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Subject and independence

Role: independent reviewer in a fresh context, commissioned for refutation. The reviewer took no part in writing the connections page, the reconstruction pages or any other page in the folder, read no earlier review or grade, and made no web search.

Subject: wiki/research/erdos_354/connections.md as it stood at 2026-09-28T13:50:48Z, section "Problem 124: mixed power sets", lines 580--609 (the H2 at line 580 through the line before the next H2 at line 611). No separate lead page was named; the page's shared opening (lines 1--58: the standing paragraph, the folder notation and the definitions of complete, strongly complete, normalized pair, dyadic rational, Hegyvári's conjecture and Fan's condition (1.5)) was read in full as input.

Reading, at depth:

• The section, in full, every sentence attacked.
• wiki/problems/diophantine_problems/E0124/_index.md, lines 1--52: frontmatter and the Statement, Status, Source, References and Formalization paragraphs. The H2 sections (Current assessment, Progress, Known Results, Linked library material) were not opened.
• The Fan card fan_2026_strongly_complete_sets_conjecture_erdos, in full (see exposures); its result page remark_4_2.md was not opened.
• The held PDF fan_2026_strongly_complete_sets_conjecture_erdos.pdf (v5; the checked-out file's SHA-256 equals the card's provenance line): pages 7 and 34 read as page images (Theorem 1.5 with (1.13), the Bergelson--Simmons sentence, Lemma 2.1 and the definition of $\Delta$; Section 7 in full); pages 6--10 and 34--36 in the text layer ((1.12) and its footnote, Theorem 2.2, Lemma 2.3, Corollary 2.4, the appendix). The v4 PDF's text layer was searched only to confirm that the labels Theorem 1.5, Lemma 2.1, Theorem 2.2 and Section 7 are the same in both versions.
• the normalization page: Definitions, Statement (items 1--6 and the Reduction) and the four proofs of item 6, as premises.
• the Remark 4.2 page: Statement and Step 3 of the proof (condition (1.5) for the two-ray set), as premises; the folder's other reconstruction pages were listed by name only.
• docs/verification.md ("Grading and claim standing", the canonical audit checklist, "Whole-claim report", "Audit checklist"), docs/approach_vetting.md, docs/evidence.md ("Source fidelity") and docs/math_authoring.md, in full.

Exposures: (1) the Fan card was read in full, beyond its provenance paragraph; its Bears-on paragraph carries Problem 354 standing text (the site-accepted part (i) proof and a quoted phrase from the site's review), which concerns another section's target and was not used here; (2) twelve lines of the page's Problem 354 section (lines 59--70) were printed while locating the page's conventions; they paraphrase that problem's Current assessment and were not used; (3) the first fifteen lines of the page's closing section "No further consumer" (from line 611) were printed with the subject section, and their description of the descent's inputs was used to read the word "descent" in S5; (4) a directory listing of wiki/research/erdos_354/evidence/verify/ showed the file names of the folder's other reviews and its grade page; none was opened. Nothing else outside the allowed set reached the reviewer.

Restatement

Notation: $P(D)=\bigcup_{d\in D}\{d^n\}$ is Fan's mixed power set, $P(d,k)$ is the problem page's set of sums of distinct powers $d^i$ with $i\ge k$, and "the descent" is the folder's mechanism as the page's closing section describes it: two interlaced base-$2$ rays with the digit recurrence $a_{i+1}=2a_i+u_i$, $u_i\in\{0,1\}$, the interlacing $b_i<a_i<2b_i\le b_{i+1}$ of item 4 of the normalization page, and a reduced ratio at the block's first layer.

