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Subject and independence

Role: independent reviewer in a fresh context, given only the assignment; the reviewer took no part in writing the connections page, the reconstruction pages or any other page of the folder. Charge: refutation of the author-recorded statements of one section.

Subject: wiki/research/erdos_49/connections.md as it stood on 2026-09-28T13:50:48Z, section "Notation and the inputs used", lines 58--111 (from its H2 heading to the line before the H2 "Problem 1004: distinct totients on a consecutive block"). No lead page is named in the assignment and the section names none; the two leads linked from the page's introduction were not read.

Read, and at what depth:

• The section, clause by clause; the page's introduction and Standing paragraph (lines 14--57) for the standing convention they set, and the list of H2 headings; no other section of the page.
• The six reconstruction pages of the folder in the same state, in full: Theorem 3.1, Theorem 3.3, Lemma 3.2, Lemma 4.1, Lemma 5.1, Theorem 1.2.
• Pollack, Pomerance and Treviño (2013): the card's provenance and contents, its Theorem 1.2 result page, and the second card's Theorem 3.1 and Theorem 3.3 result pages (statements); the held 17-page manuscript, pp. 2 and 5--10 on the text layer and pp. 6--8 on the page images.
• Ford (1998): the card's digest; the held 43-page copy, p. 2 (Theorem 1 and the constants (1.3)--(1.6)) on the page image and the text layer.
• Problem 417: the Statement.
• docs/verification.md ("Grading and claim standing", "Audit checklist: the canonical failure modes", "Whole-claim report", "Audit checklist"), docs/approach_vetting.md, docs/evidence.md ("Source fidelity"), docs/math_authoring.md.
• An independent enumeration of the preimages of $2^{18}\cdot257$, written for this review from the divisor recursion the Lemma 5.1 page describes and cross-checked against brute force over $m\le2\cdot10^5$ for every totient value up to $3000$; not filed.

Exposures: the Problem 417 page was printed from its top through its "Progress" section, so its "Current assessment" paragraph (which records no assessment) and a "Progress" paragraph naming an author-recorded transfer from another folder were seen; neither bears on this section and neither was used. The two result pages of the second card carry a "Living verification: needs review" line, seen. The folder listing showed the file names under evidence/verify/ (a grade and six reconstruction reviews); none was opened. Nothing among the private working files was read and no web search was made.

Restatement

Conventions throughout: $\varphi$ is Euler's function; $n$ and $m$ range over positive integers; $\log_j$ is the $j$-th iterated natural logarithm.

