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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 page, the folder's reconstructions or their reviews, and read no existing review or grade body.

Subject. Page wiki/research/erdos_354/connections.md as it stood at 2026-09-28T13:50:48Z, section "## Problem 254 and the threshold $M_2^*$", lines 518--579 (the H2 at line 518 to the line before the next H2 at line 580), read in the committed bytes. No separate lead page was named; the section is the lead. The section contains no new proof: it composes reconstructed results and source statements, so the review checks each composition, each quantifier and each characterization of a source.

Reading and depth. The section clause by clause. The page's preamble (lines 1--58: purpose and standing, notation, the definition of (1.5) and of dyadic rationals) in full. As inputs whose proofs another reviewer checks: the first section's Lemma 354.1, Proposition 354.2, Corollary 354.3 and the statement of Lemma 354.4 (lines 59--200), for the interface of Corollary 354.3; the second section's Target, Proposition 354.5 and conditional deduction 354.6 (lines 292--360), for the premise (P1); the Problem 124 section and the closing section (lines 580--632), for the shared description of the descent's inputs. Reconstruction pages, treated as established at their reviewed standing: Remark 4.1 at base two and Remark 4.2 in full; the normalization page in full, for item 6; the theorem page through its four proof steps, the dependency map not read. Problem pages: the statements of Problem 254 and Problem 354. Library: the Fan card and its Remark 4.2 page; the Yu--Chen theorem page. Held PDFs, read in their text layers: Fan v5, pp. 3--4 (Theorem 1.1, Corollary 1.2, (1.8), (1.9), the paragraph on Hegyvári's results), the opening of Section 2 (syndetic and thick sets, $\Delta(S)$, Lemma 2.1, Theorem 2.2), the opening of Section 3 and Lemma 3.3, the opening of Section 4 with the lemma citations inside the proof of Theorem 1.1, Remarks 4.1 and 4.2, Propositions 4.2 and 4.3 with the paragraph between them, the statement of Corollary 6.3 and the reference list; the Yu--Chen manuscript, p. 1 (the theorem and the history paragraph) and its reference list. Guidance: the named sections of docs/verification.md, docs/approach_vetting.md, docs/evidence.md "Source fidelity" and docs/math_authoring.md.

Exposures. Five, none used for a verdict.

1. The Problem 254 page's Current assessment, Progress and Known Results sections were displayed together with its statement in a single read of the page's first eighty lines; the Progress section summarizes the section under review.
2. The Problem 354 page's whole block before its Current assessment was displayed: Formulation, Status (acceptance text about the site-accepted Lean proof and the other 2026 sources), provenance of the proof file, Source, References and Formalization. Only the statement and the three reference entries [He89], [JiMa24] and [FaHe25], marked not held and site-attested, were used.
3. The Fan card, its Remark 4.2 page and the Yu--Chen theorem page were read whole, including their Bears-on standing paragraphs.
4. A listing of the folder's evidence/verify/ directory (file names) and the twelve frontmatter lines of the normalization reconstruction's review, read to learn the record format; that frontmatter's description states the summary verdict of that other review.
5. Corpus-wide searches, file names only, for the word "syndetic" and for citations of Burr and Erdős, made to test statement S4's claim about what the corpus holds.

Restatement

Throughout, $\mathbb N=\{1,2,\ldots\}$ for sets $A\subseteq\mathbb N$ as in Fan, while the rays are indexed from exponent $0$ in every convention used: $A_{\alpha,\beta}=\{\lfloor2^n\alpha\rfloor,\lfloor2^n\beta\rfloor:n\ge0\}\setminus\{0\}$. A set is complete if every sufficiently large integer is a sum of distinct elements, strongly complete if every finite deletion leaves it complete; (1.5) is $\sum_{a\in A}\|a\theta\|=\infty$ for every $\theta\in\mathbb R\setminus\mathbb Z$; $M_2^*$ is the least positive integer $M$ such that every $A\subseteq\mathbb N$ with (1.5) and $|A\cap(2^k,2^{k+1}]|\ge M$ for all sufficiently large $k$ is strongly complete; a dyadic rational is a nonzero integer times a power of $2$.

