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Subject

The repository as it stood on 2026-09-28T13:50:48Z, the head of the branch on which the reports were filed. Pages, each read whole from the committed text of that state: wiki/research/erdos_49/connections.md (637 lines, eleven H2 sections after the introduction) and wiki/research/leads/distinct_totient_blocks_from_collision_bounds/_index.md (146 lines). The working-tree copies of both pages are byte-identical to the frozen pages (the diff against that state on the two paths is empty). Reports, each read whole: the notation review, the Problem 1004 review, the Problem 1003 review, the Problem 417 review, the Problem 416 review, the Problem 694 review, the Problem 51 review, the Problem 821 review, the Problem 415 review, the Problems 850 and 931 review and the Problem 49 clause review. Each report covers one H2 section of the page, from its heading to the line before the next heading; the Problem 1004 report also covers the lead, whose route rests on that section.

Read for adjudication: the sections "Independence and the assignment", "Exact subjects and durable evidence", "Report contract", "Grading and claim standing", "Whole-claim report" and "Audit checklist" of docs/verification.md. The six reconstruction pages of the folder in that state, read whole: Theorem 3.1, Theorem 3.3, Lemma 3.2, Lemma 4.1, Lemma 5.1 and Theorem 1.2. The held author manuscript of Pollack, Pomerance and Treviño under the primes card (17 pages), physical pages 2--3 and 5--10 on the text layer: the definition of a convenient integer, Theorems 1.2 and 1.5 and the Notation paragraph (pp. 2--3), Theorem A and (3.2) (p. 5), Theorem 3.1 with its sketch, Lemma 3.2 and Theorem 3.3 with Remark 3.1 (p. 6), the proof of Theorem 3.3 and Ford's order of magnitude (p. 7), Lemma 4.1 (p. 8), the end of its proof, Lemma 5.1 and the proof of Theorem 1.2 (pp. 9--10). The held copy of Ford's paper under Ford (1998) (43 pages), pp. 2--4 (Theorems 1, 2, 4, 5 and 8, the constants (1.3)--(1.6), Table 1 and the paragraph on multiplicities $A(m)\ge m^{c_4}$) and p. 39 (the witnesses $d=2^{18}\cdot257$ and $d=54$, the sentence "I believe that obtaining the asymptotic formula ..." and "may miss whole classes of totients") on the text layer, and the card's "Results to transcribe" list in that state. Page 4 of the five-page note held on the note's card (Proposition 3.1 with its proof) on the text layer, and the card's read status. To check characterizations no reviewer could reach inside its read set: the lead distinct totient preimage ratio (its Theorem 417.3 inputs, its premise (P1) and its sentence on this page's Propositions 2--4); the Problem 417, Problem 694 and Problem 51 passages of the Problem 416 connections page and the second-question paragraph of the Problem 416 folder index; and, in that state, the problem pages Problem 1004, Problem 1003, Problem 417, Problem 416, Problem 694, Problem 51, Problem 821, Problem 415, Problem 850, Problem 931 and Problem 49 (statements, and their Status, Current assessment, Progress and Known Results text, which a grader reads by design). For the shape of this record only, the reconstruction grade and the folder's evidence index were read; nothing from them enters the adjudication. No other review, evidence folder, workspace file or web page was read.

Independence, by role: distinct grader in a fresh context, given only this assignment. The grader wrote none of the two pages, none of the eleven reports, no reconstruction page, no library card and no problem page named above, and had no communication with the author or with any reviewer. A grader is not blind: the reports, the standing text of the problem pages and the card bodies were read by design.

Grader's own checks, run in a temporary script that is not retained and is described here so that it can be rerun. (1) The preimages of $2^{18}\cdot257$, of $2^{18}\cdot257+28$, of $1$, $2$ and $54$ were enumerated by the divisor recursion (every prime $p$ dividing a preimage of $d$ has $p-1\mid d$), each result checked with $\varphi$, and the recursion cross-checked against a brute-force sieve on all $791$ totients up to $3000$: the seed has exactly the eight even preimages listed on the Lemma 5.1 page, least $2^{11}\cdot257^2=135268352$; $2^{15}\cdot257+1=8421377=1123\cdot7499$ is composite; $67371037$ is prime; the preimages of $1$, $2$ and $54$ are $\{1,2\}$, $\{3,4,6\}$ and $\{81,162\}$. (2) A sieve of $\varphi$ to $1.2\cdot10^7$ finds no totient $v\le2\cdot10^6$ with $f_{\max}(v)/f_{\min}(v)=16/15$, while $\varphi(16)=\varphi(15)=8$ and the preimages of $8$ are $15,16,20,24,30$; the algebraic proof is re-derived under C11. (3) For $k=2$ and $k=6$ and $|j|\le2\cdot10^5$, the natural numbers $j$ with $\gamma(j)=\gamma(j+k)$ are $\{2\}$ and $\{2,3,6,12,18,48\}$, and the further integer solutions are $\{-4,-1\}$ and $\{-54,-24,-18,-12,-9,-8,-4,-3,-2\}$; over $-3000\le x<y<3000$ the only pair satisfying the three conditions of Problem 850 (radicals of absolute values) is $(x,y)=(-4,2)$, and the two-condition pairs with $0<x<y<3000$ are $(2,8)$, $(6,48)$, $(14,224)$, $(30,960)$ and $(75,1215)$. (4) $\sum_{t\ge0}\exp(-\exp(0.2\cdot1.8^t))=0.72178\ldots$, so $K=\exp(4\cdot0.72178)=17.94\ldots$, and the ratio $a_{t+1}/a_t$ is $0.81$, $0.62$, $0.27$ at $t=0,1,2$. (5) The longest block of consecutive integers with pairwise distinct totients below $10^5$ and $10^6$ has length $111$ (start $73318$) and $194$ (start $900979$). (6) The largest subset of $\{1,\ldots,N\}$ on which $\varphi$ is strictly increasing (the longest strictly increasing subsequence of $\varphi(1),\ldots,\varphi(N)$, by patience sorting) has exactly $\pi(N)$ elements for every $2\le N\le10^6$. (7) The witness of C4 and the clustered configuration of C6 were recomputed by hand, and the estimate $\log_3(x/\mu)-\log_3x=O(1/(\log x\log_2x))$ behind C2 was re-derived.

