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Subject and independence

Role. Independent reviewer in a fresh context, commissioned for refutation of one section. The reviewer took no part in writing the connections page, the reconstruction page, or any other page of the folder, had no contact with the compilation that wrote them, and received only the assignment text.

Frozen subject. The source as it stood on 2026-09-28T13:50:48Z (called "the commit" below), path wiki/research/erdos_939/connections.md, section ## Shared results, lines 58 to 302 (from the H2 heading to the line before ## Problem 939 at r = 4: the two-summand case). The section holds Lemma 1 with its application paragraph and Corollary 1.1 (lines 60 to 146), Lemma 2 (lines 148 to 184), Theorem 3 and Corollary 3.1 with a coverage paragraph (lines 186 to 238), and Proposition 4 with Corollary 4.1 (lines 240 to 301). The paths named are what was read. The page's Conventions paragraph (lines 35 to 56) was read as the section's notation input.

Lead pages. The assignment names none, and the section names none. The page's $r=5$ section names a lead folder; it was not read.

Read set and depth.

• The connections page at the commit, whole, to locate the section boundaries; the section and the Conventions paragraph line by line.
• wiki/research/erdos_939/theorem_1_reconstruction.md at the commit, whole, as a premise (the identity, the data $J$, $t$, $v_\ell$, $P$, $B$, and the distinctness argument).
• The page recalled_abc_abcd_bounds.md of the source folder corvaja_zannier_2011_abcd_function_fields under library/diophantine_problems/, whole; the card _index.md of that source (provenance and reading coverage); and printed p. 438 (PDF p. 3) of the held journal PDF by text extraction, to confirm the two recalled bounds at claims level.
• wiki/research/erdos_940/finite_core_reduction.md, the section "Independent polynomial summands still reduce to fixed forms" only (statement and proof of the polynomial-factor proposition), read as the input consumed by Corollary 3.1; the proof was read because the section adapts that proof rather than the statement, so the bridge could only be checked against the proof's actual structure. The rest of that page was not read beyond its heading list.
• The Statement paragraph of wiki/problems/diophantine_problems/E0939/_index.md.
• docs/verification.md sections "Report contract", "Grading and claim standing" and "Audit checklist"; docs/approach_vetting.md; docs/evidence.md section "Source fidelity"; docs/math_authoring.md.

Exposure. Two incidental exposures, neither used: while locating the Statement paragraph of the Problem 939 page, its frontmatter (a desc line summarizing the problem's status and a status field) scrolled past; and the connections page's other sections, read only to fix the section boundaries, summarize in their own words what the target pages' assessments record. No _index.md, no evidence/ content, no review, grade, Current assessment or Known results text was opened, and no web search was made.

