$ is $\bigvee_k\|\varphi(\check k)\|$. The sets on which $\Phi=\Psi$ therefore form a $\sigma$-algebra containing the cylinders, which generate the Borel sets of $2^S$. So (R2) holds, and its "in particular" follows from $\mu_\Theta(W\times2^{\Theta\setminus S})=\mu_S(W)$. (R2) enters at both uses in Weakest steps 1 and 2.

Strongest attack

The strongest attack aimed at the general case: break the reduction by showing that the reinterpreted $\iota$ need not remain a bijection of the reinterpreted $X$ onto the reinterpreted $X'$, or that the reinterpreted $N$ need not be the preimage of $2^\omega\setminus X'$, so that $\dot G\restriction S$ could land in the reinterpreted $N$ with positive Boolean value, or $\iota^{-1}(\iota(\dot z))$ could differ from $\dot z$. The attack fails: "$\iota$ is injective", "the range of $\iota$ lies in $X'$", "$\iota^{-1}\circ\iota$ is the identity on $X$" and "$N=F_0^{-1}(2^\omega\setminus X')$" are each a universal statement over points with a Borel matrix in the codes, that is $\Pi^1_1$, true in the ground model, and $\Pi^1_1$ statements about codes of the ground model hold in the extension. What survives of the attack is that the page's (R4) does not say this, while its proof cites (R4) as though it did (F1).

Three further attacks were tried. A counterexample name that no countable $S$ reads: impossible, because a name for a point of $2^\omega$ is determined by the countably many values $b_n$, each countably supported by (R1), and every other $X$ reduces to $2^\omega$ through (R5). The $\sigma$-algebra on $2^\Theta$ for uncountable $\Theta$: the page's $\mu_\Theta$ is the completion of the product measure on the cylinder $\sigma$-algebra, matching the source's words on p. 4, and every set of that $\sigma$-algebra depends on countably many coordinates; if one reads "product measure" as the Haar measure on the compact group $2^\Theta$, every compact set lies inside a compact $G_\delta$ of arbitrarily close measure by inner regularity, so every Borel set is almost equal to a set of the cylinder $\sigma$-algebra and the measure algebra is the same, and (R1) holds either way. Boundary cases $S=\emptyset$, $N=\emptyset$, $X$ countable and $X=\emptyset$: the argument goes through or is vacuous (F6).

Premises

• Source Lemma 4.1. Held: the PDF under the library card, physical p. 4 (statement) and p. 5 (proof), read clause by clause with page images. Interface: exactly the Restatement above. Standing on the page: the reconstructed subject, author-recorded.
• (R1) Countable supports. Interface: $\mathbb B(\Theta)$ is a complete Boolean algebra with the countable chain condition, and every $\mu_\Theta$-measurable set is almost equal to $W\times2^{\Theta\setminus S}$ with $S$ countable and $W\subseteq2^S$ Borel. Source named by the page: Kunen's handbook chapter, the source's [4], not held in the read set; the fact was checked from the definition of the product $\sigma$-algebra (Strongest attack). Named as imported: yes. The countable chain condition is not used on the page.
• (R2) The generic point. Interface: for countable $S$ and Borel $W\subseteq2^S$ of the ground model, $\|\dot G\restriction S\in W\|=[W\times2^{\Theta\setminus S}]$, with the two consequences stated on the page. Source named by the page: Kunen's chapter and Jech, Chapters 14–15, not held; re-derived in Weakest step 3 from the cylinder case. Named as imported: yes.
• (R3) Forcing theorem and maximum principle. Interface: as stated on the page. Source: Jech, Chapter 14, not held. Named as imported: yes. Not cited in the proof; needed only to treat $\iota(\dot z)$ as a name (F6).
• (R4) Absoluteness. Interface as worded on the page: reinterpretation from codes, and absoluteness of Borel statements about points. Interface actually consumed: absoluteness of $\Pi^1_1$ statements about codes (F1). Source named by the page: Jech, Chapters 14–15, and Kechris; not held. From memory, the consumed fact is Chapter 25 of Jech (absoluteness for transitive models and the Borel-code lemmas); the book is outside the read set, so the chapter is unverified here. Named as imported: yes.
• (R5) Borel isomorphism. Interface: every standard Borel space is Borel isomorphic to a Borel subset of $2^\omega$. Source: Kechris (1995), the Borel isomorphism theorem, not held; standard. Named as imported: yes.
• Explicit assumptions of the page. The source's "product measure" is the fair-coin product; the generic filter is identified with the point $u_G$; the source's $\mathbb B_{\omega_2}$ is $\mathbb B(\omega_2\times\omega)$ (F4). No local claim of the repository is consumed, and there is no batch acceptance order.

Findings

F1. Severity: suggested. Location: "(R4) Absoluteness." and, in the proof, "by (R2) and (R4)" and "since $\Vdash\dot G\restriction S\notin N$". Defect: (R4) promises reinterpretation from codes and absoluteness of "Borel statements about points", which covers membership of ground-model points in coded sets; the general case uses three universal statements about all points of the extension, namely that the reinterpreted $N$ is $F_0^{-1}(2^\omega\setminus X')$, that the reinterpreted $\iota$ maps the reinterpreted $X$ into the reinterpreted $X'$, and that $\iota^{-1}\circ\iota$ is the identity there; the first case uses that the reinterpreted $F$ is the map built from the reinterpreted $W_n$. These are the absoluteness of $\Pi^1_1$ statements about Borel codes, the same fact that makes "reinterpreted from the same codes" independent of the code chosen, and the page's citations for its forcing facts (Jech, Chapters 14–15) and its descriptive set theory (Kechris) do not name it; from memory it is Chapter 25 of Jech, which is outside the read set. Witness: source p. 5 says only "modify the reading on the null set where the bitwise value falls outside that subset", so the absoluteness burden is the page's own. Proposed replacement for the second clause of (R4): "inclusions and identities between Borel sets given by codes in $M$ hold in $M[G]$ when they hold in $M$, and so do the statements that a coded Borel map is injective, carries a coded set into a coded set, or is inverse to another coded map (absoluteness of $\Pi^1_1$ statements for transitive models, Jech, Chapter 25)". In the proof, cite (R4) at the three places named and at "so $\Vdash\dot z(n)=F(\dot G\restriction S)(n)$".

