A Topological Obstruction to Shallow Boolean Circuits Collapsibility, Compatibility Complexes, and Depth Lower Bounds

This paper develops an informational framework for understanding the limits of deterministic polynomial-time computation. Building on earlier results showing that local and monotone constructions cannot generate genuine global inconsistency, it introduces the notion of irreducible global dependencies (DGNL), capturing families of instances whose inconsistency cannot be exposed by any polynomially bounded local analysis. A quantitative invariant, inconsistency flow, is defined to measure how efficiently a computation can eliminate globally coherent configurations. It is shown that polynomial-time algorithms admit only polynomial flow, while DGNL families require exponential flow. To formalize this obstruction, the paper introduces Informational Monotonicity, expressing the impossibility of destroying algorithmic information without explicit representation. Under this principle, a general impossibility theorem is obtained, reducing the P versus NP problem to a precise structural question about the nature of deterministic computation.