P ≠ NP from Operational Gradients: A Process-Primacy Approach to Computational Complexity

We present a novel approach to the P vs NP problem grounded in operational geometry: a framework where processes (operations) are ontologically prior to objects. We argue that computational complexity classes are distinguished by intrinsic operational gradients—directional asymmetries in the operational substrate itself that cannot be eliminated by algorithmic cleverness. We introduce the Axiom of Intrinsic Operational Gradient, which posits that operational space has canonical directionality encoded in the threading cost functional Iτ . Under this axiom, together with process-primacy and the free traced symmetric monoidal category (TSMC) structure, we prove that P ≠ NP. Our approach reframes P vs NP as an ontological question: Does mathematical reality have intrinsic operational gradients? We provide evidence from category theory (freeness of the operational category), information theory (Shannon bounds on distinguishability), and physical analogies (thermodynamic and causal gradients). The paper thus establishes a conditional equivalence: the existence of intrinsic operational gradients is equivalent to P ≠ NP. This framework clarifies why the intuition “search is harder than verification” reflects a deep structural principle about reality itself, and suggests that P ≠ NP may be a law of nature analogous to the second law of thermodynamics.