The Field Equations of Semantic Coherence: A Geometric Theory of Meaning, Curvature, and Reasoning in Transformer Architectures

This document presents a complete geometric field theory for semantic coherence in transformer-based language models. The framework establishes that meaning propagation in neural architectures obeys field equations analogous to those governing physical systems, where curvature constraints determine the boundaries of coherent reasoning.

The theory unifies several phenomena previously treated as unrelated: context window limitations arise from holonomy accumulation on the semantic manifold; attention head behavior reflects parallel transport of meaning vectors; and reasoning failures correspond to geodesic deviation under excessive curvature.

This reference contains 89 mathematical results (theorems, lemmas, corollaries, and propositions) with explicit dependency structure, organized into foundational definitions, energy functionals, dynamics, and system-level guarantees. Key constructs include the Davis field equations for semantic evolution, curvature-based validity bounds, holonomy budget constraints, and harmonization theorems connecting non-deterministic processes to deterministic observables.

This release establishes priority for the theoretical framework. Proofs, worked examples, and application-specific implementations are reserved for future publication.

Keywords: differential geometry, transformers, semantic coherence, field theory, curvature, holonomy, attention mechanisms, context windows, geometric deep learning