Distinct grade of the Problem 18 connections reviews

Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Sep 28

Subject

The repository as it stood at 2026-09-28T13:50:48Z, the default branch's state at the time of grading. The pages graded, each read in full as of that time:

wiki/research/erdos_18/connections.md, the connections page: its preamble and Conventions, the nine sections from ## Shared lemmas to ## Other divisor problems examined, and its closing sections "Leads created" and "Sources read for this page";
wiki/research/leads/factorial_unit_fraction_transfer/_index.md, the factorial lead;
wiki/research/leads/lcm_divisor_gap_greedy_route/_index.md, the lcm lead.

The reports graded are the nine focused reviews filed beside this record in wiki/research/erdos_18/evidence/verify/, one per section of the connections page, each read in full:

connections_shared_lemmas_review.md;
connections_problem_304_the_number_of_unit_fractions_for_a_b_review.md, with the factorial lead;
connections_problem_293_the_least_denominator_missing_from_k_review.md, with the factorial lead;
connections_problem_18_question_a_along_the_least_common_mul_review.md, with the lcm lead;
connections_problem_18_question_c_what_the_greedy_route_woul_review.md;
connections_problem_1099_sums_of_powers_of_consecutive_divis_review.md;
connections_problem_673_the_sum_of_consecutive_divisor_ratio_review.md;
connections_problem_859_the_density_of_n_representing_a_fixe_review.md;
connections_other_divisor_problems_examined_review.md.

The reports are focused reviews of author-recorded mathematics; they carry no tier and this grade assigns none.

Role and independence. The grader is a distinct grader in a fresh context: neither the author of the connections page, the leads or the reconstruction pages, nor the reviewer who wrote any of the nine reports, and not a participant in either piece of work. The grader received only the grading assignment, which named the subject date, the page, the leads, the reports and the sections of docs/verification.md to read. No working material of the author or of the reviewers reached the grader, and nothing was read from the web. The grader is not blind: the problem pages' Status and Current assessment text, the library result pages and the earlier grade of the reconstruction reviews in this folder were read where a finding needed them, and are disclosed below.

Read set and depth.

docs/verification.md: "Independence and the assignment", "Exact subjects and durable evidence", "Report contract", "Grading and claim standing", "Whole-claim report" and both "Audit checklist" sections, in full; docs/approach_vetting.md "Compare substance, not vocabulary", for the rule on novelty claims.
The connections page and the two leads in full as of that time; the retained script gap_profile.py and the evidence index of the lcm lead in full.
Reconstruction pages as of that time: the Remark 6 page in full (its Statement covers $h(n!)$ only and its proof gives $\log\tau(n!)\ll n/\log n$ ); the Lemma 3.1 page's Definitions, Statement and Proof; the Theorem 1 page's symmetry facts (i) and (ii).
Problem pages as of that time: Problem 304 (Statement, Formulation, Status, Origin, the Known results paragraphs on the van Doorn claim and on the deduced, unreviewed transfer, and the bullets on Vose and on Problem 293); Problem 293 (Statement, Formulation, Status and the "Finite values" paragraph); Problem 1099 in full.
Library: the Theorem 1.2 result page of van Doorn (2026) in full; physical p. 171 of the held Erdős (1981) paper in the layout text extraction, for the candidates sentence.
The earlier grade in this folder (grade.md) and the folder's review index, read for the record format and to update the index.
Not read: the van Doorn note, the Hughes preprint, the van Doorn–Tang paper, the Weingartner paper and the Erdős (1982) survey. The reviewers' source-fidelity checks, made on page images and agreeing wherever two reports cover the same source, are accepted as filed; no finding turned on a source reading that this grade had to re-adjudicate.

Independent computations. Two findings turn on numbers, and both were recomputed here from scratch by scripts written for this grade in a temporary directory and not retained (the assignment permits writing only this record and one index sentence); the descriptions and numbers below let a reader rerun them.

The divisor gaps of $L_x=\mathrm{lcm}(1,\dots,x)$ for $x=20,30,40,50,60,64$ : every divisor enumerated as an exact integer from the prime-power factorization and sorted, the count checked against $\tau(L_x)$ , and over the consecutive pairs $d<b$ with $x\le\sqrt{db}\le\sqrt{L_x}$ the value $\kappa(d,b)=(\log x)\log(1/\log(b/d))/\log\sqrt{db}$ , with $\log(b/d)=\log(1+(b-d)/d)$ from the exact difference. Results: $\kappa_{\min}=1.116$ , $1.257$ , $1.311$ , $1.298$ , $1.348$ , $1.358$ , the page's values; the binding pairs at positions $\log\sqrt{db}/\log\sqrt{L_x}=1.000$ , $0.956$ , $0.989$ , $0.962$ , $0.995$ , $0.996$ , that is, $\sqrt{L_x}/\sqrt{db}=1.00$ , $1.88$ , $1.23$ , $2.58$ , $1.15$ , $1.12$ (at $x=20$ the pair $15048<15470$ around $\sqrt{L_{20}}$ ); the largest $\log(b/d)$ in the top tenth of $[0,\log\sqrt{L_x}]$ , the retained script's bands, is $2.77\cdot10^{-2}$ at $x=20$ and $4.45\cdot10^{-5}$ at $x=64$ , while the fifth and sixth tenths at $x=64$ have maxima $8.8\cdot10^{-4}$ and $3.1\cdot10^{-4}$ (with tenths of $[\log x,\log\sqrt{L_x}]$ , the lead's wording, $3.3\cdot10^{-4}$ and $1.9\cdot10^{-4}$ , the reviewer's values).
The series $C(\alpha)=4\sum_{j\ge2^{16}}(4\varepsilon_j)^{\alpha-1}(\tfrac12\log j+\log2)$ with $\varepsilon_j=j^{-(\log_2j/2-\log_2\log_2j)}$ : summed in floating point over $2^{16}\le j<3\cdot10^6$ , the tail bounded by the integral of the decreasing summand. Results: $C(1.1)\approx7.9\cdot10^{4}$ , $C(5/4)\approx20.3$ , $C(3/2)\approx1.8\cdot10^{-4}$ , $C(2)\approx3.7\cdot10^{-14}$ , against $2\zeta(5/4)\approx9.19$ ; the per-term bound $(4\varepsilon_j)^{\alpha-1}\le2^{-62(\alpha-1)}$ holds with equality at $j=2^{16}$ .

