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Subject

Date and pages. The source as it stood on 2026-09-28T13:50:48Z (called "the commit" below). The page wiki/research/erdos_939/connections.md (the connections page), read whole from git at that commit with its line numbering (585 lines; H2 headings at lines 58, 303, 339, 406, 427, 466, 488, 517, 561 and 574), and the lead wiki/research/leads/quintic_powerful_identity_route/_index.md (the quintic lead) at the same commit, read whole, together with its evidence index and the header and gauge table of its retained script evidence/util/quintic_identity_search.py, read to check the two search-coverage sentences. Every line number below is a line of the connections page at this commit.

Reports graded. The eight focused reviews filed in wiki/research/erdos_939/evidence/verify/, each read whole:

• Shared results (lines 58–302);
• Problem 939 at r = 4 (lines 303–338);
• Problem 939 at r = 5 (lines 339–405, and the lead);
• Problem 939, the remaining parts (lines 406–426);
• Problem 940 (lines 427–465);
• Problem 1107 (lines 466–487);
• Problem 937 (lines 488–516);
• Other problems examined (lines 517–560).

Read, and at what depth.

• docs/verification.md, the sections "Independence and the assignment", "Exact subjects and durable evidence", "Report contract", "Grading and claim standing", "Audit checklist" and, under "Erdos-specific", "Whole-claim report" and "Audit checklist".
• wiki/research/erdos_939/theorem_1_reconstruction.md at the commit, whole (the identity, the data $J$, $t$, $v_\ell$, $P$, $B$, the distinctness fill, the Boundary paragraphs); the folder index wiki/research/erdos_939/_index.md, its Open, Mechanism and Connections paragraphs.
• Problem 939 whole, including the Status, Formalization, Current assessment and Known results text, read to check the sections' attributions to the page (the grader is not blind); the statement blocks and Current assessments of Problem 940 and Problem 937; the status lines and the Current assessment heading of Problem 1107, Problem 941 and Problem 1081; the Statement paragraphs of Problems 364, 365, 366, 367, 137 and 935.
• wiki/research/erdos_940/finite_core_reduction.md, the section "Independent polynomial summands still reduce to fixed forms", statement and proof; the theorem statements on wiki/research/erdos_940/powerful_subsum_counts.md and wiki/research/erdos_940/linear_collision_families.md, as interfaces.
• The library pages recalled_abc_abcd_bounds.md of the Corvaja–Zannier card, whole; the Bajpai–Bennett–Chan result pages for Corollary 1.3, Theorem 1.2 and Proposition 5.2, their Source and Statement paragraphs; the CrowdMath (2020) and Walsh (2024) cards, digest and result lists. No PDF was opened: every source fact ruled on below was checked by a reviewer on the held PDF, and the grader checked the reviewer's reading against the card or result page.
• The existence at the commit of the Problem 699 page that Lemma 2's cross-reference names.

Grader's recomputation. Exact arithmetic in a scratch directory outside the repository, not retained: Proposition 4 over $\mathbb Q(\omega)$ at $(a,b,c)=(1,1,-2)$ and at four rational triples with $ac=-2b^2$; the factorization of the $Q_k$, the identity $Q_0+Q_1+Q_2=3UV$ and the vanishing of $x^2+y^2+z^2+xy+yz+zx$ on $(Q_0,Q_1,Q_2)$; the hypotheses (i)–(iv) and the conclusions of Lemma 1 for the Problem 1107 review's Witness A on a sample of admissible points, and the number of its distinct totals up to $X=10^{12},10^{14},\dots,10^{20}$, which is $0.18X^{7/20}$ to two digits at every one of these $X$, against the section's $O(X^{1/5})$; the two Problem 937 witnesses $49,289,529$ and $17^3,3^4\cdot7^3,37^3$ and the scaled progressions $6^ks^k,2\cdot6^ks^k,3\cdot6^ks^k$; the five partitions of $15$ into parts at least $5$; and the gauge representatives $UV(U-V)$, $U^2V$, $U^3$ of the retained script at degree $3$. The derivations in the reviews' weakest-step parts that the corrections rely on (the Theorem 3 bridge at $r=5$, the progression bridge for Problem 937, the density-zero count for Problem 1107) were re-derived by the grader line by line.

