Contextual Structural Logic and Interpretation Moduli in Algebraic Geometry: A Unified Philosophical and Mathematical Framework

Algebraic geometry relies on a highly structured categorical language whose semantic precision masks a deep interpretative flexibility. This paper develops a unified framework that integrates a philosophically grounded account—Contextual Structural Logic (CSL)—with a rigorous mathematical formalization—Interpretation Moduli Objects (IMO). CSL analyzes how algebraic-geometric language remains rigorous despite admitting multiple legitimate interpretations of morphisms, objects, and properties, while IMO provides a moduli-theoretic structure to model interpretation variability. The framework combines contextual semantics, equivalence-preserving translation mechanisms, and structural constraints imposed by categorical behavior. Key contributions include a philosophical–mathematical account of interpretative plurality, the definition of interpretation functors and moduli spaces, and theorems demonstrating consistency, gluing, and stability under morphisms. This results in a novel conceptual system that serves as a prototype for a unified philosophy–mathematics theory, illuminating the meta-structure of algebraic geometry without replacing its foundations.