Standard applied. A finding is accepted as a correction when a mathematical statement of the page (a display, an inequality, a consequence sentence, a description of a computation) is false or unsupported as written, whether or not a weaker true statement was intended; when a result is cited outside its stated hypotheses; when an assessment sentence misdescribes what the page's own mathematics shows; or when the corpus's own rule requires the change (an undescribed search behind a novelty claim). Wording with a true reading, optional labels and locators, thresholds that sit under an explicit "for large" clause, characterizations of sources or status text that the read set cannot decide, and observations for which no exact replacement is warranted are downgraded to notes. The reviewers graded severity unevenly (the same missing bounded-seed clause is "required" in one report and "suggested" in another); this grade applies the one standard above to all nine.

Exposure ruling. Every report discloses incidental exposures of the same kinds: the Status paragraph or frontmatter status line of a problem page printed with its Statement; a library card's Read status, Bears on or Overview paragraphs printed with its provenance paragraph; the Standing paragraphs of the reconstruction pages; neighboring sections of the connections page itself; file names of sibling reviews and of the earlier grade in a directory listing. By the content test, nothing in any report could only have come from these exposures and no attack direction followed them: every finding cites the page's own text, a held source, or a derivation made in the report. All disclosed exposures are ruled immaterial.

Reports graded

Each report has a subject block that resolves (the subject date above, the page, the section by heading and line range, and the lead where one was assigned), states its role, fresh context, allowed material actually read and exposures, restates every statement of its section with quantifiers, gives an explicit verdict on each of the ten audit-checklist items, re-derives up to three weakest steps, records a strongest attack that could have succeeded, lists its premises with interfaces and reading depth, and closes with a verdict per statement that assigns no tier.

connections_shared_lemmas_review.md: pass. Lemma D's asymptotic floor is re-derived to the exact point where the proof loses the seed term, and the attack carries the witness $n_0=2^k$ , $A_0=3$ , $L_0=1$ that refutes the display as stated; Lemma B's upper-range partition, Lemma C′'s second case and the van Doorn accounting of the closing paragraph are re-derived; exhaustive exact checks of Lemmas A, C and C′ are reported as attacks, not warrants. F1 and F2 are accepted (C1).
connections_problem_304_..._review.md: pass. The three weakest steps (the Lemma 3.1 interface inside Lemma C′, the passage from $(n-1)!<b$ to $n<\log b$ , the two obstructions) are re-derived, and the attack on the extension-chain obstruction carries its own witness $n_0=m!$ , $A_0=2$ , $L_0=1$ ; the lead's (293.1) is checked against the lead's sketch, the full proof being the Problem 293 report's charge. F1 is accepted (C2); F2 (C3) and F6 (C4) are adopted under the standard above.
connections_problem_293_..._review.md: pass. Lemma C′ is re-derived with an exact finite check of the construction for $3\le b\le5040$ , the arithmetic of (293.1) and its corollary is re-derived with the exact threshold $k\ge25$ , and the two caps are attacked at the non-factorial case and at the unbounded seed. F1 is accepted (C5) and F2 (C6).
connections_problem_18_question_a_..._review.md: pass, with one limitation recorded. The proof of (18a.1) is re-derived and its slip found; the upper-range recurrence is re-derived to show the constant $2/c$ ; the window argument behind $c\le2\log2$ is re-derived; the computation was reproduced by an independent enumeration that is not retained with the report, which a focused review may do but which this grade therefore repeated (Subject, above) with the same results. F1 to F6 are accepted (C7 to C12).
connections_problem_18_question_c_..._review.md: pass. The mean-gap floor is re-derived with the mirror argument at the top window, the necessity translation is re-derived, and the strongest attack exhibits an admissible capped profile with integral $O((\log n)^2)$ that the floor does not exclude, together with the counting bound that every admissible profile obeys. F1 and F2 are accepted (C13).
connections_problem_1099_..._review.md: pass. The zone decomposition, the window bound and the corollary are re-derived clause by clause; the attack on the bound below the threshold $T$ is exact and correctly stopped by the hypothesis $n\ge T$ ; the divergence of $C(\alpha)$ as $\alpha\to1^+$ is exact and the witness $C(5/4)$ is confirmed here. F1 is accepted (C14).
connections_problem_673_..._review.md: pass. The site comment's lower bound is re-derived with its true range, the even- $n$ bound and the factorial bound are re-derived, and the attack on the density-zero obstruction goes through the ratio-two integers and the order of the average. F1 is accepted (C15).
connections_problem_859_..._review.md: pass. The transfer lemma, the existence of the density and the polynomial floor are re-derived, with the unconditional factorial and power-of-two seeds supplied; the attack seeks a transfer at logarithmic scale and explains why none is available. F1 is accepted (C16).
connections_other_divisor_problems_examined_review.md: pass. The width comparison, the strictness gap in the Problem 144 bullet and the entailment between practicality and pseudoperfection are re-derived, each attack carrying its witness. F1 is accepted (C17).

Corrections

Accepted corrections, numbered across the folder in page order. Each names the page, the location and the exact replacement. "Section" means the named H2 section of wiki/research/erdos_18/connections.md; "factorial lead" and "lcm lead" are the two lead pages named in the Subject.

C1. Section "Shared lemmas", Lemma D, two places.

(a) Statement. Replace the display and the two sentences after it,

> \sum_{j<J}L_j\ \ge\ \sum_{j<J}\frac{\log A_j}{\log\bigl(\tau(n_j)+1\bigr)} > \ \ge\ \Bigl(\frac1{2\log2}-o(1)\Bigr)(\log\log n_J)^2 > \qquad(n_J\to\infty), >

where the $o(1)$ tends to $0$ as $n_J\to\infty$ uniformly over all such chains. Hence the bound $h(n_0)+\sum_jL_j$ that such a chain certifies for $h(n_J)$ is never below $(1/(2\log2)-o(1))(\log\log n_J)^2$ .

with

> \sum_{j<J}L_j\ \ge\ \sum_{j<J}\frac{\log A_j}{\log\bigl(\tau(n_j)+1\bigr)} > \ \ge\ \frac{(\log\log n_J)^2-(\log\log n_0)^2}{2(\log2+\eta)}-C(\eta) >

for every $\eta>0$ , with $C(\eta)$ depending on $\eta$ alone. Hence, for chains from a bounded seed, $n_0\le C$ with $C$ fixed, the bound $h(n_0)+\sum_jL_j$ that such a chain certifies for $h(n_J)$ is at least $(1/(2\log2)-o(1))(\log\log n_J)^2$ , where the $o(1)$ tends to $0$ as $n_J\to\infty$ uniformly over the chains with $n_0\le C$ . Without a bound on the seed the sum $\sum_jL_j$ has no floor in $n_J$ alone: the one-step chain $n_0=2^k$ , $A_0=3$ , $L_0=1$ satisfies the hypothesis of Lemma 3.1 and has $\sum_jL_j=1$ .

