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Claim. Let $V(x)$ count the distinct totient values in $[1,x]$. With $B=\log\log x$, a constant $\lambda=\log(1/\rho)\approx0.6114$ where $\rho\in(0,1)$ is the root of $\sum_{j\ge1}a_j\rho^j=1$ for $a_j=(j+1)\log(j+1)-j\log j-1$, the integer $m=\lfloor(\log B-\log\log B)/\lambda\rfloor$ and the fractional part $\theta\in[0,1)$ of $(\log B-\log\log B)/\lambda$, the manuscript's constant $\gamma=(\sum_{j\ge1}ja_j\rho^j)^{-1}\approx0.3235$ of its (2.2) (Ford writes it as $\lambda$; it is not Euler's constant), and the recursively defined $g_j$ with $G_m=B^m/(m!\prod_{i\le m}g_i)$, Theorem 2.1 of OpenAI, An asymptotic formula for the number of totients, OpenAI Math Release preprint, 25 September 2026 (the preprint link, at the pinned revision; the manuscript's author line is "OpenAI", and the release README says the manuscripts were produced by an internal OpenAI model and stand at different stages of verification, not all with Lean formalizations), states that

$$V(x)\sim\frac{x}{\log x}\,G_m\,A(1;\theta),$$

where $A(1;\cdot)$ is the uniform limit of explicit nonnegative functions $A_H(1;\cdot)$ on $[0,1)$, each a finite inclusion–exclusion sum over bounded prime and integer data (the tail witnesses of the manuscript's Section 2), and $0<\inf A(1;s)\le\sup A(1;s)<\infty$. The factor $xG_m/\log x$ is Ford's counting scale, so the theorem replaces the bounded multiplicative uncertainty $e^{O(1)}$ of Ford's Theorem 1 by one explicit periodic function of the phase $\theta$. The manuscript defines the coefficient without reference to $V$ and asserts no regularity of $s\mapsto A(1;s)$. The release presents the result as an asymptotic equivalent for $V(x)$, which is the second question of Problem 416; together with the fixed-scale limit of the same theorem, the subject of the accepted partial claim, it would answer both questions and settle the problem, so this page is the full claim. The manuscript is carded at its intake card (Theorem 2.1 page).

Formalization. The declaration OAI.TotientAsymptotic.totient_asymptotic_formula in lean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean at the pinned revision proves, as its first three conjuncts, the uniform convergence of AH H (fun _ => 1) to A (fun _ => 1) on Set.Ico 0 1, positive lower and finite upper bounds for A (fun _ => 1) there, and Tendsto (fun x => V x / mainTerm x) atTop (nhds 1) with mainTerm x = x / Real.log x * G x (m x) * A (fun _ => 1) (theta x); the comparator challenge lean/ComparatorChallenges/TotientAsymptotic.lean defines every constant above. The same declaration was built by this corpus's verification with the axioms propext, Classical.choice and Quot.sound only, as the accepted partial page records. This corpus's verification found that the main term uses no junk value: an empty root set would make the third conjunct false, and the $g_i$ are positive.

Depends on. Nothing in this wiki: Theorem 2.1 itself contains the fixed-scale clause, so the full claim rests on the manuscript and its Lean tree alone.

Standing. Settled: the release's Lean-checked declaration proves $V(cx)/V(x)\to c$ for every $c>0$, which answers the first question, and proves $V(x)\sim(x/\log x)\,G_m\,A(1;\theta(x))$ with a main term built without $V$. Not settled: the factor $A(1;\theta)$ depends on a phase $\theta(x)$ that cycles through $[0,1)$ as $\log\log\log x$ grows, and it is a limit of finite arithmetic sums with no closed form, no computed value and no convergence rate, so whether it is the formula in elementary functions that Erdős asked for (for example $V(x)\sim K$ times Ford's elementary scale for some constant $K$) is unsettled. Erdős's own words set the bar: on p. 201 of his 1974 remarks (card) he asks for a formula "in terms of elementary functions", and in 1979 (card) he doubts a genuine asymptotic formula and offers the ratio law as the substitute. Whether the limit of Ford's $O(1)$ factor, $e^{Q(\theta)}A(1;\theta)$ (the ratio of $V(x)$ to Ford's elementary expression, with $Q$ an explicit quadratic in $\theta$), is constant is not known; a constant would give $V(x)\sim K$ times Ford's elementary scale. The release's own catalog entry names only the scaling question as answered. The claim therefore stays claimed: the site labels the problem OPEN, the formal-conjectures statement erdos_416.parts.ii has no accepted proof, and no reviewer, referee or catalog has ruled that this equivalent is the formula asked for.