Restatement

• S1 (Lemma 1). Fix integers $r\ge2$ and $n\ge3$ and nonzero $A_1,\dots,A_n\in\mathbb Z[U,V]$ with $A_1+\cdots+A_{n-1}=A_n$, together with one presentation $A_i=c_i\prod_jH_{ij}^{e_{ij}}$ ($c_i$ a nonzero integer, $H_{ij}\in\mathbb Z[U,V]$, every $e_{ij}\ge r$, empty products allowed) such that $V^r$ divides $A_i$ whenever $|c_i|>1$; $A_1=K^r$ with $K\in\mathbb Z[U,V]$, $K(U,0)=\pm U^d$ and $K(0,V)=\pm V^{d'}$; and $A_2=c_2U^aV^b$ with $a+b\ge1$, the symbol $c_2$ being that of the presentation. For every integer $B\ge1$ divisible by every prime dividing some $c_i$ ($1\le i\le n$) and every integer $m\ge1$ with $\gcd(m,B)=1$ and $A_i(m,B)>0$ for all $i$: each of $A_1(m,B),\dots,A_{n-1}(m,B)$ is $r$-powerful, their greatest common divisor is $1$, and $A_n(m,B)$ is $r$-powerful. For such a $B$: if every $A_i(U,B)\in\mathbb Z[U]$ has positive leading coefficient then the positivity holds for all large $m$; if $A_n(U,B)$ is nonconstant then the sums $A_n(m,B)$ over $m$ coprime to $B$ take infinitely many values; if $A_i(U,B)\ne A_j(U,B)$ for all $i\ne j\le n-1$ then the $n-1$ values are pairwise distinct for all but finitely many $m$.
• S2 (application paragraph). For each $r\ge6$ the Theorem 1 identity of the reconstruction page, written in $U,V$ with $X=U^r$ and $Y=V^r$, satisfies (i) to (iv) with $A_1=(U^r-V^r)^r$, $K=U^r-V^r$, $A_2=2rU^{r(r-1)}V^r$, the other binomial and split terms as monomials, $A_n=(U^r+V^r)^r$ and $n=r-1$; and with $B=B_0=\prod_{p\in P}p$ and $m=q$ a prime above $B_0$, Lemma 1 returns the reconstruction's conclusions except distinctness.
• S3 (Corollary 1.1). $r\ge6$. (a) For every positive multiple $B$ of $B_0$ and every integer $m>B$ with $\gcd(m,B)=1$, the $r-2$ right-hand terms of the identity at $(m,B)$ are pairwise distinct positive $r$-powerful integers with greatest common divisor $1$, and their sum $N=(m^r+B^r)^r$ is $r$-powerful. (b) Distinct pairs $(m,B)$ give distinct tuples. (c) There are constants $c_r,C_r>0$ depending only on $r$ such that for all large $X$ the number of these tuples with $N\le X$ is at least $c_rX^{2/r^2}/\log\log X$ and at most $C_rX^{2/r^2}$.
• S4 (Lemma 2 and its normalization remark). For nonzero pairwise coprime $a,b,c\in\mathbb C[t]$, not all constant, with $a+b=c$, $\max(\deg a,\deg b,\deg c)\le n_0(abc)-1$, where $n_0$ counts distinct complex roots. This is the library's recalled Mason–Stothers bound in genus zero: the height of a polynomial tuple without common factor is its maximal degree, the support is the root set of $abc$ together with $\infty$, and $|S|-2=n_0(abc)-1$.
