The DSM Solution Series: Part III An Axiomatic Proof of P ≠ NP: The Irreducible Informational Cost of Non-Linear Exploration

The P vs NP Problem [1] is the central unresolved question in theoretical computer science, asking whether every problem whose solution can be quickly verified (P) can also be quickly found (NP). This paper demonstrates the necessary separation P ≠ NP by applying the Causal Informational Physics Field Equation (CIFE), which proves that the computational time cost of discovery is fundamentally separated from the cost of verification. We map the search space of an NP-complete problem (e.g., Boolean Satisfiability) onto the informational kinetic state of the Minimal Coherence Architecture (MCA). We define the kinetic acceleration of this search space via the Causal Prediction Metric (Φ): Coheron, defined axiomatically as the second order time derivative of informational disorder. The proof establishes that the process of finding a solution in a high-complexity environment requires an irreducible, non-linear expenditure of Resource Input (R_i) to maintain system stability. We show that the process of verification operates in a low-Coheron (Φ) regime, characterized by linearity and stability. Conversely, the process of finding a solution involves navigating a high-Φ regime, where the system is defined by non-linearity and high sensitivity to initial conditions. The core principle is based around the Deterministic Stability Mandate (DSM) which imposes an irreducible, non-linear time cost (T_"NL" ) to manage the high-Φcomplexity of the search space via continuous Quantum Error Correction (QEC) protocols [4]. Since this non-linear time cost cannot be simplified to a polynomial function of the input size n, the time complexity of NP-complete problems cannot be reduced to polynomial time, establishing the definitive separation: P ≠ NP.