For every number of points n≥ 1, integers ell,k≥ 1 with k>ell, and configurations x_1,…,x_n∈R^ell and y_1,…,y_n∈R^k, define [ F(y_1,…,y_n)=∑_1≤ i<j≤ n(lVert y_i-y_j rVert^2-lVert x_i-x_j rVert^2)^2. ] If nabla F(y_1,…,y_n)=0 and the Hessian nabla^2F(y_1,…,y_n) is positive semidefinite, then F(y_1,…,y_n)=0. Equivalently, every second-order stationary point satisfies lVert y_i-y_j rVert^2=lVert x_i-x_j rVert^2 for all 1≤ i<j≤ n; this holds without genericity assumptions and allows repeated points and degenerate configurations.

One-extra-dimension conjecture for complete-graph squared-stress optimization

For every number of points $n\ge 1$ , integers $\ell,k\ge 1$ with $k>\ell$ , and configurations $x_1,\ldots,x_n\in\mathbb{R}^{\ell}$ and $y_1,\ldots,y_n\in\mathbb{R}^{k}$ , define

F(y_1,\ldots,y_n)=\sum_{1\le i<j\le n}\left(\lVert y_i-y_j\rVert^2-\lVert x_i-x_j\rVert^2\right)^2.

If $\nabla F(y_1,\ldots,y_n)=0$ and the Hessian $\nabla^2F(y_1,\ldots,y_n)$ is positive semidefinite, then $F(y_1,\ldots,y_n)=0$ . Equivalently, every second-order stationary point satisfies $\lVert y_i-y_j\rVert^2=\lVert x_i-x_j\rVert^2$ for all $1\le i<j\le n$ ; this holds without genericity assumptions and allows repeated points and degenerate configurations.

References

Primary source

arXiv

Additional references

One Extra Dimension Suffices for the Complete-Graph Squared-Stress — arXiv — Lilin Yan, Hongwei Zhao

Posted 1 day ago · edited 1 day ago

Progress summary

Refreshed 1 day ago

Claimed progress

A recent paper proves the conjecture after doubling the required dimension, but the one-extra-dimension case remains open.

The conjecture asserts that complete-graph squared-stress optimization has no nonglobal second-order critical points once the ambient dimension is one larger than the configuration dimension. The exact threshold remains unresolved.

Known results

The landscape can have spurious local minima when $k=\ell$ , even with $n=\ell+2$ points (Song et al., 2025; Criscitiello et al., 2026).
Benignness is proved at the conjectured threshold $k=\ell+1$ when $n\le \ell+3$ .
For arbitrary numbers of points, every second-order critical point is globally optimal when $k\ge 2(\ell+1)$ (Criscitiello, 2026).

October 2026 formalization claim

A separate arXiv item by Lilin Yan and Hongwei Zhao advertises a Lean 4 formalization, kernel check, and axiom audit for a claimed theorem concerning this setting. The retrieved material does not independently assess whether the encoded statement matches the conjecture or whether it establishes the full $k\ge\ell+1$ claim.

Current status (as of October 2026): The complete-graph conjecture for $k\ge\ell+1$ remains open; the factor-two theorem and the restricted $n\le\ell+3$ case are established, while the formalization-based full-resolution claim is unverified.

Sources

No solutions have been posted yet.