• S5 (Theorem 3 and sharpness). $r\ge2$. If $a,b,c\in\mathbb C[t]$ are nonzero, pairwise coprime, $r$-powerful as polynomials (every root of multiplicity at least $r$; constants included), not all constant, and $a+b=c$, then $r=2$ and none of $a,b,c$ is constant. Weighted form: if $a,b,c$ are $r_a$-, $r_b$-, $r_c$-powerful with $r_a,r_b,r_c\ge2$ under the same other hypotheses, then the sum of $1/r_f$ over the nonconstant $f$ exceeds $1$. The theorem is sharp: $(t^2-1)^2+(2t)^2=(t^2+1)^2$.
• S6 (Corollary 3.1). $r\ge2$; $a,b\in\mathbb Z[t]$, $c=a+b$, not all constant; there is an arithmetic progression such that for every sufficiently large integer $t$ in it the values $a(t),b(t),c(t)$ are positive $r$-powerful integers, and $\gcd(a(t),b(t))=1$ for infinitely many such $t$. Then $r=2$ and $a,b,c$ are all nonconstant. Hence for $r\ge3$ no one-parameter polynomial family of coprime $r$-powerful solutions of $a+b=c$ exists, and for $r=2$ none with a constant term. The proof consumes the author-recorded polynomial-factor proposition of the Problem 940 folder, with its proof adapted to inputs restricted to a progression.
• S7 (coverage paragraph). Corollary 3.1 covers a fixed polynomial identity evaluated on integer parameters; in the two-parameter device it applies along every line $B$ fixed and along every polynomially parametrized curve on which its hypotheses hold; it does not cover an identity whose terms share a nonconstant factor $D$ and whose members are taken only at the integer points where $D=\pm1$.
• S8 (Proposition 4). For $\omega$ a primitive cube root of unity and $Q=aU^2+bUV+cV^2\in\mathbb C[U,V]$ with $ac=-2b^2$, $\sum_{k=0}^{2}Q(\omega^kU,\omega^{-k}V)^5=(3bUV)^5$.
• S9 (Corollary 4.1). With $(a,b,c)=(1,1,-2)$: the three forms $Q_k=\omega^{2k}U^2+UV-2\omega^kV^2$ factor as $\omega^{2k}(U-\omega^kV)(U+2\omega^kV)$ and satisfy $Q_0^5+Q_1^5+Q_2^5=(3UV)^5$ in $\mathbb Q(\omega)[U,V]$; no proper subsum of the four-term relation vanishes; the four forms have no common factor; the Fermat quintic surface $x^5+y^5+z^5=w^5$ contains the conic parametrized by $(Q_0:Q_1:Q_2:3UV)$, defined over $\mathbb Q(\omega)$; no element of $\mathrm{PGL}_2(\mathbb C)$ acting on $(U,V)$ together with scalars on the four forms makes all four real; consequently the identity has no four-term version (in that sense) in $\mathbb Q[U,V]$, and Lemma 1 does not apply to it over $\mathbb Z$.
• S10 (provenance sentence). The identity was found by a numerical search at $r=5$ and then recognized and proved; the literature on conics on the Fermat quintic was not searched. This describes process and asserts no mathematics.

