Incremental Stability and the Structural Irresolubility of the P versus NP Problem

The P versus NP problem is formulated within abstract models of computation that measurecomplexity as the number of discrete steps executed by an idealized machine. While this formal-ism deliberately avoids physical constraints, it does not impose any axiom enforcing structuralcoherence in the incremental growth of computational effort across input sizes.In this work we introduce the Axiom of Incremental Stability, which formalizes the require-ment that the additional computational effort induced by increasing problem size remains poly-nomially bounded. We show that, in the absence of such an axiom, the standard computationalmodel admits structurally pathological time functions that invalidate accumulation-based argu-ments relying on growth, feasibility, or asymptotic comparison.This yields a negative structural result: without incremental stability, a broad class of prooftechniques commonly employed in complexity-theoretic separations— including hierarchical,diagonalization, and simulation arguments—cannot be formally justified within the standardabstract model. In this precise sense, the P versus NP problem appears structurally irresolublerelative to the currently assumed axiomatic framework: the properties required to formulateaccumulation-based proofs are not derivable from its axioms.This work does not present a proof of P̸ = NP, nor does it claim logical independence.Its contribution is strictly foundational: it identifies and formalizes incremental stability as aminimal structural assumption without which incremental reasoning about complexity lacks aformal basis. The analysis provides a diagnostic perspective on the limitations of the abstractcomputational model itself.