• S1 (Target; descriptive). Problem 254 asks whether every $A\subseteq\mathbb N$ with $|A\cap(x,2x]|\to\infty$ as $x\to\infty$ and $\sum_{n\in A}\|\theta n\|=\infty$ for every $\theta\in(0,1)$ is complete. The Fan card records Corollary 1.2 (every $A$ with (1.5) and at least five elements in $(2^k,2^{k+1}]$ for all large $k$ is strongly complete) as the preprint's confirmation in strong form, the proof of Theorem 1.1 unread; the folder reconstructs Remarks 4.1 and 4.2, which consume Corollary 1.2 as a statement.
• S2 (Sharpness). (a) The set $A=\{2^k+1:k\ge1\}$ satisfies (1.5), has exactly one element in $(2^k,2^{k+1}]$ for every $k\ge1$, and is not complete (premise: Remark 4.1 page). (b) Hence the statement obtained from Problem 254 by replacing $|A\cap(x,2x]|\to\infty$ with "at least one element in $(2^k,2^{k+1}]$ for all large $k$", keeping the divergence hypothesis and the conclusion, is false; Corollary 1.2 asserts the version with "at least five" and the conclusion strengthened to strong completeness. (c) $M_2^*\in\{2,3,4,5\}$, and its exact value is open in the source.
• S3 (Two-per-interval family). (a) For all $\alpha,\beta>0$ with $\alpha/\beta$ not a power of $2$ and at least one of $\alpha,\beta$ not a dyadic rational, $A_{\alpha,\beta}$ satisfies (1.5) and has at least two elements in $(2^k,2^{k+1}]$ for all large $k$ (premise: Remark 4.2 page, Steps 2--3). (b) When $\alpha/\beta$ is irrational these sets are strongly complete on the Yu--Chen theorem, if it stands; when $\alpha/\beta$ is rational and not a power of $2$, on each such line $A_{\alpha,\beta}$ is strongly complete for Lebesgue-almost every coefficient (Corollary 354.3, author-recorded). (c) The family lies outside the hypothesis of Corollary 1.2 and is consistent with $M_2^*=2$, the value that Remark 4.2 shows would imply Hegyvári's conjecture. (d) Fan's Proposition 4.3 says that Theorem 2.2 cannot yield $M_2^*\le4$ and that a different mechanism may be needed; the descent is a different mechanism but consumes the digit recurrence of two rays, not a per-interval count, so it says nothing about a general set with two elements per dyadic interval.
• S4 (Burr--Erdős instance). Item 6 of the normalization page, the bound $\operatorname{gap}(P_n)\le N$ derived from $c_{j+1}-\sum_{i\le j}c_i\le c_0$, is the two-ray instance of the Burr--Erdős lemma that Fan records as Lemma 2.1 (for infinite $S\subseteq\mathbb N$, $\Delta(S)<\infty$ makes $\operatorname{FS}(S)$ syndetic); the corpus holds neither the Burr--Erdős paper nor a general proof.
• S5 (What it does not give; obstruction). The folder holds no reconstruction of Theorem 1.1 or of Theorem 2.2, and the descent does not extend to sets without the digit structure.
• S6 (Next investigation). (a) The compilation task for Problem 254 is a reconstruction of Fan's Theorem 2.2 and Section 4, which would also fix the standing of premise (P1). (b) The corpus holds no candidate for an incomplete set with (1.5) and two elements in every large dyadic interval. (c) Every incomplete two-ray set known (Hegyvári 1989, Jiang--Ma, Fang--He, site-attested) has a ratio that is a power of $2$ and is therefore a single ray as a set.
• S7 (Outcome; standing). The section transfers the sharpness example and the family as author-recorded context for $M_2^*$ and fails as a route to Problem 254.

