Information Dynamical System: Entropy Ridge Geometry, Parameter Identifiability, and PLSA Coupling

In this paper we present an upgraded formulation of the Information Dynamical System (IDS)framework. We focus on a two-dimensional local model for the entropy geometry in terms of a temporalparameter $K$ and a spectral parameter $\sigma$. Around the structurally preferred attractor$(K_0,\sigma_0)=(1,\tfrac12)$ we approximate the entropy by a quadratic form with Hessianparameters $(a,b,c)$. This setting allows us to derive a closed-form expression for theoptimal compensation mechanism $\Delta\sigma^\*=-\tfrac{b}{c}\Delta K$ and the correspondingcompensation efficiency $\eta = b^2/(ac)$.We further show that the entropy ridge $\sigma^\*(K)$ emerges as a natural minimal--entropycurve in the $(K,\sigma)$ plane, and that the Hessian parameters $(a,b,c)$ can be recoveredfrom purely observational perturbation data via a simple linear regression.Finally, we couple the IDS geometry with a structural efficiency functional$\phi(K,\sigma)$ inspired by PLSA, and demonstrate how weak compensationregimes lead to systematic decay of structural efficiency under random perturbations.