Binary neutron star mergers (BNSMs) are unique astrophysical laboratories to explore all four fundamental interactions in their extreme regimes. The landmark detection of the gravitational wave GW170817 and its counterparts in the entire electromagnetic spectrum demonstrated the enormous impact of BNSMs observations on fundamental physics and astrophysics, including the nature of matter at supranuclear densities, the origin of high-energy cosmic photons and of heavy elements. The goal of InspiReM is to break new grounds in the theoretical modeling of BNSMs and to deliver first-principles models linking the source dynamics to the observed radiations. The programme timely addresses central open problems in the modeling of the different coalescence phases with a novel, comprehensive, general-relativistic, (3+1)D and multiscale approach. Simulations and analytical relativity methods are combined to deliver full-spectrum gravitational-wave templates for unbiased, high-precision measurements in gravitational-wave astronomy. Merger remnants and outflows are investigated on uncharted postmerger timescales and including, for the first time, all the relevant processes from the four interactions. The self-consistent secular evolution of the outflows up to days and years is further explored to directly connect the strong-gravity engine to the electromagnetic emission. Bayesian approaches with simulation-driven models are developed for the joint analyses of gravitational and electromagnetic signals. InspiReM leverages on recent breakthroughs and the unique interdisciplinary expertise of my team on all the aspects of the research, and develops novel techniques for exascale parallel computations in relativistic astrophysics. If successful, it will shape the rising field of multimessenger astronomy and drive new groundbreaking discoveries in the related fields.