Checklist

• Quantifiers and scope. Pass. Every statement carries its own quantifiers: Lemma 1 is universal over the admissible $(B,m)$; Corollary 1.1 says "for all large $X$" with constants depending only on $r$; Theorem 3 needs "not all constant"; Corollary 3.1 needs "some progression" and "infinitely many"; the non-reality claim of Corollary 4.1 quantifies over $\mathrm{PGL}_2(\mathbb C)$ and scalars, and finding F3 asks that its consequence sentence say so. No "almost all" is upgraded to "all".
• Circularity. Pass. Lemma 2 is proved from first principles; Theorem 3 uses only Lemma 2; Corollary 3.1 uses Theorem 3 and the imported proposition, whose proof (read) uses neither. Corollary 1.1 uses Lemma 1 and the reconstruction page. Nothing consumes its own conclusion.
• Model and convention changes. Pass. Integer $r$-powerfulness (positive, every prime to exponent at least $r$, $1$ included) and polynomial $r$-powerfulness (root multiplicities, constants included) are both defined in Conventions and used consistently; joint coprimality (Lemma 1, Corollary 1.1) and pairwise coprimality (Lemma 2, Theorem 3, Corollary 3.1) are kept apart, and coincide for two summands; the substitution $X=U^r$, $Y=V^r$ is declared. The normalization from $S$-unit heights to degrees in S4 was re-derived (Premises).
• Finite and statistical overreach. Pass. The only finite computation mentioned, the numerical search behind Proposition 4, is labeled exploratory and supports no statement; the identity is proved symbolically. Corollary 1.1's count is analytic.
• Uniformity. Pass. Corollary 1.1 states that its constants depend only on $r$; the coprime-count error term $2^{\omega(B)}$ is bounded uniformly for $B\le M$ by $M^{o(1)}$; the "sufficiently large" thresholds in Corollary 3.1 are per polynomial and never asserted uniform.
• Extremal conclusions. Pass. Theorem 3's sharpness sentence was checked in its own units: the inequality $k/r>1$ with $k\le3$ admits exactly $(k,r)=(3,2)$, and the Pythagorean identity attains it. Corollary 1.1 claims no sharpness of the exponent $2/r^2$.
• Consequences and composition. Pass with findings F2 and F3. Corollary 3.1 inherits the author-recorded standing of the Problem 940 proposition and discloses it in place. The heading of Theorem 3 (F2) omits the qualifier $r\ge3$ that its statement carries; the "consequently" sentence of Corollary 4.1 (F3) leaves "version" undefined.
• Computation. Pass. The reviewer recomputed, in exact arithmetic over $\mathbb Q(\omega)$ with own code that expands the fifth powers directly (a different route from the page's multinomial-coefficient argument): the three coefficients $[U^8V^2]$, $[U^5V^5]$, $[U^2V^8]$ of $Q^5$ for three unrelated triples; Proposition 4 for eight rational triples with $ac=-2b^2$; the factorization of $Q_k$; $Q_0+Q_1+Q_2=3UV$; the linear independence of $Q_0,Q_1,Q_2$ (determinant $6+12\omega$); the non-vanishing of all fourteen proper subsums; and the two involution cases. Lemma 1 and Corollary 1.1(a) were checked exactly for $r=6,7,8,9$ on sixteen pairs $(m,B)$ each, with $B\in\{B_0,2B_0,3B_0,7B_0\}$ and composite $m$ included. All agree with the section.
• Reproduction. Not applicable. The section retains no evidence program and makes no rerun claim; the search scripts it mentions are filed under a lead outside this review.
• Source and verdict fidelity. Pass with finding F4. The recalled Mason–Stothers and Brownawell–Masser bounds were read on the held PDF (printed p. 438) and match the library page and the section's use; the reconstruction page's identity, data and distinctness argument are cited exactly; the Problem 940 proposition is quoted with its hypotheses and standing. Every statement is called author-recorded, and no tier is assigned. The preamble sentence discussed in F4 lies outside the section and is ambiguous about which input Lemma 2 reproves.

