WSIG-QM: Windowed Scattering \& Information Geometry\\ for Quantum Mechanics\\ \bigskip A Unified Framework of Quantum Concept Definitions\\ and Criterion System (with Complete Proofs)

Using the de Branges--Kreĭn (DBK) canonical system and weighted Mellin space as carriers, we explicitly incorporate the finite bandwidth/time window of real instruments into spectral measures, forming a windowed readout framework; we characterize ``commit (collapse/commit)'' through KL/Bregman information geometry; use scattering phase--spectral density--Wigner--Smith delay as energy scale; close non-asymptotic errors via Nyquist--Poisson--Euler--Maclaurin (three-term decomposition); ensure realizability and stability through variational optimization of frame/sampling density and window/kernel. The core unified formula is $ \,\varphi'(E)=-\pi\,\rho_{\mathrm{rel}(E)=1{2}trQ(E)\,}\quad(a.e.) $ unifying (single/multi-channel) scattering phase derivative, relative local density of states (LDOS) and Wigner--Smith delay; under information geometry we obtain Born probability = minimal-KL projection (I-projection), and ``pointer basis'' is spectral minimum of windowed readout operator. Above criteria consistent with Herglotz--Weyl, Birman--Kreĭn, Wigner--Smith, Ky Fan, Poisson/EM and other standard results, directly interchangeable and implementable.