The Poincaré Conjecture was a central question in topology, concerning the characterization of three-dimensional spheres. It proposed that every simply connected, closed 3-manifold is homeomorphic to a 3-sphere.
Formulated by Henri Poincaré in 1904, the conjecture remained open for a century and drove major developments in geometric topology. Its resolution would confirm a fundamental property of three-dimensional spaces.
In 2003, Grigori Perelman provided a proof based on Richard Hamilton’s Ricci flow program, which was later verified and accepted by the mathematical community. Perelman declined the Millennium Prize.
This proof does not flow — it folds. Where Ricci sought to reshape, this approach traces closure. Each loop, each contraction, each surface: aligned not by curvature, but by certainty of topological behavior.
There is no metric, no differential structure — only the shape of space under its own constraints. Triangulation, decomposition, and irreducibility form the scaffolding. From them, the 3-sphere appears not as a result, but as an inevitability.
Poincaré’s statement does not require geometry to stand. It requires only the truth of loops — and their final return.
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