Information-Geometric Grammar Principle: A Statistical-Geometric Framework for Adaptive Systems

We introduce the **Information-Geometric Grammar Principle (IGGP)**, a mathematical framework that unifies concepts from formal language theory, information geometry, Bayesian inference, and complex adaptive systems. In conventional grammar-based descriptions of adaptive systems, individual agents constitute the alphabet, interaction rules define the grammar, and collective behaviors emerge as the language generated by repeated applications of these rules. We generalize this paradigm by replacing symbolic representations with probabilistic ones. Each agent is identified with a local statistical manifold endowed with a Fisher information metric, while communication acts as a geometric production operator that transforms probability distributions through Bayesian updating and information exchange. Repeated communication generates an evolving collective statistical manifold whose geometry encodes the emergent organization of the system. Within this formulation, communication assumes the role of a probabilistic grammar: the elementary statistical states of individual agents constitute the alphabet, communication operators define the production rules, and the family of admissible collective probability distributions forms the language generated by the adaptive dynamics. The resulting Fisher geometry provides a quantitative description of syntactic organization, with off-diagonal metric components measuring communication-induced statistical dependencies among agents. Emergent collective structures are therefore interpreted as geometric properties of a statistical language rather than as purely symbolic constructions. The Information-Geometric Grammar Principle naturally extends traditional grammar-based models by introducing intrinsic geometric observables, including curvature, geometric entropy, and communication-induced coupling measures, thereby linking symbolic organization with differential geometry. The framework is sufficiently general to encompass biological populations, social and communication networks, distributed robotic systems, scientific communities, and human–artificial intelligence ecosystems. More broadly, it suggests that the emergence of collective organization can be understood as the generation of an increasingly structured statistical language whose syntax is encoded by an evolving Fisher geometry. This perspective establishes a unified mathematical language for studying adaptive organization across disciplines and opens new directions for the geometric analysis of learning, communication, and emergence in complex systems.