Weakest steps

W1: joint coprimality in Lemma 1. Let $p$ divide $A_i(m,B)$ for every $i\le n-1$. Since $n\ge3$, both $A_1(m,B)$ and $A_2(m,B)$ are among these. From $A_2(m,B)=c_2m^aB^b$, $p$ divides $c_2$, $m$ or $B$; a prime of $c_2$ divides $B$ by the choice of $B$; so $p\mid mB$, and as $\gcd(m,B)=1$ exactly one of $p\mid m$, $p\mid B$ holds. From $A_1(m,B)=K(m,B)^r$, $p\mid K(m,B)$. Write $K=\sum k_{ij}U^iV^j$. If $p\mid m$, every term with $i\ge1$ vanishes modulo $p$, so $K(m,B)\equiv K(0,B)=\pm B^{d'}\pmod p$, a unit modulo $p$ since $p\nmid B$; if $p\mid B$, likewise $K(m,B)\equiv K(m,0)=\pm m^d\pmod p$, a unit since $p\nmid m$. Both contradict $p\mid K(m,B)$. The step needs $c_2$ to be the full integer coefficient of the monomial (Finding F1): the argument is exactly what F1's witness breaks when a prime of that coefficient is hidden in an $H_{2j}$ and left out of $B$.

W2: the lower bound in Corollary 1.1. Put $M=\lfloor X^{1/r^2}/2\rfloor$ and count pairs $(B,m)$ with $B$ a multiple of $B_0$, $B\le M/2$, $M/2<m\le M$, $\gcd(m,B)=1$. Then $m>B$, and $m^r+B^r\le M^r+(M/2)^r<2M^r$, so $N<2^rM^{r^2}\le2^{r-r^2}X\le X$. By Möbius inversion the number of $m$ in $(M/2,M]$ coprime to $B$ is $\sum_{d\mid B}\mu(d)(\lfloor M/d\rfloor-\lfloor M/(2d)\rfloor)$, which differs from $(M/2)\phi(B)/B$ by at most the number of squarefree divisors, $2^{\omega(B)}$. For $B\ge3$, $\phi(B)/B\gg1/\log\log B\ge1/\log\log M$, while $2^{\omega(B)}\le C_\varepsilon B^\varepsilon$ for every $\varepsilon>0$ and $B^\varepsilon\le M^\varepsilon$, so each admissible $B$ contributes $\gg M/\log\log M$ values of $m$ once $M$ is large, uniformly in $B$. There are $\asymp M/B_0$ admissible $B$, giving $\gg M^2/\log\log M\asymp X^{2/r^2}/\log\log X$ pairs, and distinct pairs give distinct tuples because $N$ and the first term recover $m^r+B^r$ and $m^r-B^r$, hence $m$ and $B$. The reviewer also checked the coprime-count estimate numerically for $B\in\{30,60,90,210,420,2310,30030\}$ and $M\in\{10^3,10^4,10^5\}$: the error never exceeded $2^{\omega(B)}/6$. The upper bound is immediate: $N\le X$ forces $m,B\le X^{1/r^2}$.

W3: Corollary 3.1's adaptation of the imported proof to a progression. The imported proof factors $f=c\prod_jg_j^{m_j}$ over $\mathbb Z$ with distinct primitive irreducible $g_j$, fixes $j$, and picks a prime $p$ that divides some value of $g_j$ and avoids the finite set of primes dividing $c$, the leading coefficients, the discriminants and the pairwise resultants of the $g_i$; a root $n_0$ of $g_j$ modulo $p$ is then simple and is not a root of any other $g_i$, and among the $p$ lifts of $n_0$ modulo $p^2$ exactly one is a root of $g_j$ modulo $p^2$, so on a different lift's class modulo $p^2$ one has $v_p(g_j(n))=1$ and $v_p(g_i(n))=0$ for $i\ne j$, hence $v_p(f(n))=m_j$. When inputs are confined to a progression $t\equiv t_0\pmod M$, add $p\nmid M$ to the finite avoidance set; $\gcd(p^2,M)=1$ lets the Chinese remainder theorem combine the class modulo $p^2$ with the progression into a class modulo $p^2M$, all of whose members $n$ satisfy $n\equiv n_0\pmod p$ and avoid the root lift modulo $p^2$, so $v_p(f(n))=m_j$ still, and the class contains arbitrarily large $n$ at which $f(n)$ is a positive $r$-powerful integer by hypothesis; thus $m_j\ge r$. Positivity on the whole line is never used, only positivity at the chosen inputs. The rest of Corollary 3.1 (separability of irreducible factors over $\mathbb Q$, Gauss's lemma for the common factor, then Theorem 3) was re-derived without difficulty.

