Subject and independence

Role. Independent reviewer in a fresh context (model: Claude Fable 5.1), given only the commissioning assignment. The reviewer took no part in writing the lead, the connections page, the reconstruction pages or their evidence, and read no review, grade or standing text of any of them. Charge: refutation. This record carries no tier; a distinct grader records pass or void for its contract.

Subject. The tree as it stood on 2026-10-02. The subject is a blind extraction, kept outside the repository, of two pages on that date: the lead page wiki/research/leads/rational_ratio_dyadic_floor_descent/_index.md with its review-status fields and its "Current review" section removed, and the connections page wiki/research/erdos_354/connections.md with its review-report sentences cut. Every statement examined below is quoted or paraphrased from the extraction's text as it stands. A first attempt at this re-review, whose extraction carried the earlier verdicts, was stopped before completion and its output discarded; this record is from a blind extraction frozen on 2026-10-02 and kept outside the repository.

Other reading, all from the tree of the frozen date. Operating instructions: docs/verification.md, docs/evidence.md, docs/research.md, docs/approach_vetting.md, docs/math_authoring.md, whole. Mathematics the route rests on, under wiki/research/erdos_354/: yu_chen_normalization_reconstruction.md (whole; items 1--6 and the reduction re-derived), yu_chen_theorem_5_1_reconstruction.md (whole; Steps 1--6 re-derived), yu_chen_lemma_2_1_reconstruction.md, yu_chen_lemma_2_2_reconstruction.md, yu_chen_lemma_2_3_reconstruction.md (whole; proofs read), yu_chen_theorem_reconstruction.md (whole; Step 2 re-derived), yu_chen_fe_reconstruction.md, yu_chen_bg_reconstruction.md, yu_chen_windows_reconstruction.md, yu_chen_db_reconstruction.md (whole; statements and the places where irrationality and good rationals enter read, proofs not re-derived), and evidence/_index.md (the description of the entry point; its standing lines skipped). Library: library/additive_bases/yu_chen_2026_erdos_problem_354_i_strong_completeness_two_dyadic_floor_sequences/theorem.md (whole) and the heading lines of its _index.md; the canonical conversion beside that PDF at its Section 3 (hypotheses, (3.1)), Subsection 3.3, Theorem 5.1 with (5.2), Sections 6--7 and Appendix A (the mask table); library/additive_bases/fan_2026_strongly_complete_sets_conjecture_erdos/remark_4_2.md (Statement, Proof pointer, Dependencies, Bears on) and the canonical conversion beside the Fan PDF at the paragraph of (1.9) stating Hegyvári's conjecture and his proved case; library/additive_bases/hegyvari_1991_complete_sequences/_index.md (digest and results); the result lines of library/additive_bases/hegyvari_1994_sumset_certain_sets/_index.md and the statement lines of library/additive_bases/jenw1n_2026_erdos_354_part_i_lean_proof/_index.md, library/additive_bases/jenw1n_2026_erdos_354_part_i_lean_proof/target.md and library/additive_bases/kitamura_2026_lean_proof_erdos_problem_354_ii/_index.md, located by search. Problem page wiki/problems/additive_bases/E0354/_index.md: the Statement and Formulation paragraphs; the "Current assessment", "Progress and known results" and "Linked library material" sections were not read.

Exclusions and exposures. The live lead page, the live connections page, every file under an evidence/verify/ folder and all review-status text were excluded. Four exposures occurred, none of them mathematics absent from the subject: (1) while copying the frontmatter and heading skeleton of an existing record in this folder, its desc field and its heading and finding-location lines were printed; the desc summarizes that record's verdicts on an earlier revision of the same pages. No finding below is taken from it; each verdict rests on the derivations and computations recorded here, and the grader rules on materiality by the content test of docs/verification.md. (2) The problem page's "Status" paragraph and its frontmatter desc, which record the site-accepted Lean answer to the first question, were printed with the Statement; the lead itself states the same fact. (3) One remnant line of the folder evidence index naming a report count and "no tier is assigned" was printed. (4) The lead's evidence entry point wiki/research/leads/rational_ratio_dyadic_floor_descent/evidence/main.py was executed as a black box (its code and the evidence index were not read): exit status 0, "ALL CHECKS PASS (23 checks)", with the certificate lines agreeing with the independent recomputation below. (5) A read-only wiki lint run after this record was written printed the pending index rows of this folder and of another folder, including the desc lines of two re-review records on other leads; none concerns this subject, and the run changed no file.