Role and independence. The grader is distinct from the author of the page and the lead (the compilation) and from each of the eight reviewers: a fresh-context session given only this assignment, which read the reports and the frozen subject and rebuilt the mathematics of every finding before ruling. Each reviewer identifies itself as an independent examiner in a fresh context given only its assignment, which took no part in writing the page, the lead or the reconstruction page and read no earlier review or grade of the folder; each records its allowed and actual reading and its exposures. The grader is not blind and read standing text on purpose, as the verifier-quotation rule assigns that check to the grader.

Exposure rulings. Each report discloses its exposures; the grader rules every one immaterial by the content test, since nothing in any report could only have come from the exposed text and no attack followed it:

• Shared results: the Problem 939 frontmatter and the other sections' summaries, seen while fixing section boundaries.
• Problem 939 at $r=4$: the Status paragraph of Problem 939. The References list, the witness of its F1, is bibliographic data that the section's own sentence is about; reading it was the source-fidelity check the charge requires, not an exposure.
• Problem 939 at $r=5$: the Status paragraph of Problem 939, and the Corvaja–Zannier card's reading-depth paragraphs.
• The remaining parts: the Status and Formalization paragraphs of Problem 939 and the library cards' digests.
• Problem 940: the scope sentences of the two Problem 940 research pages and the Formalization paragraph of Problem 940.
• Problem 1107: the one-word status entries of Problems 1107 and 941.
• Problem 937: the status line of Problem 937 and the paragraph labels of the result pages.
• Other problems: the status line and empty sections of Problem 364.

File names of parallel reviews seen in directory listings, disclosed by several reviewers, carried no content.

Reports graded

Every report is graded pass. Each opens with a subject block that resolves (the commit and the section's exact line range, which the grader confirmed against the numbered page), records the reviewer's role, independence facts, read set with depth and exposures, restates every statement of its section with quantifiers and the page's conventions, gives an explicit verdict on each of the ten checklist items, re-derives up to three weakest steps in its own words, records a strongest attack per statement that is a real attempt, lists its premises with interfaces and reading depth, and files a verdict per statement in the full verdict words. The reasons specific to each:

• Shared results: pass. All nine mathematical statements re-derived (Lemma 1's joint-coprimality step, Corollary 1.1's count with the Möbius estimate, the progression adaptation of the imported proof); the Mason–Stothers and Brownawell–Masser bounds confirmed on the held PDF; Proposition 4 and Corollary 4.1 recomputed by a different route, including an independent non-reality argument through the conic's $\mathbb Q$-model. Its four suggested findings and three notes are all sound; two become corrections (C1, C2).
• Problem 939 at $r=4$: pass. The abc sketch pushed to its edge, the line-restriction clause proved by composition, both source characterizations checked on the held PDFs; the required finding F1 is a true bibliographic defect (C4), and the grader adds what the excluded Current assessment shows.
• Problem 939 at $r=5$: pass. The strongest of the eight: it supplies the missing bridge to Theorem 3 (W1), exhibits the inhomogeneous admissible shape that defeats the closing implications (W2), finds the missing nonconstancy hypothesis (W3) and the false coverage sentence (F4), and grades the lead sentence by sentence. Its two required and four of its six suggested findings are accepted (C6–C9, C15, C16).
• The remaining parts: pass. Theorem 1's transfer under the joint reading, with the pairwise-coprimality witness at $r=6$; Walsh's Theorem 1.1 read on the held PDF; the trivial bound derived from the count of $r$-powerful numbers; the count of Theorem 1's own family computed.
• Problem 940: pass. The one-summand representation, the consistency of the two folder bounds for every $r\ge6$ by an exact inequality, and the shape comparison with the rank theorem, all re-derived; the two imported theorems checked against their pages.
• Problem 1107: pass. Two refutations with exact witnesses, both recomputed by the grader; an independent proof of the density-zero conclusion by two routes, which the corrections adopt.
• Problem 937: pass. The misapplication of Corollary 3.1 exhibited with two witnesses, the repair derived through the corollary's own two steps and Theorem 3, the dropped primitivity qualifier witnessed by scaled progressions, and Corollary 1.3 checked on the held PDF.
• Other problems examined: pass. The constant-term case of Theorem 3 re-derived from Lemma 2, the bridge for values checked with $b=1$, the Pell witness for the strongest reading exhibited, all thirteen problem descriptions compared with their statements.