(b) Proof. Replace

With $\ell_j=\log n_j$ and $f(\ell)=\log\ell/\ell$ , which decreases for $\ell>e$ , let $j(\eta)$ be the least index with $n_{j(\eta)}\ge n(\eta)$ .

with

With $\ell_j=\log n_j$ and $f(\ell)=\log\ell/\ell$ , which decreases for $\ell>e$ , let $n(\eta)$ be a threshold beyond which $\eta'(n)\le\eta$ and $\log n>e$ , and let $j(\eta)$ be the least index with $n_{j(\eta)}\ge n(\eta)$ ; if there is none, then $n_J<n(\eta)$ and the bound holds with any $C(\eta)\ge(\log\log n(\eta))^2/(2\log2)$ .

and replace

Hence $\sum_jL_j\ge(\log\ell_J)^2/(2(\log2+\eta))-O_\eta(1)$ ; let $\eta\to0$ . $\square$

with

If $j(\eta)=0$ the subtracted term is $(\log\ell_0)^2=(\log\log n_0)^2$ and nothing compensates it. In either case $\sum_jL_j\ge\bigl((\log\ell_J)^2-(\log\ell_0)^2\bigr)/(2(\log2+\eta))-O_\eta(1)$ , which is the second inequality. For $n_0\le C$ the subtracted term is $O_C(1)$ , and letting $\eta\to0$ gives the floor $(1/(2\log2)-o(1))(\log\log n_J)^2$ . $\square$

Verified: the witness meets Lemma 3.1's hypothesis ( $2^k$ is practical by binary expansion; the residues $0,1,2$ modulo $3$ are the empty sum, the divisor $1$ and the divisor $2$ , one summand each, total at most $2^k$ , none divisible by $3$ ), and its middle term is $\log3/\log(k+2)<1$ for $k\ge2$ while the stated right side tends to infinity with $k$ , so the display as written is refuted; the Problem 304 report's witness $n_0=m!$ , $A_0=2$ , $L_0=1$ refutes it likewise. The proof's case $j(\eta)\ge1$ gives $(\log\ell_J)^2/(2(\log2+\eta))-O_\eta(1)$ as the page says, which is at least the subtracted form; the case $j(\eta)=0$ gives exactly the subtracted form; and the case with no $j(\eta)$ is trivial. The page's "Hence" sentence at $J=0$ and unrestricted seed would assert $h(n_0)\ge(1/(2\log2)-o(1))(\log\log n_0)^2$ for every practical $n_0$ , which nothing on the page proves and which the page itself disclaims two paragraphs later; for bounded seeds it follows from the repaired display and $h(n_0)\ge0$ . The closing paragraph's "from a fixed seed" is exactly the restored hypothesis and needs no change. This adopts the shared-lemmas report's F1 and F2, its F3 for the definition of $n(\eta)$ , and the subtracted form proposed in the Problem 304 report's F1 and the Problem 293 report's F3.

C2. Two places.

(a) Section "Problem 304", paragraph "What it does not give, and why". Replace

Along the extension chains of the van Doorn note it is impossible too: Lemma D shows that every certificate built from Lemma 3.1 extensions is at least $(1/(2\log2)-o(1))(\log\log N)^2$ .

with

Along the extension chains of the van Doorn note it is impossible too: Lemma D shows that every certificate built from Lemma 3.1 extensions from a seed bounded independently of the scale, as the note's chain is, is at least $(1/(2\log2)-o(1))(\log\log N)^2$ .

(b) Factorial lead, section "What the route cannot give". Replace

and the counting floor (Lemma D of the connections page) shows that certificates built from Lemma 3.1 extensions are never below $(1/(2\log2)-o(1))(\log\log N)^2$ .

with

and the counting floor (Lemma D of the connections page) shows that certificates built from Lemma 3.1 extensions from a seed bounded independently of the scale are never below $(1/(2\log2)-o(1))(\log\log N)^2$ .

Verified: both sentences consume Lemma D at the interface C1 removes; the note's chain starts from $n_0=2^E(2Q(k_0))^{k_0}$ , bounded independently of $x$ and $p_*$ on the Proposition 4.1 page, so the section's conclusion about the note's mechanism stands under the repaired lemma. Adopts the Problem 304 report's F1.

C3. Section "Problem 304", the sentence after the proof of (304.1). Replace

This is below Vose's $\sqrt{\log b}$ for every $\varepsilon<\tfrac12$ and is the first consequence of the accepted part (b) outside Problem 18.

with

This is below Vose's $\sqrt{\log b}$ for every $\varepsilon<\tfrac12$ and is the first consequence of the accepted part (b) drawn in this corpus; no search for an earlier one was made.

Verified: the page's "Sources read" lists no search bearing on the priority claim, and docs/approach_vetting.md "Compare substance, not vocabulary" requires an incomplete search behind a novelty claim to be described as such; the note itself (p. 1, as both the Problem 304 and the Problem 293 reports read it) says that bounds on $h$ transfer to $N(b)$ , so the consequence is immediate to anyone holding both results. Adopts the Problem 304 report's F2.

C4. Two places.

(a) Section "Problem 304", paragraph "Next investigation". Replace

A positive answer proves the conjecture through Lemma C; Lemma D and Remark 6 show that neither mechanism of this folder can supply it, so it needs a new certificate of practicality.

with

A positive answer proves the conjecture through Lemma C. Remark 6 shows that factorials cannot supply it and Lemma D that Lemma 3.1 chains from a bounded seed cannot; the greedy count of Lemma B on another ratio-two family, such as $L_x$ under Lemma A, is excluded by neither, since its counting floor $h(N)\ge\log N/\log(\tau(N)+1)$ is only of order $\log\log N$ for $N=L_x$ , but no gap input of the required strength is known.

(b) Factorial lead, section "What the route cannot give". Replace

Both mechanisms stop at exponent $2$ ; the conjecture of Problem 304 needs a practical $N\in[b,\exp((\log b)^{O(1)})]$ with $h(N)\ll\log\log N$ , which needs a new certificate of practicality.

with

Both mechanisms stop at exponent $2$ ; the conjecture of Problem 304 needs a practical $N\in[b,\exp((\log b)^{O(1)})]$ with $h(N)\ll\log\log N$ , which neither mechanism can supply; the greedy count of Lemma B on another ratio-two family is not excluded by counting, but no gap input of the required strength is known.