Strongest attack

• S1. The reviewer tried to break joint coprimality by hiding a prime of the monomial's coefficient inside an $H_{2j}$ so that $B$ need not carry it: with $r=2$ and $(V^2-U^2)^2+(2UV)^2=(U^2+V^2)^2$, presenting $A_2=1\cdot(2UV)^2$ satisfies (i) with $c_2=1$ and (ii) vacuously, and $B=1$, $m=3$ gives the values $64,36,100$ with $\gcd(64,36)=4$. The attack fails against the statement as written because (iv) reuses the symbol $c_2$, so a single presentation must have $\prod_jH_{2j}^{e_{2j}}=U^aV^b$ and $c_2=4$, after which (ii) forces $V^2\mid A_2$ and $B$ must be even; with $B=2$, $m=3$ the values are $25,144,169$ with greatest common divisor $1$. The attack survives only as the clarification F1. A second attack, $B=1$ with all $|c_i|=1$, leaves the argument intact ($p\mid m$ and $K(m,1)\equiv\pm1$). Hiding constants in $H_{ij}$ for $i\ne2$ does not hurt powerfulness, since $p\nmid c_i$ then puts $p$ in some $H_{ij}(m,B)$ with exponent at least $r$.
• S2. Checked (i) to (iv) against the reconstruction's display (1): $K(U,0)=U^r$ and $K(0,V)=-V^r$; every coefficient other than $1$ is $2\binom rj$ with odd $j\ge1$ or a $v_\ell$, and sits on $V^{rj}$ or $V^{3r}$; $P$ is exactly the set of primes of these coefficients, so $B_0$ qualifies, and a prime $q>B_0$ is coprime to $B_0$ with $q^r>B_0^r$. No gap found.
• S3. Attacked distinctness at odd $r$, where the $j=r$ term $2B^{r^2}$ and $(m^r-B^r)^r$ both have $p$-adic valuation $0$ for $p\mid m$: equality would make $2$ a rational $r$-th power. Attacked injectivity through the tuple order: the first term is $(m^r-B^r)^r$ by the identity's order, and sets instead of tuples change the count by at most $(r-2)!$, absorbed in the constants. Attacked the count's uniformity in $B$ (W2). Nothing broke.
• S4. Attacked the case $W=0$ and the case of a constant summand: if $a/b$ is constant then coprimality makes $b$ a unit, so all three are constant, excluded; if $a$ is constant then $W=ab'$ still has degree $\deg a+\deg b-1$. The third expression $-W=bc'-b'c$, not displayed, gives the bound for $\deg a$ (note F5). The normalization remark was checked against the library interface: "$f_2/f_1$, $f_3/f_1$ not both constant" is equivalent to "not all constant" under pairwise coprimality, and $S$ must contain $\infty$ as the pole of the nonconstant members.
• S5. Attacked the weighted form at the boundary $(r_a,r_b,r_c)=(2,3,6)$, where the sum equals $1$: the proof gives $D\le D-1$, so no coprime polynomial solution of $x^2+y^3=z^6$ exists, consistent with the statement's strict inequality. Attacked $k=2$ with a constant term: the identity forces $k\ge2$ and $k/r>1$ forces $r<k$. No gap.
• S6. Attacked with a progression whose modulus shares primes with the bad set, with positivity holding only on the progression, and with a constant summand: the adapted proof (W3) avoids finitely many primes, uses positivity only at the chosen inputs, and constants are $r$-powerful polynomials by convention, so Theorem 3 applies and excludes them. The Gauss-lemma step needs only that a primitive nonconstant $D_0$ has $|D_0(t)|\ge2$ for large $t$. No gap.
• S7. Attacked the claim that the device runs "along the lines $B$ fixed": Corollary 3.1 wants one progression, while Lemma 1 ranges over all $m$ coprime to $B$; one residue class modulo $B$ coprime to $B$ is a progression on which the hypotheses hold, so the reduction stands.
• S8. Exact recomputation for eight rational triples with $ac=-2b^2$, including non-integral ones, and the degenerate $b=0$ case (both sides vanish). No gap.
• S9. The reviewer re-derived the classification of anti-holomorphic involutions preserving $\{0,\infty\}$: $\tau(z)=\lambda\bar z$ needs $|\lambda|=1$ and $\tau(z)=\lambda/\bar z$ needs $\lambda$ real for $\tau^2=\mathrm{id}$; the first fixes $\{1,-2\}$ only with $\lambda=1$ and then sends $\omega$ to $\omega^2\notin\{\omega,-2\omega\}$; the second needs $\lambda=-2$ and is then fixed-point free, while $g\circ\overline{\phantom{z}}\circ g^{-1}$ fixes the circle $g(\mathbb{RP}^1)$. An independent route confirms the conclusion for the identity as it stands: the parametrized curve lies in the plane $w=x+y+z$ (since $Q_0+Q_1+Q_2=3UV$) on the conic $x^2+y^2+z^2+xy+yz+zx=0$, a form that is positive definite over $\mathbb R$, so the conic has no real point and admits no real parametrization (note F6). The only residue is the wording "version" (F3).
• S10. Not attacked; not a claim.