Corrections

The accepted corrections, numbered across the folder. Each names the page, the location at the commit, and the exact replacement; none touches a statement: field, since the pages carry no claim cards.

• C1. Page: connections page. Location: the desc field, lines 7–8, "a proved Mason-Stothers obstruction to every polynomial family with two summands", and the H3 heading at line 186, "Theorem 3: no polynomial family with two summands". Replacement: in the desc, "a proved Mason-Stothers obstruction to every polynomial family with two summands for r at least 3"; heading, "Theorem 3: no polynomial family with two summands for r at least 3". Reason: at $r=2$ such families exist, as line 202 records; the statement carries the qualifier and the summaries do not (shared results F2).
• C2. Page: connections page. Location: lines 32–33, "the former is reproved below as Lemma 2, and the latter is used only to say what it does not exclude". 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 consumed in Corollary 3.1 with its proof adapted to a progression". Reason: with "the former" read as the proposition the sentence is false (shared results F4).
• C3. Page: connections page. Location: lines 324–325, "so $c<\kappa(\eta)c^{3(1+\eta)/4}$ bounds $c$". Replacement: "so, for any fixed $\eta<1/3$, $c^{(1-3\eta)/4}<\kappa(\eta)$ bounds $c$". Reason: the displayed inequality bounds $c$ only for $\eta<1/3$; at $\eta=1/3$ it reads $c<\kappa(1/3)\,c$ ($r=4$ review F2).
• C4. Page: connections page. Location: line 329, "makes the same remark; neither is cited on the problem page." Replacement: "makes the same remark; Walsh (2024) is entry [Wa24] of the problem page's References, cited there for its $r=3$ construction, and the CrowdMath note is cited in the page's Current assessment for this finiteness." Reason: false as written; the References list of E0939.md at the commit carries [Wa24] at lines 52–54 ($r=4$ review F1), and its Current assessment cites the CrowdMath card at lines 126–132, which the reviewer's read set excluded and the grader checked.
• C5. Page: connections page. Location: lines 336–337, "A negative outcome of either kind is not obtainable from polynomial identities, by Theorem 3." Replacement: "Neither an example nor an infinite family at $r=4$ can come from a polynomial identity, by Corollary 3.1." Reason: Theorem 3 excludes an example or an infinitude produced by a polynomial family; it says nothing about nonexistence or finiteness, which the sentence's plain reading attributes to it ($r=4$ review F4).
• C6. Page: connections page and the lead. Location: connections page lines 348–351, "that satisfies (i)–(iv) and the positivity clause would give infinitely many jointly coprime $5$-powerful triples with $5$-powerful sum and answer the second question negatively at $r=5$"; lead, section "Route", the premise "(P) ... with positive leading coefficients in $U$ for some admissible $B$." Replacement: connections page, "that satisfies (i)–(iv), the positivity clause and the nonconstancy of $A_4(U,B)$ would give infinitely many jointly coprime $5$-powerful triples with $5$-powerful sum and answer the second question, read with joint coprimality, negatively at $r=5$"; lead, "(P) ... with positive leading coefficients in $U$ and $A_4(U,B)$ nonconstant for some admissible $B$." Reason: Lemma 1 derives the infinitude from the separate hypothesis that $A_n(U,B)$ is nonconstant, and (iii) admits $K=1+UV(V-B)$ with $K(U,B)=1$; the hypothesis is automatic for homogeneous identities of positive degree and must be carried for the general premise ($r=5$ review F3 and F9).
• C7. Page: connections page. Location: lines 351–354, "(that would be a three-term identity of $5$-powerful polynomials after combining them)". Replacement: "(combining them leaves a three-term identity that still contains a rational multiple of $A_1=K^5$ beside a rational multiple of the monomial $A_2$, which are coprime by (iii) and (iv), so its three terms are pairwise coprime; positivity at an admissible point excludes the merges with a vanishing coefficient; and Theorem 3 applies after specializing $V$, or $U$ when $a=0$, to a generic complex value, which preserves $5$-powerfulness, pairwise coprimality and the nonconstancy of the monomial)". Reason: Theorem 3 needs pairwise coprime polynomials in one variable, not all constant, and the parenthetical checks none of these ($2f+3f=5f$ is a three-term identity of $5$-powerful polynomials); the grader re-derived the reviewer's bridge, case by case over the six proportional pairs ($r=5$ review F2).
