A Geometric Solution and Interpretation of the Poincaré Conjecture: The Perspective of Visual Relief and Sweep-out Protocol

Abstract

This paper presents a novel conceptualization and structural analysis of the 3-dimensional Poincaré Conjecture. By introducing the "Sweep-out Protocol" via a dynamically expanding 2-dimensional sphere trajectory and integrating an inherent "Infinite Brake Mechanism," we demystify the 100-year-old conundrum. This approach eliminates the systemic paradoxes associated with artificial spatial slicing (Dehn surgeries) and demonstrates that under complete contraction, topological twists behave merely as non-obstructive "surface relief." We verify through geometric flow that any closed, simply connected 3-manifold is inherently homeomorphic to a 3-sphere.