Fixed Points, a Predictor-Impossibility Theorem, and Applications

We introduce an activation hierarchy consisting of stage machines, stage domains, and stage languages generated by an activation operator. The central result is a Predictor-Impossibility Theorem (PIT), which shows that no effective predictor family can uniformly determine all stage languages of the hierarchy. The proof combines the semantic activation construction with the S-m-n Theorem and Kleene's Recursion Theorem to obtain a self-referential fixed point that yields a contradiction. We then define an aggregate language MIS and establish a slice theorem connecting aggregate inputs to individual stage languages. This provides a bridge from polynomial-time decidability of MIS to the existence of a predictor family. By PIT, the aggregate language is, therefore, not polynomial-time decidable. Under the aggregate growth condition defining valid aggregate objects, MIS is shown to belong to NP. Combining these two results yields MIS in NP-P. The paper is organized so that PIT stands independently as a recursion-theoretic result, while the complexity-theoretic consequences are derived from the aggregate-language framework.