• C8. Page: connections page and the lead. Location: connections page lines 362–363, "The gap is therefore: does a four-term identity of $5$-powerful binary forms with rational terms exist?", and lines 402–404, "a proof that no real four-term identity of $5$-powerful binary forms exists would close the polynomial route and leave $r=5$ to constructions of the kind that settled $r=3$."; lead, section "What is known about the premise", "Over $\mathbb C$ the premise's polynomial part is satisfiable"; section "Consequential gap", "A proof that no four-term identity of $5$-powerful binary forms with real terms exists would close this route"; section "Next step", "a proof of nonexistence in all degrees would close the route". Replacement: connections page, "The gap is therefore: does a four-term identity in $\mathbb Z[U,V]$ satisfying (i)–(iv) with $r=5$ exist? Lemma 1 does not need homogeneity; the homogeneous case, a four-term identity of $5$-powerful binary forms with rational terms, is the case the searches below and the lead treat." and "a proof that no real four-term identity of $5$-powerful binary forms exists would close the homogeneous case of the polynomial route, leaving inhomogeneous identities of the shape (i)–(iv), which nothing here examines, and constructions of the kind that settled $r=3$."; lead, "Over $\mathbb C$ a four-term identity of $5$-powerful forms with one fifth-power term and one monomial term exists", "A proof that no four-term identity of $5$-powerful binary forms with real terms exists would close the homogeneous case of this route; the transfer does not need homogeneity, and inhomogeneous identities satisfying its hypotheses are a separate possibility that nothing here examines", and "a proof of nonexistence in all degrees would close the homogeneous case of the route". Reason: (iii) is satisfied by inhomogeneous $K$, for instance $U^2+U^2V^3+V^2$, and no reduction of the inhomogeneous case to binary forms exists on either page, so the closing implications are unsupported as written ($r=5$ review F1).
• C9. Page: connections page and the lead. Location: connections page lines 378–379, "(the mixed shapes $L^6M^9$, $L^7M^8$, $L^5M^{10}$ were not searched)"; lead, section "What is known about the premise", "The mixed shapes of degree $15$ ($L^6M^9$, $L^7M^8$, $L^5M^{10}$ with $L,M$ linear) and every degree above $15$ were not searched."; the same list in the docstring of evidence/util/quintic_identity_search.py, line 12. Replacement: connections page, "(the shapes $L^6M^9$ and $L^7M^8$, and identities mixing shapes among the four terms, were not searched; $L^5M^{10}=(LM^2)^5$ lies inside the searched shape)"; lead, "The shapes $L^6M^9$ and $L^7M^8$ of degree $15$ ($L,M$ linear), identities mixing shapes among the four terms, and every degree above $15$ were not searched; $L^5M^{10}=(LM^2)^5$ lies inside the searched shape."; script docstring, "not the mixed shapes L^6 M^9, L^7 M^8". Reason: the $5$-powerful shapes of degree $15$ are the partitions $(15)$, $(10,5)$, $(5,5,5)$, $(9,6)$, $(8,7)$; the first three are fifth powers of cubics, and the search at degree $3$ fixes $Q_4$ to $UV(U-V)$, $U^2V$ and $U^3$ with $Q_1,Q_2,Q_3$ general, so $(10,5)$ was searched ($r=5$ review F4).
• C10. Page: connections page. Location: lines 463–464, "No route from this folder to either question of Problem 940 exists, and none is proposed." Replacement: "No route from this folder to either question of Problem 940 was found, and none is proposed." Reason: the section proves that the direct transport fails and that the two folder bounds are unthreatened; a universal negative about routes does not follow (Problem 940 review F1).