Exposure ruling. Every report discloses reads wider than its allowed material: Status, Source, References and Formalization paragraphs of problem pages; the Current assessment, Progress and Known Results text of Problems 417, 51 and 1004; card bodies with read-status paragraphs, "Relation to E49" sections and "Bears on" rows; a result page's proof pointer and its "Bears on" sentence; the Problem 416 folder index with its standing text; a lead's frontmatter fields; the file names of sibling reviews. Content test: nothing mathematical in any report could only have come from that text. Every finding cites the frozen page, a reconstruction page or the held sources, every re-derivation proceeds from the PDF or from the page's own displays, and the direction of each attack (the symbol $C_2$, the clustered configuration, the reach witness, the seed $16/15$, the integer domain of Lemma 3.2, the lacunary placement against (W), the iteration of convenience, the block argument against a multiplicity) follows from the subject. Two notes rest on exposed corpus-state text (Problem 417 F4 and Problem 51 F3, that a "not compiled" sentence is stale in that state); they are notes about the corpus record, not mathematics, and are handled below as downgraded findings. None of the exposed text is a review of the page or a verdict on it. The exposures are ruled immaterial for all eleven reports.

Reports graded

Notation review: pass. The subject block resolves (path, date, line range). The independence facts and exposures are stated. The restatement carries every input with its quantifiers (the absolute $x_0$ of (T1); the conditions on $\varepsilon$ and the uniformity of (T3); the order fixed pair, fixed $D$, then constants in (L4); the $\mu$-dependent threshold and absolute constant in (F)). All ten checklist items carry an explicit verdict. The three weakest steps are re-derived, not paraphrased: the limit $Z(x/\mu)/Z(x)\to1/\mu$ through the iterated-logarithm increments, the deduction of the doubling inequality with the lower half of Ford's theorem made visible, and the $D$-only labeling of the constants; the grader re-derived the first two and agrees. The strongest attack is real: a ninth or odd preimage of the seed, a wrong range or exponent in (T1), the odd-$k$ clause, a $\mu$-dependent constant, and the derivation of (F)'s third consequence from its two listed ones, which exposed C2. The premises carry interface, locator and reading depth. The verdict is stated per statement and assigns no tier.

Problem 1004 review: pass. The subject block resolves and names the lead. The independence facts and the one exposure are stated. The restatement carries Proposition 1 with its range of $L$, the corollary with its fixed $c<2$, the four steps as separate statements, the barrier, the reach clause, the conditional deduction, the Poisson prediction and the computation. All ten checklist items carry an explicit verdict. The three weakest steps are re-derived: the application of (T1) and (T3) at the shifted point with the fixed $\varepsilon(y)$ and the uniformity traced to $y$ alone; the Euler product bounding $S(L)$ by $A_0(\log L)^{49}$; and the union bound with the return to $x$; the grader re-derived all three and agrees. The strongest attack is real and succeeded twice, against the reach clause with the exact witness $L=\lfloor(\log x)^2/(\log\log x\cdot\log\log\log x)\rfloor$ and against the first-moment deduction with the clustered configuration, both verified here; the nine attacks on Proposition 1 are recorded with why each failed. The computation was reproduced from a fresh implementation, and the grader reproduced it again. The premises carry interface and reading depth. The verdict is stated per statement, in full, and assigns no tier.

Problem 1003 review: pass. The subject block resolves. The independence facts and exposures are stated, including that Yamada's paper is not held. The restatement separates the target, the transported bound, the emptiness of the structured family, the direction clause and the parity clause. All ten checklist items carry an explicit verdict. The three weakest steps are re-derived: coprimality of $j$ and $j+1$, the application of (T1) at $k=1$ with the ordering of the three bounds, and the corrected reason for the obstruction; the grader checked the ordering and agrees. The strongest attack is real and succeeded against the direction clause with the witness (L4); the attack on the parity clause went to Ford's Lemma 3 and its shift $\ell\kappa(a,b)$. The premises carry interface and reading depth. The verdict is stated and assigns no tier.

Problem 417 review: pass. The subject block resolves and names the parallel lead it read. The independence facts and three exposures are stated. The restatement carries the three propositions with their constants, the cluster interval, the conditional clause with its antecedent, and the two obstructions as statements about the recorded interface. All ten checklist items carry an explicit verdict. The three weakest steps are re-derived: the doubling consequence of (F), the dyadic pigeonhole with the bridge to $V(y)-V'(y)$, and the admissibility of the degenerate inputs $d_1=d_2=1$, $D=K$ with the absoluteness of $K$; the grader re-derived each and agrees. The strongest attack is real: the injectivity and containment in Proposition 2, a skipped value in the cluster interval, an odd preimage of the seed, the lacunary placement of convenient integers that satisfies (L4) at every scale while emptying a window, and the burst configuration against a limit inferior. Both author-recorded computations were rerun independently. The premises carry interface and reading depth, with Ford's section 5 read at the points the "Next investigation" paragraph names. The verdict is stated and assigns no tier.

Problem 416 review: pass. The subject block resolves. The independence facts and three exposures are stated. The restatement lists ten statements with the conventions $V_k$ and $V(x;k)$ supplied. All ten checklist items carry an explicit verdict. The three weakest steps are re-derived: the order-of-magnitude chain with the upper bound located in (F), the obstruction through multiplicity classes with the proof that multiplicity is unbounded and Ford's disjoint families for $d=4$ and $d=6$, and the unspecified constant; the grader re-derived the chain and the multiplicity argument and agrees. The strongest attack is real and succeeded against the "recovers from it" sentence; the attack on S5 through the near-neighbor collisions and a growing seed list is recorded. The premises carry interface and reading depth. The verdict is stated and assigns no tier.

Problem 694 review: pass. The subject block resolves. The independence facts and four exposures are stated. The restatement separates Proposition 5, its two consequence sentences and the obstruction. All ten checklist items carry an explicit verdict. The three weakest steps are re-derived: the passage from integers to totients with the injectivity through the least preimage, the passage from $Z$ to $V$, and the strengthening sentence; the grader agrees. The strongest attack is real and succeeded with the exact witness $(a,b)=(16,15)$, proved algebraically and corroborated by an enumeration whose coverage argument is stated; the grader re-proved the witness and reran the enumeration. The premises carry interface and reading depth, with the note's Proposition 3.1 read on the page image. The verdict is stated and assigns no tier.

Problem 51 review: pass. The subject block resolves. The independence facts and three exposures are stated, one of them the whole problem page. The restatement carries the transport with its interval and constants and the four clauses of the obstruction. All ten checklist items carry an explicit verdict. The three weakest steps are re-derived: least preimage and injectivity, the count with the constant $c_d/C_1$, and the obstruction under repeated application, where the reviewer proves that convenience composes and that the factor $n/\varphi(n)$ of an (L4) integer is at most $1+O(\log_3D/D)$, summable along at-least-doubling seeds; the grader re-derived the composition, the coprimality and the tail bound and agrees. The strongest attack is real: geometric growth by iteration, defeated by that bound. The premises carry interface and reading depth. The verdict is stated and assigns no tier.