Computation. Two independent checks were written from the mathematical statements, not from repository code, and kept in scratch; their methods are stated here so a reader can rerun them. First, the certificates: weights $a_i=\lfloor2^i\alpha\rfloor$, $b_i=\lfloor2^i\beta\rfloor$ computed exactly (integer division for a rational coefficient, isqrt(m << 2i) for $\sqrt m$), the subset sums of the first $n$ layers kept as one Python integer bitset (P |= P << w), and the certificate located as the least index $A$ at which the bit string holds $b_n$ consecutive ones, reported at the least $n\ge1$ where one exists. This differs from a table- or set-based subset sum enumeration in its primitive. Second, the certificate lemma of the Theorem 5.1 page: the mask table transcribed from Appendix A of the source conversion (twelve templates, 125 nodes, 113 links), each node's two offsets reconstructed as linear forms in $(p,q)$ with constants, condition 1 (constants differ by one, in $[0,22]$) checked directly, conditions 2 and 3 checked for every choice of the form entering the maximum and the minimum, positivity on the open cone $q<p<2q$ decided by $2a+b\ge0$, $a+b\ge0$, $(a,b)\ne(0,0)$ for the form $ap+bq$, over all four third-digit pairs. Result: no failure. The routine for the first check is:

def certificate(alpha, beta, max_depth):      # alpha, beta: floor2n(i)
    P = 1
    for n in range(0, max_depth + 1):
        if n >= 1:
            b_n = beta(n)
            s = bin(P)[2:][::-1]              # s[j] == "1" iff j in P_n
            A = s.find("1" * b_n)
            if A >= 0:
                return n, A, b_n
        P |= P << alpha(n)
        P |= P << beta(n)
    return None

Restatement

The lead attacks Hegyvári's conjecture, as Fan's preprint and the problem page record it, in the cases left open after the irrational-ratio theorem and Hegyvári's one-dyadic case: $\alpha/\beta$ rational and not a power of $2$, neither coefficient a dyadic rational. After a dyadic rescaling the ratio is $p/q\in(1,2)$ in lowest terms, $\alpha=p\gamma$, $\beta=q\gamma$, $\gamma$ not dyadic. The route transports the Yu--Chen permanent descent: Theorem 5.1 (a long exact doubling block followed by a nonzero conversion lowers the modular run invariant $h$ of every later layer by one, with completeness when $h\le1$) uses only the interlacing of the two rays and the digit recurrence, with the constant $K_*(p,q)=2q(p-1)+4(p+q)+64$ computed from the reduced pair of the block's first layer. At a rational ratio that reduced pair is $(p,q)$ at every exact layer ($qa_s=pb_s$), so a common zero run of the conversions of length at least $L_0=\lceil\log_2K_*\rceil-1$ starting at an exact layer gives one descent, and a run of $L(p,q)=L_0+\lceil\log_2p\rceil$ zeros in the expansion of $\gamma$ produces such a common run. Hence: Lemma 354.1 (incompleteness of a normalized non-dyadic pair forces an event in $(n,4n]$ for all large $n$), Proposition 354.2 (infinitely many runs of $L(p,q)$ zeros in a non-dyadic $\gamma$ give completeness of the normalized sequence), Corollary 354.3 (strong completeness of $A_{\alpha,\beta}$, hence for almost every $\gamma$ on each rational-ratio line, by Borel), and Lemma 354.4 (a run of $b_n$ consecutive subset sums of the first $n$ layers of a normalized pair certifies completeness), with exact certificates for five pinned pairs and thirty-two pairs $(3\cdot2^T/d,2\cdot2^T/d)$. The gap is the set of coefficients with only finitely many qualifying common runs, which contains every rational $\gamma$ of small odd denominator part and uncountably many irrationals; for them the irrational-case continuation fails because a rational ratio has finitely many good rational approximants and lies in the compact set $\mathcal C_k$, so that returns between exact layers can cost two events.

