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Claim. If $f$ is monic of degree four and all zeros of $f$, counted with multiplicity, lie in the open unit disk, then two zeros from this list can be joined inside $\{z:\lvert f(z)\rvert<1\}$ by a possibly degenerate polygonal path of length less than $2$. This is the degree-four case of Problem 1041, with a repeated zero counting as joined by the constant path. The preprint A degree-four lemniscate path theorem by Venkata Siddharth Pendyala appeared on arXiv on 2026-06-23 and was announced on the site's discussion thread on 2026-06-24, where the author reports that the result was also formalized and verified in Lean through the Aristotle system, without a public copy of the Lean files; the preprint describes its degree-four proof as elementary.

Covers. Degree four only. Together with the cubic case claimed on Borisov 2026 it would settle every degree at most four in the affirmative; the general question is claimed false in degree seven on ani 2026 (counterexample).

Depends on. No page of this wiki.

Standing. Claimed. The thread records no check of the preprint, no proof claim was registered on the site's proof-claims tab, the preprint is not refereed, and the Lean files are not public. Cook's thread post of 2026-09-11 cites the theorem as treating the general degree-four case under the original root-location hypothesis. A reader wrote on the tab of the cubic claim Borisov 2026 (2026-09-19 and 2026-09-23) that the same author had proved the cubic case in June 2026 and shared the proof privately; no public cubic statement by Pendyala exists, and a private communication gets no page.