Problem 821 review: pass. The subject block resolves. The independence facts and three exposures are stated. The restatement carries the problem's quantifier order and the four statements. All ten checklist items carry an explicit verdict. The three weakest steps are re-derived: the bridge from Lemma 4.1 to Ford's Theorem 2, the obstruction shown to defeat the implication at its own hypotheses (a witness output forces a witness seed, and a fixed seed yields finitely many), and the identification with Ford's mechanism on his pp. 24--25; the grader agrees. The strongest attack is real: non-convenient candidates, seeds growing with $x$, iteration, and the collision bounds turned on a multiplicity, which yield only $g(v)\ll v\log\log v/\log v$ (the grader checked the block argument). The premises carry interface and reading depth. The verdict is stated and assigns no tier.

Problem 415 review: pass. The subject block resolves. The independence facts and three exposures are stated. The restatement fixes the reading of "any of the $k!$ patterns" as "each", defines realization and the run maxima, and lists seven statements. All ten checklist items carry an explicit verdict. The three weakest steps are re-derived: the existence of $F(n)$ and the passage to a nonincreasing run inside $[1,n]$, the asymptotic form with the case $c=0$ isolated, and the interface of Proposition 1 with the uniqueness of the realized pattern; the grader re-derived the first two and agrees. The strongest attack is real: order information from equality-invariant inputs, the scales, the direction of monotonicity, and the literal question at $c=0$. The premises carry interface and reading depth, with Theorem 1.5 read on p. 3 and its proof's structure on pp. 12--15. The verdict is stated and assigns no tier.

Problems 850 and 931 review: pass. The subject block resolves. The independence facts and four exposures are stated. The restatement fixes the radical of a negative integer and the sets $J_k$. All ten checklist items carry an explicit verdict. The three weakest steps are re-derived: the lemma's injection extended to every integer $j\notin\{0,-k\}$ with the sets for $k=6$ exhibited and the case check for entries $0,\pm1$; the Problem 931 obstruction; and the union clause with the two-condition family from Guy's B29 and the exhaustive check below $3000$; the grader recomputed the solution sets, the family and the search and agrees. The strongest attack is real: the widened domain, the sign of $k$, a nonexistence proof from $k^\epsilon$, a fixed difference in Problem 931. The premises carry interface and reading depth (Evertse not held). The verdict is stated and assigns no tier.

Problem 49 clause review: pass. The subject block resolves and includes the page's one-line summary. The independence facts and three exposures are stated. The restatement lists the three clauses of the problem and six statements with the conventions $M^\uparrow$ and $M_<$. All ten checklist items carry an explicit verdict. The three weakest steps are re-derived: the factor tending to infinity from $W\gg Z$ and the exponent of $Z$, the comparison with Tao's rate on the same weak maximum, and the elementary strict bound together with Lemma 5.1's application to strict sets; the grader re-derived each and agrees. The strongest attack is real: an exact count of the strict maximum against $\pi(N)$ to $10^7$ (the grader reproduced it to $10^6$), the observation from Tao's Proposition 4.1 that clause (i) at $N$ implies a prime in $((p-1)^2,p^2)$ for every prime $p\le\sqrt N$, and the attempt to build a transfer from each shared input. The premises carry interface and reading depth. The verdict is stated and assigns no tier.

Corrections

C1. Page: wiki/research/erdos_49/connections.md. Location: three places. (a) Section "Notation and the inputs used", bullet (T3), after "uniformly in $k$" insert

, where $C_2=2\prod_{p>2}(1-(p-1)^{-2})$ is the twin-prime constant in the source's normalization (Theorem 3.3 page)

so that the clause reads "uniformly in $k$, where $C_2=\ldots$ (Theorem 3.3 page), and $\frac1{2k}\le c(k)\le\ldots$". (b) Bullet (F), the clause "hence $V(x/\mu)\le(C_2/\mu)V(x)$ for $x\ge x_2(\mu)$ with an absolute $C_2$": rename the constant to $C_3$ in both occurrences. (c) Section "Problem 417", proof of Proposition 3, the sentences "By (F) with $\mu=2^{J+1}$, $V(x/2^{J+1})\le C_22^{-J-1}V(x)$ for large $x$; fix $J$ with $C_22^{-J-1}\le c/2$.": rename both occurrences to $C_3$. The uses of $C_2$ in the proof of Proposition 1 (Steps 2 and 4) and in "What it does not give" keep the twin-prime meaning and are unchanged.

Basis, checked on every occurrence of the symbol on the frozen page: (T3) uses $C_2$ without defining it, in the source's sense (p. 6, with the normalization on p. 5); (F) introduces "an absolute $C_2$" that is a different constant; Proposition 1 consumes the first and Proposition 3 the second. Read with the (T3) meaning, the (F) inequality is the unproved numerical claim $V(x/\mu)\le(1.32\ldots/\mu)V(x)$; read with the (F) meaning, the sieve bound loses its constant. Filed by the notation review as F1 (required); accepted. The change touches no mathematics.

C2. Page: wiki/research/erdos_49/connections.md. Location: section "Notation and the inputs used", bullet (F), the sentence beginning "Two consequences are used:" and ending "with an absolute $C_2$." (after C1, $C_3$). Replace it with:

Three consequences are used: there are absolute $c_0,C_1>0$ with $c_0Z(x)\le V(x)\le C_1Z(x)$ for large $x$; and for each fixed $\mu\ge1$, $Z(x/\mu)/Z(x)\to1/\mu$, because $\log(x/\mu)\sim\log x$ and $\log_3(x/\mu)-\log_3x=O(1/(\log x\log_2x))$, so the exponent of $Z$, whose leading term is $C(\log_3x-\log_4x)^2$, changes by $O(\log_3x/(\log x\log_2x))\to0$; hence $V(x/\mu)\le(C_3/\mu)V(x)$ for $x\ge x_2(\mu)$ with an absolute $C_3$.

Basis, re-derived here: from the two listed facts alone the chain ends at $V(x/\mu)\le C_1Z(x/\mu)\le(2C_1/\mu)Z(x)$ and does not reach $V(x)$; the step $Z(x)\le V(x)/c_0$ is the lower half of the quoted $V(x)=Z(x)e^{O(1)}$, which the page's inventory sentence does not list. The limit needs the increment estimate, since $\log_3x-\log_4x\to\infty$ and a change of the arguments that is merely $o(1)$ would not suffice: with $\delta_2=\log_2(x/\mu)-\log_2x=O(1/\log x)$ one gets $\delta_3=O(1/(\log x\log_2x))$ and $\delta_4=O(\delta_3/\log_3x)$, so the square changes by $2(\log_3x-\log_4x)(\delta_3-\delta_4)+O(\delta_3^2)$, which tends to $0$. Filed by the notation review as F2 (suggested); promoted to a correction on the grader's verification, because the sentence states the inputs consumed, a scope surface, and states two where three are used, as the reconstruction grade treated the import counts C4 and C5. The change touches one sentence of the inputs list.

C3. Page: wiki/research/erdos_49/connections.md. Location: section "Problem 1004", paragraph "What it does not give", the clause "so whatever average bound for $c(h)$ is used, the second term of Step 2 is at least of order $L\log L\cdot x/(\log x)^2$". Replace "the second term of Step 2" with "the bound on $B(x,L)$ obtained in Step 4".

