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Claim. The infimum of $\Lambda(f)$, over all degrees and all monic $f$ whose roots lie in the closed unit disc, is $2$. Tang's note proves the upper bound on the model polynomials $z^n-1$: the open set $\{|z^n-1|<1\}$ has $n$ congruent components, each with boundary length $\frac{2^{1/n}}{n}B(\tfrac12,\tfrac1{2n})$, and this tends to $2$ as $n\to\infty$. For the lower bound $\Lambda(f)\ge2$ the note splits on $|f(0)|$. If $|f(0)|<1$, the note applies Pommerenke's Theorem 3 of 1961 to the component $U_0$ of $0$ in the open set $\{|f|<1\}$ and obtains a diameter of at least $2-r^2\ge1$, $r$ the largest modulus of a root (equality at $r=0$, where $f=z^n$ and $U_0$ is the unit disc). Pommerenke's statement bounds the component of $0$ in the closed set $\{|f|\le1\}$, which can be larger than $U_0$ when components of the open set touch at a critical point, so this step rests on Pommerenke's argument carried over to the open component rather than on his statement as printed; the formalization linked below proves that open-component version (diameter above $1$) itself. The diameter of a bounded open set equals that of its boundary, and a rectifiable closed curve has length at least twice its diameter, so $\partial U_0$ has length at least $2$. If $|f(0)|=1$, every root lies on the unit circle, $0$ is a boundary point of some component, that component contains a root and so reaches the unit circle, and its boundary has diameter at least $1$ and length at least $2$. The note also raises the fixed-degree question, whether $z^n-1$ minimizes $\Lambda$ among monic polynomials of degree $n$ with roots in the disc, proves it for $n=1$ and $n=2$, and leaves $n\ge3$ open; that question is not part of the problem.

Sources used. Pommerenke's Theorem 3 (Michigan Math. J. 8 (1961), p. 99) is the lower bound's one outside input; it is compiled on the result page theorem_3 of the card pommerenke_1961_metric_properties_complex_polynomials. Tang's note is held only at its public address, a PDF with its TeX source in the author's repository, uploaded on 2026-01-05 and announced in the site's thread the same day; there is no journal version.

Acceptance. Reviewed: Terence Tao wrote in the thread on 2026-01-14 that Tao had read the note and thought the argument works, resting on Pommerenke's Theorem 3, which Tao saw no reason to doubt; and the site's curator, T. F. Bloom, labels the problem solved and credits the resolution to Tang. No refereed publication exists. Tang disclosed in the thread on 2026-01-21 that ChatGPT-5.2 Pro was used in Step 1 for numerical exploration, which suggested $z^n-1$ as the likely extremizer, and that a purported proof it produced was discarded as incorrect. Nothing here is independently reviewed by this project.

Lean. Lorenzo Luccioli's development Erdos1044, published as a gist on 2026-04-28, declares itself a formalization of Tang's note and proves Erdos1044.erdos_problem_1044 : lambdaInf = 2, importing only Mathlib, as its header states. Luccioli wrote in the site's thread on 2026-04-28 that they had formalized Tang's solution using Aristotle; the gist's header does not name the system. A second posting of the same development, in Boris Alexeev's repository of Lean proofs, names Tang as informal author and Aristotle and Luccioli as formal authors; both postings are linked above. The development proves the open-component form of Pommerenke's Theorem 3 (the component of $0$ has diameter above $1$ when $|f(0)|<1$) within the file, which its docstring presents as a corrected statement of the cited theorem. It is the Lean qualifier of the site's label. The formal-conjectures statement file for the problem, at its revision of 2026-09-18, credits Tang, states the problem with every theorem's proof left as sorry, and carries a formal_proof attribute pointing to the copy in Alexeev's repository. This corpus has not built or audited the development, so no formalized evidence is listed.

Depends on. Nothing on the wiki; the one outside input is the 1961 theorem named under Sources used.