• S1 (elementary notation facts). With $V(x)$ the number of totients in $[1,x]$, $\mathcal W(x)=\{\varphi(n):n\le x\}$ and $W(x)=\#\mathcal W(x)$ for real $x$: (a) $W(x)\le V(x)$ for every real $x$; (b) the function Problem 417 calls $V'(x)$ is $W(x)$.
• S2 (definitions by reference). $P(x;k)$ for natural $k$ splits as $P_0(x;k)+P_1(x;k)$, with $P_0$ counting the solutions of the Theorem A form and $P_1$ the rest; $c(k)$ is the coefficient (3.2) of the source, defined for even $k$; $Z(x)$ is Ford's function, with exponent $C(\log_3x-\log_4x)^2+C'\log_3x-(C'+\tfrac12-2C)\log_4x$; and $n$ is convenient for a totient $d$ with preimages $n_1,\ldots,n_k$ when $d\varphi(n)$ is a totient whose preimages are exactly $n_1n,\ldots,n_kn$. These are definitions; what is checkable is their agreement with the pages and the source they cite.
• S3 (T1). There is an $x_0$ independent of $k$ such that for every $x>x_0$ and every natural number $k\le\exp((\log x)^{1/3})$, $P_1(x;k)<x/\exp((\log x)^{1/3})$. Standing sentence: a published theorem whose proof the source labels a sketch; the reconstruction page records the sketch and its one written deduction only.
• S4 (T3). Fix a function $\varepsilon(x)>0$ with $\varepsilon(x)\to0$ and $x^{\varepsilon(x)}\to\infty$. As $x\to\infty$, uniformly over even $k$ with $2\le k\le x^{\varepsilon(x)}$, $P_0(x;k)\le(16C_2+o(1))c(k)x/(\log x)^2$, with $C_2$ the source's twin-prime constant; and for every even $k$, $1/(2k)\le c(k)\le3\cdot7^{3+2\omega(k)}\prod_{p\mid k,\,p>2}\frac{p-1}{p-2}\cdot\frac1k$. Standing sentence: reconstructed modulo Selberg's sieve as the source states it and Evertse's bound inside Lemma 3.2. Further: for every odd $k$ and every $x$, $P_0(x;k)=0$.
• S5 (L4). For every pair of totients $d_1,d_2$ and every $D\ge\max\{d_1,d_2\}$ there are an absolute constant $K\ge1$ (the same for all $d_1,d_2,D$), a constant $c_D>0$ and a threshold $x_0(D)$ such that for every $x\ge x_0(D)$ at least $c_DZ(x)$ integers $n$ satisfy $\varphi(n)\le x/D$, $n$ convenient for $d_1$ and for $d_2$, and $n/\varphi(n)\le K$. Standing sentence: a published lemma whose two counting steps import Ford's lower-bound argument; the reconstruction is partial.
• S6 (F). Ford's Theorem 1 gives $V(x)=Z(x)e^{O(1)}$ for large $x$. Consequences: (a) there is an absolute $C_1$ with $V(x)\le C_1Z(x)$ for all large $x$; (b) for each fixed real $\mu\ge1$, $Z(x/\mu)/Z(x)\to1/\mu$ as $x\to\infty$; (c) there is an absolute constant (written $C_2$ on the page) such that for each fixed $\mu\ge1$ there is an $x_2(\mu)$ with $V(x/\mu)\le(C_2/\mu)V(x)$ for all $x\ge x_2(\mu)$.
• S7 (P). $2^{18}\cdot257=67371008$ is a totient with exactly eight preimages, the eight integers listed on the Lemma 5.1 page, and all eight are even; the enumeration was repeated for the page by the same divisor recursion, and neither run is filed as evidence.

Checklist

• Quantifiers and scope. Pass. S3 carries "absolute $x_0$" and the $k$-range as the source states them (p. 6); S4 fixes $\varepsilon$ before the uniformity claim and keeps the source's "uniformly in $k$"; S5 keeps the order fixed pair, fixed $D$, then constants; S6 states "for large $x$", "for each fixed $\mu\ge1$" and a $\mu$-dependent threshold with a $\mu$-independent constant; S7 is a finite statement.
• Circularity. Pass. The section states inputs and elementary consequences; nothing in it is assumed in its own derivation.
• Model and convention changes. Fail in one place (F1): the symbol $C_2$ names the source's twin-prime constant in S4 and a fresh absolute constant in S6, and the section never defines the S4 meaning. The conventions $V$, $W$, $V'$ and $Z$ (with $C'$ for Ford's $D$) agree with Problem 417, the Theorem 1.2 page and Ford's p. 2.
• Finite and statistical overreach. Pass. S7 is used only as a finite fact about one totient, and the section draws no infinite conclusion from it.
• Uniformity. Pass. In S6(c) the constant is absolute and the threshold depends on $\mu$, as re-derived below; in S5 the constants may be labeled by $D$ alone (W3); in S3 and S4 the uniformity is the source's.
• Extremal conclusions. Inapplicable. The section asserts no infimum, supremum or sharpness; the least and greatest preimages belong to the Lemma 5.1 page, which this review consumes as a premise.
• Consequences and composition. Pass with a remark (F2): the "hence" of S6(c) also consumes the lower half of Ford's $e^{O(1)}$, which the quoted theorem supplies but the "two consequences" sentence does not list.
• Computation. Pass. $2^{18}\cdot257=67371008$; the eight preimages were re-enumerated independently, each was checked by evaluating $\varphi$, and all are divisible by $4$.
• Reproduction. Pass. The section states no rerun command; its one computational claim (S7) was reproduced from the stated method with a cross-check that would have failed on a missed or spurious preimage.
• Source and verdict fidelity. Pass. S3, S4 and S5 match the source word for word in their hypotheses and bounds (pp. 6--8 on the page images); S6's quotation matches Ford's Theorem 1 (p. 2); the definition of convenient matches the source's p. 2; each standing sentence matches the Standing paragraph of the reconstruction page it describes; the Problem 417 notation matches that page's Statement.

