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Subject and independence

• Role: independent reviewer in a fresh context, commissioned for refutation. The reviewer took no part in writing the page, the folder's reconstruction pages or any lead, and received only the assignment text.
• Subject: path wiki/research/erdos_49/connections.md as it stood on 2026-09-28T13:50:48Z, section "## Problem 1003: the unit shift", lines 255--282 (from the H2 heading to the line before the next H2 at line 283).
• Lead pages: none. The section records a failed transfer and names no lead.
• Read in the subject state, in full unless stated: the section; the page's preamble, Standing, Outcomes-at-a-glance and "Notation and the inputs used" (lines 1--111) as inputs whose interfaces were confirmed at each use; the reconstruction pages theorem_3_1_reconstruction.md, theorem_3_3_reconstruction.md and theorem_1_2_reconstruction.md in full, and the Standing and Statement sections of lemma_4_1_reconstruction.md and lemma_5_1_reconstruction.md, as premises; wiki/problems/arithmetic_functions/E1003/_index.md lines 1--35, the preamble that holds the statement (see Exposures); the cards and result pages of Pollack, Pomerance and Treviño (both cards, the Theorem 3.1 and 3.3 pages), Graham, Holt and Pomerance (card, Theorem 2 and Theorem 4 pages), Erdős, Pomerance and Sárközy II (card, Theorem 2 page), Ford 2020 (card, Theorem 1 page) and Li 2026 (card, Corollary 1.5 and Theorem 1.4 pages); the held PDFs on their text layers: the Pollack--Pomerance--Treviño manuscript pp. 5--6, Ford 2020 pp. 1--4 (p. 2 also as a page image), Erdős--Pomerance--Sárközy II printed p. 253, Li 2026 pp. 2--3; docs/verification.md sections "Grading and claim standing", "Audit checklist -- the canonical failure modes", "Whole-claim report" and "Audit checklist", docs/approach_vetting.md, docs/evidence.md section "Source fidelity" and docs/math_authoring.md.
• Exposures: the E1003 page has no "## Statement" heading; its statement sits in the preamble, which also carries the one-line site status ("Open"), the source line, the reference list and the formalization pointer, all seen while reading the statement. Nothing below the "## Current assessment" heading of that page was read. No _index.md of the folder, no evidence/ content, no Known Results, standing or acceptance text, and no other review was read; a working-tree status listing showed the file names of other folders' untracked review files, none opened. No web search was made. Yamada's paper is not held, so the comparison with his bound was checked only through the quotations in Li 2026 (p. 2, equation (1.1)) and Ford 2020 (p. 2).

Restatement

Throughout, $\varphi$ is Euler's function, $P(x;k)=\#\{n\le x:\varphi(n)=\varphi(n+k)\}$ for a natural number $k$, and $P(x;k)=P_0(x;k)+P_1(x;k)$ is the split of the Theorem 3.3 page: $P_0(x;k)$ counts the solutions $n\le x$ that have the form of Theorem A for some admissible $j$ and $r$, and $P_1(x;k)$ the rest. Theorem A (Graham, Holt and Pomerance, Theorem 1, as quoted on the source's p. 5) supposes that $j$ and $j+k$ have the same set of prime factors, puts $g=\gcd(j,j+k)$, and supposes that for a positive integer $r$ both $(j/g)r+1$ and $((j+k)/g)r+1$ are primes not dividing $j$; then $n=j(((j+k)/g)r+1)$ satisfies $\varphi(n)=\varphi(n+k)$.

