Unified Measurement via Windowed Readout:\\ Born Probability = Minimal KL,\\ Pointer Basis = Minimal Energy Eigenbasis\\[10pt] \large (With Non-Asymptotic Error Closure and Window/Kernel Optimization)
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.
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.
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.
Under unified framework of de Branges--Kreĭn (DBK) canonical system, scattering--functional equation dictionary and Bregman/information geometry, this paper establishes ``Trinity Theorem'' for windowed measurement. Conclusions in three tiers: (I) Born = Information Projection (iff): Under orthogonal projection measurement (and its generalization to POVM), optimal probability vector induced by windowed readout equivalent to I-projection (minimal KL/Bregman cost) over family of linear alignment constraints if and only if it equals Born probability. (II) Pointer = Spectral Minimum (iff): For any distinguishable window family, let Ky Fan partial sum of ``windowed trace quadratic form'' minimize over all orthonormal bases, then if and only if that basis is spectral eigenbasis of measured observable (modulo degeneracy). (III) Windows = Minimax Optimal: Under constraint of bandlimited even window with normalization w(0)=1, taking Nyquist--Poisson--Euler--Maclaurin ``alias + Bernoulli layer + truncation'' non-asymptotic error upper bound as adversary, optimal window exists and (under Hilbert strongly convex proxy) unique, satisfying frequency-domain projection KKT condition. Key bridge is Birman--Kreĭn formula and Wigner--Smith delay giving phase derivative = spectral density, precisely connecting windowed readout with relative state density (LDOS).
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.
This paper constructs and rigorizes WSIG-QFT (Windowed Scattering \& Information-Geometry Quantum Field Theory): under weighted Mellin--logarithmic model and de Branges--Kreĭn (DBK) canonical system, uses Weyl--Heisenberg kinematic scale for ``phase--scale'', connects scattering phase derivative with (relative) spectral density via Birman--Kreĭn (BK)--Wigner--Smith (WS) chain, realizes Born probability = relative entropy minimization through Csiszár-type I-projection, implements pointer basis = spectral minimum of readout quadratic form via Ky-Fan spectral minimum, provides non-asymptotic error closure and bandlimited-sampling criterion through Nyquist--Poisson--Euler--Maclaurin (NPE). For multi-channel establishes windowed BK identity and multi-window frame--Wexler--Raz synergy conditions, giving verifiable premises, explicit statements and complete proofs (based on recognized criteria). Keywords: Windowed readout; de Branges--Kreĭn canonical system; Weyl--Heisenberg representation; Birman--Kreĭn; Wigner--Smith delay; I-projection; Wexler--Raz; Nyquist--Poisson--Euler--Maclaurin (NPE) error closure MSC: 81Txx; 47Bxx; 46E22; 42C15
Under de Branges--Kreĭn canonical systems and multi-channel scattering theory, taking scattering phase derivative, relative state density, and Wigner--Smith group delay trace as three equivalent formulations of the same scale, we construct a ``universal measure coordinate'' and prove its change-of-variables consistency under windowed readouts and non-asymptotic error closure of Nyquist--Poisson--Euler--Maclaurin (NPE) three-fold decomposition. The core unification formula (almost everywhere on absolutely continuous spectrum) is equation d\mu_\varphi(E)=\varphi'(E){\pi}\,dE=1{2\pi}tr\mathsf Q(E)\,dE=\rho_{rel}(E)\,dE, equation where \mathsf Q(E)=-i\,S(E)^\dagger S'(E) is the Wigner--Smith delay matrix, \rho_{rel}=-\xi' is the spectral shift density relative to reference operator H_0, and \det S(E)=\exp(-2\pi i\,\xi(E)) is the standard gauge of Birman--Kreĭn formula. The trinity chain is derived consistently from Birman--Kreĭn formula and Wigner--Smith delay, applicable to multi-channel unitary scattering; sub-unitary systems admit compatible extension via generalized (complex) delay. The paper provides: (i) windowed change-of-variables consistency theorem (energy \leftrightarrow phase/delay/density coordinates); (ii) maximum entropy--minimum delay duality theorem; (iii) causal monotonicity \Leftrightarrow phase monotonicity; and formulates sampling--frame thresholds, Wexler--Raz biorthogonality, and Balian--Low impossibility as invariant expressions in phase coordinates, while establishing KKT and \Gamma-limit stability for multi-window/multi-kernel optimization. All conclusions align term-by-term with BK/SSF, Wigner--Smith, Herglotz--Weyl, and Carleson--frame theory criteria, directly verifiable and implementable.
Under the unified perspective of ``discrete--static blocks (EBOC)'' and ``continuous--causal manifolds/gauge systems,'' we construct an axiomatic framework for quantum measurement: wave functions are characterized as complex-amplitude superpositions of compatible SBUs (Static Block Units) in static blocks; collapse is characterized as observer anchor switching and π-semantic collapse from operator algebras to classical record algebras; entanglement is characterized as conditional mutual information and quantum Markov property in causal networks. Mathematically, quantum instruments and POVMs serve as core objects, with Stinespring/Naimark/Kraus representations, Lüders update, relative entropy minimization principle, and Belavkin filtering closing the ``measurement--update--record'' chain; at the information-geometric level, strong subadditivity and conditional mutual information control causal propagation of entanglement and ``no-signaling.'' At the readout--geometry interface, employing windowed compression Kw,h and ``trinity master-scale'' πφm′=ρrel=2π1trQ unified readout framework, with non-asymptotic error closure under Nyquist--Poisson--Euler--Maclaurin (finite-order) discipline; this identity holds almost everywhere on absolutely continuous spectrum, thereby rigorously unifying ``phase derivative--relative spectral density--Wigner--Smith group delay.'' Davies--Lewis instruments/POVMs, postulated Lüders rule, Stinespring--Kraus--Naimark representations, and Belavkin continuous measurement stochastic filtering provide testable anchor points; Birman--Kreĭn identity, Kreĭn--Friedel--Lloyd bridge, and de Branges kernel diagonal formula provide spectral/functional-analytic basis for unified readout of ``windowed--spectral--scattering.''