Basis, checked on the page's displays: the second term of the Step 2 display is $(16C_2+1)S(L)\,y/(\log y)^2$, and with $c(h)\ge1/(2h)$ for even $h$ it is at least of order $\log L\cdot x/(\log x)^2$; the factor $L$ enters through Step 1 and appears in the Step 4 bound, which is at least of order $L\log L\cdot x/(\log x)^2$. The barrier is right and the term named is wrong by the factor $L$. Filed by the Problem 1004 review as F2 (suggested); promoted because the sentence as written is false. The change touches four words.

C4. Page: wiki/research/erdos_49/connections.md. Location: section "Problem 1004", paragraph "What it does not give", the sentence "The anonymous note's sharper average $\sum_{h\le L}c(h)\ll\log L$ would extend Proposition 1 to $L\le(\log x)^2/(\log\log x)^{1+\delta}$ and no further." Replace it with:

The anonymous note's sharper average $\sum_{h\le L}c(h)\ll\log L$ would extend Proposition 1 exactly to the $L$ with $L\log L=o((\log x)^2)$, for instance $L\le(\log x)^2/(\log\log x)^{1+\delta}$ for fixed $\delta>0$, and to no $L\ge\kappa(\log x)^2/\log\log x$ with fixed $\kappa>0$.

Basis, verified here: with $S(L)\ll\log L$ the union bound gives $B(x,L)\ll L\log L\cdot x/(\log x)^2+o(x)$, which is $o(x)$ exactly when $L\log L=o((\log x)^2)$, and it is at least of that order by the weights $(L-h)$ and $c(h)\ge1/(2h)$. The witness $L=\lfloor(\log x)^2/(\log_2x\log_3x)\rfloor$ has $L\log L\le2(\log x)^2/\log_3x=o((\log x)^2)$ and exceeds $(\log x)^2/(\log_2x)^{1+\delta}$ once $\log_3x<(\log_2x)^\delta$, so "and no further" is false. Filed by the Problem 1004 review as F3 (suggested) with the verdict refuted-as-stated; promoted because a refuted sentence is corrected. The exponent-$2$ content is unchanged.

C5. Page: wiki/research/leads/distinct_totient_blocks_from_collision_bounds/_index.md. Location: section "Relationship to the retained note", the sentence "The sharper average, if verified, would extend the range to $L\le(\log x)^2/(\log\log x)^{1+\delta}$ and no further." Replace it with:

The sharper average, if verified, would extend the range exactly to the $L$ with $L\log L=o((\log x)^2)$, for instance $L\le(\log x)^2/(\log\log x)^{1+\delta}$ for fixed $\delta>0$, and to no $L\ge\kappa(\log x)^2/\log\log x$ with fixed $\kappa>0$.

Basis: the same witness as C4; the lead carries the same clause. Filed by the Problem 1004 review as F3; accepted with C4.

C6. Page: wiki/research/erdos_49/connections.md. Location: section "Problem 1004", paragraph "What it does not give", the sentence "If that lower bound holds, as the Hardy--Littlewood heuristic predicts, the expected number of collisions inside a block of length $L$ at a random start $n\le x$ is of order $L\log L/(\log x)^2$, so for $c>2$ almost all blocks contain a collision and only atypical starts can answer the question; a Poisson model for the positions of collisions then predicts a longest collision-free block below $x$ of length $(\log x)^{3+o(1)}$, so under that model the every-$c$ form would fail for $c>3$." Replace it with:

If that lower bound holds, as the Hardy--Littlewood heuristic predicts, the expected number of collisions inside a block of length $L$ at a random start $n\le x$ is at least of order $L\log L/(\log x)^2$. A first moment does not decide how many blocks are collision-free. Under a Poisson model for the positions of collisions, for $c>2$ almost all blocks contain a collision and only atypical starts can answer the question, and the same model predicts a longest collision-free block below $x$ of length $(\log x)^{3+o(1)}$, so under it the every-$c$ form would fail for $c>3$.

The following sentence, "The model is a heuristic and is stated here as such.", stays.

Basis, verified here: the hypothesis controls the expected number of collision pairs per block; the number of collision-free starts depends on the spacing of the collision points. Place $\rho x$ points of $[1,x]$ in runs of $K$ consecutive integers, with $\rho=\log L/(\log x)^2$, $L=(\log x)^c$, $K=(\log x)^{c-1}$: at most $(\rho x/K)(L+K)=\rho x(\log x+1)\ll x\log L/\log x=o(x)$ starts have a window of length $L$ meeting a run, while the number of points, hence the first moment per block, is that of an evenly spread configuration. So the "so" is a first moment upgraded to "almost all", a canonical failure mode, and the conclusion needs the spacing input that the Poisson model supplies. The phrase "of order" also needs the retained note's unverified average for its upper half; (T3) alone gives "at least of order". Filed by the Problem 1004 review as F1 (required); accepted. Proposition 1, its proof and the outcome line are untouched.

C7. Page: wiki/research/leads/distinct_totient_blocks_from_collision_bounds/_index.md. Location: section "The consequential gap", the sentence "If that lower bound holds, as the Hardy--Littlewood heuristic predicts, a random block of length $L$ contains of order $L\log L/(\log x)^2$ collisions, so for $c>2$ almost all blocks fail and only atypical starts can answer the question; a Poisson model for the positions of collisions predicts a longest collision-free block below $x$ of length $(\log x)^{3+o(1)}$, under which the every-$c$ form would fail for $c>3$." Replace it with:

If that lower bound holds, as the Hardy--Littlewood heuristic predicts, the expected number of collisions in a block of length $L$ at a random start is at least of order $L\log L/(\log x)^2$; a first moment does not decide how many blocks are collision-free. Under a Poisson model for the positions of collisions, for $c>2$ almost all blocks fail and only atypical starts can answer the question, and the same model predicts a longest collision-free block below $x$ of length $(\log x)^{3+o(1)}$, under which the every-$c$ form would fail for $c>3$.

The following sentence, "Both are heuristics.", stays. Basis: as C6; the lead's "contains of order" also states a typical value where only an expectation is available. Filed by the Problem 1004 review as F1; accepted with C6.

C8. Page: wiki/research/erdos_49/connections.md. Location: section "Problem 1003", two sentences. (a) Paragraph "What it does not give, and why", the sentence "Infinitude is an existence statement; every reconstructed input bounds a count from above." Replace it with:

Infinitude is an existence statement. The reconstructed bounds on $P(x;k)$, (T1) and (T3), are upper bounds; the reconstructed lower bounds, (L4) and Lemma 5.1, count convenient integers and missing totient values, and neither constrains $\varphi(n+1)$ for any $n$ it counts.