Weakest steps

W1: the limit $Z(x/\mu)/Z(x)\to1/\mu$ (S6(b)). Write $Z(x)=(x/\log x)\exp E(x)$ with $E(x)=C(\log_3x-\log_4x)^2+C'\log_3x-(C'+\tfrac12-2C)\log_4x$. Then

$$\frac{Z(x/\mu)}{Z(x)}=\frac1\mu\cdot\frac{\log x}{\log x-\log\mu}\cdot \exp\bigl(E(x/\mu)-E(x)\bigr),$$

and the middle factor tends to $1$. For the exponent, put $\delta_2=\log_2(x/\mu)-\log_2x=\log(1-\log\mu/\log x)=O(1/\log x)$; then $\delta_3=\log_3(x/\mu)-\log_3x=\log(1+\delta_2/\log_2x)=O(1/(\log x\log_2x))$ and $\delta_4=\log_4(x/\mu)-\log_4x=\log(1+\delta_3/\log_3x)=O(\delta_3/\log_3x)$. With $a=\log_3x-\log_4x$ and $\delta=\delta_3-\delta_4$, the square changes by $2a\delta+\delta^2=O(\log_3x/(\log x\log_2x))\to0$, and the two linear terms change by $O(\delta_3)+O(\delta_4)\to0$. So $E(x/\mu)-E(x)\to0$ and the limit is $1/\mu$. The page's justification ("whose change under $x\mapsto x/\mu$ tends to $0$") names the conclusion rather than this estimate; the estimate is what is needed, because $a\to\infty$, so a change of the arguments that is merely $o(1)$ would not suffice.

W2: the deduction $V(x/\mu)\le(C_2/\mu)V(x)$ (S6(c)). From Ford, $c_0Z(x)\le V(x)\le C_1Z(x)$ for $x\ge x_1$ with absolute $c_0,C_1>0$. For $x\ge\mu x_1$ and large enough that $Z(x/\mu)\le2Z(x)/\mu$ (W1),

$$V(x/\mu)\le C_1Z(x/\mu)\le\frac{2C_1}{\mu}Z(x)\le\frac{2C_1}{c_0\mu}V(x),$$

so the constant $2C_1/c_0$ is absolute and only the threshold depends on $\mu$. The lower bound $c_0Z(x)\le V(x)$ is indispensable: without it the chain ends at $Z(x)$, not at $V(x)$. It is available from the quoted $V(x)=Z(x)e^{O(1)}$, which is two-sided, but it is not one of the "two consequences" the page lists (F2).

W3: the labeling $c_D$, $x_0(D)$ in S5. The source writes $\gg_D$, and the Lemma 4.1 page warns that the constants may also depend on $d_1,d_2$. The section's labeling is nevertheless correct as stated: for fixed $D$ the pairs of totients $d_1,d_2\le D$ are finitely many, so the minimum of the per-pair constants and the maximum of the per-pair thresholds depend on $D$ alone and serve every pair at once. The clause $K\ge1$ follows from $n/\varphi(n)\ge1$ on the set of $n$ the lemma supplies, which is nonempty for $x\ge x_0(D)$ because $c_DZ(x)>0$. Neither remark is used elsewhere in the section.

