The Solver's Paradox in Formal Problem Spaces

Abstract

This paper investigates how global decision problems over arithmetically represented domains acquire reflective structure through class-quantification. Arithmetization forces diagonal fixed points whose verification requires reflection beyond finitary means, producing Feferman-style obstructions independent of computational technique. We use this mechanism to analyze uniform complexity statements, including $\mathsf{P}$ vs. $\mathsf{NP}$, showing that their difficulty stems from structural impredicativity rather than methodological limitations. The focus is not on deriving separations but on clarifying the logical status of such arithmetized assertions.

Structural analysis of global decision problems by analyzing how impredicativity is transported through arithmetized problem spaces, integrating diagonalization, reflection, and uniform complexity. 18 Pages