Trinity Theorem for Windowed Measurement\\[10pt] \large Born = Information Projection (iff),\\ Pointer = Spectral Minimum (iff),\\ Windows = Minimax Optimal

Trinity Theorem for Windowed Measurement Born = Information Projection (iff), Pointer = Spectral Minimum (iff), Windows = Minimax Optimal Auric Version v1.1 (Preprint, Notation Corrected) Date: October 25, 2025 (Revised) November 24, 2025 Abstract Under unified framework of de Branges–Kre˘ın (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–Kre˘ın formula and Wigner–Smith delay giving phase derivative = spectral density, precisely connecting windowed readout with relative state density (LDOS). 1 Notation, Conventions and Basic Setup 1.1 Hilbert Space and Measurement Hseparable; pure state ψ∈ H,|ψ|= 1. PVM case take mutually exclusive complete projections {Pi}(PiPj=δijPi,PiPi=I), measurement probability pi=⟨ψ, Piψ⟩. POVM case with {Ei⪰0},PiEi=I,pi=⟨ψ, Eiψ⟩. 1.2 DBK Canonical System and Weyl–Titchmarsh For one-dimensional channel, Weyl–Titchmarsh function m(z) is Herglotz–Nevanlinna function (analytic in upper half-plane with non-negative imaginary part), boundary value 1 imaginary part gives spectral measure dρ:ℑm(E+i0+) = π dρ/dE (almost everywhere). de Branges–Kre˘ın (DBK) theory gives one-to-one correspondence between Herglotz class and canonical systems: each Herglotz function uniquely corresponds to canonical system (transfer matrix M(t, z) satisfying J-unitarity), establishing spectral representation and evaluation embedding. 1.3 Scattering–Functional Equation Dictionary and Phase–Spectral Shift det S(E) = exp −2πi ξ(E),d dE arg det S(E) = −2π ξ′(E). If normalized det S(E) = c0e−2iφ(E), then φ(E) = π ξ(E) (up to constant), thus φ′(E) = π ξ′(E) = −π ρrel(E). Convention statement: We fix det S(E) = c0e−2iφ(E)with |c0|= 1; according to Birman–Kre˘ın, det S=e−2πiξ, thus φ′=πξ′; by Q=−iS†S′and d dE arg det S= tr Q, get tr Q=−2φ′=−2πξ′. 1.4 Wigner–Smith Matrix Q(E) = −i S†(E)dS dE (E) self-adjoint,1 2πtr Q(E) = ρrel(E) = −ξ′(E). Compatible with det S(E) = e−2πiξ(E)of § 0.3, obtained from tr Q=d dE arg det S. Key identity: trace of −iS†∂ESequals ