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The gravitational waves emitted in the related bound-orbit system\u2014the binary inspiral\u2014are now routinely detected by gravitational-wave observatories<jats:sup>3<\/jats:sup>. Theoretical physics needs to provide high-precision templates to make use of unprecedented sensitivity and precision of the data from upcoming gravitational-wave observatories<jats:sup>4<\/jats:sup>. Motivated by this challenge, several analytical and numerical techniques have been developed to approximately solve this gravitational two-body problem. Although numerical relativity is accurate<jats:sup>5\u20137<\/jats:sup>, it is too time-consuming to rapidly produce large numbers of gravitational-wave templates. For this, approximate analytical results are also required<jats:sup>8\u201315<\/jats:sup>. Here we report on a new, highest-precision analytical result for the scattering angle, radiated energy and recoil of a black hole or neutron star scattering encounter at the fifth order in Newton\u2019s gravitational coupling <jats:italic>G<\/jats:italic>, assuming a hierarchy in the two masses. This is achieved by modifying state-of-the-art techniques for the scattering of elementary particles in colliders to this classical physics problem in our universe. Our results show that mathematical functions related to Calabi\u2013Yau (CY) manifolds, 2<jats:italic>n<\/jats:italic>-dimensional generalizations of tori, appear in the solution to the radiated energy in these scatterings. We anticipate that our analytical results will allow the development of a new generation of gravitational-wave models, for which the transition to the bound-state problem through analytic continuation and strong-field resummation will need to be performed.<\/jats:p>","DOI":"10.1038\/s41586-025-08984-2","type":"journal-article","created":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T15:02:11Z","timestamp":1747234931000},"page":"603-607","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":68,"title":["Emergence of Calabi\u2013Yau manifolds in high-precision black-hole scattering"],"prefix":"10.1038","volume":"641","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3983-5852","authenticated-orcid":false,"given":"Mathias","family":"Driesse","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gustav Uhre","family":"Jakobsen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Albrecht","family":"Klemm","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3070-5717","authenticated-orcid":false,"given":"Gustav","family":"Mogull","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0202-536X","authenticated-orcid":false,"given":"Christoph","family":"Nega","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2883-7825","authenticated-orcid":false,"given":"Jan","family":"Plefka","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2071-257X","authenticated-orcid":false,"given":"Benjamin","family":"Sauer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Johann","family":"Usovitsch","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,5,14]]},"reference":[{"key":"8984_CR1","first-page":"154","volume":"1918","author":"A Einstein","year":"1918","unstructured":"Einstein, A. \u00dcber Gravitationswellen. 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