KL-Geometric Structure of Observer Entropy: A Minimal Information-Theoretic Framework

This paper establishes a finite information-geometric formulation of observer entropy defined via Kullback–Leibler divergence under coarse-graining.The main result (Bridge Theorem) proves the local expansion S_obs(p_theta, eps) = (1/2) eps^2 v^T I(theta*) v + O(eps^3), showing that observer entropy is governed at leading order by the Fisher information matrix. All results are derived in a fully finite setting with explicit regularity assumptions. An exact softmax example and a resolution–information trade-off are provided.