(b) The Outcome sentence "Fails: the inputs are upper bounds, and the only constructive ingredient needs an even shift." Replace it with:

Fails: the bounds on $P(x;k)$ are upper bounds, the lower-bound inputs count other objects, and the only constructive ingredient needs an even shift.

Basis, checked against the page's own inputs list: (L4) concludes that at least $c_DZ(x)$ integers satisfy its three conditions, and Lemma 5.1 concludes $\#(\mathcal W(x)\setminus\varphi(S))\ge cW(x)$; both are lower bounds on counts, so the universal is false over the inputs the page declares. The obstruction holds for the reason the replacement gives: convenience constrains the multiples $n_in$ and Lemma 5.1 the image of a nondecreasing set, and neither reaches $n+1$. Filed by the Problem 1003 review as F1 (required); accepted. The verdict "fails" and the outcomes line are unchanged.

C9. Page: wiki/research/erdos_49/connections.md. Location: section "Problem 417", the paragraph after Premise (W), the phrase "Ford's Theorem 4, on the Ford card (short intervals $y\ge x^\theta$ for $V_k(x+y)-V_k(x)$, stated by Ford to follow from a simple modification of the proofs of his Theorems 1 and 2)". Replace it with:

Ford's Theorem 4, on p. 3 of the paper held by the Ford card and not among the results the card lists (for admissible $\theta$, $y\ge x^\theta$ and a possible multiplicity $k$, $V_k(x+y)-V_k(x)\asymp V(x+y)-V(x)\asymp\frac{y}{x+y}V(x+y)$, stated by Ford to follow from a simple modification of the proofs of his Theorems 1 and 2)

Basis, checked in that state and on Ford's p. 3: the card's "Results to transcribe" list carries Theorems 1, 2, 5--7, 9, 8 and 10--11 and no Theorem 4, and the card has no result pages, so the locator points where the result is not; Ford's statement carries the two hypotheses "$\theta$ admissible" and "the multiplicity $k$ is possible", which the page's parenthetical omits. Filed by the Problem 417 review as F1 (suggested); promoted because a locator that does not resolve is a source-fidelity defect. The characterization of the theorem is otherwise faithful.

C10. Page: wiki/research/erdos_49/connections.md. Location: section "Problem 416", paragraph "What transports", the sentence "(L4) is the lower-bound half of Ford's order of magnitude; Proposition 2 above even recovers $W(x)\asymp V(x)\asymp Z(x)$ from it." Replace it with:

(L4) is the lower-bound half of Ford's order of magnitude; the proof of Proposition 2 above recovers $W(x)\ge c_KZ(x)$ from it and, with the upper bound of (F), $W(x)\asymp V(x)\asymp Z(x)$.

Basis, checked on the page: the proof of Proposition 2 closes with "with (F), $V(x)\le C_1Z(x)\le(C_1/c_K)W(x)$"; (L4) supplies only $W(x)\ge c_KZ(x)$, the upper half is Ford's Theorem 1, and the sentence contradicts its own first clause. Filed by the Problem 416 review as F1 (required); accepted. The outcome "Fails" is unchanged.

C11. Page: wiki/research/erdos_49/connections.md. Location: section "Problem 694", the sentence after the proof of Proposition 5, "which strengthens the note's Proposition 3.1 from "infinitely many with ratio at least $a/b$" to "a positive proportion with ratio exactly $a/b$"". Replace the second quoted phrase so that the sentence reads:

which strengthens the note's Proposition 3.1 from "infinitely many with ratio at least $a/b$" to "a positive proportion with ratio exactly $f_{\max}(n)/f_{\min}(n)$, hence at least $a/b$", where $n=\varphi(a)=\varphi(b)$

Basis, re-proved here. Under Proposition 3.1's hypotheses $a>b$ are any two preimages of $n$, and Proposition 5 applied to $n$ gives ratio exactly $f_{\max}(n)/f_{\min}(n)$, which equals $a/b$ only for the extreme pair. Witness: $a=16$, $b=15$, $n=8$. Suppose $f_{\max}(v)/f_{\min}(v)=16/15$; then $f_{\max}(v)=16u$, $f_{\min}(v)=15u$ and $\varphi(16u)=\varphi(15u)=v$. Write $u=2^ew$ with $w$ odd and $w=3^a5^bw'$ with $\gcd(w',15)=1$. Then $\varphi(16u)=2^{e+3}\varphi(w)$ and $\varphi(15u)=\varphi(2^e)\cdot8r_3r_5\varphi(w)$ with $r_3\in\{1,3/2\}$ and $r_5\in\{1,5/4\}$ according as $a=0$ or $a\ge1$, $b=0$ or $b\ge1$. If $e=0$, equality forces $r_3r_5=1$, so $\gcd(u,30)=1$, and then $\varphi(30u)=v$ with $30u>16u$, contradicting maximality. If $e\ge1$, equality reads $2^{e+2}r_3r_5=2^{e+3}$, so $r_3r_5=2$, impossible. Hence no totient has extreme ratio $16/15$, as the enumeration to $2\cdot10^6$ confirms, while the note's "infinitely many with ratio at least $16/15$" is true. Filed by the Problem 694 review as F1 (required); accepted. Proposition 5 and the outcome are unchanged.

C12. Page: wiki/research/erdos_49/connections.md. Location: section "Problems 850 and 931", paragraph "What transports", the two sentences "Lemma 3.2 says that for each $k$ at most $3\cdot7^{3+2\omega(k)}$ integers $j$ have $\gamma(j)=\gamma(j+k)$. With $k=y-x$, each of the three conditions of Problem 850 says that $(j,j+k)$ is such a pair for $j=x,x+1,x+2$; so for each fixed difference $k$ there are finitely many candidates, boundedly many in terms of $\omega(k)$." Replace them with:

Lemma 3.2 says that for each natural number $k$ at most $3\cdot7^{3+2\omega(k)}$ natural numbers $j$ have $\gamma(j)=\gamma(j+k)$. Take $x<y$, the conditions being symmetric, and $k=y-x\ge1$; for positive $x$ each of the three conditions says that $j=x,x+1,x+2$ is such a $j$. Under the Statement's literal reading over all integers the same bound holds, because an entry $0$ or $\pm1$ forces $x=y$ or a failing pair, and the lemma's injection $j\mapsto((j+k)/k,-j/k)$ counts every integer $j\notin\{0,-k\}$ with $\gamma(|j|)=\gamma(|j+k|)$. So for each fixed difference $k$ there are at most $3\cdot7^{3+2\omega(k)}$ candidates $x$, boundedly many in terms of $\omega(k)$.