• S1 (target). Problem 124 asks, for integers $3\le d_1<\cdots<d_r$ with $\sum_{i=1}^r1/(d_i-1)\ge1$, whether every sufficiently large integer is $\sum_ic_ia_i$ with $c_i\in\{0,1\}$ and $a_i\in P(d_i,0)$; and, when $\gcd(d_1,\ldots,d_r)=1$, whether the same holds with $a_i\in P(d_i,k)$ for every $k\ge1$. The section words the second hypothesis as "the $d_i$ are coprime" and does not state the quantifier over $k$.
• S2 (page status). The problem page's status is open, and its progress is not compiled.
• S3 (Fan's Theorem 1.5 as the section renders it). The Fan card records Theorem 1.5 at statement level, and the theorem says: $P(D)$ is strongly complete whenever $D=D_1\cup D_2\cup C$ with $\gcd(C)=1$ and $\sum_{d\in D_i}1/(d-1)\ge1$ for $i=1,2$; this needs one component fewer than the Bergelson--Simmons theorem. No hypothesis on $D$ beyond the partition is stated.
• S4 (folder index). The folder index names Problem 124 only as a further target of the same preprint.
• S5 (obstruction). Nothing transports from the folder to Problem 124: the problem's objects are the single-coefficient rays $\{d^n\}$ at distinct integer bases $d\ge3$ without floors; the descent needs two base-$2$ rays with the digit recurrence and the interlacing $b_i<a_i<2b_i\le b_{i+1}$; a power set has none of these.
• S6 (proof route of Theorem 1.5). Theorem 1.5 is not reconstructed in the folder, and its proof (the source's Section 7) runs through the three-component criterion (Theorem 2.2), using nothing from the folder.
• S7 (shared step). The only step common to that proof and the folder is elementary: the bounded-$\Delta$ argument giving syndetic subset sums, which is Fan's Lemma 2.1 and, for two rays, item 6 of the normalization page; this is a resemblance of tools, not a transfer of a result.
• S8 (yield). The folder's mechanism gives nothing about either question of Problem 124.
• S9 (next step). No investigation follows from the folder; the compilation task is a reconstruction of Theorem 1.5 from Theorem 2.2.
• S10 (outcome). The transfer fails.

Checklist

• Quantifiers and scope. Defective in the descriptive sentences only. S1 drops "for every $k\ge1$" and blurs $\gcd(d_1,\ldots,d_r)=1$ into "the $d_i$ are coprime", which a reader can take as pairwise coprimality, a stronger hypothesis (F2). S3 drops three hypotheses of Theorem 1.5 (F3). S5 and S8 quantify correctly: "nothing transports" is a universal claim over the descent's inputs, and the Strongest attack below confirms it.
• Circularity. Not applicable: the section proves nothing from the folder's results and assumes no target.
• Model and convention changes. Two conventions meet in the section without a claimed bridge: the problem's multiset formulation ($\sum_ic_ia_i$ with $a_i\in P(d_i,0)$, so each base contributes its own $d_i^0=1$) and Fan's set union $P(D)$, which the source itself calls a "slightly stronger set-theoretic formulation" (p. 6). The section asserts no equivalence and deduces nothing across them, so no unproved transfer is made. The descent's conventions (normalized pair, item 4) are cited at their actual hypotheses.
• Finite and statistical overreach. Not applicable: no finite verification or heuristic is used.
• Uniformity. Not applicable: no constant or bound is claimed.
• Extremal conclusions. The one comparative count, "one component fewer than Bergelson--Simmons", is checked in its own units against the source (p. 7): the cited Bergelson--Simmons theorem needs a partition $D_1\cup D_2\cup D_3\cup C$ with (1.12) for each $D_i$; Theorem 1.5 needs $D_1\cup D_2\cup C$ with (1.13) for $i=1,2$. Correct as a relay of the source; the Bergelson--Simmons paper is not held (F5).
• Consequences and composition. S8 and S10 follow from S5, and S5 is established independently below. S9's "with Theorem 2.2" is accurate but incomplete as a dependency list: Section 7 also consumes Lemma 3.1 (bounded successive ratios give a countable $H_2$) and an elementary argument for $H_1(\mathcal C)=\{0\}$; nothing in the section claims otherwise.
• Computation. Not applicable in the section. The reviewer's one computation is the inequality $d_r\ge r+2$, hence $r/(d_r-1)<1$, in F1.
• Reproduction. Not applicable: no rerun instruction or coverage claim.
• Source and verdict fidelity. Defective in two places. S1's density condition is not the page's typed condition (F1). S3 attributes to the card a statement the card does not carry and renders Theorem 1.5 without its hypotheses (F3). S6 and S7 are faithful to pages 7--8 and 34 of the source, with the "only" of S7 overstated (F4). S2's second clause and S4 concern excluded text and are unverified.

