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.