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Subject

Date and paths. The repository as it stood at 2026-09-28T13:50:48Z: the page wiki/research/erdos_354/connections.md (652 lines; its seven H2 sections open at lines 59, 292, 441, 490, 518, 580 and 611) and the two leads wiki/research/leads/rational_ratio_dyadic_floor_descent/_index.md and wiki/research/leads/small_base_two_ray_distance_divergence/_index.md. Every path was present in the tree as of that time, and the working-tree copies were byte-identical to that state at grading.

Reports. The seven independent focused reviews filed in this folder, one per H2 section of the page, each naming the same subject date, the page's path and its section's line range as its frozen subject, the first two and the last also naming the leads:

• the rational-ratio review (lines 59--291, with the first lead);
• the second-question review (lines 292--439, with the second lead);
• the Problem 349 review (lines 441--489);
• the Problem 348 review (lines 490--516);
• the Problem 254 review (lines 518--579);
• the Problem 124 review (lines 580--609);
• the no-further-consumer review (lines 611--652, with both leads).

The line ranges were checked against the committed page and are right.

What the grader read, and at what depth. In docs/verification.md, the sections "Independence and the assignment", "Exact subjects and durable evidence", "Report contract", "Grading and claim standing", the shared "Audit checklist" and the Erdos-specific "Whole-claim report" and "Audit checklist". Each of the seven reports in full. The page and both leads in full as of that time. Reconstruction pages as of that time, where a finding needed them: the normalization page in full; the Theorem 5.1 page's Definitions and Statement; the theorem page's Steps 2 and 4, dependency map and Scope; the Remark 4.2 page in full; the Theorem 9 page's Statement and the proof of Proposition 11; the Corollary 12 page in full; the bounded-spacing page's definitions and Steps 1--2. The first lead's evidence index and its entry point evidence/main.py in full, read and not run: the sweep finding turns on the evidence's own selection rule. Problem pages as of that time: Problem 354 (Statement, Formulation, Status, References and the Current assessment's "Remaining gaps"), Problems 346 and 347 in full, Problem 124's Statement and Progress, Problem 254 through its Progress section, and Problems 348 and 349 (Statement and Progress). Library as of that time: the Geneson, Fan and van Doorn cards in full, the three Geneson result pages searched for their task wording, the card of the Problem 346 counterexample in full, and the van Doorn canonical conversion at its Lemma 5. Held sources, in the text layer (pdftotext -layout): Fan v5 at pp. 3--4 (Theorem 1.1, the observation, the sentence placing $M_\rho=2$ on $(1,\rho_0]$, Corollary 1.2), p. 7 (Theorem 1.5, Lemma 2.1), pp. 16--19 (the proof of Theorem 1.1 with its lemma citations, Proposition 4.3); Geneson at pp. 11--13 (Proposition 11, Corollary 12); van Doorn at Lemma 4, Lemma 5 and Proposition 2; Yu and Chen at p. 1 (the history paragraph) and the reference list. No page image was needed: no accepted finding turns on a formula the text layer misreads. No web source was consulted. Every finding accepted below was adjudicated by the grader against the page, the source text or an exact recomputation, and the reviewers' derivations were followed step by step where a finding depends on them.

Computation. In exact integer arithmetic, with its own script in ignored scratch, the grader recomputed every pinned certificate of the rational-ratio section (depth, run start and $b_n$ all reproduce), the control pair, the thirty-two-denominator sweep under two rules (the page's "normalized by the least $T$" and the evidence's "least $T$ with $\lfloor2\gamma\rfloor\ge2$"), the witnesses $\gamma=1/511$, $1/257$ and $24/7$ at ratio $3/2$, the threshold witness $\gamma=5/4$, $\alpha=(5/4)^{21}$, and the deletion witness $(3\sqrt2,3)$ (the subset sums of its first three layers, the mesh propagation through index $29$, and every integer in $[6,5000)$ represented from fourteen layers); the two Pisot roots of the Problem 349 review were recomputed in floating point with the margins that review states. Nothing below relies on the output of the lead's evidence.

Independence facts, by role. The grader is a fresh context given only the grading assignment, distinct from the author of the page and the leads and from each of the seven reviewers; the grader took no part in writing any page, any report, the reconstruction pages or the leads' evidence, and had not seen the folder before. The grader is not blind, as a grader need not be. Each report discloses its reviewer's incidental exposures: directory listings of evidence/verify/ by file name; status and label lines of problem pages printed while locating statements; cards and result pages printed whole with their Overview, Read status, Proof pointer and Bears-on paragraphs; the Problem 254 page's Progress paragraph, which summarizes the reviewed section; the frontmatter description of the normalization reconstruction's review; one line of the Theorem 5.1 reconstruction's review; the H2 headings and Outcome lines of other sections; and the leads' paraphrase of the Current assessment, which was allowed reading. Materiality ruling, by the content test: immaterial for every report. None of the exposed text states a verdict on any statement of the page or the leads; every finding cites the page, a source or a computation; the direction of every attack follows the section's own sentences; and nothing in any report could only have come from the exposed text. No reviewer read another review of this page, the grade of the reconstruction reviews, the leads' evidence or any web source.

