Windowed Path Integrals:\\ Spectral ``Window--Kernel'' Formulation\\ and Rigorous Equivalence to Propagators

Under WSIG-QM framework composed of de Branges--Kreĭn (DBK) canonical system and Weyl--Heisenberg (including logarithmic/Mellin) representation, this paper takes spectral theorem + analytic Fourier duality as main thread, giving rigorous mathematical characterization of path integral = propagator kernel, proving windowed path integral theorem: any realizable path integral-type observation equivalent to ``window--kernel--density'' convolution in energy domain; time domain precisely propagator time trace (or state-weighted kernel) Fourier dual under same window/kernel. For numerical implementation, discretization error non-asymptotically closes as ``alias (Poisson) + Bernoulli layer (Euler--Maclaurin) + truncation'' three-term decomposition; under bandlimited + Nyquist conditions alias term strictly zero. For phase scale, on absolutely continuous spectrum almost everywhere holds $ \varphi'(E)=1{2}trQ(E),\qquad \rho_{rel}(E)=s_{\mathrm{BK}}{2\pi}trQ(E),\qquad \varphi(E)=s_{BK}\,\pi\,\xi(E)\pi, where Q(E)=-i\,S^\dagger(E)\, dS{dE}(E) is Wigner--Smith delay matrix, \rho_{rel}=\xi' spectral shift density, s_{BK} BK notation version parameter (this paper adopts s_{BK}=+1$); this given by Birman--Kreĭn formula and relative scattering delay unification, closing path weight action phase with measurable energy scale unified. On information geometry side, Born probability = minimal-KL (I-projection) gives log-sum-exp soft potential convex dual semantics; single-window and multi-window synergy of window/kernel expressible as strongly convex/sparse optimization interfacing with frame--dual window theory. All above anchor standard criteria: spectral theorem and Stone theorem, Birman--Kreĭn formula, Wigner--Smith delay, Poisson summation and Euler--Maclaurin formula, Nyquist--Shannon sampling, Wexler--Raz biorthogonality and ``painless'' expansion etc.