• S1 (target). Problem 1003 asks whether there are infinitely many positive integers $n$ with $\varphi(n)=\varphi(n+1)$. The section also relays what the problem page's current assessment records: the question is open, the strongest upper bound is Li's Corollary 1.5, and Ford's infinitude results concern even shifts divisible by a fixed large integer.
• S2 (the transported bound). $P_0(x;1)=0$ for every $x$, so $P(x;1)=P_1(x;1)$, and there is an absolute $x_0$ such that $P(x;1)<x/\exp((\log x)^{1/3})$ for every $x>x_0$. This is the 1987 bound of Erdős, Pomerance and Sárközy, and it is weaker than Yamada's bound and than Li's.
• S3 (emptiness of the structured family). No positive integer $j$ has the same set of prime factors as $j+1$; hence no integer has the Theorem A form with $k=1$.
• S4 (no construction). No statement reconstructed in the folder (Theorems 3.1, 3.3 and 1.2, Lemmas 3.2, 4.1 and 5.1) produces an integer $n$ with $\varphi(n)=\varphi(n+1)$.
• S5a (obstruction, direction clause, as written). "Every reconstructed input bounds a count from above"; in the Outcome, "the inputs are upper bounds". Under its strongest reading, the universal ranges over the inputs the page declares as consumed ((T1), (T3), (L4), (F), (P)) and over the reconstructed statements.
• S5b (obstruction, substance). Infinitude is an existence statement, and no reconstructed input yields the existence of even one $n$ with $\varphi(n)=\varphi(n+1)$, let alone infinitely many.
• S6 (obstruction, parity clause). Theorem A, the only construction of shift solutions in the source chain, produces solutions of $\varphi(n)=\varphi(n+k)$ only from pairs $j$, $j+k$ with equal radicals; equal radicals force $k$ even; this requirement is the structural reason Ford's (2020) infinitude results concern even shifts and do not reach $k=1$; the reconstruction supplies no way around it.
• S7 (outcome). There is no next investigation from this mechanism, and the connection fails.

Checklist

• Quantifiers and scope. Pass. (T1) is applied at the single value $k=1$, which lies in its range $k\le\exp((\log x)^{1/3})$ for every $x>1$; its threshold $x_0$ is absolute, matching "for large $x$"; the strict inequality matches the source's Theorem 3.1. $P_0(x;1)=0$ holds for every $x$, not only eventually. No almost-all or limit statement occurs.
• Circularity. Pass. Nothing about Problem 1003 is assumed; the section concludes that the mechanism gives nothing.
• Model and convention changes. Pass. $P$, $P_0$, $P_1$ and Theorem A are used with the conventions of the Theorem 3.3 page, which match the source's p. 5; "same prime factors" and "equal radicals" are the same condition ($\gamma(j)=\gamma(j+k)$ on the Theorem 3.3 page).
• Finite and statistical overreach. Not applicable: no finite verification or heuristic enters. The explicit totient (P) is not used here.
• Uniformity. Pass. Only the fixed instance $k=1$ of (T1) is consumed, so the uniformity of (T1) in $k$ carries no weight; the absolute threshold is what the sentence "for large $x$" needs.
• Extremal conclusions. Not applicable: no infimum, supremum or sharpness claim is made. The comparison "weaker than Yamada's and Li's" is an ordering of three explicit bounds and is checked under Weakest step W2.
• Consequences and composition. One defect. $P(x;1)=P_1(x;1)$ composes correctly with (T1). The consequence sentence "every reconstructed input bounds a count from above", read as the universal it is, fails on (L4) and Lemma 5.1, which are lower bounds (Finding F1). The obstruction itself survives under the corrected reason (W3).
• Computation. Not applicable: the section runs no computation.
• Reproduction. Not applicable: the section states no rerun command or coverage claim.
• Source and verdict fidelity. Pass with two limitations. Theorem 3.1 and Theorem A were checked against the manuscript's pp. 5--6; the Erdős--Pomerance--Sárközy bound against printed p. 253; Li's Corollary 1.5 against p. 3; Ford's parity remark and his Lemma 3 construction against pp. 2 and 4. Yamada's bound was checked only as quoted by Li and by Ford, since his paper is not held. The sentence relaying the problem page's current assessment could not be checked against that text, which is outside this review's read set; the mathematical content it relays (Li's Corollary 1.5, Ford's Theorem 1) matches the held sources.

Weakest steps

W1 (S3, and $P_0(x;1)=0$ in S2). Suppose $j\ge1$ and the sets of prime factors of $j$ and $j+1$ coincide; call the common set $S$. Every $p\in S$ divides both $j$ and $j+1$, hence divides $(j+1)-j=1$, so $S$ is empty. Then $j$ has no prime factor, so $j=1$, and $j+1=2$ has the prime factor $2$, so $S$ is not empty: a contradiction. Hence no $j$ is admissible for $k=1$, the set of integers of the Theorem A form with $k=1$ is empty whatever $r$ is, and $P_0(x;1)=0$ for every $x$. The Theorem 1.2 page, step 2, reaches the same conclusion for every odd $k$ by parity (exactly one of $j$, $j+k$ is even); the section's route through coprimality is a different valid argument, and both use that $j+k\ge2$ has a prime factor. Composition: with $P_0(x;1)=0$ the split gives $P(x;1)=P_1(x;1)$, which is what S2 needs.

