Curvature Templates in Clinical Information Geometry: Fisher Manifolds, Relational Embeddings, and Worked Examples

Clinical prediction models commonly represent patients as points in high-dimensional feature spaces, but they rarely examine the information geometry generated by the statistical organization of clinical states. This study develops a theoretical and computational framework in which clinical states are represented as coarse-grained relational ensembles and analyzed through Fisher information geometry. We use two established statistical manifolds as controlled curvature templates: the Gaussian location-scale family, which has negative Fisher scalar curvature and represents fluctuation-dominated organization, and the categorical/simplex family, which has positive Fisher scalar curvature and represents normalized finite-capacity organization. The main contribution is the clinical-relational operationalization of these templates, together with worked examples showing how candidate clinical observables may be mapped to fluctuation-dominated, compositional, and graph-based settings. We further distinguish exact Fisher-geometric results from covariance-based diagnostic proxies used in finite simulations, and we introduce a reproducible protocol including metric-conditioning checks, sensitivity analysis, and a Riemannian-Laplace approximation for higher-dimensional tractability. The framework is not presented as a validated diagnostic or prognostic model. Rather, it provides a hypothesis-generating geometric layer for studying instability, constraint, and state-space organization in complex clinical systems.