Basis, checked on the Lemma 3.2 page and on manuscript p. 6 ("The number of natural numbers $j$ for which $j$ and $j+k$ have the same set of prime factors"): the page widens the lemma's domain to all integers, the same widening that C1 of the reconstruction grade removed from that lemma's desc, and then applies it to a Statement over "integers" without fixing $x<y$ or the sign of $x$. The widened sets differ from the natural ones (for $k=6$, six natural and nine further negative solutions, recomputed here), and the bound on the widened domain is a derivation the page does not give: a prime dividing $j$ divides $j+k$ and hence $k$, so $u=(j+k)/k$ and $v=-j/k$ are nonzero $S$-units with $u+v=1$ for $S$ the infinite place and the primes dividing $k$, and Evertse's count covers solutions of either sign. The case check for entries $0$ or $\pm1$ was re-derived here under either convention for the prime factors of $0$. Filed by the Problems 850 and 931 review as F1 (required); accepted. The outcome "Fails" is unchanged.

Rejected and downgraded findings

Notation review.

• F3 (note: name the recorded gap of the Theorem 3.1 sketch at (T1)). No change required. The standing sentence is accurate and the Theorem 3.1 page, linked from the page's introduction, records the gap; the sentence may name it.
• F4 (note: the letter $k$ counts preimages in the definition of convenient and is the shift elsewhere). No change required; the source and the Lemma 5.1 page use the same letter.

Problem 1004 review.

• F4 (note: "the $\binom L2$ shifts" where a block has $\binom L2$ pairs and $L-1$ shifts; page and lead). Downgraded to optional; the phrase is loose, not false in its use, and the proof groups pairs by shift.
• F5 (note: "rests on (T1) and (T3) alone" omits Mertens' bound and the parity of Theorem A's shape; page and lead). Downgraded to optional. Both omitted inputs are classical and elementary, and the sentence's content, that no average estimate beyond (T3)'s own bound is needed, holds.
• F6 (note; lead: the restatement of Theorem 3.3 omits the two conditions on $\varepsilon$, "is" states an upper bound as an equality, and the sums over $h$ are not restricted to even $h$). Downgraded to optional with a recommendation to adopt all three: the lead links the reconstruction, whose statement carries the conditions, and its route rests on Proposition 1 as proved on the connections page, not on the summary; none of the three affects the route.
• F7 (note: the hypothesis of Proposition 1 is read for all $x$; the proof gives uniformity in $L$). Downgraded to optional; $B(x,L)=o(x)$ is an asymptotic statement insensitive to $L$ on a bounded range, and the stronger uniform form may be recorded.
• F8 (the statements about the site's thread, the retained note's pages and the problem page's current assessment): no verdict was given; the grader checked the last against Problem 1004 in that state and found the characterization faithful (see the checks below).

Problem 1003 review.

• F2 (note: link the cards of Li (2026) and Ford (2020); Yamada's bound is checkable in the corpus only through their quotations). Downgraded to optional; a page-mechanics improvement.
• F3 (note: say that (T1) at $k=1$ recovers Theorem 2 of Erdős, Pomerance and Sárközy). Downgraded to optional. The section already names that theorem; the sentence rests on (T1) as published, at the scope the page's Standing allows.

Problem 417 review.

• F2 (suggested: "the two pages are independent checks of one another"). Downgraded to optional. Under the page's Standing paragraph ("not independently reviewed") the sentence can only mean two derivations from different recorded inputs, which is what they are; the reviewer's point that both rest at the root on Ford's section 5 construction is right and the proposed wording may be adopted.
• F3 (note: "(P1) ... is the premise (W) above in multiplicity form"). Downgraded to optional. Checked on the parallel lead: (P1) counts totients in $(2x/3,x]$ all of whose preimages are divisible by $3$, and (W) at $d=54$, $\lambda=3/2$ implies it, while the converse is not claimed; "multiplicity" is also a misnomer for a divisibility class. The proposed wording ("follows from (W) at $d=54$ and $\lambda=3/2$") may be adopted.
• F4 (note, from a disclosed exposure: "The corpus page is not yet compiled beyond the statement" is stale in that state). Downgraded; not mathematics. In that state the problem page carries Progress and Known Results text that cites this section, landed in the same change series, so the sentence describes the corpus as it was when the page was written; the integrator may date it or drop it.
• F5 (note: "(W) is the intermediate statement that modification produces" asserts what an unwritten argument would yield). Downgraded to optional; the next sentence labels (W) a premise, so no standing is overstated, and "would produce" may be adopted.

Problem 416 review.

• F2 (suggested: the paraphrase of Ford's p. 39 drops "I believe" and "may" and merges two sentences). Downgraded to optional. Checked on p. 39: Ford writes "I believe that obtaining the asymptotic formula for $V(x)$ will require simultaneously determining the asymptotics of $V_k(x)/V(x)$ ... and $V(x;k)/V(x)$ for each $k$" and, three sentences later, that the Section 5 construction "may miss whole classes of totients with multiplicity $k$". The page's "would require" keeps a hedge, and its "miss whole classes" is a true statement the review proves (a seed with $k$ preimages produces only totients of multiplicity $k$, and multiplicity is unbounded because $V(x)=o(x)$; the grader re-derived it), so the attribution is loose rather than strengthening; the fuller wording may be adopted.
• F3 (suggested: "miss whole classes" blocks a formula only with the positive-proportion clause of Ford's Theorem 2; "never a formula" is a universal). Downgraded to optional. The outcome follows from the unspecified constant alone, as the review itself records; the clause and the scoping to this mechanism may be added.
• F4 (note: $V_k(x)$ and $V(x;k)$ are undefined on the page). Downgraded to optional; a definition or a pointer to the card's rows may be added.

Problem 694 review.

• F2 (note: "grows like $\log\log x$" is the note's Theorem 2.1 at the note's standing). Downgraded to optional. The section's Target paragraph names the note and its asymptotic as the source of the solved status, and the obstruction needs only that (L4) fixes the seed; an attribution may be added.
• F3 (note: the identification of the note's device with multiplication by convenient integers is attributed to the Problem 416 connections page). Downgraded; the characterization is faithful. Checked in that state: that page's Problem 694 section says the note's Proposition 3.1 "is the multiplication by convenient integers behind Ford's Theorem 8 and the Problem 49 folder's Lemma 4.1". The reviewer's remark that the note multiplies by a prime $r\nmid ab$ without the exactness clause of convenience, which is why it obtains "at least $a/b$" where Proposition 5 obtains "exactly", is right and concerns the other folder's page.

Problem 51 review.

• F1 (suggested: the obstruction covers one application of (L4); under iteration the bound $Kn_1/d$ compounds). Downgraded to optional. The mechanism as the page defines it is one application to a fixed seed, and the section's argument is right for it. The reviewer's derivation that the statement survives iteration is correct (the grader re-derived the composition $Nn$ convenient for $d$, the coprimality from $q\le d\varphi(N)+1<p_L$, and the bound $n/\varphi(n)\le1+O(\log_3D/D)$ with the tail of $\sum_t\exp(-\exp(0.2\cdot1.8^t))$) and may be added as the proposed sentence.
• F2 (suggested: "would break the convenience count, which is Ford's argument for exactly that set" describes a modification of an argument the corpus has not read). Downgraded to optional; the sentence is a reason, not a result, and the reviewer's weaker wording may be adopted.
• F3 (note, from a disclosed exposure: "The corpus page is not compiled beyond the statement (open)" is stale in that state). Downgraded; as Problem 417 F4.
• F4 (note: the statement of Proposition 5 exposes neither the form $v=d\varphi(n)$ nor convenience). Downgraded to optional; the proposed wording may be adopted.