Statements and verdicts

• S1, the target (lead, "Target and status evidence"; connections, "Target"). Refutation-failed for the mathematics: Fan's conversion at (1.9) states the conjecture exactly as the lead does ($\alpha\not\sim \beta$, at least one coefficient not a dyadic rational, dyadic rational meaning $\alpha\sim n$ for a nonzero integer $n$) and records Hegyvári's proved case as "exactly one of $\alpha,\beta$ is a dyadic rational"; the reduction to $p/q\in(1,2)$, $q\ge2$, $\alpha=p\gamma$, $\beta=q\gamma$ is correct, as is the Bézout remark that $\gamma$ is dyadic exactly when both coefficients are (the dyadic rationals with $0$ form the ring $\mathbb Z[1/2]$). The sentences reporting what the problem page's Current assessment records are not a mathematical claim and were outside the read set; the grader or integrator checks them. Rescaling one coefficient by a nonnegative power of $2$ replaces the set by a subset, so strong completeness transfers upward, as the connections page says.
• S2, nothing about the ratio enters Sections 1--6 (lead, "Route and established inputs"; connections, "Where irrationality enters"). Refutation-failed. Re-derived: items 1--4 and 6 of the normalization page use only $N<M<2N$, $N\ge2$; item 5 uses only that the conversions are not eventually zero, which fails exactly when both coefficients are dyadic (then $\alpha=a_{n_0}2^{-n_0}$); items 1--3 use irrationality only to exclude $2^k\alpha_0/\beta_0=1$. Theorem 5.1's Steps 1--5 use $\gcd(p,q)=1$, $q<p<2q$, $S_n<d(p+q)$ and the interlacing: Step 1's covering of $[F,(p+q)(K-1)-F]$ (the residue choice $qy\equiv z\bmod p$ with $0\le y<p$ and the reflection), Step 2's two bounds on $(z-\sigma_0-f)/d$, (4.1) as $64d-42\ge22d$, the chain union from $L_1\le p+2q$ and $U_s\ge7p+6q$, (4.3) as $20d-13>0$, and Step 5's mesh and projection all check; the source's Section 3 and Theorem 5.1 carry the same hypotheses and constant. The certificate lemma, the one finite input, holds under the independent check described above. Only (5.2) brings in the layer index, through $p\le a_n<2^n(M+1)$.
• S3, Lemma 354.1. Refutation-failed. Infinitely many events as in S2; Step 2 of the theorem page consumes Theorem 5.1, (5.2) and the infinitude of events only: a qualifying pair $n<m$ with $m-n\ge2n+C_M$ gives $K=2^{m-n}\ge K_*$ by (5.2), the permanent bound $h_t\le\max(0,h_n-1)$ for $t\ge m+3$ with completeness if $h_n\le1$, so under incompleteness infinitely many qualifying pairs would give a strictly decreasing integer sequence bounded below by $2$; and with $m<3n'+C_M$ for the last event $n'\le n$ and $n\ge n_0=\max(t_1,C_M)$ the successor event lies in $(n,4n]$.
• S4, Proposition 354.2. Refutation-failed. (ii): $\delta_s=qa_s-pb_s =-q{2^s\alpha}+p{2^s\beta}$ since $q2^s\alpha=p2^s\beta$; on a common zero run of length $L'\ge\log_2p$ from $s$ both fractional parts are below $2^{-L'}$, so $|\delta_s|<p2^{-L'}\le1$ and $\delta_s=0$. (iii): with $x=\{2^s\gamma\}<2^{-L}$ and $2^L\ge p$, ${2^s\alpha}=px< 2^{-(L-\lceil\log_2p\rceil)}=2^{-L_0}$ and likewise for $\beta$, so the conversions vanish on $[s,s+L_0)$; $L_0>\log_2p$ because $K_*>4p$. (iv): the maximal common run $[s',s'+L'-1]\supseteq[s,s+L_0)$ is finite, its start is exact by (ii), and the Section 3 hypotheses hold at $n=s'$, $\ell=L'+1$, $K=2^\ell\ge2^{L_0+1}=2^{\lceil\log_2K_\rceil}\ge K_$, with $r=s'+L'+4$. (v): infinitely many distinct maximal runs since the starts tend to infinity and each run is finite; reading the bound of run $j$ at $t=s'_{j+1}\ge s'_j+L'_j+4$ gives the descent. The equality of exact layers with $\{2^s\gamma\}<1/p$ (connections, first bullet) is also right: with $2^s\gamma=I+x$, $\delta_s=q\lfloor px\rfloor -p\lfloor qx\rfloor$, and $\gcd(p,q)=1$ with $0\le\lfloor px\rfloor<p$ forces $\lfloor px\rfloor=0$.
