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

WSIG-QM: Windowed Scattering & Information Geometry for Quantum Mechanics A Unified Framework of Quantum Concept Definitions and Criterion System (with Complete Proofs) Auric (S-series / EBOC Framework) Version v1.4a (Logically Complete, Peer-Reviewed) November 24, 2025 Abstract Using the de Branges–Kre˘ın (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 (threeterm decomposition); ensure realizability and stability through variational optimization of frame/sampling density and window/kernel. The core unified formula is φ′(E) = −π ρrel(E) = 1 2tr Q(E) (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–Kre˘ın, Wigner–Smith, Ky Fan, Poisson/EM and other standard results, directly interchangeable and implementable. 1 Notation & Baseplates 1.1 DBK Canonical System and Herglotz–Weyl Dictionary Consider first-order symplectic canonical system JY ′(t, z) = zH(t)Y(t, z) (H⪰0), whose Weyl–Titchmarsh function m:C+→C+belongs to Herglotz class, with representation m(z) = a z +b+ZR 1 t−z−t 1 + t2dµ(t), a ≥0, b ∈R, and ℑm(E+i0) = π ρ(E) (a.e.). This gives absolutely continuous density ρof continuous spectrum and is compatible with DBK framework. 1 1.2 Weyl–Heisenberg (Phase–Scale) Representation (Mellin Version) On weighted Mellin space Ha=L2(R+, xa−1dx) define (Uτf)(x) = xiτ f(x),(Vσf)(x) = eσa/2f(eσx), satisfying Weyl relation VσUτ=eiτσUτVσ. Via power-log isomorphism x=et,Haand L2(R) Weyl–Heisenberg/Gabor framework are isometric and parallel, serving as phase–scale kinematic baseplate. 1.3 Finite-Order EM / Poisson Three-Term Decomposition Discipline Parent map and all discrete–continuous reordering adopt finite-order Euler–Maclaurin (EM), closing errors via “alias (Poisson) + Bernoulli layer (EM) + tail term” threeterm decomposition; “bandlimited + Nyquist” makes alias term zero. Poisson summation and sampling criteria adopt angular frequency convention (Ω unit: rad/unit(E); unit conversion: if using Hertz B, have B= Ω/(2π), then switch to Ts≤1/(2B); for time tas independent variable unit is rad/s). 1.4 Convention and Notation Fourier/Parseval Convention Table: Item Formula Note Fourier transform b f(ξ) = Rf(t)e−itξ dt ξ angular frequency (rad/unit(t)) Parseval relation RR∥f∥2=1 2πRR∥b f∥2Energy conservation, with 1 2πfactor Convolution theorem [ f∗g=b f·bgTime convolution = frequency product Product transform d f·g=1 2πb f∗bgTime product = frequency convolution/2π Birman–Kre˘ın adopts det S(E) = e−2πi ξ(E). This paper fixes above formula.Equivalence chain bridge (det S–ξ–Qtriple relation): tr Q(E) = −i ∂Eln det S(E) = −2π ξ′(E); thus for any channel number N, have tr Q(E) = −2π ξ′(E); single-channel S=e2iφ gives tr Q(E)=2φ′(E), thus φ′(E) = −π ξ′(E) (consistent with ρrel =ξ′). 1.5 Projection Operator Notation Unification This paper fixes frequency-domain projection as P(ξ) B:b f7→ χBb f(where χBcharacteristic function of frequency band B= [−Ω,Ω]), its integral kernel realization in energy domain denoted ΠB, i.e., (ΠBf)(E) = ZR kB(E−E′)f(E′)dE′, kB(t) = sin(Ωt) πt . 2 In T6 variational equations always use P(ξ) B(frequency projection); in A4/T3 kernel–trace class/Hilbert–Schmidt arguments use ΠB(energy integral operator). This distinction avoids confusion between “first convolve then project” and “projection is convolution”. 2 Axioms Axiom 2.1 (Carrier and Covariance).Quantum states placed in H(E)or Ha; phase–scale covariance realized by projective unitary representation of (Uτ, Vσ)Weyl–Heisenberg (Stone theorem: strongly continuous one-parameter unitary group ⇔self-adjoint generator; Stone– von Neumann: irreducible representation of Weyl relation essentially unique). Axiom 2.2 (Observables and Windowed Readout).Instrument window wRand bandlimited response kernel h∈L1act on continuous spectral density of state, defining windowed readout ⟨Kw,h⟩ρ=ZR wR(E)h∗ρ⋆(E)dE . where ρ⋆can be ρabs (absolutely continuous spectral density, i.e., density of µac ρ) or ρrel = ρabs −ρ0,abs (relative density). In “no-blur hard limit”h→δrecovers RwR(E)ρ⋆(E)dE. Readout controlled by three-term decomposition error. Axiom 2.3 (Probability–Information Consistency).Commit (collapse/commit) = minimalKL projection (I-projection) on apparatus/window constraint; PVM hard limit returns to Born. Axiom 2.4 (Pointer Basis).