• C11. Page: connections page. Location: lines 477–481, from "In the two-variable device of Lemma 1 the total $A_n(U,V)$" to "whose values also have density zero." Replacement: "In any device of Lemma 1 every total $A_n(m,B)$ is an $r$-powerful integer, and the $r$-powerful integers up to $X$ number $O_r(X^{1/r})$, so the totals have density zero for every $r\ge2$, whatever the degree of $A_n$; for the Theorem 1 total $(U^r+V^r)^r$ the count is $O(X^{2/r^2})$, since $N\le X$ forces $m,B\le X^{1/r^2}$." Reason: the device $(U^2-V^5)^2+4U^2V^5=(U^2+V^5)^2$ at $r=2$ satisfies (i)–(iv), its total is not a form, and its distinct totals up to $X$ number $\gg X^{3/10}/\log X$ against the asserted $O(X^{1/5})$; the grader recomputed the count. The replacement also removes the word "form", so Problem 1107 review F3 is subsumed (Problem 1107 review F1).
• C12. Page: connections page. Location: lines 481–484, from "With $s$ parameters the count is $O(X^{s/D})$" to "and not an identity." Replacement: "An identity in any number of parameters whose total is $c\prod_jH_j^{e_j}$ with every $e_j\ge r$, as (i) requires, has totals of density zero as well: its positive values are $|c|\prod_j|h_j|^{e_j}$ with nonzero integers $h_j$, and there are $O(X^{1/r}(\log X)^{k-1})$ such products up to $X$. Without that shape the total is only the sum of its $r$-powerful summands, and representing a positive proportion of the integers by it is the Waring-type problem for $r$-powerful numbers itself, not a consequence of an identity." Reason: the degree of the total does not control the count; $U_1^2+U_2^2+(U_3U_4U_5)^2$ has five parameters, degree six, and totals of lower density at least $5/6-\pi/4$ by the three-square theorem, while under the $r$-powerful reading no number of parameters reaches a positive proportion (Problem 1107 review F2).
• C13. Page: connections page. Location: lines 500–502, "so Corollary 3.1 applies to any polynomial family of pairwise coprime three-term progressions and excludes it for every $r\ge3$, whatever the shape of the terms". Replacement: "so, once $a$, $b$ and $c$ are $r$-powerful as polynomials and $\gcd(a,c)=1$ in $\mathbb Q[t]$, which the first two steps of the proof of Corollary 3.1 give from the values on the progression, Theorem 3 applies to the triple $(a,c,2b)$ and excludes every polynomial family of pairwise coprime three-term progressions, in the sense of Corollary 3.1, for every $r\ge3$, whatever the shape of the terms". Reason: Corollary 3.1's statement requires the values of the sum to be $r$-powerful, and $2b(t)$ is not whenever $b(t)$ is odd, as at $49,289,529$ ($2\cdot289=2\cdot17^2$) and at $17^3,3^4\cdot7^3,37^3$; the polynomial-level remark cannot supply a value-level hypothesis. The statement is true by the reviewer's derivation, which the grader checked (Problem 937 review F1 and F4).
• C14. Page: connections page. Location: lines 511–513, "three-term progressions of $k$-powerful numbers with $k\ge4$ are conditionally finite". Replacement: "primitive three-term progressions of $k$-powerful numbers with $k\ge4$ are finite under abc". Reason: $6^ks^k$, $2\cdot6^ks^k$, $3\cdot6^ks^k$ is a three-term progression of $k$-full numbers for every $k\ge2$ and $s\ge1$; the source's $A^\infty(k)$ counts primitive progressions, as the result page states (Problem 937 review F2).
• C15. Page: the lead. Location: section "Route", the bullet "The three-term obstruction", "No identity $a+b=c$ of pairwise coprime $r$-powerful polynomials with $r\ge3$ exists". Replacement: "No identity $a+b=c$ of pairwise coprime $r$-powerful polynomials, not all constant, with $r\ge3$ exists". Reason: constants are $r$-powerful under the page's convention, so $1+1=2$ refutes the sentence as written; Theorem 3 carries the hypothesis ($r=5$ review F6).
• C16. Page: the lead. Location: section "Target and status evidence", "its $r=4$ case is excluded from every polynomial route by the theorem recalled below". Replacement: "its $r=4$ case is excluded from the polynomial-identity route of Corollary 3.1 of the connections page, one-parameter families and polynomially parametrized parameter curves, by the theorem recalled below". Reason: Corollary 3.1 and its scope paragraph name what the recorded result does not cover; "every polynomial route" claims more ($r=5$ review F7).