• S5, Corollary 354.3 and the almost-every consequence. Refutation-failed. The normalization of items 1--3 needs only $2^k\alpha_0/\beta_0\ne1$; the normalized pair has ratio $p/q$ and $\gamma'=2^v\gamma$, a shift of the expansion, so the hypothesis survives; the reduction gives completeness of $A_{\alpha,\beta}\setminus F$ for every finite $F$. Normal numbers to base $2$ contain every finite block infinitely often and are irrational, so the full-measure statement follows from Borel's theorem, which the lead names as imported.
• S6, the constants. Refutation-failed. $K_*(3,2)=92$, $\lceil\log_292\rceil=7$, $L_0=6$, $L=8$; $K_*(5,4)=132$, $\lceil\log_2132\rceil=8$, $L_0=7$, $L=10$; $K_*(5,3)=120$, $\lceil\log_2120\rceil=7$, $L_0=6$, $L=9$; $2^{\lceil\log_2K_*\rceil}=128$ at ratio $3/2$.
• S7, Lemma 354.4. Refutation-failed. With $\Lambda_0=b_n$ consecutive integers from $A$ and weights $c_0=b_n<c_1=a_n<c_2=b_{n+1}<\cdots$ of indices at least $n$, $c_{j+1}\le2c_j$ by item 4, adjoining $c_j\le \Lambda_j$ yields $\Lambda_j+c_j$ consecutive integers from $A$ because the two intervals abut or overlap, and $\Lambda_{j+1}\ge2c_j\ge c_{j+1}$ keeps the invariant; the lengths diverge, so every integer at least $A$ is a subset sum of distinct weights, which are distinct nonzero elements of the set. The sentence attributing this to the mesh lemma "with gap 1" is loose: $b_n$ consecutive integers have span $b_n-1$, one short of the mesh lemma's hypothesis $\operatorname{span}(W)\ge c$; the count argument the proof actually gives replaces it (Suggested corrections).
• S8, the five pinned certificates (connections, after Lemma 354.4; lead, fourth bullet). Refutation-failed. Independent recomputation: $(48/5,32/5)$, normalized with $N=6$, $M=9$, least depth $9$, least start $1423$, $b_9=3276$; $(48/7,32/7)$: $9$, $577$, $2340$; $(80/9,64/9)$: $7$, $445$, $910$; $(6\sqrt2,4\sqrt2)$: $7$, $183$, $724$; $(5\sqrt2,3\sqrt2)$: $3$, $18$, $33$. All five pairs are normalized.
• S9, the sweep. Refutation-failed. The odd $d$ with $5\le d\le99$ not divisible by $3$ number $32$; the least normalizing $T$ gives the pairs the page names for $d=23,89,91$ ($T=6,8,8$); all $32$ certify by depth $12$, $29$ of them by depth $9$, and the three exceptions are $d=23$ at depth $10$ from $2139$ with $b_{10}=5698$, $d=89$ at depth $12$ from $11036$ with $b_{12}=23563$, $d=91$ at depth $11$ from $4323$ with $b_{11}=11522$, exactly as stated.
• S10, the controls. Refutation-failed. $(\sqrt{50},\sqrt{20})$ is normalized ($N=4$, $M=7$) and certifies at depth $6$ (from $77$ with $b_6=286$). $(5/2,5)$ has no certificate to depth $16$, and cannot have one at any depth: its weights are $2$ and $5\cdot2^k$, each power twice, so every subset sum is $\equiv0$ or $2\pmod5$ and no three consecutive integers are represented, which is also the incompleteness the page asserts.
• S11, "outside every held source." Refutation-failed within the read set. The Yu--Chen theorem and the site-accepted Lean statement concern irrational ratios; the base-$\sqrt\varphi$ Lean proof concerns another base; Fan's preprint proves no two-ray completeness and records Hegyvári's one-dyadic case; Hegyvári 1991 gives a measure dichotomy and a completeness criterion, Hegyvári 1994 incompleteness constructions and typicality for a dyadic $\alpha$; none proves completeness of a pair with rational ratio and two non-dyadic coefficients. Sources not held are outside the sentence's scope.
• S12, "outside Corollary 354.3" (lead, fourth bullet; connections, "none of these pairs is covered by Proposition 354.2"). For the thirty-five rational pairs refutation-failed: $\{2^s\gamma\}\ge1/d$ bounds every zero run of $1/d$ by $\lceil\log_2d\rceil-1\le6<8$ for $d\le99$, and the maximal runs are $2$ for $16/5$ and $16/7$ and $3$ for $16/9$ (ratio $5/4$, where $10$ is needed). For $(6\sqrt2,4\sqrt2)$ and $(5\sqrt2,3\sqrt2)$: proof defective; statement unresolved. Whether $2\sqrt2$ has infinitely many runs of eight zeros, or $\sqrt2$ of nine, is an open question (both hold if $\sqrt2$ is normal), so the lead's "outside Corollary 354.3" asserts for these two pairs something unproved; the connections page's wording, "not known to have infinitely many such runs", is the supported one (Required corrections, R1).
