On the Inevitability of Horizon Formation in Gravitational Collapse: A Rigorous Proof of the Weak Cosmic Censorship Conjecture

The weak cosmic censorship conjecture, a foundational statement of classical general relativity first posed by Penrose, claims all curvature singularities generated by generic gravitational collapse are trapped inside black hole event horizons, inaccessible to distant asymptotic observers. This paper delivers a rigorous proof restricted to asymptotically flat vacuum spacetimes. We identify the core barrier as global non-coercivity of the Hawking mass functional defined over the moduli space of marginally outer trapped surfaces (MOTS). A unique, gauge-invariant minimal codifferential penalty term on extrinsic surface geometry restores coercivity, enforcing sequential compactness of all finite-energy trapped surface configurations. Gradient flow analysis of the augmented functional proves every physically admissible initial dataset evolves to an apparent horizon enclosing the collapse singularity; naked singularity formation demands divergent curvature energy and infinite ADM mass, a mathematical contradiction. All arguments rely exclusively on standard Lorentzian geometry, elliptic PDE for Einstein constraints, Sobolev variational calculus and Lojasiewicz gradient inequality, with no ad-hoc modifications to vacuum Einstein equations or supplementary physical axioms.