We consider the system of N (≥ 2) elastically colliding hard balls of masses m1, ⋯ ,Nand radius r on the flat unit torus Tν, ν≥2. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i.e. the full hyperbolicity and ergodicity of such systems for every selection (m1, ⋯ ,mN; r) of the external parameters, provided that almost every singular orbit is geometrically hyperbolic (sufficient), i.e. the so called Chernov-Sinai Ansatz is true. The present proof does not use the formerly developed, rather involved algebraic techniques, instead it employs exclusively dynamical methods and tools from geometric analysis. © 2009 Springer-Verlag.