Part II: The Geometric Core🔗

This part constructs the reparametrization u demanded by the reduction of Section 2: from a handle family (Section 3), which is in turn assembled from a family of vertex labellings on a fine subdivision (Sections 4–5). The quantile labelling of Section 5 is the general-position heart of the whole proof.

1. 3. The Reparametrization from a Handle Family
2. 4. From Vertex Labellings to Handles
3. 5.a. The Quantile Vertex
4. 5.b. The Quantile Labelling Family