Geometric optimal transport for generative modeling without hallucinations

The classification of 3-manifolds relies on Ian Agol's proof of the Virtual Haken Conjecture, which lifts local isometries to the Salvetti complex into globally injective embeddings within finitesheeted covers. Despite its theoretical importance, this framework lacks a constructive geometric realization. This work presents an algorithmic pipeline for virtual retractions on 2D discrete quadrilateral meshes. We compute the canonical completion of immersed subcomplexes by enforcing Gromov's link condition relative to the Salvetti complex. A Breadth-First Search (BFS) is then employed to extract the connected component that isolates the lifted embedding. This approach transforms an abstract existence theorem into a computational tool for resolving topological self-intersections in geometric data.