Periodic Reporting for period 1 - TOPUP (Theory of particle collider processes at ultimate precision) Reporting period: 2021-09-01 to 2023-02-28 Summary of the context and overall objectives of the project The Large Hadron Collider (LHC) at CERN probes the interaction of elementary particles at unprecedented energy and to very high precision. The full exploitation of the upcoming data from the LHC relies on a close interplay between theory and experiment, which calls in particular for highly accurate theoretical predictions. This theoretical accuracy can be achieved only though the expansion of the fundamental scattering amplitudes to sufficiently high order in perturbation theory.This project aims to meet this challenge for modern collider physics by providing the conceptual and technical foundations for theory predictions at ultimate precision. It will develop and establish a new standard of theoretical precision in the description of physical observables at the LHC based on perturbation theory expanded to the third non-trivial order (N3LO). We will achieve this ambitious goal by targeting the main obstacles in present-day methods, and by developing novel ways for the computation of multi-loop scattering amplitudes and in the understanding and handling of unresolved multi-particle emission.The concrete goal of the project is to enable theoretical predictions at ultimate precision for multiple processes in high-energy particle collisions with full final state kinematical information. This will lead to a more precise extraction of fundamental physics parameters, such as couplings and particle masses, and will shape the LHC precision physics program. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far The project has started in the fall of 2021. Its early results include several conceptual and methodological developments that will subsequently enable high-precision predictions for processes at the CERN LHC collider. As a first application, we computed fully differential observables in lepton pair production at the CERN LHC, which are immediately relevant to precision measurements of gauge boson masses and electroweak coupling parameters at the LHC. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) Highly precise predictions in particle theory require an intricate combination of exact algebraic calculations and accurate numerical algorithms. In this project, we are aiming for major innovations in both directions. Early project results have overcome several long-standing limitations of established methodologies, and have produced innovative approaches to address the complexity challenges posed by modern precision applications of particle theory and beyond. Momentum spectrum of lepton pairs at LHC, compared to precision theory prediction (blue).