Scattering experiments are the most elaborate way to analyse the laws of nature. In quantum field theory, the S-matrix encompasses all scattering processes, and describes physical phenomena as varied as collision experiments, gravitational waves, the strong nuclear force, string theoretic and quantum gravitational effects such as black hole production in high-energy scattering. There are two approaches to compute the S-matrix: perturbative and non-perturbative. The first has progressed a lot, propelled by the modern colliders’ needs, but captures only restricted physical regimes. Interestingly, it gave rise to new concepts, such as the double-copy construction of gravity, whose origin is elusive. The non-perturbative approach aims to compute the S-matrix directly, by combining the powerful principles it must obey: unitarity, causality, and crossing. Its main bottlenecks are our lack of control on multi-particle processes, and the non-linear nature of unitarity. These constraints are so stringent that today we still have not obtained a single fully consistent S-matrix. This project aims to fill this gap. It will (1) provide the first fully consistent S-matrices and produce the most accurate up-to-date pion S-matrix model, (2) produce models of unitary quantum-gravity scattering with high-energy black hole production. Even more ambitiously, for (1) and (2), the team will explore the space of all such consistent S-matrices. Finally, (3) SPARTA will investigate the non-perturbative nature of double-copy and explore its role as a new S-matrix principle. SPARTA will reach these goals thanks to an original approach to non-perturbative unitarity, "scattering-from-production", which tackles the bottlenecks of the non-perturbative approach, while making use of modern perturbation theory, and using powerful numerical tools. By this unique approach, SPARTA will lay the foundations of a grand programme, at the interface of scattering amplitudes and non-perturbative S-matrix.