P ≠ NP from the Intrinsic Operational Gradient Theorem

We demonstrate that P ≠ NP follows from applying the Intrinsic Operational Gradient Theorem (IOGT) to computational complexity. IOGT proves that in any infinite composable system with non-invertibility, construction and reconstruction are generically asymmetric, inducing intrinsic gradients of difficulty. We show that computation instantiates such a system, and therefore inherits these structural constraints. The result is conditional on accepting that computation falls within IOGT’s scope, but this acceptance follows naturally from recognizing computation as a physically realizable operational process. For abstract computation divorced from physical reality, the result remains formally conditional; for all physically realizable computation, the constraints are unavoidable consequences of thermodynamics, information theory, and categorical structure. Yet even in pure mathematics asymmetry, irreversibility, and directionality exist. The only remaining possibility that P = NP is under the condition of exponential parallelism but which must include the meta-level structure organizing the parallelism itself. The framework suggests that computational hardness reflects not algorithmic limitations but fundamental structural properties of operational reality itself.