Small regions, persistent homology: an illustrated companion to the construction

This companion follows the construction in Homological obstructions to local contractibility in noncollapsed Ricci limits. We follow a class carried by one inserted core, show how quotient maps preserve infinitely many such classes, and explain why representative spheres can avoid every set below the relevant Hausdorff-dimension threshold. The geometric construction is then developed through circle regularization, Ricci compensation, transition annuli, and exact retention of the inserted domains. A separate metric comparison explains how an isolated point can have persistent neighborhood homology and a unique product-cone tangent. Figures accompany the maps, curvature estimates, and choices of scale. Illustrated companion to Homological obstructions to local contractibility in noncollapsed Ricci limits. The deposit contains the companion PDF and an archive of its editable LaTeX source, figures, and reproducible figure generators.