P vs NP: Finite-Grain Blindness, Completion, and Canonical Realization

Why do so many promising approaches to P vs NP fail before they even begin to touch the real difficulty? This paper proposes a new structural answer. Instead of treating P vs NP only as a question about algorithms or proof techniques, it studies the entire space of decision problems through a topology induced by finite-grain computational observability. In this setting, P and NP appear as stable regions, while the true difficulty is localized on their boundary. The first main result is a no-go theorem: every monotone separator determined at a finite observational level is necessarily blind exactly where the separation problem lives. This isolates a common structural core behind broad families of limited approaches. The paper then takes the next natural step: if no finite level is enough, what happens when one passes to the full inverse-limit completion of the observational tower? The answer is striking. The obstruction does not automatically disappear. At the completion level, the problem becomes one of canonicity: even when all compatible finite observations are assembled, the resulting object may still fail to admit a canonical realization. The contribution is therefore twofold: it identifies finite-grain blindness as a topological obstruction; it shows that the natural limit object of the filtration may preserve the difficulty as failure of canonical realization. This does not resolve P vs NP. It does something different and important: it sharpens the structural description of what any successful approach would have to overcome.