Premises

• Theorem 1 reconstruction page (the reconstruction page), read whole. Interface used: for $r\ge6$, the identity $(X+Y)^r=(X-Y)^r+\sum_{j\in J\setminus\{3\}}2\binom rjX^{r-j}Y^j+\sum_{\ell}v_\ell X^{r-3}Y^3$ with $J$ the odd integers in $[1,r]$, $t=\lfloor r/2\rfloor-2\ge1$, $v_1,\dots,v_t$ distinct positive integers summing to $2\binom r3$, $P$ the primes of the $v_\ell$ and of the $2\binom rj$ ($j\in J\setminus\{3\}$), $B=\prod_{p\in P}p\ge2$, and the $q$-adic distinctness argument. Standing taken as the assignment states (independently reviewed at an earlier commit, corrections applied); the section uses it within these hypotheses.
• Recalled abc and abcd bounds (library result page), read whole, with the card's provenance paragraph and the held PDF at printed p. 438 by text extraction. Interface: for $S$-units $u,v$ on a curve of genus $g$ with support $S$, not both constant, $1+u+v=0$ gives $H(1:u:v)\le2g-2+|S|$; for $S$-units with $z=1+u+v$ and no vanishing subsum, $H(1:u:v:z)\le3(2g-2+|S|)$. Claims-checked on the page and confirmed on the PDF; used by the section only for the normalization remark of Lemma 2, which the section proves itself.
• Polynomial-factor proposition (the Problem 940 finite-core page, section "Independent polynomial summands still reduce to fixed forms"), statement and proof read. Interface: a nonconstant $f\in\mathbb Z[T]$, eventually positive and $r$-powerful at every sufficiently large integer input, has every nonconstant irreducible factor over $\mathbb Q$ with multiplicity at least $r$. Author-recorded on its page, as the section states; Corollary 3.1 adapts its proof (W3), so Corollary 3.1 stands at author-recorded standing and says so.
• Problem 939 statement (Problem 939), Statement paragraph only: the $r\ge4$ questions on sums of $r-2$ coprime $r$-powerful numbers and the $r=3$ triple question. The section's Corollary 1.1 concerns the second question at $r\ge6$.
• Conventions paragraph of the connections page (lines 35 to 56): definitions of integer and polynomial $r$-powerfulness, joint coprimality, $n_0$, and the substitution $X=U^r$, $Y=V^r$.