Checklist

• Quantifiers and scope. One failure: S6(c) quantifies over "every incomplete two-ray set known" without the hypothesis that one coefficient is not a dyadic rational, and the pair $(2,6)$ refutes the literal clause (F3). S3(b)'s "each rational-ratio line" is read within the family (ratio not a power of $2$), which the sentence's "these sets" supplies; outside it the phrase would be false (F5). S2(b)'s weakening is a genuine weakening: $|A\cap(x,2x]|\to\infty$ at $x=2^k$ gives any fixed count for large $k$. The problem's $\theta\in(0,1)$ and Fan's $\theta\in\mathbb R\setminus\mathbb Z$ agree by periodicity.
• Circularity. None. The family's strong completeness in S3 comes from the Yu--Chen theorem and Corollary 354.3, not from $M_2^*=2$; the lower bound $M_2^*\ge2$ comes from a witness, not from Corollary 1.2.
• Model and convention changes. None found. Ray indices start at $0$ in the preamble, in Fan's (1.9) and in the normalization page; strong completeness (deletion of a finite $B\subseteq A$ in Fan, of a finite $F\subseteq\mathbb Z$ on the Yu--Chen pages) is the same property; Problem 354's multiset reading is not used by the section, which speaks of sets throughout, and "a single ray as a set" in S6(c) names the convention it uses.
• Finite and statistical overreach. None. The almost-every statement of Corollary 354.3 is used as almost-every; no finite check is cited as coverage.
• Uniformity. Not at issue. No constant in the section is claimed uniform over a family; the exceptional null set in S3(b) may depend on the line, and the section claims nothing across lines.
• Extremal conclusions. S2(c) is checked in the claim's own unit, the count per half-open dyadic interval: the witness has count exactly $1$ for every $k\ge1$, so $M=1$ fails and $M_2^*\ge2$. The upper bound and the existence of $M_2^*$ rest on the preprint's Corollary 1.2, which the sentence does not mark (F1).
• Consequences and composition. Each "so" and "hence" was attacked separately: S2(b) from S2(a) holds; S2(c)'s lower half holds; S3(c) holds since the family has exactly two elements per large interval; S6(c)'s "hence a single ray as a set" holds. S6(a) composes wrongly: the proof of Theorem 1.1 consumes Lemmas 3.1--3.3 of Section 3, so a reconstruction of Theorem 2.2 and Section 4 alone would not fix (P1)'s standing (F2).
• Computation. Inapplicable: the section computes nothing; the constants $u_2=2$, $v_2=3$, $M_2=5$ used on the Remark 4.2 page were recomputed from (1.6) and agree.
• Reproduction. Inapplicable: the section states no rerun command and claims no evidence coverage.
• Source and verdict fidelity. Fan's Lemma 2.1, Proposition 4.3 and the paragraph before it, Remark 4.1's openness sentence and the card's read-status sentence are characterized faithfully. The attribution of non-completeness results to Hegyvári 1989 cannot be checked from held sources, and the held Yu--Chen manuscript attributes them to Jiang--Ma and Fang--He only (F4). S6(a)'s scope of Fan's proof is misstated (F2).

Weakest steps

W1: from the witness to the threshold (S2). Let $A=\{2^k+1:k\ge1\}$. Membership: $2^k<2^k+1\le2^{k+1}$ for $k\ge1$, and $2^j+1$ with $j\ne k$ lies in a different interval, so the count in $(2^k,2^{k+1}]$ is exactly $1$. Divergence: if $\sum_{a\in A}\|a\theta\|<\infty$ then $\|(2^k+1)\theta\|\to0$, and $\|\theta\|=\|2(2^k+1)\theta-(2^{k+1}+1)\theta\|\le2\|(2^k+1)\theta\|+\|(2^{k+1}+1)\theta\|\to0$, so $\theta\in\mathbb Z$. Incompleteness: the elements at most $2^{k+1}$ are $2^j+1$ for $1\le j\le k$, whose nonempty subset sums number at most $2^k-1$, leaving at least $2^k+1$ integers of $[1,2^{k+1}]$ outside $\operatorname{FS}(A)$, unboundedly many. So the threshold definition fails at $M=1$: $A$ has (1.5) and one element in every large interval but is not even complete, hence $M_2^*\ne1$, and since larger $M$ only shrink the set of $A$ considered, $M_2^*\ge2$ whenever $M_2^*$ exists. Existence and the bound $5$ are exactly Corollary 1.2. Problem 254's growth hypothesis implies any fixed count in every large dyadic interval, so replacing it by "at least one" is a weakening, and the witness refutes the weakened statement: S2(b) holds.