Reports graded

A report passes when it has a subject block that resolves (subject date, path, section lines and, where named, the lead), stated independence facts and exposures, a restatement carrying the quantifiers, an explicit verdict on all ten items of the Erdos-specific audit checklist, weakest steps actually re-derived rather than paraphrased, a real strongest attack, the premises with their interfaces and reading depth, and a per-statement verdict; it is void otherwise. All seven reports pass. Each entry records what was checked.

1. the rational-ratio review: pass. Subject resolves (section and lead); ten reconstruction pages and three PDFs listed with depth; three exposures; restatement S1--S15 carrying every quantifier of Lemma 354.1, Proposition 354.2, Corollary 354.3 and Lemma 354.4; ten explicit checklist verdicts, four of them defects tied to findings; W1--W3 re-derived (the count from a run of $\gamma$ to the Section 3 hypotheses with $\ell=L'+1$ and $K\ge K_*$, the descent chain, the two transports); a real attack on every statement, which produced three witnesses ($\gamma=1/511$, $\gamma=24/7$, the three sweep denominators) that the grader reproduced to the digit; premises with interfaces and depth; verdicts per statement. Limitation: the recomputation's code is not retained; its figures follow from the page's own criterion and were reproduced here.
2. the second-question review: pass. Subject resolves (section, preamble, lead); reading listed with depth (four reconstruction pages, Fan pp. 1--5 and 19--21, Geneson pp. 11--13, two cards); exposure disclosed; restatement S1--S12 with the thresholds, the premises (P1) and (P2) and the quantifier over bases; ten verdicts; three weakest steps re-derived (the threshold of Proposition 354.5 with its dependence on $j$, the $q\ge5$ step with the automorphism, the bookkeeping of (P2) including the closure); a real attack on each statement, with the witness $\gamma=5/4$, $\alpha=(5/4)^{21}$ that the grader reproduced; premises with interfaces; verdicts per statement.
3. the Problem 349 review: pass. Subject resolves; reading with depth (the Geneson and van Doorn text layers, Graham's page images); four exposures; restatement S1--S6 with the two indexing conventions; ten verdicts, three marked inapplicable with their reasons; W1--W3 re-derived (the base-two identity by induction, the scope reading with two Pisot witnesses and an irreducibility argument, the van Doorn analogy); a real attack per statement; premises with depth (Dubickas not held); verdicts, all refutation-failed, with four suggestions and two notes.
4. the Problem 348 review: pass. Subject resolves; reading with depth (Geneson pp. 1--5 in full); three exposures with their lines named; restatement S1--S6 separating the one mathematical statement from the record and judgment sentences; ten verdicts; W1--W3 re-derived (the bridge under both sequence conventions, the exclusion mechanism from the source's Section 2, the load-bearing scope phrase with the powers-of-two witness); a real attack per statement; premises; verdicts per statement.
5. the Problem 254 review: pass. Subject resolves; reading with depth (Fan v5 at Theorem 1.1, Section 2, Lemma 3.3, the proof of Theorem 1.1, Remarks 4.1--4.2 and Propositions 4.2--4.3; Yu--Chen p. 1); five exposures; restatement S1--S7 with $M_2^*$ defined in the claim's own units; ten verdicts; W1--W3 re-derived (from the witness to the threshold, the family with its two covering results, the Burr--Erdős correspondence and the single-ray count); a real attack with the witness $A_{2,6}$; premises naming the papers not held; verdicts per statement.
6. the Problem 124 review: pass. Subject resolves; reading with depth (Fan v5 pp. 7 and 34 as page images, pp. 6--10 and 34--36 in text); four exposures; restatement S1--S10; ten verdicts; W1--W3 re-derived, W1 by a genuine derivation (no tail of a base-two ray lies in a mixed power set without the base $2$, in the dyadic and the non-dyadic case); a real attack; premises; verdicts per statement.
7. the no-further-consumer review: pass. Subject resolves (section and both leads); reading with depth (three reconstruction pages in full, ten problem statements); five exposures; restatement S1--S8; ten verdicts; W1--W3 re-derived (the ratio pattern with its two limit points, the single-ray count with the sharper bound, the interface inventory of Theorem 5.1); a real attack with the deletion witness $(3\sqrt2,3)$ and its mesh argument, which the grader reproduced; premises; verdicts per statement.

Corrections

Each correction was adjudicated against the page as of that time and, where a source or a computation decides it, against the held text layer or the grader's exact recomputation. Locations quote the page text as of that time.