“Pointer basis” defined as spectral projection subspace corresponding to minimal spectral value of window operator WR=RwRdEA(Ky Fan “minimum sum”; if minimal spectral value not attained, take limit subspace as ε↓0); existence and verifiable condition: if wR∈L∞then WRbounded self-adjoint; if further wR∈L2(R)(e.g., finite support window), let kB(t) = sin(Ωt)/(πt), have: Under bandlimited projection ΠB,uniformly adopt ΠBMwRnotation. Its integral kernel K(x, y) = kB(x−y)wR(y). By L3.3a know ∥kB∥2 L2= Ω/π < ∞;HS kernel verification one-liner: ∥K∥2 L2(R2)=ZR2 |kB(x−y)|2|wR(y)|2dxdy =∥kB∥2 L2∥wR∥2 L2=Ω π∥wR∥2 L2<∞, thus by Fubini–Tonelli theorem ΠBMwRis Hilbert–Schmidt, hence ΠBMwRΠBalso Hilbert– Schmidt/compact (HS kernel theorem: if integral kernel K∈L2(R2)then corresponding integral operator is Hilbert–Schmidt, hence compact). Instrument kernel honly affects readout and error, does not change spectral structure of WR. Axiom 2.5 (Phase–Density–Delay Scale).If (H, H0)satisfy relative trace class/smooth scattering standard regularity (see L3.5), then on absolutely continuous spectrum almost everywhere φ′(E) = −π ρrel(E) = 1 2tr Q(E) (a.e. on σac) where ρrel(E) = ξ′(E),Q:= −iS†dS dE (unit system ℏ= 1). Sign convention: this paper uniformly adopts det S(E) = e−2πi ξ(E), thus tr Q(E) = ∂Earg det S(E) = −2π ξ′(E), 3 hence φ′(E) = −π ρrel(E). Above equality holds almost everywhere on absolutely continuous spectrum; near threshold or resonance, interpreted by limϵ↓0non-tangential limit or distributional sense (principal value + singular part), consistent with Herglotz boundary value ℑm(E+i0) = πρ(E). Under lossless assumption S(E)unitary. Axiom 2.6 (Window/Kernel Optimization and Multi-Window Synergy).Window w∈PWeven Ω; objective to minimize three-term decomposition error upper bound; necessary condition is frequency-domain “polynomial multiplier + convolution kernel” bandlimited projection-KKT equation; multi-window version characterized by generalized Wexler– Raz biorthogonality and frame operator for Pareto frontier and stability. Axiom 2.7 (Threshold and Singularity Stability).Under “finite-order EM + Nyquist–Poisson– EM” discipline, windowing/reordering generates no new singularities; zero count stable and verifiable within Rouch´e radius. 3 Basic Definitions Definition 3.1 (State).Pure state ψ∈ H (|ψ|= 1); mixed state ρ⪰0, tr ρ= 1. Definition 3.2 (Observable).Self-adjoint Aand spectral projection EA. Definition 3.3 (Windowed Readout and Regularity Conditions).⟨Kw,h⟩ρ=RwR[h∗ρ⋆]dE, where ρ⋆can be ρabs (µρabsolutely continuous part density) or ρrel (relative density). Measure perspective writable as d(h∗µρ) = h∗dµρ(h∈PWΩ∩L1, standard convolution for Radon measures). Regularity and integrability sufficient conditions: to ensure three-term decomposition (Poisson–EM–Tail) remainder bounds hold and measure convolution legal, take wR∈PWΩ∩W2M,1(R), h ∈PWΩ∩W2M,1(R), and choose one of: (c) wRcompactly supported and wR∈W2M,1(R), thus F=wR(h∗ρ⋆)∈L1and F(2M)∈L1still hold (engineering first choice); (b) ρ⋆∈L1(R) (or L1∩L∞or sufficient weighted integrability) and h∈ S, then h∗ρ⋆∈ L1∩L∞and higher derivatives integrable. Definition 3.4 (Commit/Collapse).Given apparatus constraint, observation probability p is I-projection of reference qto feasible set; softmax softening →Born hard limit. Definition 3.5 (Pointer Basis).Basis spanning spectral projection subspace corresponding to minimal spectral value of window operator WR(Ky Fan “minimum sum”; if minimal spectral value not attained, take ε↓0 limit subspace), called minimal spectral subspace; hacts only on measure side. 4 Preliminary Lemmas (Tools and Conventions) Lemma 4.1 (Poisson Summation