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Claim. Kenneth A. Mendoza's Zenodo deposits Computational verification of the Erdős–Herzog–Piranian conjecture, six versions from 2026-03-23 to 2026-05-08, assert that for every degree $n$ from $3$ up to the version's top degree ($10$, then $12$, $13$ and $14$) every monic polynomial of degree $n$ has a lemniscate no longer than that of $z^n-1$, the instances of Problem 114 in those degrees. The method is a branch-and-bound search over a bounded box of coefficients, reduced by symmetry to $2n-3$ real parameters, in which interval arithmetic (the Rust library inari, IEEE 1788) carries the sums, a Lipschitz margin bounds each box, and a marching-squares computation in floating point estimates each lemniscate's length; the deposits also give the closed form $L(z^n-1)=2^{1/n}B(\tfrac1{2n},\tfrac12)$ and a numerical Hessian at $z^n-1$. The author announced the search on the site's discussion thread on 2026-03-27 (degrees $3$ to $12$) and on 2026-05-08 (degrees $3$ to $14$, with release v3.1.0 of the code, which replaced a degree-13 row whose run had evaluated no boxes), presenting it as the finite complement of Tao's theorem for large degrees and claiming nothing for $n\ge15$.

Submission note. Posted to the site's forum by Kenneth A. Mendoza on 27 March 2026:

We have computationally verified the EHP conjecture for all degrees $3 \leq n \leq 12$ using dual independent implementations (Python/mpmath and Rust/inari) with IEEE 1788 certified interval arithmetic and branch-and-bound optimization. Both implementations produce identical certified enclosures across x86_64 and arm64 architectures. The dominance margins increase monotonically from $17.1\%$ ($n=3$) to $71.4\%$ ($n=10$); all non-extremizer boxes are eliminated at level $0$ for $n \geq 5$. We also derive a new closed-form expression for $L(z^n - 1)$ valid for all $n$.

Combined with Tao's result [Ta25] for sufficiently large $n$, the conjecture is now verified for all $n \leq 12$ and all $n \geq N_0$. The remaining gap is $13 \leq n < N_0$.

Preprint and certified result JSON files with SHA-256 checksums: doi.org/10.5281/zenodo.19229245

Code: github.com/MendozaLab/erdos-experiments

Withdrawal. Issue #4 of the code repository, opened on 2026-08-25 by Bertrand Chatelet, objects that the lemniscate lengths come from floating-point marching squares with an added margin of five percent, that the Lipschitz constant is estimated by finite differences at several scales, and that the Hessian is a central-difference approximation, so that none of the three is a bound and the chain of certification is broken; Chatelet pointed to the issue on the site's thread on 2026-09-05. The author's reply of 2026-10-01 accepts the three objections and reports an audit of the engine that found more: for $n\ge6$ a closed-interval test marked half of the coefficient box, the half containing $z^n-1$, as extremizer boxes that were never evaluated, and the outer region of coefficient space was covered by pseudo-random samples, not by a bound. The author states that the deposits are a computational census consistent with the conjecture, and not a proof, a certificate or an independently checked result for any degree from $3$ to $14$, and announces a corrected Zenodo version and a reply on the site's thread. The claim is recorded as withdrawn by its author.

Covers. Degrees $3$ to $14$, as claimed; nothing is settled. Degree $2$ is settled by Wang 1998 and Eremenko–Hayman 1999, and all sufficiently large degrees are claimed, pending, by Tao 2025.

Depends on. No page of this wiki.