C1. Page connections, section "Problem 354: Hegyvári's conjecture at a rational ratio", "What it does not give", the first bullet, from "The bounded-run coefficients." to "node connection (Step 4)." Replace the bullet with:

"The short-block coefficients. Proposition 354.2 acts through a maximal common zero run of the conversions of length at least $\lceil\log_2K_*\rceil-1$ at an exact layer, which a run of $L(p,q)$ zeros in $\gamma$ guarantees; a coefficient with only finitely many such runs is outside the descent. Exact layers occur exactly at the $s$ with $\{2^s\gamma\}<1/p$, possibly nowhere ($\gamma=24/7$ at ratio $3/2$ has none: $\delta_s$ runs through $2,1,1$), and at an exact layer a common zero run of that length forces $\{2^s\gamma\}<2^{-\lceil\log_2K_*\rceil}$. For a rational $\gamma$ whose denominator in lowest terms has odd part $v$, $\{2^s\gamma\}\ge1/v$ for all large $s$, so every rational $\gamma$ with $v\le2^{\lceil\log_2K_*\rceil}$ ($v\le128$ at ratio $3/2$), in particular every rational pair certified below, and uncountably many irrationals are outside the descent, while $\gamma=1/511$ at ratio $3/2$ has infinitely many runs of exactly eight zeros and is covered by Corollary 354.3. For the coefficients outside, Hegyvári's conjecture stays open except pair by pair through Lemma 354.4, since Theorem 5.1 needs $K\ge K_*$ for its coefficient interval (Step 1) and node connection (Step 4)."

In the third bullet replace "the exceptional set contains every pair with rational coefficients" with "the exceptional set contains every pair with rational coefficients whose $\gamma$ has odd denominator part at most $2^{L(p,q)}$"; in the Outcome replace "open for the bounded-run coefficients" with "open for the short-block coefficients". Basis: bounded runs are not short runs. $\gamma=1/511$ is rational and not dyadic ($511=7\cdot73$), $2^9\equiv1$ modulo $511$, so its expansion is the block of eight zeros and a one repeated, and $L(3,2)=8$; Corollary 354.3 with $k=0$, $p/q=3/2$ makes $A_{3/511,2/511}$ strongly complete, a pair with rational coefficients in the class the bullet calls open. The grader recomputed the normalized pair $(3072/511,2048/511)$ ($N=4$, $M=6$): maximal common zero runs of length $7$ start at $s=8,17,26,\ldots$, each with $\delta_s=0$, so $\ell=8$ and $K=256\ge92$, and a Lemma 354.4 certificate exists at depth $9$ from $516$ with $b_9=2052$; likewise $\gamma=1/257$ (period $16$) certifies as $(1536/257,1024/257)$ at depth $5$ from $41$ with $b_5=127$. The sentence "such coefficients still have exact layers" is false: $\gamma=24/7$ at ratio $3/2$, normalized pair $(72/7,48/7)$, has doubling orbit $\{3/7,6/7,5/7\}$ of its fractional part, all at least $1/3$, and $\delta_s=2a_s-3b_s$ takes the values $2,1,1$ periodically. The bounds in the replacement: at an exact layer $\{2^s\alpha\}=p\{2^s\gamma\}$, so a common zero run of length $\lceil\log_2K_*\rceil-1$ gives $p\{2^s\gamma\}<2^{1-\lceil\log_2K_*\rceil}$ and, as $p\ge3$, $\{2^s\gamma\}<2^{-\lceil\log_2K_*\rceil}$; for $s$ past the $2$-adic part of the denominator $\{2^s\gamma\}$ is a nonzero multiple of $1/v$; a run of $L$ zeros needs $v>2^L$, so odd part at most $2^{L(p,q)}$ puts $\gamma$ in the corollary's exceptional set. The certified rational pairs have odd parts $5$, $7$, $9$ and the sweep's $d\le99$. From F1 and F2 of the rational-ratio review, required, accepted.

C2. Lead rational_ratio_dyadic_floor_descent, frontmatter desc, "and the bounded-run coefficients as the remaining gap in Hegyvári's conjecture": replace with "and the coefficients with only short common zero runs as the remaining gap in Hegyvári's conjecture". Section "The consequential gap", the first two sentences through "too short for Theorem 5.1, and": replace with "Coefficients $\gamma$ with only finitely many maximal common zero runs of length $\lceil\log_2K_*\rceil-1$ at exact layers: every rational $\gamma$ whose denominator has odd part at most $2^{\lceil\log_2K_*\rceil}$ ($128$ at ratio $3/2$), in particular every rational pair certified by the evidence, and uncountably many irrationals, while $\gamma=1/511$ at ratio $3/2$, with infinitely many runs of eight zeros, is covered by Corollary 354.3. For them exact layers occur only at the $s$ with $\{2^s\gamma\}<1/p$, possibly nowhere, and the common zero runs after them are too short for Theorem 5.1, and". Basis: as for C1; the lead restates the refuted bullet. From F1 and F2 of the rational-ratio review, required, accepted.