Verified: Remark 6 concerns $n!$ through $\tau(n!)$ and Lemma D concerns Lemma 3.1 chains, while for $L_x$ one has $\log\tau(L_x)\asymp x/\log x$ and $\log L_x\asymp x$ , so the subset-count floor $\log L_x/\log(\tau(L_x)+1)$ is of order $\log x=\log\log L_x+o(1)$ , and the pigeonhole at the top window forces on Lemma B only $\int d\ell/\sigma\ge(\tfrac12\log L_x-O(\log x)-\log(2T))/\log\tau(L_x)$ , again of order $\log x$ ; neither obstruction reaches exponent $1$ there. Adopts the Problem 304 report's F6, with the replacement supplied here.

C5. Section "Problem 293", paragraph "What it does not give, and why". Replace

The factorial route is capped at $\exp\exp(C\sqrt k)$ : Remark 6 gives $2h(N)+1\ge2c(\log n)^2\approx2c(\log\log b)^2$ for the $N$ of Lemma C′, so it can only place $b$ in $D_k$ when $\log\log b\le\sqrt{k/(2c)}$ .

with

The factorial route is capped at $\exp\exp(C\sqrt k)$ : the counting argument of Remark 6, applied to $N$ (for $N=p^{j}(n-1)!$ one has $\tau(N)\le(j+1)\tau((n-1)!)$ and $\log N\ge\log(n-1)!$ ), gives $2h(N)+1\ge2c(\log n)^2=(2c+o(1))(\log\log b)^2$ for the $N$ of Lemma C′, so it can only place $b$ in $D_k$ when $\log\log b\le(1+o(1))\sqrt{k/(2c)}$ .

Verified: the Remark 6 page states $h(n!)\gg(\log n)^2$ for factorials only, and the second case of Lemma C′ has $N=p^j(n-1)!$ ; with $v=v_p((n-1)!)\ge1$ , $\tau(N)=\tau((n-1)!)\,(v+j+1)/(v+1)\le(j+1)\tau((n-1)!)$ , so $\log(\tau(N)+1)\ll n/\log n$ by the divisor count on the Remark 6 page, while $\log N\ge\log(n-1)!\gg n\log n$ , and the page's subset counting gives $h(N)\gg(\log n)^2$ in both of its cases; and $(\log n)^2=(1+o(1))(\log\log b)^2$ from $(n-1)!<b\le n!$ . Adopts the Problem 293 report's F1.

C6. Section "Problem 293", same paragraph. Replace

The extension chains are capped at the same shape by Lemma D, which is why the claimed Theorem 1.3 has the form it has.

with

Certificates built from Lemma 3.1 extensions of a bounded seed are capped at the same shape by Lemma D, which is why the claimed Theorem 1.3 has the form it has.

Verified as for C2. Adopts the Problem 293 report's F2.

C7. Section "Problem 18, question (a)", proof of (18a.1). Replace " $\delta\le2x^{-4}\le1/8$ " with " $\delta\le2e^{-4}<1/8$ ".

Verified: $\delta$ is nonincreasing with $\delta(T)=2T^{-c/\log x}$ and $T=x^{4/c}$ , so $\delta(T)=2x^{-4/\log x}=2e^{-4}\approx0.037$ , which exceeds $2x^{-4}$ for every $x\ge3$ ; the needed conclusion $\delta\le1/8$ holds. Adopts the question (a) report's F4.

C8. Two places.

(a) Section "Problem 18, question (a)", paragraph "The comment's form of the hypothesis, and where it fails". Replace

the comment's recurrence $R'\le R^{1-c/\log x}$ is exactly the lower-range step of Lemma B and gives the constant $1/c$ there, so the constant $3/c$ above is a bookkeeping loss, not a mathematical one.

with

the comment's recurrence $R'\le R^{1-c/\log x}$ is exactly the lower-range step of Lemma B and gives the constant $1/c$ there; the mirror recurrence on $U=L_x/R$ gives another $1/c$ in the upper range, so the direct count gives $2/c$ , and only the factor $3/2$ between it and the $3/c$ above is a bookkeeping loss.

(b) Lcm lead, section "Consequential gap". Replace

with a constant $3/c$ (or $1/c$ with the comment's recurrence in the lower range and the same mirror argument)

with

with a constant $3/c$ (or $2/c$ when the upper range is counted by the recurrence on $U=L_x/R$ instead of Lemma B's dyadic partition)

Verified: in the lower range $R'\le R\,\delta(R/2)=2^{1+\kappa}R^{1-\kappa}$ with $\kappa=c/\log x$ , so $\log R$ falls by the factor $1-\kappa$ per step, from $\tfrac12\log L_x\sim x/2$ to $\log(2T)$ , in $(1/c+o(1))(\log x)^2$ steps; in the upper range the mirror pair gives $R'\le R\,\delta(U/2)$ , so $\log U$ grows by the factor $1+\kappa$ per step over the same range, again $(1/c+o(1))(\log x)^2$ steps. The comment's form, stated up to $Y=L_x$ , would give $1/c$ over the whole range, but it is false in the top range, as the page itself shows; under the lower-half hypothesis the upper range costs its own $1/c$ . Adopts the question (a) report's F1.

C9. Two places.

(a) Section "Problem 18, question (a)", paragraph "Scale of the hypothesis". Replace

so necessarily $c\le2\log2+o(1)$ , and $(\mathrm H_c)$ with $c$ near $2\log2$ says that the largest gap matches the mean gap up to a polynomial factor at every scale above $x$ : a statement at the mean-gap scale, of the kind no factorial gap estimate reaches (Berend and Harmse's Remark 2 says their bound is the best their method gives up to a bounded power of $\log_2n$ ).

with

so necessarily $c\le2\log2+o(1)$ . With $c$ near $2\log2$ , $(\mathrm H_c)$ is at the mean-gap scale near $\sqrt{L_x}$ , where the exponent $-(c/2)x/\log x$ it allows and the exponent $-(\log2)x/\log x$ of the mean gap agree to first order; at scale $x^m$ it allows gaps $e^{-cm}$ of constant size against a mean gap of order $x^{-m}$ up to powers of $\log x$ , so below the top it asks much less than mean-gap matching. Near the top it is a statement of the kind no factorial gap estimate reaches (Berend and Harmse's Remark 2 says their bound is the best their method gives up to a bounded power of $\log_2n$ ).

(b) Lcm lead, section "Provisional premise", second bullet. Replace

and with $c$ near $2\log2$ it says that the largest gap equals the mean gap up to a polynomial factor at every scale above $x$ .

with

and with $c$ near $2\log2$ it is at the mean-gap scale near $\sqrt{L_x}$ , where the exponents of its allowance and of the mean gap agree to first order, while at scale $x^m$ it allows gaps $e^{-cm}$ of constant size against a mean gap of order $x^{-m}$ up to powers of $\log x$ .