Rejected and downgraded findings

No finding is rejected as wrong; each finding below is sound as a note and is downgraded from a correction for the reason given.

• Shared results F1 (the constant $c_2$ in (iv)): downgraded to a note. The statement is correct as written, since (iv) reuses the symbol $c_2$ of the presentation (i), which forces $\prod_jH_{2j}^{e_{2j}}=U^aV^b$; the reviewer's rewording is optional.
• Shared results F3 ("version" in Corollary 4.1): downgraded to a note. The preceding sentence fixes the scope (projective change of variables and rescaling), and the $r=5$ section restates the corollary as "this one is not real"; the more precise sentence is optional.
• Shared results F5, F6, F7: notes. The third expression $-W=bc'-b'c$ gives the bound for $\deg a$; the conic of Corollary 4.1 has the $\mathbb Q$-model $x^2+y^2+z^2+xy+yz+zx=0$, $w=x+y+z$, positive definite, which the grader verified and which strengthens the page without correcting it; the Problem 699 page named at line 163 exists at the commit, and its use of the statement was not checked.
• $r=4$ review F3 ("sporadic"): downgraded to a note. The sentence is a picture, true under the reading "no solution lies in a polynomial family" that its second clause makes natural; at $r=3$ the same theorem coexists with infinitely many solutions, so the reviewer's rewording is the clearer one and is recommended, not required.
• $r=4$ review F5 (unverified attributions): resolved by the grader. The Current assessment of Problem 939 is read 2026-09-27 and its Known results record no example at $r=4$ and the forum-reported bound $\max(a,b)>10^{14}$; the folder index records the summand count $\lceil r/2\rceil+1>r-2$ for $r\le5$.
• $r=5$ review F5 (minimal degree): downgraded to a note. "Smallest possible degree, $10$" is true: in degrees $5$ to $9$ a $5$-powerful form is $cL^D$, $D$-th powers of distinct linear forms are independent, and the degenerate cases are a vanishing proper subsum or four proportional forms with a common factor; the parenthetical's sketch treats degree $5$ and the first exception only, and the fuller sentence is recommended.
• $r=5$ review F6, the Lemma 1 part ($a+b\ge1$ and $m\ge1$ dropped in the lead's restatement): note; neither omission changes a deduction.
• $r=5$ review F8, F9, F10: notes. The symbol $d$ at line 344 is the problem page's name for the sum and is undefined on the connections page; the joint reading is now named by C6; the degree-$10$ plan in the lead needs a real-algebraic step after the Gröbner or resultant computation, which the lead should name when the computation is run.
• Remaining parts F1 ("Settled negatively by Theorem 1" without the coprimality convention): downgraded to a note. The page's Conventions define joint coprimality and its $r=4$ section states that the site question's coprimality is joint; "settled" describes the mathematics and the preamble disclaims any status change. The pairwise-coprimality witness ($12X^5Y$ and $40X^3Y^3$ share $X^3Y$) is correct and worth recording.
• Remaining parts F2 (the pronoun "it"): downgraded. In "Corollary 1.1 records the size of the family it produces" the antecedent is the subject, Corollary 1.1, under which the sentence is correct; the reviewer's count of Theorem 1's one-parameter family, of order $r^2X^{1/r^2}/\log X$ and $78{,}488$ at $r=6$, $X=10^{216}$, is right and may be added.
• Remaining parts F3, F4, F5: notes (the warrant for Nitaj and Cohn is Corollary 3.1, whose texts are not held; the first question at $r\ge6$ goes unmentioned; the trivial bound rests on the $O_r(X^{1/r})$ count).
• Problem 940 F2 ("$r$-th powers of monomials"): downgraded to a note. The parenthetical describes the shape up to the coefficients $2\binom rj$ and $v_\ell$, which is what the comparison with the rank theorem's shape needs; "positive integer multiples of $r$-th powers of monomials" is more exact and optional.
• Problem 940 F3, F4, F5: notes; the imported theorem's "for fixed $r$, $k$, $\varepsilon$" and the sharper $O(X^{2/r^2})$ are correct and optional; the outcome at $r=3,4,5$ follows from the reconstruction page's Boundary.
• Problem 1107 F3 ("form"): subsumed by C11.
• Problem 1107 F4: resolved by the grader; the Problem 1107 page at the commit records "No current assessment is recorded".
• Problem 937 F3 ("needed the elliptic curve"): downgraded to a note. The sentence reads as narrative of the source's construction; the necessity reading is unproved, and "the source's four-term construction uses an elliptic curve" is recommended.
• Problem 937 F5 (pairwise versus joint coprimality): note; for a three-term progression of $r$-powerful numbers the two readings coincide, as the reviewer's argument shows and the grader checked.
• Other problems F1 (the Pell witness): downgraded to a note. The sentence holds in the page's sense of a polynomial family, which its $r=4$ section defines and which the sentence's own clause about the Pell families presupposes; citing Corollary 3.1 with $b=1$ rather than Theorem 3 for the values statement is recommended.
• Other problems F2, F3, F4: notes; the grader checked F4 against the Statement paragraphs of Problems 935, 367 and 137 and confirms the parenthetical fits Problem 935 exactly and the other two loosely.

