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Claim. Let $g(n)=\#\{m\ge1:\phi(m)=n\}$. For every $\varepsilon>0$ there are infinitely many positive integers $n$ with

$$g(n)>n^{1-\varepsilon}$$

(Theorem 1.1 of the release's manuscript Weighted dilation graphs, smooth shifted primes and totient fibers, 2026-09-24, filed as the library's intake card). This is the question of Problem 821 answered yes, and no exponent above $1$ is possible, since $g(n)\ll_\eta n^{1+\eta}$ for every $\eta>0$ (Lemma 8.1 of the manuscript). The arithmetic input is the manuscript's Theorem 1.2: for every fixed $0<\delta<1/4$, the number of primes $p$ with $2x<p\le5x$ and $P^+(p-1)\le x^\delta$ is at least $x^{1-o(1)}$ as $x\to\infty$, where $P^+$ is the largest prime factor and the $o(1)$ may depend on $\delta$; the manuscript notes that the same count holds for every fixed $\delta>0$ by using a smaller exponent, so there are infinitely many primes $p$ with $P^+(p-1)\le p^\varepsilon$ for every $\varepsilon>0$. The step from many primes with smooth predecessors to large fibers is the product-and-pigeonhole argument of Erdős and Pomerance (Theorem B of Pomerance's account), which the manuscript proves in full: products of many primes whose predecessors draw their prime factors from a small pool collide under $\phi$. The analytic mechanism is a transference theorem for weighted dilation graphs, whose vertices are integers carrying lists of marked prime divisors and whose edges record shared prime factors; comparing the graph with an operator of independent labels yields cancellation in shifted correlations, a Type II estimate for $mn=2u+1$ with $u$ of prescribed factorization, and a sieve that removes the composite values.

The records before the release fixed one smoothness exponent, each from $x/(\log x)^{O(1)}$ primes: Baker and Harman's $0.2961$, with fiber exponent $0.7039$ (Theorem 1 and Corollary 1 on the card Baker and Harman 1998; the partial claim Baker and Harman 1998), and Lichtman's $0.2844$, giving $g(n)>n^{0.7156}$ infinitely often (Theorem 1.1 and Corollary 1.3 on the card Lichtman 2022; the partial claim Lichtman 2022), which the site's commentary gives as the best known bound; Erdős had proved $g(n)>n^c$ infinitely often for some $c>0$. The site's commentary notes that the conjecture follows from $\gg_\varepsilon x/\log x$ primes $p<x$ whose predecessor has every prime factor below $p^\varepsilon$. The release's companion manuscript The Poisson–Dirichlet law for prime predecessors (2026-09-24), the second link, filed as the library's companion card, claims that stronger conclusion: for a prime $p$ drawn uniformly from $3\le p\le x$, the logarithms of the prime factors of $p-1$, in decreasing order with multiplicity and divided by $\log(p-1)$, converge in every finite joint distribution to the Poisson–Dirichlet law of parameter one (its Theorem 1.1, the conjecture of Ford, Konyagin and Luca), so that for every fixed $u\ge1$ the proportion of primes $p\le x$ with $P^+(p-1)\le x^{1/u}$ tends to Dickman's $\rho(u)$. The companion consumes the dilation-graph theorems of the first manuscript in a different prime-extraction argument; neither manuscript states a totient result beyond Theorem 1.1.

Depends on. No page of this wiki.

Standing. The claim is a manuscript statement and is claimed. The release's README says that its manuscripts were produced by an internal OpenAI model and that the collection includes results at different stages of verification, not all with Lean formalizations; this family has none, so there is no formalized evidence, and no outside review, referee report or acceptance by the site is known (the site labels the problem OPEN; page last edited 1 October 2025). This page records the two manuscripts' statements; no outside review of either proof is known.