Weakest steps

W1: the obstruction, S5. The section argues by shape only: a power set lacks the descent's inputs. Since completeness passes to supersets, the attack that matters is whether some two-ray set $A_{\alpha,\beta}\setminus F$ sits inside $P(D)$, or inside the multiset $\biguplus_i\{d_i^n\}$, so that the descent applied to that subset would transfer. The derivation under Strongest attack shows that for finite $D$ with $2\notin D$ no tail $\{\lfloor2^n\alpha\rfloor:n\ge n_0\}$ of any base-$2$ ray lies in $P(D)$, and that for finite $D$ with $2\in D$ only the ray $\{2^n\}$ can, so that two such rays have a ratio that is a power of $2$ and fall outside every hypothesis the descent carries. The obstruction therefore holds for subsets as well as for the sets themselves, which the section does not say but does not need.

W2: the restatement of Theorem 1.5, S3. Re-read on the page image of p. 7: "Let $D\subseteq\mathbb N_{\ge2}$ be a nonempty finite subset with no two elements being powers of the same integer. Suppose that $D=D_1\cup D_2\cup C$ is a partition of $D$ into nonempty subsets such that $\gcd(C)=1$ and (1.13) [$\sum_{d\in D_i}1/(d-1)\ge1$] for every $i=1,2$. Then $P(D)$ is strongly complete." The section keeps the partition, the gcd and (1.13) and drops finiteness, the no-common-power condition and the nonemptiness of the parts. Finiteness is used in Section 7's bound $b-\sum_{b'<b}b'\le\sum_{d\in D_i}d^k/(d-1)$, whose right side is finite only for finite $D_i$; the no-common-power condition is what makes $P_k(D)=\mathcal B_1\cup\mathcal B_2\cup\mathcal C$ a partition, as Section 7 says, and without it (1.13) can be met by proper powers whose union of power sets is smaller than the count suggests (for instance $D_1=\{4,8,\ldots,2^m\}$ has $P(D_1)\subsetneq\{2^n\}$). The weakened statement is not shown false here; it is simply not the source's theorem.

W3: the "only shared step", S7. Section 7 has four steps: $\Delta(\mathcal B_i)<\infty$ from (1.13) by a geometric sum; $H_2(\mathcal C)$ countable from bounded successive ratios (Lemma 3.1); $H_1(\mathcal C)=\{0\}$ from the exact recurrence $r_{n+1}(c)=c\,r_n(c)$ and $\gcd(C)=1$; and Theorem 2.2. Item 6 of the normalization page proves $\operatorname{gap}(P_n)\le N$ from $c_{j+1}-\sum_{i\le j}c_i\le N$ on the merged list, a bounded-$\Delta$ argument as the section says, and Lemma 2.1 is the same principle for infinite sets, so the first resemblance is real. But Step 3 of the Remark 4.2 page proves (1.5) for $A_{\alpha,\beta}$ from $c_{k+1}=2c_k+1$ infinitely often, which is the same kind of step as Section 7's $H_1(\mathcal C)=\{0\}$ (a base recurrence pushed through $\|\cdot\,\theta\|$), by a different argument. By the section's own standard for "shared" (the two bounded-$\Delta$ arguments also differ in detail), there are two shared steps, both tools and neither a result.

Strongest attack

On S5 and S8 (the obstruction). The attack: find $\alpha,\beta>0$ with $\alpha/\beta$ not a power of $2$ and a finite $F$ such that $A_{\alpha,\beta}\setminus F\subseteq P(D)$ for some $D$ of Problem 124, or the analogous inclusion in the multiset $\biguplus_i\{d_i^n\}$; the descent on the subset would then give completeness of the larger set. It fails, by the following derivation.

