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Claim. For every monic $f$ with all roots in the open unit disk, two roots of $f$ (a repeated root counting as joined by the constant path) are connected by a path of length less than $2$ inside $\{z:\lvert f(z)\rvert<1\}$, which would answer Problem 1041 in the affirmative. The user ani posted the write-up on the site's discussion thread on 2026-03-17, produced after exchanges with GPT-5.4 and checked, the author said, by Opus 4.6, Gemini 3.1 and the author; revisions followed on 2026-03-22 and 2026-03-24 after AI checks by Nat Sothanaphan. In Terence Tao's summary of 2026-03-24 the argument has four parts: the Erdős–Herzog–Piranian lemma that some component $U$ of the lemniscate holds at least two roots; the bound $\int_U\lvert f'/f\rvert\,dA\le2\pi m$ for a component with $m$ roots; a tree of flow lines of $\log\lvert f\rvert$ joining the $m$ roots of $U$ with total length at most $(1/2\pi)\int_U\lvert f'/f\rvert\,dA$, under non-degeneracy hypotheses obtained by perturbation; and the lemma that a tree contains a path of length at most twice its total length divided by its number of vertices. Tao gave a shorter proof of the integral bound through the area formula, a Pólya-type inequality of Dubinin and the Pólya area bound, and noted the kinship with a paper of Crane.

Depends on. No page of this wiki.

Standing. Rejected. Tao wrote on 2026-03-25 that the topological claims of the tree step (Proposition 12) are not correct: the invocation of the strip lemma is unjustified because the boundary of the regions removed meets more than two level sets, and a plot of the flow lines of $z^2-1$ shows that the flow does not organize into connected trees, so the integral quantity does not isolate the geometry the problem needs. After further discussion the author wrote on 2026-03-26 that even the statement of Proposition 12 is incorrect and that the strategy is doomed to fail. Sothanaphan's AI checks had flagged the proposition as the main risk without finding the gap. The same user later posted the degree-seven counterexample on ani 2026 (counterexample), which, if correct, shows that the claimed statement is false.