Information Entropy--Geometric Unification and Windowed Generation of Cosmological Terms: From Relative Entropy Hessian to Effective Action, Poisson--Euler--Maclaurin Finite-Order Discipline, and Geometric Entropy Decomposition of Friedmann Equations

Within a unified ``operator--measure--function'' framework, we establish an organic assembly connecting multi-order responses of relative entropy, master-scale calibrations of scattering spectra, windowed readout Toeplitz/Berezin compressions, and Nyquist--Poisson--Euler--Maclaurin (NPE) finite-order discipline to closed derivations of geometric effective action and cosmological terms. First, under Eguchi regularized divergence and Amari \alpha-geometry, we prove construction of the Fisher--Rao