Findings

• F1 (suggested). Location: Lemma 1, hypothesis (iv), line 82, "$A_2=c_2U^aV^b$ is a monomial with $a+b\ge1$". Defect: (i) lets a presentation hide integer constants inside the $H_{ij}$, and (iv) does not say in words that its $c_2$ is the constant of that presentation; a reader who takes (iv) as a shape condition can choose $B$ without the primes of the monomial's coefficient, and the joint-coprimality step then fails. Witness: $r=2$, $(V^2-U^2)^2+(2UV)^2=(U^2+V^2)^2$, $A_2=1\cdot(2UV)^2$, $B=1$, $m=3$: values $64,36,100$ with $\gcd(64,36)=4$. Proposed replacement: "(iv) $A_2=c_2U^aV^b$ is a monomial with $a+b\ge1$, $c_2$ being the constant of (i), so that every prime of the monomial's integer coefficient divides $B$".
• F2 (suggested). Location: heading line 186, "Theorem 3: no polynomial family with two summands", and the page's desc (lines 7 to 8, outside the section), "a proved Mason-Stothers obstruction to every polynomial family with two summands". Defect: at $r=2$ such families exist, as the page itself records at line 202; the statement carries the qualifier and the heading and desc do not. Proposed replacement: "Theorem 3: no polynomial family with two summands for $r\ge3$", and in the desc "to every polynomial family with two summands for r at least 3".
• F3 (suggested). Location: Corollary 4.1, lines 279 to 281, "Consequently the identity has no version with four terms in $\mathbb Q[U,V]$". Defect: "version" is undefined; the proof covers images under a projective change of variables over $\mathbb C$ and rescaling of the four forms, and a hostile reading takes the sentence as the universal statement that the page's $r=5$ section leaves open. Proposed replacement: "Consequently no projective change of variables over $\mathbb C$ and rescaling of the four forms turns the identity into one with four terms in $\mathbb Q[U,V]$".
• F4 (suggested; outside the section's lines). Location: preamble, lines 28 to 33, "the former is reproved below as Lemma 2, and the latter is used only to say what it does not exclude". Defect: read with "the former" as the polynomial-factor proposition and "the latter" as the two bounds, the sentence says the Problem 940 proposition is reproved as Lemma 2, which is false: Lemma 2 reproves the Mason–Stothers bound, the Brownawell–Masser bound is used only negatively, and the proposition is consumed author-recorded in Corollary 3.1 with its proof adapted. Proposed replacement: "the Mason–Stothers bound is reproved below as Lemma 2, the Brownawell–Masser bound is used only to say what it does not exclude, and the proposition is used in Corollary 3.1 with its proof adapted to a progression".
• F5 (note). Location: Lemma 2, proof, line 183, "the other two expressions of $W$". Only two expressions are displayed; the third is $bc'-b'c=-W$, which gives the bound for $\deg a$. Proposed replacement: add "and $-W=bc'-b'c$" after the second expression.
• F6 (note). Location: Corollary 4.1, lines 276 to 277, "the conic $(Q_0:Q_1:Q_2:3UV)$ defined over $\mathbb Q(\omega)$". The conic is defined over $\mathbb Q$: $Q_0+Q_1+Q_2=3UV$ places it in the plane $w=x+y+z$, and $Q_0^2+Q_1^2+Q_2^2+Q_0Q_1+Q_1Q_2+Q_2Q_0=0$ identically (checked in exact arithmetic), matching the factorization $(x+y+z)^5-x^5-y^5-z^5=5(x+y)(y+z)(z+x)(x^2+y^2+z^2+xy+yz+zx)$. The quadratic form is positive definite, so the conic has no real point, which explains and independently confirms the non-reality of every reparametrization. The page's wording is correct but weaker than what holds; recording the $\mathbb Q$-model would strengthen the page.
• F7 (note). Location: lines 162 to 163, "The same statement is used on the cubic-curves page of the Problem 699 folder". This cross-reference lies outside the commissioned read set and was not verified.

Verdict

• S1 (Lemma 1): refutation-failed. The proof is complete under the statement's shared symbol $c_2$; F1 asks that the sharing be said in words.
• S2 (application to the Theorem 1 identity): refutation-failed.
• S3 (Corollary 1.1): refutation-failed. Structure, injectivity and both count bounds re-derived; constants depend only on $r$ as stated.
• S4 (Lemma 2 with its normalization remark): refutation-failed.
• S5 (Theorem 3, weighted form and sharpness): refutation-failed.
• S6 (Corollary 3.1): refutation-failed. It stands at the author-recorded standing of the Problem 940 proposition it adapts, which the section discloses.
• S7 (coverage paragraph): refutation-failed.
• S8 (Proposition 4): refutation-failed.
• S9 (Corollary 4.1): refutation-failed. The consequence sentence needs the definition of "version" in F3; with it, every clause is proved.
• S10 (provenance sentence): not a claim.

Required corrections: none. Suggested: F1, F2, F3, F4. Notes: F5, F6, F7.

Limitations: the reviewer checked the imported proposition's proof only as far as needed to validate the progression bridge (W3) and did not review that proposition itself; the standing of Corollary 3.1 is therefore bounded by that page's author-recorded standing. The library bounds were confirmed at claims level on the held PDF and not proved. The computational checks are the reviewer's scratch arithmetic, retained here only as described, not as evidence. The other sections of the connections page, the lead folder and every review or grade in the folder were outside this review.

This focused review assigns no tier and changes no status.