P vs NP: A Motion-Based Solution Framework for Complexity Separation

This paper presents a structural resolution of the P vs NP problem using a motion-based recursion framework. Computational problems are modeled as recursive systems characterized by directional deviation (∆m), cumulative motion persistence (Σ∆m), and compression thresholds (Ct). In this model, polynomial-time solvable problems (P) are those whose resolution pathways maintain bounded compression across recursive layers. NP problems are modeled as configurations where the required motion either exceeds Ct or triggers instability through unbounded strain (∆∆m). The central result proves that no general-purpose compression path exists for all NP instances within polynomial motion bounds. This asymmetry is formalized through deterministic collapse logic. For representative NP-complete problems, any polynomial compression attempt results in structural failure, flagged by a collapse variable Ke = 1. Under this framework, identity persistence Ψ(t) fails for such instances, meaning no bounded recursive pathway exists. This implies P̸ = N P . All variables and constructs are formally defined. The collapse behavior is demonstrated through discrete simulation, independent of machine implementation or time-based mechanics. The result is falsifiable, structurally deterministic, and resolves the P vs NP problem as posed by the Clay Mathematics Institute.