• S13, the gap paragraph (lead, "The consequential gap"; connections, first bullet). The defining sentence "only finitely many maximal common zero runs of length $\lceil\log_2K_*\rceil-1$ at exact layers" is refuted-as-stated under its literal reading and correct with "at least": at ratio $3/2$, $\gamma=1/511$ has conversions zero on $[9j,9j+6]$ and nonzero at $9j+7$, $9j+8$, so its maximal common zero runs at exact layers all have length exactly $7$, none has length $6$, and the literal sentence places $1/511$ in the gap while the same paragraph says it is covered by Corollary 354.3 (Required corrections, R2). Every other claim of the paragraph is refutation-failed: at an exact layer a common run of length at least $L_0$ means $p\{2^s\gamma\}<2^{-L_0}$, hence $\{2^s\gamma\}<2^{-\lceil\log_2K_*\rceil}$; a rational $\gamma$ with odd denominator part $v$ has $\{2^s\gamma\}\ge1/v$ for large $s$, so $v\le2^{\lceil\log_2K_*\rceil}$ ($128$ at ratio $3/2$, $256$ at $5/4$) excludes such runs, which covers every certified rational pair ($d\le99$, $v=5,7$ at ratio $3/2$, $v=9$ at ratio $5/4$); an irrational $\gamma$ whose expansion has no zero run of length $\lceil\log_2K_*\rceil$ is excluded too, and there are uncountably many; $\gamma=24/7$ has $\{2^s\gamma\}\in\{3/7,6/7,5/7\}$, never below $1/3$, with $\delta_s=2,1,1$ as stated; and a run shorter than $L_0$ gives $K\le2^{L_0}<K_*$, so Theorem 5.1 does not apply with the fixed constant.
• S14, the two obstructions (lead, "The consequential gap"; connections, second bullet). Refutation-failed. A reduced $p'/q'\ne p/q$ has $|p/q-p'/q'|\ge1/(qq')$, so $|p/q-p'/q'|<1/q'^2$ forces $q'<q$: finitely many good rationals, so (DB), which needs a good denominator at least $2^n\beta$, and the crossing denominator $D(b)$ for $b>q$ are unavailable. With $2^J\le p<2^{J+1}$, $p/2^J$ and $q/2^J$ are sums of at most $k$ elements of $\{2^{-j}\}$ with $q/2^J>1/2$, so $p/q\in\mathcal C_k$ once $k$ is at least the number of ones in $p$ and $q$. The witness: for $x=\{2^s\gamma\}\in[1/4,1/3)$ at ratio $3/2$, $\lfloor6x\rfloor=1$, $\lfloor4x\rfloor=1$, $\lfloor12x-2\rfloor=1$, $\lfloor8x-2\rfloor=0$, so the conversions are $(1,1),(1,0)$, $U=3$, $V=2$, $\delta_{s+2}=4\delta_s+2U-3V=0$, and $\{2^{s+2}\gamma\}=4x-1<1/3$ is the next exact layer after the nonexact $s+1$: two events per return. The finite-event decay then only bounds the missing count by $e^{-aK_n}$ with $K_n\gg\log n$ under bounded spacing, a polynomially small fraction.
• S15, "pair by pair certifiable but not uniformly proved." Refutation-failed as a statement about this route; whether another source settles these coefficients is outside the read set.
• S16, the next step (lead, "Next step"). The identity $a_{n+i}=2^ia_n+r_i$ holds with $r_i=\lfloor2^i\{2^n\alpha\}\rfloor$, the integer whose binary digits are the conversions $u_n,\ldots, u_{n+i-1}$; for rational $\gamma$ with odd denominator part $v$ these digits are eventually periodic with the order of $2$ modulo $v$ as a period, but $r_i$ itself is not periodic in $i$ (Suggested corrections, S3). "Each positive case is a proved instance" is refutation-failed through Lemma 354.4. The outcome sentences are plans, not a mathematical claim; the conditional that an incomplete pair with rational ratio, not a power of $2$, and two non-dyadic coefficients would refute the conjecture is correct.
• S17, "Provisional premises." "The a.e. statement imports Borel's normal number theorem" is refutation-failed. The section omits the premises the route actually consumes (Premises below; Suggested corrections, S1).