Let $D$ be finite with $\min D\ge2$, let $\alpha>0$, and write $x_n=\lfloor2^n\alpha\rfloor$ and $u_n=x_{n+1}-2x_n\in\{0,1\}$. Suppose $x_n\in P(D)$ for all $n\ge n_0$, each $x_n$ being some $d^m$ with $d\in D$ and $m\ge1$ (the value $1=d^0$ is excluded once $x_n>1$).

Case A, $\alpha$ not a dyadic rational, so $u_n=1$ for infinitely many $n$ (if $u_n=0$ for all $n\ge n_1$ then $x_n=2^{n-n_1}x_{n_1}$ and $\alpha=\lim2^{-n}x_n=x_{n_1}/2^{n_1}$). Take $n\ge n_0$ with $u_n=1$. Then $x_{n+1}=2x_n+1$ is odd, so $x_{n+1}=d'^{m'}$ with $d'\in D$ odd. If $u_{n+1}=0$, then $x_{n+2}=2d'^{m'}=d''^{m''}$ with $d''\in D$; comparing $2$-adic valuations gives $m''v_2(d'')=1$, so $m''=1$ and $d''=2d'^{m'}=2x_{n+1}$, whence $x_{n+1}\le\max D/2$, which fails once $n$ is large. So for all large $n$, $u_n=1$ forces $u_{n+1}=1$; hence $u_n=1$ for all $n\ge n_1$ for some $n_1$, and then $x_n+1=2^{n-n_1}(x_{n_1}+1)$ gives $\alpha=(x_{n_1}+1)/2^{n_1}$, so $x_{n_1}=\lfloor2^{n_1}\alpha\rfloor=x_{n_1}+1$, a contradiction. No tail of such a ray lies in $P(D)$.

Case B, $\alpha=c/2^s$ with $c$ an odd positive integer. Then $x_n=2^{n-s}c$ for $n\ge s$. If $c>1$, then $2^{n-s}c=d^m$ forces $d=2^eg$ with $g$ odd, $g^m=c$ and $em=n-s$; for each $d\in D$ the equation $g^m=c$ has at most one solution $m$, so only finitely many $n$ qualify. If $c=1$, the ray is $\{2^n\}$, and $2^{n-s}=d^m$ forces $d=2^e$; the exponents $E=\{e\ge1:2^e\in D\}$ form a finite set, and $n-s$ must be a positive multiple of some $e\in E$ for every large $n$, which fails at a prime $n-s$ exceeding $\max E$ unless $1\in E$, that is, unless $2\in D$.

Hence for the problem's sets ($d_i\ge3$, so $2\notin D$) no base-$2$ ray has a tail in $P(D)$, and the same holds elementwise in the multiset formulation, since each element is still some $d_i^m$. No two-ray set, and no set with a two-ray subset of the kind the descent takes, occurs. Under Fan's wider hypothesis $D\subseteq\mathbb N_{\ge2}$, the only ray tail available is $\{2^n\}$ itself, and two such rays have a ratio that is a power of $2$, outside the descent's hypothesis of an irrational ratio (normalization item 2) and outside the rational-ratio target of the page's first section (neither coefficient a dyadic rational). S5 survives, and S8 and S10 with it.

On S3. The attack: check the rendered statement against the source. It succeeds against the sentence as stated: the card's only mention of the theorem is "Theorem 1.5, mixed power sets; all at statement level only here", and the source's theorem carries three hypotheses the section omits (W2). The mathematical content of the weakened statement was not disproved; see Findings.

On S6 and S7. The attack: find a step of Section 7 that consumes a folder result, or a second shared step. The first fails: Section 7 cites Theorem 2.2, Lemma 3.1 and (1.13) only, neither Remark 4.1 nor Remark 4.2 nor any Yu--Chen or Geneson result, and the folder holds no reconstruction of Theorem 1.5 (its pages are the two Fan remarks, the two Geneson results and the Yu--Chen pages). The second succeeds against the word "only" (W3, F4) but not against the sentence's operative content, since the second shared step is also a tool and not a result.

