Phase Derivative, Spectral Density\\and Windowed Readout:\\Unified Measurement Framework

Establish unified framework connecting scattering phase derivative, spectral density, and windowed readout. Core formula holding a.e.: $\ \frac{\varphi'(E){\pi}=\rho_{rel}(E)=1{2\pi}\tr Q(E)\ } where \varphi scattering phase, \rho_{rel} relative spectral density from Birman--Kreĭn \det S=e^{-2\pi i\xi} with \rho_{rel}=-\xi', Q=-iS^\dagger\partial_E S Wigner--Smith delay matrix. Windowed readout R_w=\int w(E)[h\ast\rho_{rel}](E)dE$ with NPE three-term error decomposition. Applications: quantum metrology, scattering theory, condensed matter.