W2: the family and its two covering results (S3). Rescale to $\alpha'=2^{-s}\alpha$, $\beta'=2^{-t}\beta$ in $(1/2,1]$. For $m\ge1$, $2^m\alpha'\in(2^{m-1},2^m]$, so $\lfloor2^m\alpha'\rfloor\in[2^{m-1},2^m]$, and it lies in $(2^k,2^{k+1}]$ only for $m=k+1$ (value $>2^k$ once $\alpha'\ge1/2+2^{-k-1}$, true for large $k$ as $\alpha'>1/2$) or for $m=k+2$ with value exactly $2^{k+1}$, excluded for large $k$ by the same inequality. So each ray contributes exactly one element to every large interval, and the two differ for large $k$ because $2^{k+1}|\alpha'-\beta'|\ge2$: the count is exactly $2$, which is why the family is outside Corollary 1.2's hypothesis (S3(c)). Divergence follows from the digit argument of the Remark 4.2 page, which needs only that one coefficient is not a dyadic rational. Coverage: an irrational ratio is not a power of $2$, and two dyadic rationals have a rational ratio, so every irrational-ratio pair is in the family and the Yu--Chen theorem's hypothesis is exactly "irrational ratio"; on a line with rational ratio not a power of $2$, Corollary 354.3's hypotheses (the parameter $\gamma$ not dyadic and its expansion containing $0^{L(p,q)}$ infinitely often) hold for almost every $\gamma$, and both coefficients $p\gamma,q\gamma$ are then non-dyadic, so the covered sets are in the family. "Consistent with $M_2^*=2$" claims only that no covered member is a counterexample.

W3: the Burr--Erdős correspondence (S4) and the single-ray count (S6(c)). Fan defines $\Delta(S)=\sup_{s\in S}(s-\sum_{t\in S,t<s}t)$. For a normalized pair the merged list $c_0<c_1<\cdots$ has $c_{j+1}\le2c_j$ (item 4), so $c_{j+1}-\sum_{i\le j}c_i=(c_{j+1}-2c_j)+(c_j-\sum_{i<j}c_i)\le c_j-\sum_{i<j}c_i\le\cdots\le c_0=N$, and the term at $c_0$ is $N$ itself: $\Delta(S)=N<\infty$ with $S$ infinite, so Lemma 2.1 applies and item 6 is its quantitative two-ray instance. For a single ray $c_n=\lfloor2^n\alpha'\rfloor$ with $\alpha'\in(1/2,1]$ and $d_n=c_{n+1}-2c_n\in\{0,1\}$, induction gives $c_{n+1}-\sum_{i\le n}c_i=c_0+\sum_{i\le n}d_i$, which is unbounded when $\alpha'$ is not a dyadic rational; the integers strictly between $\sum_{i\le n}c_i$ and $c_{n+1}$ are then never subset sums, so such a ray is incomplete with unbounded gaps, while it satisfies (1.5) by the same digit argument and has one element per large interval. A two-ray set with ratio $2^j$ ($j\ge0$) equals $\{\lfloor2^n\beta\rfloor:n\ge0\}\setminus\{0\}$ for the smaller coefficient $\beta$, since $\lfloor2^n\alpha\rfloor=\lfloor2^{n+j}\beta\rfloor$: a single ray as a set, hence one element per large interval and no candidate for S6(b).