On S1. The attack: compare with the page's typed statement. The page reads $\sum_{1\le i\le r}1/(d_r-1)\ge1$, that is, $r/(d_r-1)\ge1$; with $3\le d_1<\cdots<d_r$ one has $d_r\ge r+2$, so $r/(d_r-1)\le r/(r+1)<1$ and the typed condition is never satisfied. The section's $\sum_i1/(d_i-1)\ge1$ is the source's (1.12), attributed there to the Burr--Erdős--Graham--Li paper. The section therefore describes the intended problem and silently departs from the page it links (F1); the mathematics of the sentence is not refuted.

Premises

• Fan, Strongly complete sets and a conjecture of Erdős, arXiv:2607.14071v5, held as the card's folder-name PDF (36 pages; the checked-out file's SHA-256 matches the card's provenance line). Interfaces used: (1.12) and the footnote on its necessity (p. 6, text layer); Theorem 1.5 and the Bergelson--Simmons sentence (p. 7, page image); Lemma 2.1 and $\Delta(S)$ (p. 7, page image); Theorem 2.2, Lemma 2.3 and Corollary 2.4 (pp. 8--10, text layer); Section 7 (p. 34, page image); the appendix (pp. 34--35, text layer). None of its proofs was reviewed. The v4 PDF (held) was consulted in its text layer only for label stability.
• Bergelson and Simmons, New examples of complete sets, with connections to a Diophantine theorem of Furstenberg, Acta Arith. 177 (2017), Theorem 1.23: not held; its content is known here only as Fan states it on p. 7.
• Burr, Erdős, Graham and Li, Complete sequences of sets of integer powers, Acta Arith. (1996): not held; the problem page lists it as [BEGL96] and Fan cites its Theorem 1.
• the normalization page as of 2026-09-28T13:50:48Z (author-recorded reconstruction, independently reviewed as of 2026-09-28T05:03:27Z per the assignment): items 4 and 6 for a normalized pair ($N=\lfloor\beta\rfloor<M=\lfloor\alpha\rfloor<2N$, $N\ge2$), read with their proofs of item 6; used as the interface the section cites.
• the Remark 4.2 page as of the same time: Statement and Step 3 (condition (1.5) for $A_{\alpha,\beta}$), used only to test S7.
• Problem 124: the Statement paragraph and the frontmatter status: open; nothing below the first H2.
• The connections page's shared opening (lines 1--58) for notation and the standing paragraph; the opening of its "No further consumer" section for its description of the descent's inputs, which the reading of S5 relies on.
• No explicit assumption beyond these; no batch acceptance order.