Verified: at $\sqrt{db}=x^m$ the hypothesis reads $\log(b/d)\le x^{-mc/\log x}=e^{-cm}$ , while the products of $m$ distinct primes in $(x/2^{1/m},x]$ alone put $\gg_m x^m/(\log x)^m$ divisors of $L_x$ in $(x^m/2,x^m]$ , so the mean logarithmic gap there is $\ll_m(\log x)^m/x^m$ ; at the top the allowance is $\exp(-(c/2)\psi(x)/\log x)$ against the mean gap $\exp(-(\log2)\pi(x)+O(\sqrt x\log\log x))$ , whose exponents agree to first order exactly when $c=2\log2$ . Read with "polynomial in $x$ " the page's sentence is true at scales $x^{O(1)}$ and false at the top, where the ratio is $\exp(\Theta(x/(\log x)^2))$ ; read with "polynomial in the scale" it holds for every hypothesis, since every gap is at most $\log2$ and the mean gap at scale $g$ is at least $1/g$ . Adopts the question (a) report's F5.

C10. Two places.

(a) Section "Problem 18, question (a)", paragraph "Exploratory computation". Replace

It does not decrease over this range and approaches $2\log2=1.386\ldots$ from below; the binding pairs sit in the middle of the range, where the largest $\log(b/d)$ falls from $2.8\cdot10^{-2}$ at $x=20$ to $4.5\cdot10^{-5}$ at $x=64$ .

with

It shows no downward trend over this range (it dips once, from $1.311$ at $x=40$ to $1.298$ at $x=50$ ) and approaches $2\log2=1.386\ldots$ from below; the binding pairs sit at the top of the range, within a factor of about $2.6$ of $\sqrt{L_x}$ (the middle of the full divisor range of $L_x$ ), where the largest $\log(b/d)$ in the top tenth falls from $2.8\cdot10^{-2}$ at $x=20$ to $4.5\cdot10^{-5}$ at $x=64$ .

(b) Lcm lead, section "Exploratory computation", the two sentences after the table. Replace

The largest $\log(b/d)$ in the middle tenth falls from $2.8\cdot10^{-2}$ at $x=20$ to $4.5\cdot10^{-5}$ at $x=64$ , and the binding pairs sit in the middle of the range. $\kappa_{\min}(x)$ does not decrease over the sample and approaches $2\log2=1.386\ldots$ from below, the largest value the mean-gap bound allows; this was observed after the fact, not predicted.

with

The largest $\log(b/d)$ in the top tenth falls from $2.8\cdot10^{-2}$ at $x=20$ to $4.5\cdot10^{-5}$ at $x=64$ , and the binding pairs sit at the top of the range, within a factor of about $2.6$ of $\sqrt{L_x}$ . $\kappa_{\min}(x)$ shows no downward trend over the sample (it dips once, from $1.311$ at $x=40$ to $1.298$ at $x=50$ ) and approaches $2\log2=1.386\ldots$ from below, the largest value the mean-gap bound allows; this was observed after the fact, not predicted.

Verified by the independent enumeration described in the Subject: the six values of $\kappa_{\min}$ are reproduced, the table itself shows the dip from $1.311$ to $1.298$ , the binding pairs lie within a factor $2.58$ of $\sqrt{L_x}$ , and the two quoted maxima are those of the top tenth under both the retained script's bands and the lead's wording, the middle tenths at $x=64$ being an order of magnitude larger. Adopts the question (a) report's F3.

C11. Three places.

(a) Section "Problem 18, question (a)", paragraph "Next investigation". Replace

a disproof (a sequence of $x$ with $\kappa_{\min}(x)\to0$ ) shows that the greedy route along $L_x$ gives only $(\log x)^2/\kappa_{\min}(x)$ .

with

a disproof (a sequence of $x$ with $\kappa_{\min}(x)\to0$ ) shows that no hypothesis of the form $(\mathrm H_c)$ with a fixed $c$ holds, so (18a.1) as stated gives nothing along that sequence; whether a scale-dependent profile, or one that tolerates a few exceptional pairs, still gives $(\log x)^2$ through Lemma B would remain open.

(b) Lcm lead, section "Consequential gap". Replace

A disproof, a sequence of $x$ along which $\kappa_{\min}(x)\to0$ , would show that the greedy route along $L_x$ cannot give $(\log x)^2$ , leaving Lemma B's $x/(\log x)^{3/2-\eta}$ from Tenenbaum's input as the route's best.

with

A disproof, a sequence of $x$ along which $\kappa_{\min}(x)\to0$ , would show that no hypothesis of the form $(\mathrm H_c)$ with a fixed $c$ holds, so (18a.1) as stated gives nothing along that sequence; whether a scale-dependent profile, or one that tolerates a few exceptional pairs, still gives $(\log x)^2$ through Lemma B would remain open, with Lemma B's $x/(\log x)^{3/2-\eta}$ from Tenenbaum's input the route's best bound without $(\mathrm H_c)$ .

The computation distinguishes a quickly vanishing constant, which would show the route gives only $(\log x)^2/\kappa_{\min}(x)$ , from a stable one; it cannot prove $(\mathrm H_c)$ , and $x\le64$ is far from asymptotic.

with

The computation can only begin to distinguish a quickly vanishing constant, which would show that no fixed $c$ satisfies $(\mathrm H_c)$ , from a stable one; it cannot prove $(\mathrm H_c)$ , and $x\le64$ is far from asymptotic.

Verified: a single pair with small $\kappa$ costs the greedy expansion at most one nonterminal step in the lower range, since the remainder after that step lies below the pair's lower divisor and remainders decrease, and at most one in the upper range through its mirror (after the step $N/R'>N/d$ , so the mirror pair never brackets $U$ again); the charging argument of Lemma B counts such steps separately at cost $1$ each and keeps the disjointness of the remaining intervals. So $\kappa_{\min}(x)\to0$ bounds nothing from below; it shows only that no fixed $c$ satisfies $(\mathrm H_c)$ . Adopts the question (a) report's F2 and the third part of its F7.

C12. Lcm lead, section "Next step", last sentence. Replace

A result for the middle alone would already cover the range that dominates the integral of Lemma B.

with

Under $(\mathrm H_c)$ every factor-of- $e$ range of $\ell=\log(2g)$ contributes $\log x/c$ to the integral of Lemma B, so a result near $\sqrt{L_x}$ alone covers only $O(\log x)$ of the $(\log x)^2$ count; it is nevertheless the range where $(\mathrm H_c)$ is tightest and where the computation finds the binding pairs.

