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

We formulate an Information-Geometric Grammar Principle (IGGP) for adaptive systems in which grammatical admissibility, probabilistic realization, and statistical geometry are treated as distinct but connected levels of description. A grammar is represented by a state space together with a set of admissible productions and a probabilistic rule assigning weights to those productions. The resulting family of probability distributions defines a statistical manifold endowed with the Fisher information metric. We distinguish probabilistic adaptation, which changes probabilities while preserving the admissible grammar, from structural adaptation, which changes the set of admissible productions itself. For a fixed statistical manifold, an information-geometric action leads to geodesic equations and provides a coordinate-invariant criterion for locally economical statistical evolution. Interacting components generate a collective statistical manifold, where off-diagonal Fisher components quantify statistical coupling. We further identify the additional convexity and integrability conditions required for a Legendre representation and emphasize that a thermodynamic interpretation is conditional rather than automatic. The resulting framework provides a hierarchy from grammar to statistics, Fisher geometry, adaptive dynamics, collective organization, and, when additional conditions hold, information-geometric thermodynamics.