A Theory of Information Structure A Geometric Complement to Shannon's Theory

Shannon's theory measures information but leaves its internal structure invisible. We demonstrate that information possesses an intrinsic geometry - a latent organization that exists prior to observation. By defining the information manifold as a Riemannian space with an explicit metric tensor, we prove that any information-bearing signal separates into orthogonal components: a Discriminant (angular) and a Variance (radial). We establish three fundamental results: (1) the gradient of any angular loss with respect to magnitude is strictly zero via Jacobian nullity; (2) this separation is coordinate-invariant - a topological property of projective geometry; (3) the components are statistically orthogonal as proven by the diagonal structure of the Fisher Information Matrix. This geometric framework, originally developed for privacy-preserving computation, provides the necessary condition for deterministic structure isolation - enabling the geometric distinction between semantic content and uncontrolled variance.