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Standard Big Bang cosmology encounters fundamental breakdown at its initial boundary due to primordial singularities, the unverified physics of scalar inflation fields, and the unexplained absence of empirical supersymmetry. This paper introduces the **Big Crash Theory** as a deterministic, non-singular geometric alternative founded on the mechanics of higher-dimensional manifold intersections. We model the genesis of the observable cosmos as a soft asymmetric collision between a compact 10-dimensional basis space ($\mathcal{M}_{10}$) and an expansive 4-dimensional hyper-spherical foundation space ($\mathcal{N}_4$). The macroscopic 3-dimensional physical universe is derived precisely as the topological intersection domain ($\mathcal{V}_3 \equiv \mathcal{M}_{10} \cap \mathcal{N}_4$) realized post-collision, enclosed by an ultra-thin local proper time thickness ($\tau$). We demonstrate that the post-impact sliding velocity of the basis space relative to the foundation space generates a virtual temporal axis—the global monotonic parameter ($t$)—exerting a uniform global time pressure that drives positive rest mass generation and dictates the thermodynamic arrow of time. Concurrently, the intrinsic left-handed holonomic rotation of the weak sub-manifold synthesizes a Higgs-like topological potential on the $Z$ plane, naturally accounting for electroweak gauge boson masses and maximal parity violation. Furthermore, we show that a minute $0.5^\circ$non-orthogonal coordinate tilt separating the temporal drive$t$ and the internal strong color planes induces an asymmetric polar energy bifurcation during the impact, forcing an intrinsic surplus of positive vector energy that resolves the cosmological matter-antimatter asymmetry problem. This framework completely eliminates singularity thresholds, restoring strict geometric causality to cosmic evolution.