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Claim. The answer to Problem 326 is yes. Theorem 1.1 of Aron Bhalla's manuscript A Regularly Thin Minimal Asymptotic Basis of Order Two (Bhalla's manuscript) asserts a constant $C>0$ and a minimal asymptotic basis $A=\{a_1<a_2<\cdots\}\subset\mathbb N$ of order $2$ whose counting function satisfies $A(x)=C\sqrt x+O(1)$; consequently $a_k/k^2\to C^{-2}\neq0$. That is the basis the problem asks for, which Erdős and Graham expected not to exist. The construction places one private witness at a time: a large integer whose only representation as a sum of two elements of $A$ uses a prescribed earlier element, so that removing any element destroys infinitely many representations and $A$ is minimal, while covering blocks and smoothing steps keep the counting function within a bounded distance of $C\sqrt x$. The manuscript was posted to the problem's thread on 2026-05-20, and a revised version with minor corrections on 2026-06-14, the version the library card describes.

Submission note. Posted to the site's forum by Aron Bhalla on 20 May 2026:

I have resolved this problem here. [EDIT: this version of the manuscript has now been superseded by later version; the latest public version of the manuscript is available here.]

The main theorem shows that there exists a minimal asymptotic basis $A\subset\mathbb N$ of order $2$ such that $A(x)=C\sqrt{x}+O(1)$ for a sufficiently large constant $C>0$. Most of the work is my own; however, the end product was achieved with the assistance of GPT 5.5 to stress-test ideas, suggest revisions, and identify possible gaps in earlier versions of this argument. At the moment, some of the proofs and exposition in the manuscript have been written up by GPT. I have personally verified that these correspond to our discussions and take full responsibility for the content.

Posted to the site's forum by Aron Bhalla on 14 June 2026:

After weeks of work using Aristotle, Codex, and GPT 5.5, I have now formalised the solution, confirming all of the claims in the manuscript. The full formalisation is currently almost 15,000 lines long! You can type-check it online here.

Additionally, here is the latest version of the manuscript. It has not changed significantly, but now incorporates several minor changes needed for completeness (including addressing the minor issues picked up by Nat's standard check).

AI assistance and formalization. The author writes that most of the work is the author's own, that GPT 5.5 helped to stress-test ideas, suggest revisions and find gaps in earlier versions of the argument, and that some proofs and exposition in the manuscript were written up by GPT; the author takes responsibility for the content. On 2026-06-14 the author reported a Lean formalization of almost 15,000 lines, made with Aristotle, Codex and GPT 5.5, which the author says confirms every claim of the manuscript. The thread post of that date (post 6969, linked above) carries the development as a Lean web-editor link holding the whole file, and no repository for it is recorded. This corpus has not built or audited that development, so it gives no formalized evidence.

Standing. Claimed. There is no refereed publication. The site's curator, Thomas Bloom, commented on 2026-06-15 that the exposition is hard to follow and that the manuscript does not clearly define the set $A$ it constructs, adding that this is not a criticism of the proof's correctness; the author answered that the manuscript is being rewritten. Those comments are not an acceptance, and the site labels the problem OPEN (page last edited 2026-04-17).

Depends on. No page of this wiki.