APG Structural Foundation Models

APG Structural Foundation Models (ASFM) introduce a mathematically interpretable framework for incorporating structural representations into foundation models. Instead of relying solely on high-dimensional latent embeddings, ASFM constructs deterministic structural signatures using the Arithmetic Power Geometry (APG) framework, including entropy, concentration, deformation, and trajectory descriptors. The monograph develops a unified mathematical theory covering structural representations, structural attention, structural dynamics, structural memory, structural audit, and structural generalization. Theoretical results establish conditional guarantees for stability, invariance, compression, generalization, and learning under explicit regularity assumptions while avoiding claims of universal superiority over conventional neural representations. The work also presents an executable architectural framework, deterministic numerical verification, computational complexity analysis, implementation guidelines, limitations, and future research directions. The manuscript is intended as a theoretical research monograph that provides a rigorous foundation for future empirical investigation of structurally informed foundation models.