W2 (S2). (T1), as the source states it (Theorem 3.1, manuscript p. 6) and as the Theorem 3.1 page records it: there is an absolute $x_0$ such that $P_1(x;k)<x/\exp((\log x)^{1/3})$ for $x>x_0$, uniformly for natural numbers $k\le\exp((\log x)^{1/3})$. At $k=1$ the range condition holds for every $x>1$ because $\exp((\log x)^{1/3})>1$. With W1, $P(x;1)<x/\exp((\log x)^{1/3})$ for $x>x_0$. Standing of the input: the source proves Theorem 3.1 by a sketch, and the Theorem 3.1 page records an unreconstructed case (solutions with $p\mid m$); the page's Standing paragraph allows the use of source theorems as published, and at $k=1$ the same bound (with $\le$) is Theorem 2 of Erdős, Pomerance and Sárközy (1987), a published theorem on its own, which the section names. The ordering of the three bounds: Yamada's bound, as quoted by Li (p. 2, equation (1.1)) and by Ford (p. 2), is $x\exp\{-(2^{-1/2}+o(1))(\log x\log_3x)^{1/2}\}$ for each fixed odd shift; Li's Corollary 1.5 is $x\exp\{-(\tfrac12-o(1))(\log x\log_2x)^{1/2}\}$. Since

$$\frac{(\log x\log_3x)^{1/2}}{(\log x)^{1/3}} =(\log x)^{1/6}(\log_3x)^{1/2}\to\infty, \qquad \frac{\tfrac12(\log x\log_2x)^{1/2}}{2^{-1/2}(\log x\log_3x)^{1/2}} =2^{-1/2}\Bigl(\frac{\log_2x}{\log_3x}\Bigr)^{1/2}\to\infty,$$

the exponent of (T1) is the smallest of the three for large $x$: the section's "weaker than Yamada's and Li's" is correct, and Li's is the strongest of the three, consistent with the relayed assessment.

W3 (S5a and S5b). The direction clause fails under its strongest reading. (L4), as the page's Notation lists it and as the Lemma 4.1 page states it, concludes that at least $c_DZ(x)$ integers $n$ satisfy $\varphi(n)\le x/D$, convenience for $d_1$ and $d_2$, and $n/\varphi(n)\le K$, for $x\ge x_0(D)$: a lower bound on a count. Lemma 5.1 concludes $\#(\mathcal W(x)\setminus\varphi(S))\ge cW(x)$: a lower bound. (F) is two-sided. So "every reconstructed input bounds a count from above" is false as stated. The obstruction nevertheless holds, for a reason the section does not give: the lower bounds count other objects. Convenience of $n$ for a totient $d$ with preimages $n_1,\ldots,n_k$ says that $d\varphi(n)$ is a totient whose preimages are exactly $n_1n,\ldots,n_kn$; this constrains the multiples $n_in$ and says nothing about $\varphi(n+1)$ for any $n$ it counts. Lemma 5.1 says which totient values a nondecreasing set misses and constrains no pair of consecutive integers. Theorem 1.2 bounds a maximum from above. Hence no reconstructed statement has, or implies, a conclusion of the form "there is $n$ with $\varphi(n)=\varphi(n+1)$", which is exactly what S5b asserts.