F2. Severity: suggested. Location: Statement, "We say that such an $S$ reads $\dot z$ through $F$. If $S$ reads ... because $(\dot G\restriction S')\restriction S=\dot G\restriction S$." Defect: a supplied definition and a supplied remark stand inside the Statement section without a label. Witness: the source's Lemma 4.1 (p. 4) contains neither; the source's word is "support" (p. 4, opening paragraph of Section 4), and enlargement appears only in the proof of Proposition 4.4 (p. 5, "choose a countable support $S_\alpha$ for $\dot w_\alpha$ and enlarge it"). Proposed replacement: open the passage with "Supplied terminology and remark. The source says that $S$ supports $\dot z$ (p. 4) and enlarges supports in the proof of Proposition 4.4 (p. 5); this page says that $S$ reads $\dot z$ through $F$ ..." and keep the rest.

F3. Severity: note. Location: Source paragraph, "The source gives a six-line proof". Defect: the proof occupies five typeset lines, in four sentences. Witness: physical p. 5, the paragraph from "Proof. It is enough to treat $X=2^\omega$" to the end-of-proof mark. Proposed replacement: "The source gives a five-line proof".

F4. Severity: note. Location: Conventions, "The source writes $\mathbb B_{\omega_2}$ for $\mathbb B(\omega_2\times\omega)$." Defect: the source displays $\mathbb B_{\omega_2}$ in (1.1) on p. 1 and (5.1) on p. 6 and never defines it; the identification is the reader's inference from the proof of Theorem 5.1. Witness: p. 6, "Put $\kappa=(\omega_2)^M$, $\Theta=\kappa\times\omega$, and $\mathbb B=\mathbb B(\Theta)$". Proposed replacement: "The source's $\mathbb B_{\omega_2}$ of (1.1) and (5.1) is read here as $\mathbb B(\omega_2\times\omega)$, the algebra $\mathbb B(\Theta)$ with $\Theta=\kappa\times\omega$ and $\kappa=(\omega_2)^M$ that its proof of Theorem 5.1 (p. 6) sets up."

F5. Severity: note. Location: Boundary, "applied in Proposition 4.4 to names for elements of $\mathcal O^{\mathbb Z}$". Defect: in the source, Proposition 4.4 applies Lemma 4.1 to names for elements of an arbitrary standard Borel space $X$; $X=\mathcal O^{\mathbb Z}$ is the instance that Theorem 5.1 feeds to it. Witness: p. 5, the statement of Proposition 4.4 ("let $X$ be a standard Borel space, and for each $\alpha<\kappa$ let $\dot w_\alpha$ be a name for an element of $X$"); p. 7, the bundling of the codes into $\dot w_\alpha\in\mathcal O^{\mathbb Z}$ followed by "Apply theorem 4.4." The linked reconstruction page is outside this review's read set, so whether it specializes to $\mathcal O^{\mathbb Z}$ was not checked. Proposed replacement: "applied in Proposition 4.4 to names for elements of a standard Borel space $X$, instantiated at $X=\mathcal O^{\mathbb Z}$ in Theorem 5.1, and in Lemma 4.5 to a name for a Borel code."

F6. Severity: note. Location: General $X$, "Then $\iota(\dot z)$ is a name for an element of $2^\omega$" and "Fix $x_0\in X$". Defect: two unstated small steps. $\iota(\dot z)$ is a term, not a name; a name $\dot y$ with $\Vdash\dot y=\iota(\dot z)$ comes from the maximum principle (R3), which the page lists and never cites. $X\ne\emptyset$ is needed for $x_0$ and follows from the hypothesis, since $\|\dot z\in X\|=1\ne0$ while $\|\dot z\in\emptyset\|=0$ in the nontrivial algebra $\mathbb B(\Theta)$. Witness: the page's own text; the source (p. 5) is silent on both. Proposed replacement: "By (R3) fix a name $\dot y$ with $\Vdash\dot y=\iota(\dot z)$; it is a name for an element of $2^\omega$" and "Fix $x_0\in X$, which is nonempty because $\Vdash\dot z\in X$".

Verdict

Source fidelity: faithful. The statement, its hypotheses, quantifiers and conclusion, and the locators match the source's Lemma 4.1 on physical pp. 4–5 of the held PDF; the two inexact characterizations (F3, F4) touch neither the statement nor a locator.

The argument as reconstructed: sound. Both cases were re-derived (Weakest steps 1 and 2) and the imported identity (R2) was re-derived from the cylinder case (Weakest step 3). No required correction. Two suggested corrections: the wording and citation of the import (R4), which the general case uses above its stated strength (F1), and the unlabeled supplied remark in the Statement section (F2). Four notes (F3–F6).

Limitations: the books named for (R1)–(R5) are outside the read set, so the imports were checked by derivation and from memory, not against held text; the two consumer pages named in the Boundary paragraph were not read, so F5 is stated against the source alone; the reviewer's exposure to the library card beyond its provenance paragraph is disclosed above.

This focused review assigns no tier and changes no status.