Phase-Variation Operators, Essential Spectrum, and Galerkin Bounds for Fixed-Region Fourier Energy

We study how aperture phase changes Fourier energy within a fixed receiving region. For uniform amplitude and a finite centrally symmetric receiver, a bounded self-adjoint operator represents the quadratic variation. Its compact-minus-multiplication structure gives an exact characterization of the essential spectral interval on the radial disk space. The lower spectral endpoint is unattained at every positive receiver radius, while nested Galerkin spaces recover both endpoints. Rational certificates establish isolated upper eigenvalues at radii 11/10 and 2, and operator-norm continuity extends this separation to explicit intervals. A mean-zero L4 identity gives uniform quartic remainder bounds, with finite-dimensional coefficients obtained from an exact polynomial embedding constant. On real L2, the energy has a globally Lipschitz first Fréchet derivative and a bounded second Gâteaux derivative at zero in the dual norm. At every positive disk-receiver radius, concentrating phases preclude a second-order Fréchet Taylor expansion at zero on the radial and full two-dimensional piston-free disk spaces. The resulting distinction between controlled phase classes and the full RMS sphere determines where spectral bounds yield uniform nonlinear approximations.