Strongest attack

• S1. Compared clause by clause with the problem statement (the displayed growth condition is $|A\cap(x,2x]|\to\infty$, the distance function is the nearest-integer distance) and with the card's read status ("Theorem 1.1 and the rest of the paper were read at statement level only"). No discrepancy.
• S2. Attempted to break the witness (a wrong count at the interval's endpoints, a convergent distance sum, or completeness after all): each property re-derived in W1. Attempted to show the weakening is not a weakening: it is. Attempted to obtain $M_2^*\le5$ from anything in the corpus other than the preprint's statement: nothing exists, so the unconditional sentence overstates what is author-recorded (F1); the mathematics of S2(a)--(b) and the lower bound stand.
• S3. Attempted to find a family member with five elements per interval (inside Corollary 1.2's reach): impossible, the count is exactly two. Attempted to find a covered member outside the family: the Yu--Chen hypothesis and Corollary 354.3's almost-every set both lie inside it (W2). Attempted to read "each rational-ratio line" as covering ratios that are powers of $2$: there the set is a single ray, incomplete for almost every coefficient, so the phrase is true only within the family, which the sentence names (F5). Checked Fan's own words for S3(d): "there is no hope of deducing $M_2^*\le4$ from Theorem 2.2, suggesting that a genuinely different mechanism for strong completeness may be required", and Proposition 4.3 exhibits a strongly complete set with exactly four elements per interval for $k\ge3$ admitting no partition of the Theorem 2.2 shape; the section's paraphrase does not strengthen this.
• S4. Attempted a mismatch between Fan's $\Delta$ and the display: the sup is attained at $c_0$ and equals $N$ (W3). Attempted to find the lemma proved in the corpus: Fan records it with citations only, no corpus page proves it, and no card for the 1981 paper exists (F6 for the limit of this check).
• S5. The folder listing contains no reconstruction of Theorem 1.1 or Theorem 2.2. The obstruction is stated at its actual scope: the descent's inputs (interlacing, digit recurrence, reduced ratio at a block) are absent for a general set, and the section claims silence, not impossibility.
• S6. S6(a): the proof of Theorem 1.1 in Section 4 cites Lemma 3.1 (countable spectra), Lemma 3.2 (the selection lemma) and Lemma 3.3 (the bound on $\Delta$), so the named reconstruction would leave (P1) resting on unreconstructed lemmas (F2). S6(b): the only sets in the read set with (1.5) and few elements per interval are the witness of S2, the single rays and Fan's Proposition 4.3 set (strongly complete): no candidate. S6(c): the pair $(\alpha,\beta)=(2,6)$ gives $A_{2,6}=\{2^{n+1}\}\cup\{3\cdot2^{n+1}\}$, a two-ray set whose ratio $1/3$ is not a power of $2$ and whose elements are all even, so it is incomplete; the literal universal clause is refuted. The witness fails (1.5) at $\theta=1/2$, so S6(b) is untouched (F3).
• S7. The standing sentence claims author-recorded context only, which is what the section provides.

Premises

• Remark 4.1 at base two (reconstruction page; reviewed at its stated standing). Interface used: the three properties of $\{2^k+1:k\ge1\}$ and $M_2^*\ge2$; its $M_2^*\le5$ is explicitly attributed there to Corollary 1.2. Read in full.
• Remark 4.2 (reconstruction page; reviewed). Interface used: Steps 2--3 (two elements per large interval and (1.5)) for every pair with ratio not a power of $2$ and one non-dyadic coefficient; the implication $M_2^*=2$ implies Hegyvári's conjecture; the statements of Theorem 1.1 and Corollary 1.2 with $u_2=2$, $v_2=3$, $M_2=5$. Read in full.
• Normalization page (reconstruction; reviewed). Interface used: items 4 and 6 and their proofs, valid for every normalized pair. Read in full.
• Yu--Chen theorem (reconstruction page and library theorem page). Interface: $\alpha,\beta>0$, $\alpha/\beta$ irrational, then strongly complete; an unrefereed manuscript claim, author-recorded reconstruction. Statement and proof steps read; the section's "if it stands" matches.
• Corollary 354.3 (first section of the page; author-recorded; its proof is another reviewer's subject). Interface: rational ratio not a power of $2$, $\gamma=\beta/q$ not dyadic with infinitely many runs of $L(p,q)$ zeros, then strongly complete; in particular almost every $\gamma$ on each such line. Statement and proof read for the interface only.
• Premise (P1) (second section): Fan's Theorem 1.1 as a preprint claim whose proof, Sections 2--4, is not reconstructed. Read at statement.
• Fan (2026), arXiv:2607.14071v5, held; text layer read at the pages listed above. Used: (1.5), (1.8), (1.9), Theorem 1.1, Corollary 1.2, the definition of $\Delta(S)$, Lemma 2.1 (recorded without proof, with citations to Burr--Erdős [8, Lemma 3.2] and Bergelson--Simmons [1, Lemma 2.11]), Theorem 2.2, Lemma 3.3, the lemma citations in the proof of Theorem 1.1, Remark 4.1, Remark 4.2, Propositions 4.2 and 4.3.
• Yu and Chen (2026), manuscript, held; text layer read at p. 1 and the references. Used: the theorem's statement and the history sentence attributing non-completeness results for ratios that are powers of two to Jiang--Ma and Fang--He.
• Burr and Erdős (1981), Hegyvári (1989), Jiang and Ma (2024), Fang and He (2025): not held; the last three are site-attested through the Problem 354 page's reference list. Nothing about their contents was verified.
• Problem 254 and Problem 354: statements read.