Strongest attack

• S1. The section's phrasing was compared with the problem page's statement, "Are there infinitely many solutions to $\phi(n)=\phi(n+1)$"; they agree. The relayed descriptions of Li's Corollary 1.5 (an upper bound at the unit shift) and of Ford's Theorem 1(a) (infinitude whenever $442720643463713815200\mid k$, an even modulus) match the held sources. The claim about what the assessment "records", and its date, cannot be attacked inside the read set. Attack failed.
• S2. Three attacks: exhibit a solution of $\varphi(n)=\varphi(n+1)$ of the Theorem A form, which W1 rules out; place $k=1$ outside the range of (T1), impossible since the range contains every natural number up to a quantity exceeding $1$; reverse the ordering with Yamada's or Li's bound, which W2 rules out. The strict inequality was checked against the source (strict in Theorem 3.1; the 1987 theorem has $\le$, the weaker form). Attack failed.
• S3. W1 is the attack and its failure: a common prime factor of $j$ and $j+1$ would divide $1$.
• S4. Each reconstructed statement was inspected for an existence conclusion about consecutive integers: Theorems 3.1 and 3.3 bound $P_1$ and $P_0$ from above; Lemma 3.2 bounds the number of admissible $j$; Lemma 4.1 counts convenient integers; Lemma 5.1 counts missing totient values; Theorem 1.2 bounds $M^\uparrow(x)$. The two constructions inside them, the candidate set of Lemma 4.1 and the reversed pairs of Lemma 5.1, produce totient values with prescribed preimages, never a pair $n$, $n+1$; the only shift construction in the source chain, Theorem A, produces $n$, $n+k$ with $k$ even (S6). Attack failed.
• S5a. The witness (L4) defeats the sentence as written: it is a lower bound on a count, declared on the same page as a consumed input. Attack succeeded; Finding F1.
• S5b. The attack was to build a solution of $\varphi(n)=\varphi(n+1)$ from the reconstruction's lower bounds. Convenience controls $\varphi(n_in)$, a multiple of $n$; Lemma 5.1 controls the image of a nondecreasing set; neither reaches $n+1$ (W3). Attack failed.
• S6. First, equal radicals force $k$ even: if $k$ is odd then $j$ and $j+k$ have different parity, so $2$ divides exactly one of them and their prime supports differ; this uses only $j\ge1$. Second, the attribution to Ford was checked against the held PDF. Ford's Lemma 1 (p. 2) is Theorem A, and the text after it says that for odd $k$ there are no such $j$. Theorem 1 is proved (p. 4) through Lemma 3: with $a'=a/\gcd(a,b)$, $b'=b/\gcd(a,b)$ and $s$ the product of the primes dividing $a'b'$, the solutions are $m_1=b'\ell s(a'r+1)$ and $m_2=a'\ell s(b'r+1)$, whose shift is $m_1-m_2=\ell\kappa(a,b)$ with $\kappa(a,b)=(b'-a')s$. The pair $b'\ell s$, $a'\ell s$ has equal radicals, and $\kappa(a,b)$ is even: if $a'$ and $b'$ are both odd then $b'-a'$ is even, and otherwise $2\mid s$. So every shift Ford's Theorem 1 reaches is even, for exactly the equal-radical reason the section names; his Theorems 2 and 3 also produce only multiples of even numbers or simultaneous equations at shifts $\ell h_i$, none at $k=1$. Attack failed.
• S7. Follows from S4 and S6. Attack failed.

Premises

• (T1), Theorem 3.1 of Pollack, Pomerance and Treviño. Interface: an absolute $x_0$; $P_1(x;k)<x/\exp((\log x)^{1/3})$ for $x>x_0$, uniformly for natural numbers $k\le\exp((\log x)^{1/3})$. Held: the 17-page author manuscript, statement and sketch on p. 6, read on the text layer. The reconstruction page was read in full; it records the sketch, one written deduction, and an unreconstructed case; treated as established at that standing. Used here at $k=1$ only.
• Theorem A (Graham, Holt and Pomerance, Theorem 1) and the split $P=P_0+P_1$. Interface as restated above. Held: quoted on the manuscript's p. 5, read on the text layer; the corpus's verification on the Theorem 3.3 page was read. The Graham--Holt--Pomerance PDF was not opened; its card and its Theorem 2 and 4 pages were read.
• Theorem 1.2 page, step 2. $P_0(x;k)=0$ for odd $k$; re-derived in W1. Read in full.
• (L4), Lemma 4.1, and Lemma 5.1. Not consumed by the section; their Standing and Statement sections were read because their direction is the witness of F1. Interface of (L4): for fixed totients $d_1,d_2$ and fixed $D\ge\max\{d_1,d_2\}$, at least $c_DZ(x)$ integers $n$ with $\varphi(n)\le x/D$, convenient for both, and $n/\varphi(n)\le K$, for $x\ge x_0(D)$. Interface of Lemma 5.1: absolute $c>0$, $x_0$ with $\#(\mathcal W(x)\setminus\varphi(S))\ge cW(x)$ for every nondecreasing $S\subseteq[1,x]$, $x\ge x_0$.
• (T3), (F), (P). Not consumed by the section; their listed scopes were read on the page.
• Erdős, Pomerance and Sárközy (1987), Theorem 2. Interface: for large $x$, at most $x/\exp\{(\log x)^{1/3}\}$ solutions $n\le x$ of $\varphi(n)=\varphi(n+1)$. Held: printed p. 253 read on the text layer; card and Theorem 2 page read.
• Li (2026), Corollary 1.5. Interface: $\#\{n\le x:\varphi(n)=\varphi(n+1)\}\ll x\exp\{-(\tfrac12-o(1))(\log x\log_2x)^{1/2}\}$ as $x\to\infty$. Held: arXiv v2, p. 3 read on the text layer; p. 2 read for its quotation (1.1) of Yamada's bound; card, Corollary 1.5 and Theorem 1.4 pages read. A preprint; used only as the stated bound.
• Ford (2020), Theorem 1, Lemma 1 and Lemma 3. Interfaces as used in the attack on S6. Held: arXiv v5, pp. 1--4 read on the text layer and p. 2 as a page image; card and Theorem 1 page read.
• Yamada. Not held. His bound is taken from Li's (1.1) and Ford's p. 2, which agree.
• Problem 1003. Statement read from the page preamble.
• No batch acceptance order applies; no explicit assumption beyond the standings above was needed.

