Thermodynamic Geometry and Hypersurfaces

From the perspective of thermodynamic geometry, we study the hypersurface properties by considering the fluctuation theory and material formations for a class of finite parameter filters and arbitrary irregular shaped circuits. Given a constant mismatch factor, the Gaussian fluctuations over an equilibrium statistical basis accomplish a well-defined, non-degenerate, flat and regular intrinsic surface. For a variable mismatch factor ensemble, the long rang global correlation function is given by the ratio of two ordinary summations. Given covariant intrinsic description of a definite physical configuration, both the local and global correlations reduce to finite polynomials in the system parameters. As per the theory of statistical correlations, we determine the ensemble stability conditions and analyze the limiting thermodynamic geometric fluctuations as a set of invertible evolution maps by defining the underlying tangent manifold and connection functions. By invoking the joint role of the Riemannian geometry and coding theory, we describe the nature of correlations and phase-transitions for arbitrary finite parameter configurations.