This paper reformulates a topological approach to computational complexity into a dynamic, algorithm-relative framework based on computationally coherent filtrations. Building on earlier results showing that shallow and logarithmic-depth computation enforces low-dimensional topological simplicity and that NP-complete problems require high assembly complexity, it defines structural conditions under which algorithmic execution incrementally exposes compatibility relations without creating higher-dimensional homology. A conditional theorem reduces the P versus NP problem to the existence of such filtrations for all polynomial-time algorithms. The framework reconciles topological obstruction methods with classical complexity barriers and isolates a precise falsification target for future research.
Michael Arias (Wed,) studied this question.