Findings

• F1 (required). Location: lines 271--272, "Infinitude is an existence statement; every reconstructed input bounds a count from above.", and lines 279--281, "Fails: the inputs are upper bounds, and the only constructive ingredient needs an even shift." Defect: the universal is false over the inputs the page declares. Witness: (L4), listed under "the inputs consumed below", concludes that at least $c_DZ(x)$ integers satisfy its three conditions, a lower bound; Lemma 5.1 concludes $\#(\mathcal W(x)\setminus\varphi(S))\ge cW(x)$, a lower bound; (F) is two-sided. The obstruction survives for the reason given in W3, which the section does not state. Proposed replacement for lines 271--272: "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." Proposed replacement for the Outcome: "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." The verdict "fails" and the Outcomes-at-a-glance entry are unaffected.
• F2 (note). Location: line 260, "Li's Corollary 1.5"; lines 260--261, "Ford's infinitude results"; line 266, "weaker than Yamada's and Li's"; line 276, "Ford (2020)". The three sources are named without links. The library holds a card for Li (2026) with a Corollary 1.5 page and a card for Ford (2020) with a Theorem 1 page; Yamada's paper is not held, so the comparison with his bound is checkable inside the corpus only through Li's quotation (p. 2, equation (1.1)) and Ford's (p. 2). Proposed: link the two cards, and either state Yamada's bound with that in-corpus locator or drop the comparison with it.
• F3 (note). Location: lines 263--266, "(T1) with $k=1$ gives ...". The bound is anchored on (T1), whose proof in the source is a sketch with an unreconstructed case recorded on the Theorem 3.1 page; at $k=1$ the identical bound is Theorem 2 of Erdős, Pomerance and Sárközy (1987), a published theorem the section already names. Proposed: say that (T1) at $k=1$ recovers that theorem, so that the sentence rests on the published unit-shift theorem rather than on the sketch. No conclusion changes.

Verdict

• S1: refutation-failed. The statement matches the problem page; the relayed assessment content is outside the read set except for its mathematical descriptions, which match the held sources.
• S2: refutation-failed. $P(x;1)=P_1(x;1)<x/\exp((\log x)^{1/3})$ for $x>x_0$ follows from (T1) at $k=1$ and W1; the ordering against Yamada's and Li's bounds holds (W2).
• S3: refutation-failed (W1).
• S4: refutation-failed.
• S5a: refuted-as-stated. The direction clause is false as a universal over the page's declared inputs; witness (L4). Finding F1 gives the replacement.
• S5b: refutation-failed. The obstruction holds for the corrected reason (W3).
• S6: refutation-failed. Equal radicals force an even shift, and the attribution to Ford (2020) is faithful to his Lemma 1 and Lemma 3.
• S7: refutation-failed.

Limitations. The sentence relaying the problem page's current assessment was not checked against that text, which this review was not allowed to read. Yamada's paper is not held; his bound was taken from two held quotations that agree. The proof sketch behind (T1) was not re-derived; at $k=1$ the bound rests independently on the 1987 theorem. The Graham--Holt--Pomerance PDF was not opened, since Theorem A enters only through its hypothesis, which the manuscript's p. 5 states. This focused review assigns no tier and changes no status.