Findings

• F1 (suggested). Location: "the exact threshold $M_2^*$ lies in $\{2,3,4,5\}$ and is open in the source" (lines 538--539), and the Outcome's "context for $M_2^*$". Defect: the upper bound and the existence of $M_2^*$ rest on the preprint's Corollary 1.2, whose proof no corpus page has read, while the sentence reads unconditionally; the page's second section labels the same theorem premise (P1). Witness: the Target paragraph itself ("the proof of Theorem 1.1 unread"). Replacement: "on the preprint's Corollary 1.2 the exact threshold $M_2^*$ lies in $\{2,3,4,5\}$; only $M_2^*\ge2$ is reconstructed, and the value is open in the source".
• F2 (suggested). Location: "a reconstruction of Fan's Theorem 2.2 and Section 4, which would also fix the standing of premise (P1) above" (lines 569--570). Defect: the proof of Theorem 1.1 in Section 4 consumes Lemmas 3.1, 3.2 and 3.3 of Section 3, so the named reconstruction would not fix (P1)'s standing. Witness: the citations of Lemma 3.1, Lemma 3.2 and Lemma 3.3 inside the proof of Theorem 1.1 (v5, pp. 17--19), and the Remark 4.2 page's own scope, "Sections 2--4". Replacement: "a reconstruction of Fan's Theorem 2.2 with Sections 3--4".
• F3 (required). Location: "every incomplete two-ray set known (Hegyvári 1989, Jiang--Ma, Fang--He, all site-attested) having a ratio that is a power of $2$ and hence a single ray as a set" (lines 572--574). Defect: the universal clause omits the hypothesis that one coefficient is not a dyadic rational. Witness: $A_{2,6}$, whose elements are all even, is an incomplete two-ray set with ratio $1/3$; any pair of even integers with ratio not a power of $2$ gives such a set. The conclusion of the sentence survives because these sets fail (1.5) at $\theta=1/2$. Replacement: "every incomplete two-ray set known with one coefficient not a dyadic rational (...) having a ratio that is a power of $2$ and hence a single ray as a set, while a pair of dyadic rationals fails (1.5)".
• F4 (note). Location: the same clause's attribution to Hegyvári 1989. The held Yu--Chen manuscript credits Hegyvári 1989 with the completeness case (one coefficient dyadic, the other not) and the non-completeness results for ratios that are powers of two to Jiang--Ma and Fang--He; the 1989 paper is not held, so the attribution is unverified here. No replacement proposed; a reader should not take the parenthesis as a verified census.
• F5 (note). Location: "for almost every coefficient on each rational-ratio line" (lines 546--547). The phrase is true within the family the sentence names (ratio not a power of $2$) and false for the lines with ratio a power of $2$, where the set is a single ray. Optional replacement: "on each rational-ratio line of the family".
• F6 (note). Location: "the corpus holds neither Burr--Erdős nor a general proof" (line 561). Checked against the corpus's own pages (no page proves the lemma) and Fan's text (the lemma is recorded with citations only); held PDFs outside this review's read set were not examined for a proof. The claim stands as far as checked.

Verdict

• S1: not a claim; its description of the problem and of the card is faithful.
• S2: refutation-failed for (a) and (b) and for the lower bound in (c); the upper bound in (c) is the preprint's Corollary 1.2 and should be marked as such (F1).
• S3: refutation-failed (F5 is a reading note).
• S4: refutation-failed (F6 states the limit of the corpus check).
• S5: refutation-failed; the obstruction is stated at its actual scope.
• S6: (b) refutation-failed; (c) refuted-as-stated in its literal universal reading, with the witness $A_{2,6}$, its conclusion surviving under the missing hypothesis (F3); (a) refuted-as-stated as a description of what would fix (P1), since Section 3 is also consumed (F2).
• S7: refutation-failed; the standing sentence claims exactly author-recorded context.

Limitations: the reconstructions and Corollary 354.3 were consumed at their stated standing and not re-derived beyond their interfaces; the three site-attested papers of S6(c) are not held, so nothing about their contents was verified; no computation was run; the corpus check behind S4 covered corpus pages and Fan's text, not every held PDF. This focused review assigns no tier and changes no status.