The Operational Gradient: A Framework for P ≠ NP

We present a comprehensive argument that P≠NP follows from physically and mathematically unavoidable properties of computation. The argument rests on the Intrinsic Operational Gradient Axiom, which we justify through converging principles from information theory, thermodynamics, category theory, and empirical observation. While the proof is formally conditional on accepting this axiom, we demonstrate that the axiom emerges naturally from fundamental asymmetries in counting, irreversible physical constraints on computation, and universal patterns across all known computational systems. The resulting framework suggests that the separation between P and NP reflects not algorithmic limitations but fundamental structural properties of operational reality itself.