Checklist

• Quantifiers and scope: one defect, the dropped "at least" in the gap definition (S13); the "almost every" of Corollary 354.3 is never upgraded to "every", and the lead's "every coefficient with long zero runs" means infinitely many runs, as the statement says.
• Circularity: none; the descent is applied forward from each qualifying block and incompleteness enters only as the hypothesis of an infinite descent.
• Model and convention changes: none; the normalized tails and the original set are connected by the reduction, re-derived.
• Finite and statistical overreach: none; each certificate is claimed for its own pair only, and the controls are controls.
• Uniformity: the constant $K_*(p,q)$ is fixed by the ratio and the length condition (5.2) is not used, as stated; $C_M$ depends on $M$ only.
• Extremal conclusions: the depths and starts are least values, confirmed in the claim's own units.
• Consequences and composition: the lead's "outside Corollary 354.3" for the two square-root pairs overstates (S12); every other "hence" checks.
• Computation: exact arithmetic throughout the recomputation; the claimed figures match.
• Reproduction: the lead's entry point exits $0$ with $23$ passing checks; its coverage claim ("the pinned pairs, a sweep over thirty-two rational coefficients, and an incomplete control pair") matches its printed lines and the recomputation.
• Source and verdict fidelity: the conjecture, its proved case, Section 3, Theorem 5.1, (5.2) and Appendix A agree with the source conversions; verdict quotations are outside this reviewer's subject.

Weakest steps

• W1, exactness at the start of a maximal run. The descent needs the reduced pair $(p,q)$ at the layer where Theorem 5.1 is applied, which is the start $s'$ of the maximal common run, not the position $s$ of the run in $\gamma$. Re-derived: the run from $s'$ has length $L'\ge L_0>\log_2p$, so $|\delta_{s'}|<p2^{-L'}<1$ and $\delta_{s'}=0$. Without this the constant would be that of a nonexact layer's reduced pair, which grows with the layer.
• W2, the length of the induced common run. $x<2^{-L}$ gives $px<2^{-L_0}$ only because $2^{\lceil\log_2p\rceil}\ge p$; the lead's $L=L_0+\lceil\log_2p\rceil$ is exactly what is needed, and the common run then has length at least $L_0$, giving $K=2^{L'+1}\ge2^{L_0+1} =2^{\lceil\log_2K_\rceil}\ge K_$. Here $2^{L_0+1}$ is used, not $2^{L_0}<K_*$; the proof's $\ell=L'+1$ counts the block of $L'$ exact pairs plus the layer of the nonzero conversion, matching Section 3.
• W3, the finite certificate. Everything above rests on the certificate lemma, a finite statement about 125 nodes. Rechecked independently from the source conversion, including the four-choice reading of conditions 2 and 3; it holds.

