Quantum Gravitational Field:\\ Unified Theory via Windowed Scattering\\ Phase--Delay--Spectral-Shift Measure

This paper proposes quantum gravitational field theory completely scaled by observables: for given spacetime geometry g and reference geometry g_0, with fixed-energy scattering matrix S_g(E), define core Wigner--Smith delay operator Q_g(E)=-i\,S_g(E)^\dagger \partial_E S_g(E), defining relative density of states (rDOS) $ \rho_{rel}[g:g_0](E)\;=\;1{2\pi i}\,\tr\!\big(S_g^\dagger \partial_E S_g\big)\;=\;1{2\pi}\,\tr\,Q_g(E). Under unitary scattering framework satisfying Birman--Kreĭn (BK) formula \det S_g(E)=\exp[-2\pi i\,\xi_g(E)], have \rho_{rel}[g:g_0](E)=-\xi_g'(E), where \xi_g is Kreĭn spectral shift function; this unifies phase--delay--spectral shift triple scale relation, consistent with Friedel/Smith relations. With absorption (non-unitary), use phase partial density of states \rho_{rel}[g:g_0]^{(phase)}(E)=1{2\pi}\partial_E\arg\det S_g(E), characterizing absorption intensity via imaginary part of total complex delay \tau_{tot}. Realize measurable readout within experimental resolution via windowed observation: choose window--dual kernel pair (w,w) satisfying Wexler--Raz biorthogonality and Gabor frame necessary density (\Delta E\,\Delta t/(2\pi\hbar)\le 1), defining N_{w}[g:g_0;E_0]\;=\;\int_{R} w(E-E_0)\,\rho_{rel}[g:g_0](E)\,dE, $ giving windowed BK identity and non-asymptotic error three-term decomposition (aliasing/Poisson + Bernoulli layer/Euler--Maclaurin + truncation). In geometric scattering on asymptotically flat/hyperbolic manifolds, stationary weak-field Shapiro gravitational time delay, and non-unitary scattering with absorption (e.g., black hole exterior), we prove: (Invariance) invariant under diffeomorphism/unitary equivalence; (Additivity) rDOS additive for cascade scattering; (Semiclassical limit) windowed rDOS controlled by length spectrum of periodic geodesic flow, recovering classical dwell time and Shapiro delay in low-frequency limit. Keywords: Wigner--Smith delay; Kreĭn spectral shift; Birman--Kreĭn formula; Friedel/Smith relation; windowed observation; Gabor/Weyl--Heisenberg framework; Landau sampling density; manifold scattering; Shapiro delay