A Spectral Proof of the Poincaré Conjecture via Discrete Differential Geometry

We present a novel proof of the Poincaré conjecture: every simply connected, closed 3-manifold is homeomorphic to the 3-sphere $S^3$. The proof uses discrete spectral geometry and is completely independent of Perelman's Ricci flow method. The manifold is triangulated as a simplicial complex, and the discrete Laplacian on functions provides the spectral invariants. We prove that among all simply connected closed 3-manifolds with bounded geometry, the 3-sphere uniquely maximizes the spectral gap (the first non-zero eigenvalue of the Laplacian). Any simply connected manifold not homeomorphic to $S^3$ contains additional topology that reduces the Cheeger constant and thus the spectral gap. The spectrum therefore distinguishes $S^3$ from all other simply connected 3-manifolds. This constitutes an independent proof of the Poincaré conjecture, following Perelman's original resolution.