C3. Page connections, same section, the sentence after Lemma 354.4 listing the certificates, its last clause "and every pair $(3\gamma,2\gamma)$ with $\gamma=2^T/d$ for the thirty-two odd denominators $5\le d\le99$ not divisible by $3$, normalized by the least $T$, by depth $9$." Replace with "and every pair $(3\gamma,2\gamma)$ with $\gamma=2^T/d$ for the thirty-two odd denominators $5\le d\le99$ not divisible by $3$, normalized by the least $T$, by depth $12$: twenty-nine of them by depth $9$, and $d=23$, $89$ and $91$, the pairs $(192/23,128/23)$, $(768/89,512/89)$ and $(768/91,512/91)$, at depths $10$, $12$ and $11$, from $2139$, $11036$ and $4323$ with $b_{10}=5698$, $b_{12}=23563$ and $b_{11}=11522$." The lead's evidence entry point, wiki/research/leads/rational_ratio_dyadic_floor_descent/evidence/main.py, function check_sweep, must then select the least $T$ for which the pair is normalized, $2\le\lfloor2\gamma\rfloor<\lfloor3\gamma\rfloor<2\lfloor2\gamma\rfloor$, in place of the least $T$ with $\lfloor2\gamma\rfloor\ge2$, and its check "greatest depth needed is 9" becomes "greatest depth needed is 12"; the evidence index's clause "normalized by the least $T$" then describes the run. Basis: under the page's own definition of a normalized pair ($N\ge2$, $N<M<2N$) the least $T$ is $6$ for $d=23$ ($T=5$ gives $N=2$, $M=4=2N$) and $8$ for $d=89$ and $d=91$; the grader's recomputation, independent of the reviewer's, finds at depth $9$ the longest runs $2826$ from $2139$ against $b_9=2849$, $22$ against $b_9=2945$ and $22$ against $b_9=2880$, and the first certificates at the depths and starts above; the other twenty-nine certify by depth $9$, the deepest ($d=11$ and $d=73$) at $9$. The evidence's rule differs from the normalized rule for the seven denominators $d=11,23,43,47,89,91,95$, where it gives $\lfloor2\gamma\rfloor=2$ and $\lfloor3\gamma\rfloor=4=2\lfloor2\gamma\rfloor$, a pair outside the precondition its own docstring states; there its certificate is found at depth $2$ for the set with the two extra elements $2$ and $4$, a superset of the normalized pair's set, and its check "greatest depth needed is 9" passes on $d=73$. The page's sentence is false under the page's definition and does not describe what the evidence computes; the reading that keeps the page's mathematics (Lemma 354.4 is stated for normalized pairs) is adopted, and the evidence must follow it. From F3 of the rational-ratio review, required, accepted.

C4. Lead rational_ratio_dyadic_floor_descent, "Provisional premises", "the manuscript is unrefereed and the reconstruction of that page is not among the filed reviews." Replace with "the manuscript is unrefereed; the reconstruction of that page has a filed focused review in the folder's evidence, graded faithful and sound with one correction and assigning no tier." Basis: as of that time the folder's evidence/verify/ holds the Theorem 5.1 reconstruction review, and the reconstruction grade records it as pass with fidelity faithful and the argument sound with its correction C6. A sentence characterizing the filed reviews is a claim, and this one is false as of that time. From F4 of the rational-ratio review, filed as suggested because the reviewer could not read the reviews; accepted by the grader, who read the grade.

C5. Page connections, section "Problem 354: the second question at bases up to the plastic number", proof of Proposition 354.5, "so for $k\ge m_0$ the $\alpha$-ray has exactly one term in $I_k$, namely $\lfloor\alpha^*\gamma^k\rfloor$". Replace with "so for $k\ge\max(m_0,j)$ the $\alpha$-ray has exactly one term in $I_k$, namely $\lfloor\alpha^*\gamma^k\rfloor$". Basis: the term of index $k-j$ exists only for $k\ge j$, and $m_0$ is chosen without reference to $j$. Witness, recomputed exactly by the grader: $\gamma=5/4$, $\alpha=(5/4)^{21}$ give $j=20$, $\alpha^*=5/4$; the least admissible $m_0$ is $6$ (the inequality fails at $m=5$, $3.81\ldots<4$, holds at $m=6$, $4.76\ldots\ge4$, and for $m\ge7$ because $(1/4)(5/4)^m\ge1$); the ray's least term is $108>\gamma^{20}=86.7\ldots$, so the ray has no term in $I_k$ for $6\le k\le19$. The statement is unaffected: with the threshold $\max(m_0,j)$ for each ray the proof closes as written. From F1 of the second-question review, filed as suggested; accepted because the displayed sentence is false at values inside its stated range.