Problem 821 review.

• F1 (suggested: the "so" from multiplicity preservation to a positive proportion has its bridge unwritten). Downgraded to optional. The sentence attributes the statement to Ford's published Theorem 2, which it names, and the bridge (Lemma 4.1 at $d_1=d_2=D=d$, injectivity through the least preimage, (F)) is valid, re-derived by the reviewer and the grader; it may be written in.
• F2 (suggested: the section argues the failure for the convenient-integer construction only). Downgraded to optional. The outcome is true for both mechanisms: the collision bounds are upper bounds on coincidences and, turned on a multiplicity through blocks of length $\log x$, give only $g(v)\ll v\log\log v/\log v$, which the grader checked; the proposed sentence may be added.
• F3 (note: the bibliographic clause), F4 (note: the proportion depends on the seed), F5 (note: non-convenient candidates). No change required.

Problem 415 review.

• F1 (suggested: "lengths of order $\log\log\log n$" overstates the proved scale). Downgraded to optional. The phrase names the problem's own conjectured scale, and the section's background note gives $F(n)\le(1+o(1))\log_3n/\log_6n$; the proposed wording may be adopted, and the separation from $(\log x)^c$ only widens.
• F2 (note: "for some constant $c$" is dropped from the target). Downgraded to optional with a recommendation to restore the phrase, checked against the problem page's Statement.
• F3 (note: name the degenerate case $c=0$), F4 (note: define $\log_k$), F5 (note: only the first question asks that every pattern occur). No change required; each may be adopted.

Problems 850 and 931 review.

• F2 (suggested: "the union over all $k$ is unbounded" has an unnamed referent). Downgraded to optional. In context, after "for each fixed difference $k$ there are finitely many candidates", the referent is the candidate sets, whose union is infinite (every $j\ge2$ lies in some $J_k$); the proposed wording may be adopted. Grader's remark: under the Statement's literal reading over all integers, with radicals of absolute values, $(x,y)=(-4,2)$ satisfies all three conditions ($\{2\}$, $\{3\}$, $\{2\}$), so that reading trivializes the question and the intended one is over positive integers; this concerns the problem page's formulation, outside this subject, and does not affect C12, whose bound holds on either domain.
• F3 (suggested: "the lemma's hypothesis is not even met" is imprecise). Downgraded to optional; the proposed wording may be adopted.
• F4 (note: the paraphrase of Problem 931 omits that the block lengths are fixed first). Downgraded to optional with a recommendation to restore "for fixed $k_1\ge k_2\ge3$"; not load-bearing, since nothing transfers.
• F5 (note: "not compiled beyond the statement"). Downgraded; in that state both pages carry "Progress. Not yet compiled." and a Known Results sentence pointing back at this page; as Problem 417 F4.
• F6 (note: $\gamma$ and $\omega$ undefined on the page), F7 (note: "standard input" is a literature characterization). No change required.

Problem 49 clause review.

• F1 (suggested: the contrast between the reversed pairs and the elementary strict bound is inverted). Downgraded to optional. Each clause of the sentence is true; the implied division of labor is misleading, since Lemma 5.1's only hypothesis is that $\varphi$ is nondecreasing on $S$, which every strict set satisfies, giving $|A|\le(1-c)W(N)$, while the Theorem 1.2 proof assigns the repeated values to the collision bounds and the distinct values to Lemma 5.1 (checked on the reconstruction and on manuscript p. 10). The proposed wording may be adopted; the conclusion for clause (i) is unchanged.
• F2 (note: the range $N\ge2$ is unstated). No change required; the problem page's Status paragraph already records the $N=1$ exception, and the phrase may be added.
• F3 (note: link Tao's Theorem 1.1 and state the elementary reason in place). Downgraded to optional; a page-mechanics improvement.

Checks of the grader's own that produced no correction. The Target paragraphs' characterizations of excluded text hold in that state: Problem 1004's current assessment (2026-09-09) records the question as open, names the uniform bounds among the inspected pages, says that no deduction of a growing-block existence theorem from the collision bounds is claimed, and retains the anonymous note as an unreviewed candidate lead; Problem 416's assessment is dated 2026-09-27 and records the first question answered by the accepted Lean proof and the second open; Problem 694 is recorded as solved with the note's $(e^\gamma+o(1))\log\log x$; Problem 415's Statement asks the first question "for some constant $c$"; Problem 49's Status records one remaining clause. The "Parallel derivation" paragraph's account of the Problem 416 connections page is faithful (its Propositions 417.2 and 417.4 and Theorem 417.3, the seed $54$ with preimages $81$ and $162$, a geometric sequence of scales), and that page and the parallel lead in turn describe this page's Propositions 2--4 as derived from Lemma 4.1 with the seeds $1$ and $2^{18}\cdot257$ and the lead's premise (W) as (P1) in convenient-integer form. Ford's p. 39 lists $d=2^{18}\cdot257$ for $k\in\{2,4,8\}$ and $d=54$ for $k\in\{3,9\}$, as the Problem 417 review and the parallel lead say. The Problem 416 folder index states the second-question gap as the Problem 416 section says. Every locator on the page that the reviews checked (manuscript pp. 2, 5--10; Ford pp. 2--4 and 39; the note's p. 4) matches the held sources.

Graded verdicts

Notation and the inputs used.

• S1 (elementary notation facts): refutation-failed.
• S2 (definitions by reference): not a claim; each agrees with its cited page and with the source.
• S3 ((T1) transcription and standing sentence): refutation-failed.
• S4 ((T3) transcription, the bounds on $c(k)$, the standing sentence, odd $k$): refutation-failed, with C1.
• S5 ((L4) transcription with $K\ge1$ and standing sentence): refutation-failed.
• S6 (Ford's theorem and its consequences): refutation-failed, with C1 and C2.
• S7 (the eight even preimages): refutation-failed; reproduced by the reviewer and the grader.

Problem 1004: distinct totients on a consecutive block.

