P-neq-NP as Structural Necessity: A Cognitional Mechanics Perspective on Computational Irreducibility

This paper presents a structural reinterpretation of the classical P–NP formulation through the axiomatic framework of Cognitional Mechanics (CM). Rather than pursuing a proof within the traditional complexity-theoretic setting, the paper treats P-neq-NP as a necessary consequence of non-commutative operational structure. In CM, computational processes are not modeled primarily by time or space bounds, but by mappings constrained by non-commutativity, bounded constructive capacity, and the existence of unreachable regions in configuration space. The core argument rests on a strict separation between construction and verification as distinct operations. Solution construction and solution verification are formalized as mappings over a Turing configuration space, where their non-commutativity is structural rather than algorithmic. Verification necessarily eliminates information required for construction, making the inversion of verification into efficient construction structurally inadmissible. This asymmetry is not contingent on resource bounds, but follows directly from the axioms of CM. Within this framework, complexity classes are reformulated as reachable sets under bounded sequences of operations. P corresponds to the set reachable by construction alone, while NP corresponds to the set reachable by verification following construction. From this perspective, P-neq-NP is not an unresolved conjecture but a structural consequence of non-commutative reachability. The persistent resistance of the classical P–NP formulation to resolution is thus explained as a consequence of implicit commutative assumptions embedded in standard complexity theory. By removing those assumptions, the problem is not solved in the traditional sense, but dissolved at the level of its foundational structure.