Strongest attack

• S1. Attack: find a real $x$ with $W(x)>V(x)$, or a mismatch with Problem 417's symbols. Failed: every $\varphi(n)$ with $n\le x$ is a totient at most $n\le x$, so $\mathcal W(x)\subseteq\mathcal V\cap[1,x]$; Problem 417 defines $V'(x)=\#\{\varphi(m):1\le m\le x\}$, which is $W(x)$ literally, and its $V(x)$ is the section's.
• S2. Attack: a definition that differs from its cited page or from the source. Failed: the split and $c(k)$ are those of the Theorem 3.3 page and of the source's (3.2); $Z(x)$ is that of the Theorem 1.2 page, which is the source's p. 7 display with $C'$ where Ford writes $D$, both $2.17696874\ldots$; "convenient" is the source's p. 2 definition ("$n_1n,\ldots,n_kn$ and no others"). Residual: the letter $k$ counts preimages here and is the shift elsewhere (F4, note).
• S3. Attack: a strengthened quantifier, a changed exponent or range. Failed: the source's Theorem 3.1 (p. 6) reads "For $x>x_0$, we have $P_1(x;k)<x/\exp((\log x)^{1/3})$, uniformly for natural numbers $k\le\exp((\log x)^{1/3})$"; the section's "absolute $x_0$" is what "uniformly" means. Second attack: the standing sentence claims a reconstruction that does not exist. Failed: it says the opposite, and it matches the Theorem 3.1 page. Residual: that page also records a gap in the source's sketch (F3, note).
• S4. Attack: a wrong constant or range in the transcription. Failed for the bound, the range and the two-sided estimate of $c(k)$ (p. 6, page image). Attack on the odd-$k$ clause: exhibit odd $k$ and $j$ with $\gamma(j)=\gamma(j+k)$. Failed: $j$ and $j+k$ have opposite parity, so exactly one is even and their prime supports differ; hence no solution has the Theorem A form and $P_0(x;k)=0$. Attack on the symbol: $C_2$ is used before it is defined and is redefined in S6 (F1).
• S5. Attack: the section drops a dependence (on $d_1,d_2$) or invents one ($K\ge1$). Failed by W3.
• S6. Attack on (b): choose $\mu$ so that the exponent's change does not vanish. Failed: the change is $O(\log_3x/(\log x\log_2x))$ for every fixed $\mu$ (W1). Attack on (c): make the constant depend on $\mu$. Failed: W2 gives the absolute constant $2C_1/c_0$. Attack on the stated inputs: derive (c) from (a) and (b) alone. This succeeds only in showing that a third input, the lower half of Ford's theorem, is consumed (F2); the page quotes that theorem two-sided, so the conclusion stands.
• S7. Attack: a ninth preimage, or an odd one. Failed: the independent recursion over the primes $p$ with $p-1\mid67371008$ returns exactly the eight listed integers, each with $\varphi$ equal to $67371008$ and each divisible by $4$; on all totient values up to $3000$ the same recursion agrees with brute force over $m\le2\cdot10^5$, and the base case that adjoins $2m$ to an odd preimage $m$ was exercised there. By hand: an odd preimage would need $257^2$ in it and $2^{10}$ from the Fermat primes below $257$, which supply at most $2\cdot4\cdot16=2^7$; so none exists.

Premises

• Theorem 3.1 reconstruction: interface, the statement of S3 and the standing "record of a sketch"; read in full.
• Theorem 3.3 reconstruction: interface, Theorem A, the split $P_0+P_1$, $c(k)$ for even $k$, $C_2$, the statement of S4 and its standing; read in full.
• Lemma 3.2 reconstruction: interface, the count $3\cdot7^{3+2\omega(k)}$ behind the upper bound on $c(k)$; read in full.
• Lemma 4.1 reconstruction: interface, the statement of S5 with its caveat on the constants and its standing "partial"; read in full.
• Lemma 5.1 reconstruction: interface, the definition of convenient and the enumeration of the preimages of $d_1$ and $d_2$; read in full.
• Theorem 1.2 reconstruction: interface, the definition of $Z(x)$ and step 2 (odd $k$); read in full.
• Pollack, Pomerance and Treviño (2013): held, the 17-page author manuscript; p. 2 (convenient), p. 6 (Theorems 3.1 and 3.3, Lemma 3.2, Remark 3.1), p. 7 (proof of Theorem 3.3; Ford's order of magnitude and $Z(x)$), p. 8 (constants (4.1)--(4.2), Lemma 4.1), pp. 9--10 (Lemma 5.1, proof of Theorem 1.2), on the text layer, with pp. 6--8 also on the page images.
• Ford (1998): held, a 43-page copy whose acknowledgment records a corrected lower-bound proof; interface used, Theorem 1 with (1.3)--(1.6): p. 2 on the page image and the text layer, nothing else read.
• Problem 417: Statement only.
• No local claim card is consumed; the section carries no depends_on.