Graded verdicts

Shared results. S1 (Lemma 1): refutation-failed. S2 (application to the Theorem 1 identity): refutation-failed. S3 (Corollary 1.1): refutation-failed. S4 (Lemma 2 with its normalization remark): refutation-failed. S5 (Theorem 3, weighted form and sharpness): refutation-failed. S6 (Corollary 3.1): refutation-failed, standing at the author-recorded standing of the Problem 940 proposition it adapts. S7 (coverage paragraph): refutation-failed. S8 (Proposition 4): refutation-failed. S9 (Corollary 4.1): refutation-failed. S10 (provenance sentence): not a claim.

Problem 939 at $r=4$. S1 (target): refutation-failed; the attributions to the Current assessment are verified by the grader. S2 (the transfer is excluded): refutation-failed. S3 (interpretation): refutation-failed; the folder index's count is verified. S4 (literature): the mathematical claim that abc implies finiteness at $r=4$ and both source characterizations are refutation-failed; the sketch's inference needs C3; the bibliographic clause is refuted-as-stated and replaced by C4. S5 (picture): refutation-failed under the reading "no solution lies in a polynomial family" noted above. S6 (next investigation): refutation-failed for its content, with its last sentence replaced by C5.

Problem 939 at $r=5$ (section). S1 (target): not a claim; the two examples and the $10^{16}$ bound are on the problem page, as the grader checked. S2a (values of the transfer): refutation-failed. S2b (infinitude): proof defective; statement unresolved as written, for want of the nonconstancy hypothesis; refutation-failed after C6. S3 (no two summands proportional): refutation-failed, with its justification supplied by C7. S4 (minimal complex degree $10$): refutation-failed. S5 (non-reality): refutation-failed, as graded under the shared results. S6 (the gap is the binary-form question): proof defective; statement unresolved as written; refutation-failed after C8. S7a (degree-$10$ normalization): refutation-failed. S7b (degree-$15$ coverage): refuted-as-stated, corrected by C9. S7c (search outcomes): not a claim. S8 (what it does not give): refutation-failed as a labeled heuristic. S9a (a found identity refutes finiteness): refutation-failed. S9b (a nonexistence proof for forms closes the route): proof defective; statement unresolved as written; refutation-failed after C8.

