Correction to: Nature https://doi.org/10.1038/s41586-025-08878-3 Published online 9 April 2025

In the version of the article initially published, the fourth equation in the “Loss functions” section of the Methods was a duplicate of the fifth equation. In the fourth equation, “Ty” and both instances of “Sy” have been corrected to “Tx” and “Sx” so that the equation now reads:

$${T}_{x}=\frac{1}{m}\sum _{i\varepsilon N}\sum _{j\varepsilon N\,,j\ne i}\sum _{k\varepsilon N\,,k\ne i\,,j}{S}_{x}(i\,,j)(1\,-\,{S}_{x}(i,k))\times {\rm{\text{max}}}(D({{\bf{z}}}_{i},{{\bf{z}}}_{j})-D({{\bf{z}}}_{i},{{\bf{z}}}_{k})+\varepsilon ,0)$$

in the HTML and PDF versions of the article.