Exploration and Observability Limits in Locally Reasoning Algorithms for NP

We investigate structural limitations of algorithms for NP problems whose decision process relies on local reasoning. We introduce a notion of local observability that captures algorithms which, despite reading the full input instance, extract information only through bounded-scope inference steps. Using an information-theoretic framework, we show that for NP problems whose witness space has linear entropy, such locally reasoning algorithms are forced to perform exponential exploration in the worst case. Our results provide a principled explanation for the persistent exponential behavior of a broad class of backtracking, branching, and local search algorithms, and clarify why algorithmic breakthroughs for NP problems must rely on mechanisms that exploit global information, such as learning or non-local inference. Rather than addressing the P versus NP question directly, this work identifies an informational locality barrier that delineates the limits of local reasoning and helps explain both algorithmic failures and successes observed in practice.