Local Topological Obstructions, Homological Persistence, and the Structural Boundary of NC1

This paper extends a topological approach to circuit complexity from constant depth to logarithmic depth by analyzing how second homology evolves across computational layers. Building on earlier results showing that shallow circuits enforce global topological flatness, the work introduces filtrations of compatibility complexes and studies the persistence of homological features. It proves that in NC1 circuits, any nontrivial two-dimensional topology can persist only for bounded depth. The main contribution is the Local H₂-Triviality Axiom, which reduces global homological growth to a local geometric condition on vertex links. This framework clarifies why topological obstructions characterize constant-depth and logarithmic-depth computation but cannot extend further. While it does not resolve P versus NP, it identifies NC1 as the maximal class controlled by local topological methods.