Evaluating Constructive Utility a Framework for Extractive Accessibility in Mathematical Theorems

This paper presents a diagnostic framework for evaluating the operational viability of existence theorems. It defines the condition of extractive inaccessibility, where a result formally proves existence but resists all known methods of algorithmic reconstruction or structural realization. The Gowers inverse theorems are examined as a central case study. For higher uniformity norms, the associated bounds and structural components exceed practical computation and, in some instances, measurable definition. The framework is designed to aid computational mathematicians, algorithm designers, and applied theorists in identifying results whose extractive content is either viable, limited, or inaccessible. Connections to proof mining, reverse mathematics, and constructive analysis are included to align the framework with existing foundational tools.