Verified: with $\sigma(\ell)=c\ell/\log x$ the integral over $[\ell,e\ell]$ is $\log x/c$ for every $\ell$ , and the range from $L_x^{1/4}$ to $\sqrt{L_x}$ contributes $(\log2)\log x/c$ of the total $(\log x)^2/c$ . Adopts the question (a) report's F6.

C13. Section "Problem 18, question (c)", paragraph "What transports", the sentence beginning "Hence $C=2$ is impossible". Replace

Hence $C=2$ is impossible for the greedy route, consistent with Remark 6, and $C=3$ asks the largest gap to match this mean gap up to a constant in the exponent at every scale: the factorial analog of $(\mathrm H_c)$ , with the same mean-gap character.

with

Hence the natural profile with $C=2$ is inadmissible, and no admissible profile does better than exponent $2$ : the pair above forces $\sigma\le\log\tau(n!)$ on all but the last $\tfrac12\log n+O(1)$ of the range, so the integral is at least $(\tfrac12\log n!-O(\log n)-\log(2T))/\log\tau(n!)\gg(\log n)^2$ whenever $\log T\le(\tfrac12-\eta)\log n!$ , and otherwise the term $2\log_2(2T)$ is $\gg n\log n$ ; this is consistent with Remark 6. Whether a profile capped at $\sigma=cn/\log n$ for a small constant $c$ , whose integral is $O((\log n)^2)$ , satisfies the hypothesis of Lemma B is a maximal-gap question that this counting leaves open. The natural profile with $C=3$ asks the largest gap to match this mean gap up to a constant in the exponent throughout the windows of index $j\asymp n$ , while asking much less below them: the factorial analog of $(\mathrm H_c)$ , with the same mean-gap character near the top.

Verified: the page's argument shows only that the natural profile $\sigma(\ell)=\ell/\log n$ violates Lemma B's hypothesis at the window of index $n$ . The profile $\sigma(\ell)=\min(\ell/\log n,\,cn/\log n)$ with $\log(2T)=\log8\cdot\log n$ is admissible (nondecreasing, at least $\log8$ ), has integral at most $(1+1/(2c))(\log n)^2$ , and demands at the top windows only $\log(b/d)\le\tfrac12\exp(-cn/\log n)$ , which the floor $\tfrac12\log n/\tau(n!)=\exp(-\Theta(n/\log n))$ does not contradict for small $c$ ; so "impossible for the greedy route" is unproved as written. The counting bound for every admissible profile is as stated: the pair found at the top window forces $\delta(g)\ge\log n/\tau(n!)$ at some $g\ge\sqrt{(n-1)!/2}$ , hence $\sigma\le\log\tau(n!)$ up to $\log(2g)$ . For $C=3$ the demand $\exp(-\ell/(\log n)^2)$ at $\ell\asymp n\log n$ is $\exp(-\Theta(n/\log n))$ , the mean-gap order, while at $\ell=n$ it is $\exp(-n/(\log n)^2)$ against a mean gap of $\exp(-\Theta(n/\log n))$ (the subsets of the primes in $(n/2,n]$ alone give $\exp(\Theta(n/\log n))$ divisors in every $[e^\ell,2e^\ell]$ with $\ell\ge\theta(n)-\theta(n/2)$ ). Adopts the question (c) report's F1, with the constant $c_\tau$ of its replacement left out so that the page keeps Chebyshev as its only prime-count import, and its F2.

C14. Section "Problem 1099", two places.

(a) The sentence after the display of the two-sided bound. Replace

For $\alpha\ge1+\eta$ the correction is tiny: $(4\varepsilon_j)^{\alpha-1}\le2^{-62(\alpha-1)}$ , and $C(2)<10^{-10}$ .

with

The correction is small only when $\alpha$ is bounded away from $1$ : for every $j$ , $(4\varepsilon_j)^{\alpha-1}\le2^{-62(\alpha-1)}$ , so $C(\alpha)\le2^{62\eta}C(1+\eta)\,2^{-62(\alpha-1)}$ for $\alpha\ge1+\eta$ , with $C(2)<10^{-10}$ and $C(3/2)<10^{-3}$ ; but $C(5/4)\approx20$ exceeds $2\zeta(5/4)$ , and $C(\alpha)\to\infty$ as $\alpha\to1^+$ .

(b) Paragraph "What it does not give", last sentence. Replace

Only the explicit value $2\zeta(\alpha)+O(2^{-62(\alpha-1)})$ appears to be new, and no search for it in the later literature was made.

with

Only the explicit bound $2\zeta(\alpha)+C(\alpha)$ , with $C(\alpha)=O_\eta(2^{-62(\alpha-1)})$ for $\alpha\ge1+\eta$ , appears to be new, and no search for it in the later literature was made.

Verified by the independent evaluation described in the Subject: $C(5/4)\approx20.3>9.19\approx2\zeta(5/4)$ , $C(3/2)\approx1.8\cdot10^{-4}$ , $C(2)\approx3.7\cdot10^{-14}$ , and every term of $C(\alpha)$ increases to $4(\tfrac12\log j+\log2)$ as $\alpha\to1^+$ , a divergent series, so $C(\alpha)\to\infty$ by monotone convergence; the factor $\tfrac12\log j+\log2$ is not summable, so the per-term bound says nothing uniform in $\alpha$ . The inequality $C(\alpha)\le2^{62\eta}C(1+\eta)2^{-62(\alpha-1)}$ follows from $4\varepsilon_j\le2^{-62}$ . Adopts the Problem 1099 report's F1.

C15. Section "Problem 673", Target paragraph. Replace

a site comment's bounds $\tau(n/m)/m\le G(n)\le\tau(n)$ for $m\mid n$

with

a site comment's bounds $\tau(n/m)/m\le G(n)\le\tau(n)$ for $1<m\mid n$ (the lower bound fails at $m=1$ , since $G(n)<\tau(n)$ for every $n$ )

Verified: $G(n)$ is a sum of $\tau(n)-1$ terms each below $1$ , so $G(n)<\tau(n)-1$ for $n\ge2$ and $G(1)=0$ ; at $n=2$ , $m=1$ the bound would read $2\le\tfrac12$ . For $m\ge2$ every divisor $e$ of $n/m$ has a successor at most $em$ among the divisors of $n$ , which gives the bound; the section's own use is at $m=2$ . If the site comment states the bound without the restriction, the page still carries the true range. Adopts the Problem 673 report's F1.

