Where Exactly Does NP Emerge? A Structural Re-Reading of Cook's Original Formulation

This paper re-examines Cook's original formulation of the P versus NP problem from a structural and interpretative perspective. Rather than addressing the classical question of whether P equals NP, the paper investigates a logically prior question: at what exact point does NP emerge as a distinct computational class within Cook's original construction? The analysis reconstructs Cook's formulation as the sequence P → P_rel → NP_Cook, where NP_Cook is defined through existential extension of deterministic polynomial relations. The paper argues that Cook's construction consists entirely of finite strings, finite certificates, deterministic computation, and polynomially decidable relations. The transition R(x,y) → ∃y R(x,y) introduces an existential quantifier over a finite parameter but does not explicitly introduce a new computational mechanism. The work further develops the notions of parameter symmetry, representation dependence, and interpretative complexity, proposing that the emergence of NP may depend not only on computation itself but also on choices of representation, size measurement, and semantic role assignment. This paper does not attempt to prove either P = NP or P ≠ NP. Its objective is to examine the logical and ontological assumptions underlying Cook's original definition of NP.Keywords: P versus NP, NP_Cook, Cook's formulation, polynomial relations, computational complexity, finite structures, certificates, parameter symmetry, representation dependence, foundations of computation.