Topological Assembly Complexity and the Structural Boundary Between P and NP

This paper extends a topological approach to computational complexity from shallow circuits and logarithmic-depth computation to the boundary between polynomial time and NP. Building on earlier results showing that constant-depth and NC1 circuits enforce local and global topological simplicity, it introduces a new invariant, Hierarchical Topological Assembly Width, which measures the difficulty of assembling compatibility complexes from locally trivial components. The paper proves that explicit NP-complete families generate exponentially many globally independent two-dimensional cycles, forcing exponential assembly width, while many polynomial-time problems admit polynomial assembly. A sufficient certification condition is isolated under which polynomial-time decidability implies topological flatness. Although the framework does not yield an unconditional proof of P ≠ NP, it provides a structural explanation of computational hardness in terms of irreducible global topology.