C16. Section "Problem 859", Outcome paragraph. Replace

which is $\ge1/t^{2}$ at best (Proposition 4.1 places $N$ in $[t,t^2)$ ) and says nothing at the scale $(\log t)^{-\delta}$ .

with

which is at least $1/t^{2}$ (Proposition 4.1 places $N$ in $[t,t^2)$ once $t\ge x_0$ ) and at most $1/t$ for any seed $N\ge t$ , so polynomial in $1/t$ whatever the construction, and says nothing at the scale $(\log t)^{-\delta}$ .

Verified: $N\in[t,t^2)$ gives $1/N>1/t^2$ , so "at best" names the wrong extremum; the folder's other seeds do better (the least $n!\ge t$ has $n!<nt$ , and $2^{\lceil\log_2t\rceil}<2t$ is practical), and every seed $N\ge t$ gives at most $1/t$ ; the conclusion holds for every polynomial floor since $t^{-1}=o((\log t)^{-\delta})$ . Adopts the Problem 859 report's F1.

C17. Section "Other divisor problems examined", first bullet. Replace

Lemma A is the trivial instance for $n!$ and $L_x$ ;

with

for $n,x\ge3$ the divisors $2<3<4$ of $n!$ and $L_x$ give the property outright (Lemma A's non-strict ratio bound alone does not, as $N=2^k$ shows);

Verified: Lemma A concludes $b\le2d$ for consecutive divisors, and the Problem 144 property needs $d_2<2d_1$ ; $N=2^k$ satisfies Lemma A's hypothesis vacuously, has every consecutive ratio equal to $2$ , and has no such pair; $2!$ and $L_2$ fail the property likewise. Adopts the other divisor problems report's F1.

Rejected and downgraded findings

Findings are cited by report and label. "Downgraded to note" means the observation is correct and the change is optional, or no exact replacement is warranted; "rejected" means the claimed defect is not one.

Shared lemmas report.

F3 (suggested; $n(\eta)$ undefined): accepted inside C1(b).
F4 (note; the aside "whence $p\ll\log n$ " imports a Chebyshev-type bound unnamed): downgraded to note; the proof uses only $p\le n-1$ , which the reviewer shows holds for every $n\ge3$ , and the aside is true.
F5 (note; "as the Problem 304 page records" outside the read set): downgraded to note, verified here: the Problem 304 page's Origin paragraph records Erdős's 1950 route through $n!$ with $(n-1)!<b\le n!$ , and its Upper bounds bullet records Vose's bound as restated by van Doorn and Tang, with denominators dividing, or $b$ times divisors of, a number $N_K$ ; the attribution is right.

Problem 304 report.

F3 (note; "once $b>(n_0-1)!$ " omits $n\ge14$ ): downgraded to note; the statement asserts an existential threshold $b_0(\varepsilon)$ , the proof says "for large $b$ " before naming the threshold, and $n\ge10$ already suffices for $n<\log b$ ; the reviewer's " $b>\max(n_0-1,13)!$ " is the exact threshold and is optional.
F4 (note; "which is the note's Theorem 1.2" for a bound stronger by $2$ ): downgraded to note; the result page states $N(b)\le2c_0(\log\log b)^2$ , which the derived bound implies, and "is" reads as "recovers".
F5 (note; "Its standing is that of part (b)"): downgraded to note; the sentence goes on to list that standing and to name the transfer as Lemma C, which the page's preamble labels author-recorded.
F6 (note; the greedy count on non-factorial ratio-two integers is not blocked): accepted as C4 with the replacement supplied here.
F7 (note; "exact site questions" while Problem 293 is read with $m>1$ ): downgraded to note; the Problem 293 page's Formulation records the $m>1$ reading as the intended statement of the monograph, the site's discussion and the van Doorn–Tang paper, and every bound on that page uses it.

Problem 293 report.

F3 (note; outside the section): accepted as C1.
F4 (note; "so $v(k)\ge\exp(k^{A})$ " needs $\varepsilon<1/A$ ): downgraded to note; the deduction is immediate with $\varepsilon=1/(2A)$ and $k\ge25$ , and the page does not say $\varepsilon=1/A$ .
F5 (note; the consistency sentence is vacuous and the listing was not checked): downgraded to note; the consistency claim is indeed empty for bounds with unspecified thresholds, and the Problem 293 page does list $v(3),\dots,v(8)$ in its "Finite values (unreviewed leads)" paragraph, as values reported in a site comment and not reviewed, which "known" on the connections page overstates by a word.

Problem 18, question (a) report.

F7 (note; "nothing unconditional beyond", "Three facts ... are established", "The computation distinguishes"): the third part is accepted inside C11(c); the first two are downgraded to note, since "unconditional" reads as "not conditional on $(\mathrm H_c)$ " while the same paragraph marks the Tenenbaum reading as a text-layer reading, and since the lead's third fact names Tenenbaum's paper, whose page on the connections page carries the mark.

Problem 18, question (c) report.

F3 (suggested; "does not run through a gap profile" outruns the read record): downgraded to note; the same sentence says the structure was "read only at the level of theorem statements", which is the qualification the reviewer asks for.
F4 (suggested; "at least as hard as $(\mathrm H_c)$ " without a reduction): downgraded to note; the clause is an informal assessment of difficulty, not a claimed implication, and the reviewer's own transfer (a consecutive pair of $n!$ sits inside a consecutive pair of $L_n$ of comparable geometric mean, so $(\mathrm H_c)$ at $x=n$ reaches only the bottom of the factorial range) supports rather than refutes it; the replacement "of the same mean-gap character; neither hypothesis is known to imply the other" is optional.
N1 to N4 (notes): downgraded to notes; the dropped $(1\pm o(1))$ factors do not affect the comparisons, $\ell_0$ is Lemma B's $\log(2T)$ , "the only mechanism aimed at $n!$ " means the only one that could reach question (c), and the lower bound $\log\tau(n!)\gg n/\log n$ is unused.

Problem 1099 report.

F2 (note; a sharper form of the corollary): downgraded to note; the stated interval is correct.
F3 (note; "would be good candidates" against the held wording): downgraded to note; the section attributes the quotation to the site page, which was not read here, and the held paper (p. 171, checked here in the layout text) reads "seem to be good candidates for integers with (1.1) bounded above"; quoting the held wording is preferable.
F4 (note; "has no compiled assessment" while the page carries a Current assessment heading): rejected. As of 2026-09-28T13:50:48Z the Problem 1099 page's Current assessment section reads "No current assessment is recorded", so the sentence and the Next investigation item are accurate; the later updated stamp reflects the page's Progress paragraph, which records this very transfer.
F5 (note; unheld-source characterizations): downgraded to note, a limitation the section itself states.