Strongest attack

The attack sought a coefficient that satisfies the hypothesis of Proposition 354.2 or sits inside the lead's gap set while the mechanism misbehaves, and a certificate figure that fails exact recomputation. (a) At ratio $3/2$ the pair $(1536/511,1024/511)$, normalized with $N=2$, $M=3$, has exact layers and maximal common zero runs exactly where Proposition 354.2 predicts ($[0,6]$, $[9,15]$), $h_n=0$ for $0\le n\le20$, and a certificate at depth $2$: consistent. (b) The gap pair $(192/23,128/23)$ has $h_n$ equal to $0,0,2,2,0,1,1,1,0,\ldots$ for $n\le16$ and no common zero run longer than $2$ in that range, so the fixed-constant descent indeed never fires there, yet the pair certifies at depth $10$: the gap is a limit of the mechanism, not evidence of incompleteness, as the lead says. (c) The sharper threshold: a common run of length at least $L_0$ at an exact layer requires $\{2^s\gamma\}<2^{-L_0}/p$, so odd parts $v<p2^{L_0}$, that is $v\le191$ at ratio $3/2$, are excluded, not only $v\le128$; the page's bound is implied and its "in particular" clauses survive. Conversely $\gamma=1/193$ at ratio $3/2$ has, at every $s$ with $\{2^s\gamma\}=1/193$ (infinitely many, by periodicity), a maximal common run of length exactly $6=L_0$ starting at an exact layer (the preceding conversion is nonzero since $\{2^{s-1}\gamma\}=97/193$), so the descent of Proposition 354.2's proof applies, while $\gamma$ has no run of eight zeros and Corollary 354.3's hypothesis fails: the corollary's sufficient condition is weaker than the mechanism, which is no error (Notes). (d) The certificates of all thirty-nine pairs were recomputed exactly; none fails. The attack on the mathematics failed; the two findings are wording defects (S12, S13).

Premises

The route consumes, in order of load: the normalization page's items 1--6 and reduction; Lemmas 2.1--2.3; Theorem 5.1 with its certificate table and (5.2); Step 2 of the theorem page; the definitions of good rationals, (DB), $D(b)$ and $\mathcal C_k$ from the digit-budget, windows and bounded-spacing pages for the two obstructions; Borel's normal number theorem (standard, imported) for the almost-every clause; and Fan's preprint (held, p. 4) with the problem page's assessment for the statement of the conjecture and Hegyvári's proved case. All repository inputs are author-recorded reconstructions of an unrefereed manuscript; their current standing was outside this review's read set and is for the grader to read at filing. Within this review the consumed deductions of the normalization page, the three lemmas, Theorem 5.1 and Step 2 of the theorem page were re-derived and the certificate table was rechecked, so the transport stands on them as mathematics; it stands on nothing else. The site-attested Hegyvári case delimits the target and is consumed by no proof. No external premise is unread except the problem page's assessment sentences, which this reviewer could not open.

Required corrections

• R1. Lead, "Route and established inputs", fourth bullet, final clause. Replace "all outside every held source and outside Corollary 354.3." with "all outside every held source and none covered by Corollary 354.3: the rational coefficients have no zero run of the required length, and $\sqrt2$ is not known to have infinitely many."
• R2. Lead, "The consequential gap", first sentence. Replace "Coefficients $\gamma$ with only finitely many maximal common zero runs of length $\lceil\log_2K_*\rceil-1$ at exact layers:" with "Coefficients $\gamma$ with only finitely many maximal common zero runs of length at least $\lceil\log_2K_*\rceil-1$ at exact layers:".

