On Structural Residuals in the P versus NP Problem

The P versus NP problem occupies a central position in theoretical computer science and mathematical logic, traditionally formulated as a binary question concerning the equivalence or separation of deterministic and nondeterministic polynomial-time computation. While exten- sive efforts have been devoted to resolving this question within standard complexity-theoretic frameworks, comparatively little attention has been paid to the structural features that any such resolution must necessarily exhibit. In this paper, we argue that any formal resolution of the P versus NP problem, regardless of whether it asserts P = NP or P ̸= NP, inevitably entails an irreducible residual structure. This residual does not arise from technical limitations of current proof methods, but from the asymptotic and formal nature of computational complexity itself. We present a structural analysis showing that the binary formulation of the problem conceals an inherent incompleteness, manifested as a non-eliminable residual term accompanying any definitive answer. The purpose of this work is not to resolve the P versus NP problem in the classical sense, but to clarify why its resolution cannot be expected to be semantically closed. By framing computational complexity within a residual-based structural perspective, we aim to provide a mathematical interpretation of the persistent instability observed in attempts to fully settle the P versus NP question.