Findings

• F1 (required). Location: "(T3) ... $P_0(x;k)\le(16C_2+o(1))c(k)x/(\log x)^2$" and "(F) ... hence $V(x/\mu)\le(C_2/\mu)V(x)$ for $x\ge x_2(\mu)$ with an absolute $C_2$". Defect: one symbol, two constants. In (T3) $C_2$ is the source's twin-prime constant $2\prod_{p>2}(1-(p-1)^{-2})=1.32\ldots$, which the section never defines; in (F) "an absolute $C_2$" introduces a different constant. Read with the (T3) meaning, the (F) inequality becomes the unproved numerical claim $V(x/\mu)\le(1.32\ldots/\mu)V(x)$; read with the (F) meaning, the (T3) bound loses its constant. Witness: the two displays quoted. Replacement: in (T3) add "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)", and in (F) rename the constant, for instance to $C_3$, carrying the rename to every later use on the page.
• F2 (suggested). Location: "(F) ... Two consequences are used: ...; hence $V(x/\mu)\le(C_2/\mu)V(x)$". Defect: the "hence" consumes a third fact, $Z(x)\le c_0^{-1}V(x)$ for large $x$ (the lower half of Ford's $e^{O(1)}$), which the quoted theorem supplies but the sentence does not list; and the reason given for $Z(x/\mu)/Z(x)\to1/\mu$ states the conclusion about the exponent rather than the estimate that proves it. Witness: W1 and W2. Replacement: "Three consequences are used: absolute $c_0,C_1$ 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 ...".
• F3 (note). Location: "(T1) ... the reconstruction records the sketch and its one written deduction only." The Theorem 3.1 page also records that the source's sketch does not dispose of the solutions with $p\mid m$ (for $k=2$: $n=4,8,32$), which $P_1(x;k)$ counts. The sentence is accurate, and the page consumes the theorem as published, but a consumer weighing (T1) would want the recorded gap named here.
• F4 (note). Location: "convenient for a totient $d$ with preimages $n_1,\ldots,n_k$". The letter $k$ is the shift everywhere else in the section. Harmless, and the Lemma 5.1 page and the source's p. 2 use the same letter; "$n_1,\ldots,n_r$" would remove the overload.

Verdict

• S1 (elementary notation facts): refutation-failed.
• S2 (definitions by reference): not a claim; each definition agrees with its cited page and with the source.
• S3 (T1, transcription and standing sentence): refutation-failed.
• S4 (T3, transcription, $c(k)$ bounds, standing sentence, odd $k$): refutation-failed; F1 required, a rename and a definition, with no change to the mathematics.
• S5 (L4, transcription with $K\ge1$ and standing sentence): refutation-failed.
• S6 (Ford's theorem and its three consequences): refutation-failed; F2 suggested.
• S7 (the eight even preimages): refutation-failed.

Limitations. The proofs of Theorems 3.1 and 3.3 and of Lemma 4.1 were not re-examined: the section consumes them as published results at the scope the reconstruction pages record, and those pages are premises of this review. Ford's Theorem 1 was checked as a statement against the held copy; its proof was not read. The enumeration behind S7 is the reviewer's own run of the same method, not a filed evidence record. Whether the later sections of the page use $C_2$ in one sense or both was not examined, since those sections lie outside this review's read set.

This focused review assigns no tier and changes no status.