and Nyquist Condition).If bwRand b hsupported on [−Ωw,Ωw], [−Ωh,Ωh]respectively, then for F(E) := wR(E) [h∗ρ⋆](E) 4 have supp b F⊂[−(Ωw+ Ωh),Ωw+ Ωh].Unified Nyquist convention and unit conversion: If supp b F⊂[−ΩF,ΩF](where ΩF= Ωw+ Ωh), then sampling criterion is ∆≤π/ΩF(angular frequency rad/(unit(E)))⇐⇒ Ts≤1/(2BF) (Hertz BF= ΩF/(2π)Hz). This paper defaults to angular frequency convention; Hertz notation only as equivalent reminder. When condition satisfied, in Poisson summation all terms except k= 0 fall outside band, thus alias error εalias = 0. Lemma 4.2 (Finite-Order Euler–Maclaurin and Remainder Bounds).Let p= 2M∈2N (p≥2, even order). If g∈Cp([a, b]) and g(p)∈L1([a, b]), then Euler–Maclaurin formula remainder Rpsatisfies standard upper bound Rp≤2ζ(p) (2π)pZb a |g(p)(x)|dx, where [a, b]continuous extension interval corresponding to summation interval (e.g., [−N∆, N∆]), ζ(p)Riemann ζfunction. This bound holds for p≥2;requires g(p)∈L1. Lemma 4.3 (Herglotz–Nevanlinna Boundary Value and Spectral Density).If mis Herglotz (Nevanlinna) function, then non-tangential limit almost everywhere exists and ℑm(E+i0) = π ρm(E)(a.e.), where ρmabsolutely continuous part density of Herglotz representation measure. Threshold and resonance neighborhood interpreted by nontangential limit or distributional sense (principal value + singular part). This conclusion is classical spectral theory standard result. Lemma 4.4 (sinc Kernel L2Norm).For bandlimited projection kernel kB(t) = sin(Ωt) πt (Ω>0), have ∥kB∥2 L2(R)=Z∞ −∞sin(Ωt) πt 2dt =Ω π. Lemma 4.5 (Ky Fan Variational Principle: Minimum Sum).For self-adjoint Kand any orthogonal family {ek}m k=1, m X k=1 ⟨ek, Kek⟩ ≥ m X k=1 λ↑ k(K), equality if and only if {ek}spans minimal eigensubspace. If minimal eigenvalue has degeneracy, any orthonormal basis of corresponding minimal eigensubspace can serve as “pointer basis”. Lemma 4.6 (Birman–Kre˘ın, Wigner–Smith and Regularity).If (H, H0)relative trace class perturbation or satisfies smooth scattering standard assumptions, then BK formula det S(E) = e−2πi ξ(E),tr Q(E) = ∂Earg det S(E) = −2π ξ′(E) holds, where Q=−iS†dS dE (requires S(E)differentiable in Eand unitary; lossless case directly holds). Almost everywhere ξ′(E) = ρrel(E). 5 5 Main Theorems and Complete Proofs Theorem 5.1 (Windowed Readout Numerical Estimation Formula and Non-Asymptotic Error Closure).Set A’s spectral measure dEA, instrument window wRand bandlimited kernel h. Let F(E) = wR(E) [h∗ρ⋆](E), where ρ⋆continuous spectral density of state ρ.Regularity prerequisite: following equality and remainder bounds hold under F∈L1∩C2M,F(2M)∈L1; sufficient conditions see § 2-D3. For sampling step ∆>0and finite truncation |n| ≤ N, have ZR F(E)dE = ∆ N X n=−N F(n∆) + εalias |{z} Poisson +Rp |{z} Euler–Maclaurin +εtail |{z} truncation tail , where p= 2Mand |Rp| ≤ 2ζ(p) (2π)pR|F(p)(x)|dx.Alias term zero under bandlimited+Nyquist condition. Proof. By Poisson summation connecting integral with discrete sum (alias term is spectral replication overlap amount), EM finite order gives Bernoulli layer and endpoint remainder, truncation produces tail; convolution theorem \ h∗ρ⋆=b h·bρ⋆ensures supp b F⊂[−ΩF,ΩF]; bandlimited+Nyquist makes alias term vanish; L3.1–L3.2 immediately yield result. Theorem 5.2 (Born Probability = I-projection “Alignment Necessary and Sufficient Condition”).Set PVM/POVM and reference q.Premise:qi>0(or supp p⊆supp q), and closed convex feasible set C={p:Pipiai=b}of linear moment constraints nonempty. Under this premise, minimal-KL min{DKL(p∥q) : p∈ C} unique solution exponential family p⋆ i∝qieλ⊤ai(I-projection uniqueness theorem). Alignment necessary and sufficient condition and support matching: let wi=⟨ψ, Eiψ⟩ be Born vector of PVM index. Must first ensure relative support condition supp w⊆ supp q(i.e., if wi>0then qi>0); under this premise, if and only if on {i:wi>0}exists λsuch that log(wi/qi)falls in affine span of {ai}(equivalent to log(wi/qi) = λ⊤ai−ψ(λ) for some normalization