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Claim. There is a transcendental entire function $f$ such that for every nonempty open $U\subseteq\mathbb{C}$ the derivative $f^{(n)}$ has a zero in $U$ for all sufficiently large $n$. For every increasing sequence $n_1<n_2<\cdots$ the zeros of the $f^{(n_k)}$ are then dense, which is the affirmative answer to the corrected Statement of Problem 906, which asks for a transcendental $f$; the tab entry's notes call the polynomial witnesses of the site's wording trivial. The function is a random Fock series

$$f(z)=\sum_{k\ge0}\xi_k\frac{z^k}{\sqrt{k!}},$$

with independent coefficients $\xi_k$ uniformly distributed on the closed unit disk, and the claim is that a realization of it has the property. As the tab entry describes the argument, a saddle-point estimate for the coefficients of $f^{(n)}$ and a small-ball estimate in one coordinate give an exponential lower-tail bound for $\log\lvert f^{(n)}(z)\rvert$; for a fixed disk, Jensen's formula turns the event that $f^{(n)}$ has no zero in the disk into a large logarithmic deficit on an interior circle, of order $\sqrt n$ because the circular average of $\lvert z\rvert$ strictly exceeds its value at the center, so the probability of such a hole is bounded by an exponentially small quantity. The manuscript also proves the growth bound $\lvert f(z)\rvert\le\sqrt2\exp(\lvert z\rvert^2)$; the entry's notes say this conflicts with a 1973 theorem of Boas and Reddy as printed and that the manuscript explains the quantifier conflict for a fixed disk. The claimant is Eric Hou, who filed the claim and declares assistance from the AI system named as ChatGPT 5.6 Sol. This page's account of the manuscript follows the tab entry's summary and notes.

Submission note. Posted to erdosproblems.com as a proof claim by Eric Hou (account erichou) on 21 July 2026, giving "ChatGPT 5.6 Sol" as the AI used:

We construct a transcendental entire function $f$ such that, for every nonempty open set $U\subset\mathbb C$, the derivative $f^{(n)}$ has a zero in $U$ for all sufficiently large $n$. Equivalently, for every increasing sequence $n_1<n_2<\cdots$, the union of the zero sets of the functions $f^{(n_k)}$ is dense in $\mathbb C$. The construction is the random Fock series

$$f(z)=\sum_{k\geq0}\xi_k\frac{z^k}{\sqrt{k!}},$$

where the

coefficients are independent and uniformly distributed on the closed unit disk. A saddle estimate for the coefficients of $f^{(n)}$ and a one-coordinate small ball estimate give an exponential lower-tail bound for $\log|f^{(n)}(z)|$. For a fixed disk, Jensen's formula converts the event that $f^{(n)}$ has no zero in the disk into a large logarithmic deficit on an interior circle. The strict inequality between the circular average of $|z|$ and its value at the center makes this deficit of order $\sqrt n$, giving a hole probability bounded by $C\exp(-c Notes: The literal formulation allows nonzero polynomials and is therefore trivial, since every sufficiently high derivative of a polynomial vanishes identically. The paper proves the intended transcendental version and, more strongly, shows that every fixed nonempty open set contains a zero of every sufficiently high derivative. It also proves the growth bound

$$|f(z)|\leq \sqrt{2}\exp(|z|^2).$$

This conflicts with

Theorem 1 of Boas and Reddy, Bull. Amer. Math. Soc. 79 (1973), as printed; the paper explains the precise fixed-disk quantifier conflict. Verify with

$$lake > build CofiniteDerivatives$$

followed by

$$lake env lean Audit.lean.$$

The

axiom report for

$$> CofiniteDerivatives.exists\_transcendental\_entire\_with\_explicit\_derivative\_zeros\_and\_growth >$$

is exactly

$$[propext,\ Classical.choice,\ Quot.sound].$$

The project

contains no $\mathtt{sorry}$, $\mathtt{admit}$, or custom axioms.

Standing. The claim was filed on the site's proof-claims tab on 21 July 2026 as a full proof, with no comments. The site's label is OPEN with the page last edited 1 October 2025, before the claim, and its commentary does not mention it. No refereed publication and no outside review was found. The later claim on He's page constructs a different witness and says it does not claim to be the first solution. The manuscript's revision of August 2026, the version at the linked address, credits the earlier Gaussian proposals of Almeida and Chojecki, posted on 25 April 2026 and recorded on Almeida's page and Chojecki's page, and says its author learned of them after completing and circulating the proof; the version of 21 July 2026 in the linked repository does not mention them. The claim stays claimed.

Formalization. The linked repository, at the commit of its release tag v1.0.0, declares a Lean development whose theorem CofiniteDerivatives.exists_transcendental_entire_with_explicit_derivative_zeros_and_growth is reported by the entry's notes to have the axiom report propext, Classical.choice and Quot.sound, with no sorry, admit or custom axiom, checked by lake build CofiniteDerivatives and an audit file. This corpus has not built or audited it, so the self-report awards nothing here and no formalized evidence is listed.

Depends on. Nothing beyond the cited manuscript.