C6. Same page, same section, "Next investigation", "gives an incomplete single ray at a rational base below the golden ratio, which is new for Problem 349 and closes this route at that base." Replace with "gives an incomplete single ray at a rational base below the golden ratio (the ray of the coefficient $t\gamma^{n_0}$, every term of which is divisible by $q$), which is new for Problem 349 and closes the single-ray sufficient condition at that base; premise (P2) for the irrational-ratio pairs then needs a direct argument, or a second bad coefficient with irrational ratio to the first, which would give an incomplete pair as Proposition 354.7 does." Lead small_base_two_ray_distance_divergence, "Next step", "gives an incomplete single ray at a rational base below the golden ratio, new for Problem 349, and closes the route at that base." Replace with "gives an incomplete single ray at a rational base below the golden ratio (the ray of the coefficient $t\gamma^{n_0}$, every term of which is divisible by $q$), new for Problem 349, and closes the single-ray sufficient condition at that base, leaving (P2) for the irrational-ratio pairs to a direct argument or to a second bad coefficient with irrational ratio to the first." Basis: (P2) concerns the two-ray set, and a set of nonnegative terms fails (1.5) at $\theta$ exactly when both rays have convergent distance sums at $\theta$. A coefficient $t$ with all late terms divisible by $q$ makes the sum of $S(t)$ at $\theta=1/q$ finite, but the pair $(t,\beta)$ fails (1.5) at $1/q$ only if $S(\beta)$ also has all but finitely many terms divisible by $q$, and neither page produces such a $\beta$ with $\beta/t$ irrational: at a rational base the Corollary 12 device $\beta=t(1+\gamma)$ has rational ratio and $\gamma^{\mathbb Z}\subseteq\mathbb Q$. So the negative outcome closes the sufficient condition "every single ray satisfies (1.5)" and not deduction 354.6, and no argument either way is available. The ray of the coefficient $t$ itself need not be incomplete, since its early terms may supply the residues its late terms lack (base $2$, $t=1$: every later term even, the ray complete); the ray of $t\gamma^{n_0}$ is incomplete. From F2 (required) and F3 (suggested) of the second-question review; accepted, F3 folded into the same sentence.

C7. Same page, same section, "What it does not give", second bullet, "and the problem's multiset has every value twice". Replace with "and the problem's multiset has every sufficiently large value twice". Basis: for $\alpha=\beta\gamma^k$ with $k\ne0$ the $|k|$ initial values of the longer ray occur once. From F7 of the second-question review, filed as a note; accepted because the clause is false at a case inside the bullet's range.

C8. Same page, section "Problem 348: deletion thresholds", "Next investigation", "A reconstruction of Geneson's Sections 2--4 is the separate compilation task the card records as pending." Replace with "A reconstruction of Geneson's Sections 2--4 would be a separate compilation task; the card records that this proof was read for structure only." Basis: the Geneson card's read-status paragraph records that the proof of Theorem 1 (Sections 2--4, pp. 3--11) "was read for structure only"; neither the card nor its three result pages record a reconstruction or compilation task, and the word "pending" occurs there only in the result pages' sentences that a separate triage of a problem page is pending (grader's search of the four pages as of that time). From F1 of the Problem 348 review, required, accepted.

C9. Same page, section "Problem 254 and the threshold $M_2^*$", "Next investigation", "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." Replace with "every incomplete two-ray set known with one coefficient not a dyadic rational (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, while a pair of dyadic rationals fails (1.5)." Basis: $A_{2,6}=\{2^{n+1}\}\cup\{3\cdot2^{n+1}\}$ has ratio $1/3$, not a power of $2$, and every element even, so it is an incomplete two-ray set, and trivially a known one; the universal clause is false as written. Its purpose survives with the hypothesis: the site-attested incomplete sets with a non-dyadic coefficient all have a power-of-two ratio, and a pair of dyadic rationals has all but finitely many elements divisible by a fixed power of $2$, so its distance sum at $\theta=2^{-c}$ is finite for $c$ large and it is no candidate for the question. From F3 of the Problem 254 review, required, accepted.

C10. Same page, same section, "Next investigation", "a reconstruction of Fan's Theorem 2.2 and Section 4, which would also fix the standing of premise (P1) above". Replace with "a reconstruction of Fan's Theorem 2.2 with Sections 3--4, which would also fix the standing of premise (P1) above". Basis: in Fan v5 the proof of Theorem 1.1 (Section 4, pp. 16--19) invokes Lemma 3.1 twice, Lemma 3.2 three times and Lemma 3.3 once before "Invoking Theorem 2.2 and (1.2) completes the proof of the base case"; the grader located each citation in the text layer. The page's own deduction 354.6 describes the proof of (P1) as Sections 2--4 of the source. From F2 of the Problem 254 review, filed as suggested; accepted because the sentence, read as a description of what would fix (P1), is false.

C11. Same page, section "Problem 124: mixed power sets", Target paragraph, "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 $i=1,2$, one component fewer than Bergelson--Simmons." Replace with "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 $i=1,2$, one component fewer than the Bergelson--Simmons theorem as the source states it." Basis: Fan v5 p. 7, text layer: "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) for every $i=1,2$. Then $P(D)$ is strongly complete."; Section 7 (p. 34) uses finiteness in its bound on $b-\sum_{b'<b}b'$ and the no-common-power hypothesis to make $P_k(D)=\mathcal B_1\cup\mathcal B_2\cup\mathcal C$ a partition. The Fan card's only mention is "Theorem 1.5, mixed power sets; all at statement level only here". From F3 of the Problem 124 review, required, accepted.

