  This paper provides the first rigorous derivation of a binary-ternary
Boltzmann equation describing the kinetic properties of a dense hard-spheres
gas, where particles undergo either binary or ternary instantaneous
interactions, while preserving momentum and energy. An important challenge we
overcome in deriving this equation is related to providing a mathematical
framework that allows us to detect both binary and ternary interactions.
Furthermore, this paper introduces new algebraic and geometric techniques in
order to eventually decouple binary and ternary interactions and understand the
way they could succeed one another in time.