Findings

• F1 (suggested). Location: "asks whether, for integers $3\le d_1<\cdots<d_r$ with $\sum_i1/(d_i-1)\ge1$". Defect: the linked page types the condition as $\sum_{1\le i\le r}1/(d_r-1)\ge1$, which no admissible $d_1<\cdots<d_r$ satisfies (witness: $d_r\ge r+2$, so the sum is $r/(d_r-1)\le r/(r+1)<1$); the section uses the source's (1.12) without saying that it departs from the page. Replacement: after the condition add "(the source's (1.12); the problem page types the denominator as $d_r-1$, a condition no admissible tuple meets)", and flag the page's statement to its owner for a check against the catalog entry, which this review could not consult.
• F2 (note). Location: "and the same with $P(d_i,k)$ when the $d_i$ are coprime". Defect: the page's hypothesis is $\gcd(d_1,\ldots,d_r)=1$, which "coprime" can be read as pairwise coprimality, and the page's "for any $k\ge1$" is absent. Witness: the page's second question paragraph. Replacement: "and, when $\gcd(d_1,\ldots,d_r)=1$, the same with $P(d_i,k)$ for every $k\ge1$".
• F3 (required). Location: "the Fan card records Theorem 1.5 at statement level: $\bigcup_{d\in D}\{d^n\}$ is strongly complete when $D=D_1\cup D_2\cup C$ with $\gcd(C)=1$ and $\sum_{d\in D_i}1/(d-1)\ge1$ for $i=1,2$". Defect: the card carries no statement of Theorem 1.5 (its only mention is "Theorem 1.5, mixed power sets; all at statement level only here"), so the content is the section author's own reading of the source; and the content omits three hypotheses of the theorem as printed on p. 7 of v5: $D$ is a nonempty finite subset of $\mathbb N_{\ge2}$ with no two elements being powers of the same integer, and the three parts are nonempty. Section 7 uses finiteness (the bound on $b-\sum_{b'<b}b'$) and the no-common-power condition (to make $\mathcal B_1,\mathcal B_2,\mathcal C$ a partition). Witness: the page image of p. 7 and the card's Overview paragraph. Replacement: "the Fan card records Theorem 1.5 (v5, p. 7) at statement level only; the theorem says that for a nonempty finite $D\subseteq\mathbb N_{\ge2}$ with no two elements powers of the same integer, $\bigcup_{d\in D}\{d^n\}$ is strongly complete whenever $D=D_1\cup D_2\cup C$ is a partition into nonempty parts with $\gcd(C)=1$ and $\sum_{d\in D_i}1/(d-1)\ge1$ for $i=1,2$". The consequence for the section is confined to this sentence: nothing is deduced from the theorem.
• F4 (suggested). Location: "The only shared step is elementary: the bounded-$\Delta$ argument that makes subset sums syndetic". Defect: a second step of the same kind is shared, the verification of Fan's condition (1.5): Section 7 proves $H_1(\mathcal C)=\{0\}$ from $r_{n+1}(c)=c\,r_n(c)$ and $\gcd(C)=1$, and Step 3 of the Remark 4.2 page proves (1.5) for $A_{\alpha,\beta}$ from $c_{k+1}=2c_k+1$ infinitely often; the arguments differ as much as the two bounded-$\Delta$ arguments do. Witness: p. 34 of v5 and the Remark 4.2 page's Step 3. Replacement: "The shared steps are elementary: the bounded-$\Delta$ argument that makes subset sums syndetic, which is Fan's Lemma 2.1 and, for two rays, item 6 of the normalization page, and the verification of condition (1.5), which the Remark 4.2 page does for two rays and Section 7 does for $P_k(C)$ by a different argument; both are resemblances of tools, not transfers of results."
• F5 (note). Location: "one component fewer than Bergelson--Simmons". Defect: none in substance; the comparison is relayed from p. 7 of the source, and the Bergelson--Simmons paper is not held, so the section should say whose statement of that theorem it compares. Replacement: "one component fewer than the Bergelson--Simmons theorem as the source states it (p. 7; that paper is not held)".

Unverified, not findings: S2's "with progress not compiled" and S4 (the folder index's wording) concern text outside the allowed reading.

Verdict

• S1: refutation-failed. The described problem is the intended one and the source's; the section departs silently from the page's defective typed condition (F1, F2).
• S2: refutation-failed for "status open" (frontmatter); the clause "progress not compiled" was not examined.
• S3: refuted-as-stated. The card does not record the statement, and the statement is not Theorem 1.5: three hypotheses are missing (F3). The weakened mathematical statement itself is unresolved here.
• S4: no verdict; the sentence describes excluded content and was not examined.
• S5: refutation-failed. The obstruction holds, and holds for two-ray subsets as well, by the derivation under Strongest attack.
• S6: refutation-failed.
• S7: refutation-failed for its operative content (a resemblance of tools, not a transfer); the word "only" does not survive (F4).
• S8: refutation-failed.
• S9: refutation-failed; the reconstruction would also consume Lemma 3.1.
• S10: refutation-failed.

Limitations: the catalog entry for Problem 124 and the Bergelson--Simmons paper were not consulted, so F1 cannot say whether the typed condition is the catalog's or the page's; the folder index and the problem page's Progress section were excluded; Fan's proofs, including Section 7, were read for their dependencies and not reviewed; the derivation under Strongest attack is the reviewer's own and is author-recorded like the section.

This focused review assigns no tier and changes no status.