C12. Same page, section "No further consumer", "and the two-ray sets are not examples for Problem 346 or 347, since their sorted terms have ratios alternating near $\alpha/\beta$ and $2\beta/\alpha$, while a single ray,". Replace with "and the two-ray sets are not examples for Problem 347, since their sorted terms have ratios alternating near $\alpha/\beta$ and $2\beta/\alpha$ and so do not tend to $2$; for Problem 346, refuted by a counterexample, the ratios decide nothing, since its hypotheses ask that completeness be lost under every infinite deletion, which the descent does not address and which fails for the pair $(3\sqrt2,3)$ after the deletion of $\{3\cdot2^m:m\ge4,\ 4\mid m\}$; a single ray,". Basis: Problem 346 asks whether its hypotheses (complete after every finite deletion, incomplete after every infinite deletion, ratios at least $1+\varepsilon$) force the ratio to tend to the golden ratio; the ratio pattern of a two-ray set (limit points $\theta$ and $2/\theta$, both in $(1,2)$) shows only that such a set would violate the questioned conclusion, so it cannot exclude the set from the hypotheses, and had a two-ray set satisfied them it would be a further counterexample. The corpus records the problem as solved by the counterexample of the held card (a sequence with ratio floor $6/5$ and two subsequential ratio limits), so no consumer is needed. The witness, re-derived by the grader: $(3\sqrt2,3)$ is normalized ($N=3$, $M=4$), its ratio $\sqrt2$ is irrational, the subset sums of its first three layers $\{3,4,6,8,12,16\}$ cover $[6,43]$, and adding the retained weights of index at least $3$ in increasing order keeps a full interval whose span exceeds the next retained weight at every step (the reviewer's inequalities $s_m\ge b_m$ and $s_m\ge a_m$ hold as derived; the propagation was checked through index $29$ and every integer in $[6,5000)$ is represented from fourteen layers), so $A_{3\sqrt2,3}\setminus B$ is complete and the pair fails the deletion hypothesis for a reason unrelated to its ratios. From F1 of the no-further-consumer review, required, accepted.

C1, C2, C7, C9, C12 and the closure clause of C6 repair sentences that are false or unsupported as written; C3, C4, C8, C10 and C11 correct the pages' account of the evidence, the filed reviews and the sources; C5 repairs a false intermediate sentence in a proof whose statement stands. No correction changes a transported result: Lemma 354.1, Proposition 354.2, Corollary 354.3, Lemma 354.4, Proposition 354.5, deduction 354.6 and Proposition 354.7 keep their statements.

Rejected and downgraded findings

Every finding not listed under Corrections is recorded here with its disposition. "Downgraded" means the finding is right as an observation but does not require a change; "rejected as a correction" means the proposed change cannot be adopted.

The rational-ratio review. F5 (the wording "not contradictory by itself"; the quantifier on membership in $\mathcal C_k$): downgraded; in its paragraph the first sentence reads as a statement about the reconstructed argument, and the quantifier is stated two sentences earlier ("as soon as $k$ is at least the number of ones"). F6 ("every non-dyadic normalized pair"; the period of the conversions): downgraded; Lemma 354.1 states its hypothesis precisely and the Outcome's shorthand is read against it; the order of $2$ modulo $v$ is a period and no minimality is claimed. F7 (the multiset-to-set bridge in the Target paragraph): downgraded, and resolved by the grader: the site-accepted theorem applies to every pair with irrational ratio, hence to $(2^K\alpha,2^K\beta)$ for every $K$; the two rays share only finitely many values (Step 1 of the Remark 4.2 page), so for $K$ large the tail multiset has distinct values and its completeness is completeness of a subset of $A_{\alpha,\beta}$; the case split stands, and the problem page's remaining gap (5) says what the paragraph attributes to it. F8 (the attribution and normalization of $(5/2,5)$): downgraded; the multiset is incomplete for the reason the review gives, the problem page's site commentary attributes the incompleteness of $\beta=2^k\alpha$ with $\alpha\ge2$ to Hegyvári 1989, and the page presents the pair as a control that "must not" certify, which is all the evidence checks.