Problem 673 report.

F2 (suggested; cite the held 1982 survey instead of the site): downgraded to note; the section reports what the site page records, and the reviewer verified both attributions against the held survey, so nothing is false; citing the survey is preferable.
F3 (note; the notation $2^EV_j$ ): downgraded to note.

Problem 859 report.

F2 (suggested; "No transfer exists" is a universal negative): downgraded to note; the sentence is the page's outcome vocabulary for a section whose Outcome line reads "fails", and the reviewer's own attack finds no transfer bearing on the target.
F3 (note; status text unverified): downgraded to note, a limitation.

Graded verdicts

Per statement, with the reports' labels, as graded, with the corrections that apply. "Refutation-failed" and "refuted-as-stated" are written in full throughout.

Shared lemmas.

S1 (Lemma A): refutation-failed.
S2 (Lemma B): refutation-failed.
S3 (Lemma C): refutation-failed.
S4 (Lemma C′): refutation-failed.
S5 (Lemma D, first inequality): refutation-failed.
S6 (Lemma D, second inequality, uniform over all chains): refuted-as-stated, by the witness of C1; the subtracted form and the bounded-seed floor that C1 writes are refutation-failed.
S7 (Lemma D, "Hence" sentence, unrestricted seeds): proof defective; statement unresolved; for bounded seeds, as C1 writes it, refutation-failed.
S8 (closing paragraph): refutation-failed.

Problem 304.

S1 (Lemma C): refutation-failed.
S2 (Lemma C′): refutation-failed.
S3 ((304.1)): refutation-failed, as a deduction conditional on the accepted part (b) at its recorded standing.
S4 ((304.2)): refutation-failed, conditional on Theorem 1 at its standing.
S5 ((304.3)): refutation-failed as a deduction from Proposition 4.1, which stays claimed.
S6 (reduction): refutation-failed.
S7 (factorial obstruction): refutation-failed.
S8 (extension-chain obstruction, as written): proof defective; statement unresolved; with the bounded-seed clause of C2, refutation-failed.
S9 (consequence sentences): refutation-failed at the two reconstructed mechanisms, the scope C4 writes.
S10 ((293.1) on the lead): refutation-failed, conditional on part (b).
S11 (standing and input sentences): refutation-failed for the input sentences; the priority sentence is replaced by C3.

Problem 293.

S1 (transfer observation): refutation-failed.
S2 ((293.1)): refutation-failed, conditional on part (b).
S3 (corollary and calibration): refutation-failed.
S4 ((293.2)): refutation-failed at the standing of Theorem 1.
S5 ((293.3)): refutation-failed as a deduction from the claimed Proposition 4.1.
S6 (exponent-one obstruction): refutation-failed.
S7 (factorial-route cap): refutation-failed, with the citation repaired by C5.
S8 (extension-chain cap): refutation-failed for bounded seeds, as C6 writes it.
S9 (converse flow): refutation-failed.
S10 (falsification paragraph): refutation-failed.
S11 (closing sentence): not a mathematical claim; its consistency clause is vacuous (note).
T (target paragraph): descriptive; consistent with the Problem 293 page read here.

Problem 18, question (a).

S1 (Lemma A at $L_x$ ): refutation-failed.
S2 ((18a.1)): refutation-failed; its proof carried the slip repaired by C7.
S3 (the comment's form fails in the top range): refutation-failed.
S4 (the lower-range constant $1/c$ ): refutation-failed.
S5 (the constant $1/c$ from the mirror argument): proof defective; statement unresolved as written; the constant $2/c$ that C8 writes is refutation-failed.
S6 ((18a.2)): refutation-failed as a conditional deduction from the quoted admissibility statement, which is unverified here.
S7 (the necessary bound $c\le2\log2$ ): refutation-failed.
S8 (the mean-gap gloss): refuted-as-stated under the reading in which it has content; replaced by C9.
S9 (the computation): refutation-failed for every printed value; refuted-as-stated for the location of the binding pairs, the labeling of the quoted maxima and "does not decrease"; replaced by C10.
S10 (the obstruction drawn from a disproof): proof defective; statement unresolved; replaced by C11.
S11 (the lead's "dominates"): refuted-as-stated; replaced by C12.
S12 (what it does not give): refutation-failed.

Problem 18, question (c).

S1 (target): refutation-failed for the two bounds.
S2 (necessary condition and natural profile): refutation-failed.
S3 (demand at the middle): refutation-failed.
S4 (mean-gap floor): refutation-failed.
S5 (" $C=2$ is impossible for the greedy route"): proof defective; statement unresolved as written; refutation-failed on the narrow reading and for the counting bound that C13 writes.
S6 ( $C=3$ at every scale): refuted-as-stated for "at every scale"; refutation-failed at the windows of index $j\asymp n$ , as C13 writes it.
S7 (Berend–Harmse comparison): refutation-failed.
S8 (the accepted proof's structure): refutation-failed as the hedged description the sentence gives.
S9 (outcome): "open" refutation-failed; the hardness comparison is an informal assessment (note).
S10: not a claim.

Problem 1099.

S1 (convergence of $C(\alpha)$ ): refutation-failed.
S2 (two-sided bound for $n\ge T$ ): refutation-failed.
S3 (limits along $n!$ ): refutation-failed.
S4(a), S4(b): refutation-failed.
S4(c) (the correction uniformly tiny): refuted-as-stated; replaced by C14.
S5 (corollary on (1.2)): refutation-failed.
S6 (scope): refutation-failed.
S7 (attributions): the parts checkable against held sources agree; the rest is not a mathematical claim.

Problem 673.

S1 (target): refutation-failed.
S2a (records): refutation-failed for the two attributions.
S2b (the site comment's bound for $m\mid n$ ): refuted-as-stated at $m=1$ ; refutation-failed for $m\ge2$ , as C15 writes it.
S3 (density-zero obstruction): refutation-failed.
S4 (factorial bound): refutation-failed.
S5 (even- $n$ bound): refutation-failed.
S6 (non-transfer verdict): refutation-failed.

Problem 859.

S1 (target): refutation-failed.
S2 (recorded status): a report of status text; the deduction it carries is refutation-failed.
S3 (outcome): refutation-failed.
S4 (transfer lemma): refutation-failed.
S5 (consequence): refutation-failed.
S6 (polynomial floor): refutation-failed for the inequality; the wording is repaired by C16.
S7 (not through $h$ ): refutation-failed.
S8 ("No transfer exists"): refutation-failed as the section's conclusion.