Following the recent establishment of an exact kinetic theory realized by the Master kinetic equation which describes the statistical behavior of the Boltzmann-Sinai Classical Dynamical System (CDS), in this paper the problem is posed of the constructionof the related global existence and regularity theorems. For this purpose, based on the global prescription of the same CDS for arbitrary single-and multiple-collision events, first global existence is extablished for the N-body Liouville equation whichis written in Lagrangian differential and integral forms. This permits to reach the proof of global existence both of generic N-body probability density functions (PDF) as well as of particular solutions which maximize the statistical Boltzmann-Shannonentropy and are factorized in terms of the corresponding 1-body PDF. The latter PDF is shown to be uniquely defined and to satisfy the Master kinetic equation globally in the extended 1-body phase space. Implications concerning the global