• S1 (Proposition 1): refutation-failed, relative to (T1) and (T3) at the standing their reconstruction pages record.
• S2 (the corollary: an affirmative answer for every fixed $c<2$ in the almost-all form): refutation-failed, relative to (T1) and (T3).
• S3, S4, S5, S6 (Steps 1--4): refutation-failed.
• S7 (the barrier at exponent $2$): refutation-failed, with C3.
• S8 (the reach with the sharper average, "and no further"): refuted-as-stated in its "no further" clause; corrected by C4 on the page and C5 on the lead.
• S9 (the barrier is not a proved obstruction without a lower bound): refutation-failed as the necessity claim it states.
• S10 (the deduction "so for $c>2$ almost all blocks contain a collision" from the lower bound): proof defective; statement unresolved; corrected by C6 on the page and C7 on the lead, which withdraw the deduction and place the conclusion under the Poisson model.
• S11 (the Poisson prediction, as a model statement): refutation-failed.
• S12 (the exploratory computation): refutation-failed; reproduced by the reviewer and the grader.
• S13 (relation to the retained note): refutation-failed.
• S14 (the lead's route summary and provenance): refutation-failed.
• S15 (the structured shape of a collision): refutation-failed.
• S16 (the standing sentences): refutation-failed.

Problem 1003: the unit shift.

• S1 (target and relayed assessment): refutation-failed.
• S2 (the unit-shift bound from (T1) and its ordering against Yamada's and Li's): refutation-failed.
• S3 (emptiness of the structured family at $k=1$): refutation-failed.
• S4 (no reconstructed statement constructs a solution): refutation-failed.
• S5a (the direction clause "every reconstructed input bounds a count from above"): refuted-as-stated; corrected by C8.
• S5b (the obstruction's substance): refutation-failed.
• S6 (equal radicals force an even shift; the attribution to Ford): refutation-failed.
• S7 (outcome): refutation-failed.

Problem 417: totients below $x$ against totients of integers below $x$.

• S1 (Proposition 2): refutation-failed.
• S2 (the cluster interval; an infinite limit excluded): refutation-failed.
• S3 (Proposition 3): refutation-failed.
• S4 ($\ell_2>1$; a limit, if it exists, exceeds $1$): refutation-failed.
• S5 (every totient $\le10^6$ has an odd preimage): refutation-failed; reproduced by the reviewer.
• S6 ((W) is not deducible from (L4) as stated at any fixed $\lambda$): refutation-failed, as a statement about the recorded interface.
• S7 (Proposition 4, conditional on (W) at $d=2$, $\lambda=3/2$): refutation-failed; the antecedent is assumed, not certified.
• S8 (the method of Proposition 3 gives no limit inferior without (W)): refutation-failed, as a statement about that method's inputs.
• S9 (characterizations): not mathematical claims; C9 corrects the locator of Ford's Theorem 4.

Problem 416: an asymptotic formula for the totient count.

• S1 (target): refutation-failed.
• S2 (the problem page's assessment): descriptive; checked by the grader and faithful.
• S3 ((L4) is the lower-bound half of Ford's order): refutation-failed.
• S4 (Proposition 2 recovers the two-sided order from (L4)): refuted-as-stated; corrected by C10.
• S5 (nothing more transports): refutation-failed.
• S6 (an unspecified constant gives no formula): refutation-failed.
• S7 (finitely many seed families miss whole classes): refutation-failed, by the review's proof.
• S8 (Ford's remark in section 7): refutation-failed.
• S9 (the Problem 416 folder index states the same gap): refutation-failed.
• S10 (outcome): refutation-failed.

Problem 694: the ratio of the largest to the smallest preimage.

• S1 (target and the note's Proposition 3.1): refutation-failed.
• S2 (what transports): refutation-failed.
• S3 (Proposition 5): refutation-failed.
• S4 (every attained ratio is attained on a positive proportion): refutation-failed.
• S5 ("a positive proportion with ratio exactly $a/b$" for the collision of Proposition 3.1): refuted-as-stated; corrected by C11.
• S6 (the Problem 416 connections page's identification): a characterization, checked by the grader and faithful.
• S7 (nothing about the maximum): refutation-failed.
• S8 (outcome): refutation-failed for the transfer clause; the status clause records the problem page.

Problem 51: totients whose smallest preimage is large.

• S1 (the transport within $[n_1/d,Kn_1/d]$ on $\gg_dV(x)$ totients): refutation-failed.
• S2 (the obstruction and the outcome): refutation-failed; under repeated application the statement holds by the review's derivation.
• S3 (target): refutation-failed.
• S4 (corpus state): not a claim; stale in that state.

Problem 821: totients of very large multiplicity.

• S1 (target and bibliographic sentences): refutation-failed.
• S2 (multiplicity preservation; Ford's Theorem 2): refutation-failed.
• S3 (the obstruction): refutation-failed.
• S4 (outcome): refutation-failed for both mechanisms.

Problem 415: ordering patterns of consecutive totients.

• S1 (target): not a claim; faithful apart from the dropped "for some constant $c$".
• S2 (what transports from Proposition 1): refutation-failed.
• S3 (the obstruction): refutation-failed.
• S4 (Theorem 1.5 restated): refutation-failed.
• S5 ($F(n)=o(\log_3n)$; a negative answer for every $c>0$): refutation-failed, with Theorem 1.5 consumed as published.
• S6 (provenance), S7 (outcome): not claims.

Problems 850 and 931: pairs with the same prime factors.

• S1, S2 (target paraphrases), S3 (corpus state): not claims.
• S4 (the quoted lemma): refutation-failed, with C12.
• S5 (the per-difference bound for Problem 850): refutation-failed, with C12.
• S6 (the obstruction for Problem 850): refutation-failed.
• S7 (the obstruction for Problem 931): refutation-failed.
• S8 (assessment): not a claim.
• S9 (outcome): refutation-failed.

Problem 49: the remaining clause.

• S1 (clause identification): refutation-failed.
• S2 (nothing on the page bears on clause (i)): refutation-failed.
• S3 (the reconstructed bound exceeds $\pi(x)$ by a factor tending to infinity): refutation-failed.
• S4 (superseded by Tao's rate): refutation-failed.
• S5 (the reversed pairs and the elementary strict bound): refutation-failed for each clause and for the conclusion.
• S6 (an exact finite question, unaddressed by asymptotic mechanisms): refutation-failed.
• S7 (the folder index maps the three clauses): not a claim.

Lead outcomes.

• distinct totient blocks from collision bounds: reviewed. Its route rests on Proposition 1 (S1--S6 of the Problem 1004 section), on (T1) and (T3) at the standing their reconstruction pages record, on Mertens' bound and on the parity of Theorem A's shape; every one is refutation-failed after the corrections. C5 and C7 correct two sentences of the lead's "consequential gap" and "relationship" sections that the route does not rest on. The lead inherits, as it states, the sketch status of Theorem 3.1 and the sieve constant and uniformity imported inside Theorem 3.3; this focused review of the deduction does not discharge either.
• The parallel lead distinct totient preimage ratio is outside this subject (its route is the Problem 416 connections page). The statements of this page that it cites, Propositions 2, 3 and 4 of the Problem 417 section, are refutation-failed after C1, C2 and C9, so nothing here requires its update.

No tier is assigned and no status changes.