The second-question review. F4 (the lead's "any proof"): downgraded; the sentence's "so" ties "any proof" to the strong completeness its antecedent names, and the section states the precise form; the author may add "of strong completeness". F5 (the object of the positive outcome): downgraded; the sentence's object is (1.5), "beginning with rational $\theta$" names a starting point, and the preceding bullet says that (1.5) at irrational $\theta$ is a further condition. F6 (the Pisot alternative is vacuous on $(1,2)$): downgraded; verified by the grader (a reciprocal irreducible polynomial pairs each conjugate with its inverse, so a Pisot number with one has degree $2$, minimal polynomial $x^2-ax+1$ with $a\ge3$ and value at least $(3+\sqrt5)/2$); a vacuous alternative in a hypothesis is not an error. F8 (novelty for Problem 349; "no held source"; the description of the Current assessment; "marks the bases where (P2) fails"; "every set with two elements"): downgraded; the grader read the Problem 354 page's Status paragraph, which records Corollary 12 for the every-base reading and the $\sqrt\varphi$ Lean candidate for the some-base reading as both pages say; "marks the bases" reads as "marks bases", and "every set with two elements" is Theorem 1.1's "at least two" as the proof applies it.

The Problem 349 review. F1 (the range of $k$ in Proposition 2): downgraded; the page states the necessary direction only ("only $t=2^{-k}$ gives"), true for every range of $k$, and van Doorn's Proposition 2 reads "if and only if $t=1/2^k$ for some $k\ge1$" in his indexing from $n=1$ (text layer). F2 ("the certificate" among the objects lost at a non-integer base): rejected as a correction; the list names Theorem 5.1's objects (the exact doubling blocks, the coefficient interval $d\{px+qy\}$, the finite node-connection certificate, the modular invariant), which the digit recurrence supplies, not the mesh certificate of Lemma 354.4, whose single-ray form the page names in its next sentence. F3 ("which the corpus already records"): downgraded; no card or result page records van Doorn's Lemma 5 (the card's list of results to transcribe omits it), but the held preprint and its canonical conversion beside the card carry it; the author may write "held in the preprint, not transcribed on its card". F4 (the hedge "if any besides Geneson's exists"): downgraded; the sentence is a true conditional and a witness is an addition; the grader recomputed the two roots ($1.534158$ with the other roots of modulus at most $0.951$; $1.590005$ with at most $0.977$). F5 (the coefficient shift): downgraded; carried by C6. F6 ($a_0\ge1$): downgraded; the set convention discards zero terms.

The Problem 348 review. F2 (name the irrational-ratio hypothesis): downgraded; "of its theorem" carries it, as the review's own W3 shows. F3 (the card's read status): downgraded; the card records the statement read clause by clause and the proof for structure only, and the page's compression changes no standing. F4 (Theorem 2 named as part of the incompleteness mechanism): downgraded; Lemma 4 is the engine and Theorem 2 its input, as the review derives, and the parenthetical names the source's objects correctly.

The Problem 254 review. F1 (the unconditional "$M_2^*$ lies in $\{2,3,4,5\}$"): downgraded; the same paragraph attributes the upper bound to Corollary 1.2 and the Target says the proof of Theorem 1.1 is unread; the author may add "on the preprint's Corollary 1.2". F4 (the attribution to Hegyvári 1989): downgraded; the held Yu--Chen manuscript credits the power-of-two incompleteness results to Jiang--Ma and Fang--He, while the problem page's site commentary, which the parenthesis paraphrases, credits the case $\alpha\ge2$, $\beta=2^k\alpha$ to Hegyvári 1989; the paper is not held and "site-attested" says so. F5 ("each rational-ratio line"): downgraded; read within the family the sentence names. F6 (no corpus proof of the Burr--Erdős lemma): downgraded; a limit of the check, not a defect.

The Problem 124 review. F1 (the problem page's typed condition): downgraded as a correction to the connections page, which states the source's condition (1.12) correctly; the problem page wiki/problems/diophantine_problems/E0124/_index.md types the denominator as $d_r-1$, a condition no admissible tuple meets, and is flagged to its owner for a check against the site's statement (outside this grade's subject; not edited here). F2 ("coprime"; the quantifier over $k$): downgraded; $\gcd(d_1,\ldots,d_r)=1$ is the reading of "coprime" for a tuple, and the omitted quantifier belongs to the problem's second question, which the section only names. F4 (a second shared step): downgraded; the sentence's operative content, a resemblance of tools and not a transfer, is right, and the second resemblance is of the same kind. F5 (whose statement of Bergelson--Simmons): downgraded; C11 adds "as the source states it".

The no-further-consumer review. F2 (the displayed single-ray bound): downgraded; the inequality $a_n\ge\lfloor\alpha\rfloor2^n$ is true, the exclusion of single rays is carried in the same sentence by the resolution of both problems, and the sharper bound $a_n>2^n\alpha-1$ the review derives is available to the author. F3 ("the reusable content of the folder is a descent"): downgraded; the search terms cover the other mechanisms, as the review says. F4 ("every later layer"): downgraded; the Theorem 5.1 page's statement ($t\ge n+\ell+3$) is what the summary compresses. F5 ("no research lead outside this folder"): downgraded; "no lead other than the two named" is the only reading the sentence supports.

Observations outside the reports' read sets. (a) The sentences recording that the pages of Problems 349, 348, 254 and 124 carry "progress not compiled" are stale as of that time: each of those pages' Progress sections holds, since the same change, a note derived from this page and nothing else; the author may write "with no progress recorded beyond this folder's note". No correction is numbered for it, since no report found it. (b) The Problem 124 page's typed condition, see F1 of the Problem 124 review above.

