Establish energy and spectral conservation laws under finite-window measurement. Core result: for windowed readout with window w and spectral density \rho, windowed energy E_w=\int Ew(E)\rho(E)dE satisfies: (i) Conservation: under unitary evolution \partial_tE_w=0 when w time-independent; (ii) Covariance: gauge-invariant under phase transformations; (iii) NPE closure: discrete approximation with alias+EM+tail error decomposition.
Establish energy and spectral conservation laws under finite-window measurement. Core result: for windowed readout with window w and spectral density \rho, windowed energy E_w=\int Ew(E)\rho(E)dE satisfies: (i) Conservation: under unitary evolution \partial_tE_w=0 when w time-independent; (ii) Covariance: gauge-invariant under phase transformations; (iii) NPE closure: discrete approximation with alias+EM+tail error decomposition.
For multi-channel scattering with N(E) channels at energy E, establish unified framework connecting scattering phase derivative, relative spectral density, and Wigner--Smith delay. Core formula holding a.e. on absolutely continuous spectrum: $ \ \frac{1{N(E)}d{dE}\Arg\det S(E)=1{2\pi}\langle\tr Q(E)\rangle_N=\rho_{rel}(E)\ } where Q(E)=-iS^\dagger(E)\partial_E S(E) is Wigner--Smith delay matrix, \langle\cdot\rangle_N denotes per-channel average, \rho_{rel} relative spectral density from Birman--Kreĭn formula \det S=e^{-2\pi i\xi}$. Establish: (i) Channel covariance: formula invariant under unitary channel basis transformations; (ii) Threshold regularity: handle channel number jumps via proper boundary conditions; (iii) Windowed readout: finite-bandwidth measurement protocol with NPE error closure; (iv) Information geometry: Born probability as I-projection, pointer basis as Ky Fan minimum.
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.
Within unified framework of mirror kernel--de Branges--Kreĭn canonical system--information geometry, this paper proposes and rigorously proves three main theorems: 1. Windowed Readout Theorem: Any realizable quantum measurement readout equivalent to weighting of (relative or absolute) local density of states (LDOS) by ``energy window w_R and frontend kernel h''; when adopting realistic discrete sampling--finite truncation procedure, error can be non-asymptotically closed by Nyquist (alias)--Poisson (sampling)--Euler--Maclaurin (EM, sum--integral difference) three terms, with alias term strictly zero under bandlimited + Nyquist conditions. Conclusion based on Herglotz property and boundary value dictionary (\Im m(E+i0)=\pi\rho(E)) of Weyl--Titchmarsh m-function and its equivalent formulation with canonical systems. 2. Born Probability = Minimal KL (Information Projection): When readout dictionary aligns with log-partition potential \Lambda(\rho)=\log\!\sum_j w_j e^{\langle\beta_j,\rho\rangle}, minimal energy projection with unit response equivalent to minimal Kullback--Leibler (KL) divergence under linear moment constraints; softmax probability precisely minimal-KL projection weights, converging via \Gamma-limit to hard projection (Hilbert orthogonal) as softening parameter \tau\!\downarrow\!0 (equivalently inverse temperature \kappa=1/\tau\!\uparrow\!\infty). Equivalently using Fenchel--Legendre duality / Bregman--KL identity / Csiszár I-projection. 3. Pointer Basis = Eigenbasis of Minimal Energy/Information Projection: Under finite dictionary, coefficient vector of minimal energy mollifier \beta^\star=G^{-1c}{c^\ast G^{-1}c}; in Gram spectral decomposition G=U\Lambda U^\ast, \beta^\star expanded along \{u_k\} weighted by \lambda_k^{-1}, thus direction contributing strongest to \beta^\star realized by $ \arg\max_k\ |\langle u_k,c\rangle|^2{\lambda_k}; small eigenvalue trend amplifies that direction, but whether dominates depends on simultaneously having sufficiently large projection |\langle u_k,c\rangle|. Soft version information Hessian \nabla^2\Lambda spectral basis isomorphic to this. On scattering side, via Birman--Kreĭn and Wigner--Smith standard construction, single-channel phase derivative and (relative) spectral density satisfy \varphi'(E)=\pi\,\rho_{rel}(E)=\pi\,\xi'(E),\qquad Q(E)=-i\,S(E)^\daggerdS{dE}, hence 1{2\pi}trQ(E)=\rho_{rel}(E)$. This interprets ``negative delay'' as result of reference choice and relative counting, not causality violation. Keywords: Windowed readout; Weyl--Titchmarsh; spectral shift function; Wigner--Smith time delay; de Branges space; BN--Bregman; minimal KL; PSWF; Nyquist--Poisson--EM; non-asymptotic error.
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.
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.