The lead. L1 (target and conventions): proof defective; statement unresolved as written, for "every polynomial route"; refutation-failed after C16. L2 (transfer restated): refutation-failed for the value statement; the infinitude phrase follows S2b and C6. L3 (obstruction restated): refuted-as-stated as written, by $1+1=2$; refutation-failed after C15. L4 (complex identity restated): as S4 and S5. L5 (premise and deduction): (P) is a provisional premise, not a claim; the deduction under (P) is proof defective; statement unresolved as written and refutation-failed after C6; the distinctness clause is refutation-failed. L6 (known about the premise): refutation-failed after C8 and C9. L7 (exploratory computation): not a claim. L8 (consequential gap): proof defective; statement unresolved as written; refutation-failed after C8. L9 (next step): the closing implication as L8, refutation-failed after C8 and C9; the remaining sentences refutation-failed, with the real-algebraic step noted. L10 (standing): refutation-failed.

Problem 939, the remaining parts. S1 (third question, literature): refutation-failed; the clause "as the problem page records" is verified by the grader against the page's Status and Current assessment. S2 (Corollary 3.1 at $r=3$): refutation-failed. S3 (the constructions are non-polynomial): refutation-failed. S4 (limitation): refutation-failed. S5 (second question at $r\ge6$, coprimality joint): refutation-failed. S6 (count of the family): refutation-failed. S7 (trivial bound and comparison): refutation-failed. S8 (cross-reference): not a claim.

Problem 940. S1 (target): refutation-failed. S2 (status record): refutation-failed; the two "open" clauses are verified by the grader against the Current assessment. S3 (transfer failure): refutation-failed. S4 (no consequence): refutation-failed. S5 (L18 and L19): refutation-failed. S6 (first comparison): refutation-failed. S7 (second comparison): refutation-failed. S8 (closing assessment): the universal negative as written is unsupported; refutation-failed after C10.

Problem 1107. S1 (target facts): refutation-failed; "no compiled assessment" is verified by the grader. S2 (totals of an identity): refutation-failed. S3 (count bound): refuted-as-stated, replaced by C11. S4 (density zero): refutation-failed as a statement; its proof on the page is replaced by C11. S5 (parameters): refuted-as-stated, replaced by C12. S6 (outcome): refutation-failed.

Problem 937. S1 (target and status): refutation-failed; the status is verified by the grader against the Current assessment. S2 (reduction): refutation-failed. S3 (obstruction): proof defective; statement unresolved as written; refutation-failed after C13, which writes the reviewer's derivation into the page. S4 (the case $r=2$): refutation-failed. S5 (four terms at $r=2$): refutation-failed on the reading that the source's construction uses an elliptic curve. S6 (Corollary 1.3): refutation-failed. S7 (finiteness sentence): refuted-as-stated as written; refutation-failed after C14. S8 (closure): not a claim; the folder index at the commit records the outcome the section reports.

Other powerful-number problems examined. S0 (opening sentences): not a claim. S1 (Theorem 3 with a constant term): refutation-failed. S2 (the mixed case for Problem 366): refutation-failed. S3 (no polynomial family of consecutive powerful numbers, in the page's sense): refutation-failed. S4 (Pell consistency): refutation-failed. S5 (specific pairs and triples): refutation-failed for its conclusion. S6 (Problem 938): not a claim. S7 (Problem 943 at $r=2$): refutation-failed. S8 (Problems 1081 and 941): refutation-failed; the status words are verified by the grader. S9 (Problems 942, 935, 367, 137, 936, 1108): not a claim.

Lead outcomes. quintic_powerful_identity_route: reviewed. The route rests on Lemma 1 (S1 of the shared results), on the transfer at $r=5$ (S2a and S2b of the $r=5$ section, L2 and L5 of the lead) under the joint-coprimality reading, and, for its shape constraints, on Theorem 3, Proposition 4 and Corollary 4.1 (S5, S8 and S9 of the shared results); after C6, C7, C8, C9, C15 and C16 every one of these statements is refutation-failed. The premise (P) stays a provisional premise that nothing establishes, the searches stay exploratory, and the corrections C6, C8, C9, C15 and C16 change the lead's own text and are to be applied to it.

No tier is assigned and no status changes.