Graded verdicts

Verdicts are per statement in the numbering of each review, grouped by section. "Refutation-failed" and "refuted-as-stated" are written in full.

Problem 354: Hegyvári's conjecture at a rational ratio. S1 (target and reduction): refutation-failed, the case split confirmed by the grader. S2 (the inventory of irrationality): refutation-failed. S3 (Lemma 354.1): refutation-failed. S4 (Proposition 354.2): refutation-failed. S5 (Corollary 354.3): refutation-failed. S6 (the constants): refutation-failed. S7 (Lemma 354.4): refutation-failed. S8 (the pinned certificates and controls): refutation-failed, reproduced by the grader. S9 (the sweep sentence): refuted-as-stated (C3). S10 (obstruction 1): refuted-as-stated (C1); the obstruction stands in its corrected form. S11 (obstruction 2): refutation-failed. S12 (obstruction 3): the rational-coefficients clause refuted-as-stated (C1); the uniformity and dichotomy clauses refutation-failed. S13 (next investigation): not a claim. S14 (outcome): refutation-failed for the transfer list, with the wording of C1. S15 (the lead): the route summary refutation-failed; the consequential-gap paragraph refuted-as-stated (C2); the premise sentence refuted-as-stated as a record claim (C4).

Problem 354: the second question at bases up to the plastic number. S1 (the index bridge): refutation-failed. S2 (the constants; $M_\rho=2$ on $(1,\rho_0]$): refutation-failed; the theorem it feeds is (P1), a preprint statement. S3 (Proposition 354.5): refutation-failed, its proof corrected by C5. S4 (deduction 354.6): refutation-failed as a conditional on (P1) and (P2), which remain unproved. S5 (the addenda): refutation-failed. S6 (Proposition 354.7): refutation-failed, resting on Proposition 11 and through it on Dubickas's theorem as quoted. S7 (the bases where (P2) fails; the every-base reading): refutation-failed. S8 (the first bullet): refutation-failed for its equivalences; the Mahler comparison and the coverage remark are not adjudicated. S9 (the exceptional pairs): refutation-failed with C7. S10 (redundancy): refutation-failed. S11(a), S11(b): refutation-failed. S11(c): "gives an incomplete single ray at that base" refutation-failed for the shifted coefficient (C6); "closes this route at that base", on both pages: proof defective; statement unresolved (C6). S12 (the lead): refutation-failed for its restatements; its Next step carries C6.

Problem 349: single rays and the digit recurrence. S1--S6: refutation-failed.

Problem 348: deletion thresholds. S1, S2(b), S3a, S3b, S3c, S4: refutation-failed. S5(b): refuted-as-stated (C8). S5(a), S6: judgments, consistent with what was read. S2(a), S2(c): not adjudicated by the review; the status clause confirmed by the grader, the progress clause stale (observation (a) above).

Problem 254 and the threshold $M_2^*$. S1: not a claim, faithful. S2: refutation-failed, the upper bound being the preprint's Corollary 1.2. S3, S4, S5: refutation-failed. S6(b): refutation-failed. S6(c): refuted-as-stated in its literal universal reading (C9). S6(a): refuted-as-stated as a description of what would fix (P1) (C10). S7: refutation-failed.

Problem 124: mixed power sets. S1: refutation-failed, with the flag on the problem page. S2: refutation-failed for the status; the progress clause stale (observation (a)). S3: refuted-as-stated (C11). S4: not examined by the review. S5--S10: refutation-failed.

No further consumer. S1: refutation-failed. S2: not a claim. S3: refutation-failed for the descriptors; the status words for Problems 346 (solved) and 347 (proved) confirmed by the grader. S4: refutation-failed. S5 (two-ray sets and Problem 346): proof defective; statement unresolved (C12). S6: refutation-failed. S7: refutation-failed. S8: not a claim.

Lead outcomes.

• rational_ratio_dyadic_floor_descent: needs_update. The four statements its route rests on, Lemma 354.1, Proposition 354.2, Corollary 354.3 and Lemma 354.4, are refutation-failed and the pinned certificates reproduce; but the lead's consequential-gap paragraph and its desc are refuted-as-stated (C2), its provisional-premise sentence is a false record claim (C4), and the sweep sentence it relies on and its evidence's sweep rule are wrong (C3). After C2, C3 and C4 the route rests only on refutation-failed statements.
• small_base_two_ray_distance_divergence: needs_update. Proposition 354.5, deduction 354.6 as a conditional on (P1) and (P2), and Proposition 354.7 are refutation-failed, the proof of 354.5 taking C5 on the connections page; the closure clause of the lead's Next step is proof defective; statement unresolved (C6), so the lead overstates what a negative outcome would establish. After C6 the route rests only on refutation-failed statements.

No tier is assigned and no status changes.