Regular simplex tensor robust-eigenvector conjecture

Let $n\ge 3$ and $m\ge 3$ , let $\{v_1,\ldots,v_n\}\subset\mathbb{R}^{n-1}$ be the regular simplex frame, and let $T_{m,n}$ be the associated regular simplex tensor of order $m$ . The conjecture asserts that every robust eigenvector of $T_{m,n}$ is sign-equivalent to one of the frame vectors $v_i$ ; equivalently, up to sign equivalence, the robust eigenvectors are exactly $\{v_1,\ldots,v_n\}$ . The claimed resolution further states that for $(m,n)\in\{(3,3),(3,4),(4,3)\}$ there are no robust eigenpairs.

References

Primary source

arXiv

Additional references

Regular simplex tensors: optimization landscape, conjecture proof, and beyond — arXiv — Lei Wang

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A 2026 preprint claims to settle the conjecture, except for three low-dimensional cases that it says have no robust eigenpairs.

The conjecture says that the robust eigenvectors of a regular simplex tensor are exactly the vectors in its defining frame. Earlier work established only restricted cases and numerical evidence; a later preprint claimed a complete proof.

Known results

The case $n=2$ was known before the general conjecture.
Frame vectors were proved robust for $n\ge 2$ , $d\ge 3$ , and $n+d\ge 7$ .
A 2022 analysis found additional non-robust eigenvectors and low-dimensional exceptions, without proving the general characterization.

September 2026 claimed resolution

On September 28, 2026, a report on Lei Wang's paper Regular simplex tensors: optimization landscape, conjecture proof, and beyond stated that the characterization holds for all higher $n$ and $m$ , while $(m,n)=(3,3),(3,4),(4,3)$ have no robust eigenpairs. The claim is presented as a resolution but has no independent verification in the retrieved sources.

Current status (as of September 2026): A preprint claims the conjecture is settled for all higher $n$ and $m$ , with three exceptional pairs excluded, but the result remains unverified.

Sources

No solutions have been posted yet.