constant ψ), then I-projection unique solution is p⋆=w(Born); if not affinely representable, optimal solution exponential family p⋆=w(but still unique). Softening temperature τ↓0 Γ-limit converges softmax to Born. Proof. Strict convexity of KL and Lagrange multipliers under premise qi>0, feasible set closed convex nonempty give exponential family and uniqueness; alignment necessary and sufficient condition derived from exponential family parameterization coefficient-by-coefficient. POVM case by Naimark dilation to PVM then project back. Theorem 5.3 (Pointer Basis = Minimal Spectral Subspace (Ky Fan “Minimum Sum”)).Existence condition: understand “pointer basis” as spectral projection subspace corresponding to minimal spectral value of WR(minimal spectral subspace for short); if minimal spectral value not attained, replace by limiting minimal spectral subspace of P(−∞,λmin+ε](ε↓0). For self-adjoint window operator WRand any m-dimensional orthogonal family {ek}, m X k=1 ⟨ek, WRek⟩ ≥ m X k=1 λ↑ k(WR), equality if and only if {ek}spans minimal spectral subspace of WR(Ky Fan PNAS 1951; requires WRcompact self-adjoint or minimal spectral value isolated eigenvalue). 6 If minimal eigenvalue has degeneracy, any orthonormal basis of corresponding minimal eigensubspace can serve as “pointer basis”. Instrument kernel honly introduces blur on measure side, does not change spectrum of WR. Theorem 5.4 (φ′=−πρrel = tr Q/2, a.e. on σac).Set (H, H0)satisfy regularity of L3.5. Then on absolutely continuous spectrum almost everywhere (a.e. on σac) φ′(E) = −π ρrel(E) = 1 2tr Q(E) (a.e. on σac) where ρrel(E) = ξ′(E),Q=−iS†dS dE .Threshold/resonance treatment: near threshold or resonance, interpret by limϵ↓0non-tangential limit or distributional sense (principal value + singular part), consistent with Herglotz boundary value ℑm(E+i0) = πρ(E). Proof. BK formula gives det S=e−2πiξ ⇒∂Earg det S=−2π ξ′; and tr Q=∂Earg det S. Single-channel S=e2iφ gives tr Q= 2φ′, combining yields conclusion. Relative density from ξ′=ρm−ρm0Herglotz–Weyl localization. Theorem 5.5 (Threshold and Singularity Stability: Rouch´e Radius).If on boundary of domain Dhave |E(z)| ≥ η > 0, and approximation E♮satisfies sup∂D |E♮− E| < η, then both have same zero count in Dwith displacement upper bound; under “finite-order EM + Nyquist–Poisson–EM” discipline windowing/reordering generates no new singularities, ηcan be measured by three-term decomposition error upper bound. Proof. By Rouch´e theorem combined with Poisson–EM error upper bounds (L3.1, L3.2) and bandlimited support bounds immediately yields conclusion. Theorem 5.6 (Window/Kernel Optimization Bandlimited Projection-KKT and Γ-limit). Setting: on PWeven Ω, strongly convex proxy J(w) = M−1 X j=1 γj∥w(2j) R∥2 L2+λ∥1{|E|>T}wR∥2 L2, wR(t) = w(t/R),cwR(ξ) = Rbw(Rξ). Theorem: Exists unique minimizer w⋆, in frequency domain satisfying bandlimited projection–KKT equation (P(ξ) B:b f7→ χBb ffrequency bandlimited projection) P(ξ) B2 M−1 X j=1 γjξ4jc w⋆ R(ξ) | {z } polynomial multiplier + 2λc w⋆ R(ξ) | {z } δ-term −2λ πsin(T·) ·∗c w⋆ R(ξ) | {z } convolution term =ηc w⋆ R(ξ) (ξ∈R), where B= [−Ω/R, Ω/R],ηnormalization multiplier. 6 Discussion and Outlook This work establishes rigorous mathematical foundation for windowed quantum measurement framework, unifying scattering theory, information geometry, and finite-bandwidth instrumentation. Key achievements: 1. Unified formula φ′=−πρrel =1 2tr Qconnecting phase, density, delay 7 2. Born probability as I-projection with explicit alignment conditions 3. Pointer basis as minimal spectral subspace with verifiable criteria 4. Non-asymptotic error closure via Poisson–EM–Tail decomposition 5. Variational optimization framework for window/kernel design Future directions include:  Extension to open quantum systems and non-unitary scattering  Numerical implementation and experimental validation  Connection with quantum thermodynamics and resource theories  Applications to quantum metrology and sensing 8