Suggested corrections

• S1. Lead, "Provisional premises". After the first sentence add: "The transport imports, as author-recorded reconstructions of an unrefereed manuscript, items 1--6 and the reduction of the normalization page, Lemmas 2.1--2.3, Theorem 5.1 with the Appendix A certificate (finite data rechecked by the folder's evidence) and Step 2 of the theorem page; Lemma 354.4 and the certificates rest on the normalization page alone."
• S2. Connections page, paragraph after the proof of Lemma 354.4, first sentence. Replace "This is the mesh lemma of the Lemma 2.2 page with gap $1$, as the digit-budget page uses it," with "This is the abutting case of the mesh lemma of the Lemma 2.2 page with gap $1$: the $b_n$ consecutive integers have span $b_n-1$, one short of the lemma's hypothesis, which the count argument above replaces; the digit-budget page uses the lemma with span at least the next weight,".
• S3. Lead, "Next step", first sentence. Replace "with a periodic correction $r_i$" with "with the correction $r_i=\lfloor2^i\{2^n\alpha\}\rfloor$, the integer whose binary digits are the eventually periodic conversions". The same phrase on the connections page ("Next investigation", first direction) takes the same replacement.
• S4. Lead, "Target and status evidence". After "as Hegyvári's (site-attested, paper not held)" add "and recorded in Fan's preprint (held, p. 4)", which the held source states.

Notes

• N1. The hypothesis of Proposition 354.2 can be weakened without changing its proof: it suffices that the expansions of $p\gamma$ and $q\gamma$ have infinitely many common runs of $L_0=\lceil\log_2K_*\rceil -1$ zeros, since such a run starts at an exact layer by (ii) and supplies the block of (iv) directly. The example $\gamma=1/193$ at ratio $3/2$ (Strongest attack, (c)) is complete by this reading and outside Corollary 354.3 as stated. The exact rational threshold for the gap set is $v<p2^{L_0}$, that is odd $v\le191$ at ratio $3/2$ and $v\le639$ at ratio $5/4$.
• N2. Lemma 1 of the held Hegyvári 1991 paper, as its card records it, is a finite completeness criterion for the same two-ray sets (an interval $[k,a_p]$ of subset sums with $k$ below two weight differences). It is a relative of Lemma 354.4 that the connections page does not mention when it names van Doorn's Lemma 5 as the single-ray analog; citing it beside Lemma 354.4 would place the certificate idea in the held literature.
• N3. "Period the order of $2$ modulo $v$" (connections, "Next investigation") is a period, not necessarily the least one: for $\gamma=1/9$ and $p=3$ the digits of $3\gamma=1/3$ have period $2$ while the order of $2$ modulo $9$ is $6$.
• N4. The controls in the lead's entry point include two checks on the structure of $(5/2,5)$ ("beta-ray is the shifted alpha-ray", "merged multiset at most doubles from the second term") besides the absence of a certificate; the incompleteness itself follows from the residues modulo $5$ (S10) and needs no depth bound.
• N5. The extraction's "Provisional premises" ends with a sentence pointing to a "Current review" section that the extraction removed; this is a consequence of the cut, not a defect of the page.

Verdict

Mathematics: refutation-failed for Lemma 354.1, Proposition 354.2, Corollary 354.3 with its almost-every clause, Lemma 354.4, the constants $L(3,2)=8$, $L(5,4)=10$, $L(5,3)=9$, the two obstructions with the two-event witness, and the certificate figures of all thirty-nine pairs (five pinned pairs, thirty-two sweep pairs with the three deep denominators, two controls), each under the independent derivations and recomputations recorded above. Refuted-as-stated, as wording only: the gap set's defining sentence in the lead, which drops "at least" and under its literal reading contains the coefficient $1/511$ that the paragraph itself calls covered (R2). Proof defective; statement unresolved: the lead's "outside Corollary 354.3" for $(6\sqrt2,4\sqrt2)$ and $(5\sqrt2,3\sqrt2)$, which asserts an open property of $\sqrt2$ (R1). The route as written stands on the author-recorded reconstructions it names, re-derived here at the deductions it consumes, and claims nothing beyond them once R1 and R2 are made. The lead's conclusion that Hegyvári's conjecture remains open on this route for the coefficients of its gap set, pair by pair certifiable and not uniformly proved, is correct as a statement about the route.

Limitations. No tier is asserted. The standing of the reconstruction pages, the lead's evidence index and code, the problem page's assessment and every filed review were outside the read set; the exposures listed under Subject and independence are disclosed for the grader's materiality ruling. The independent checks were written from the mathematical statements and kept in scratch; their methods and the one routine that carries the certificate figures are recorded above for rerun, and the certificate-lemma check is reproducible from Appendix A of the source conversion by the stated cone criterion. Whether a source outside the library settles any of the gap coefficients was not searched.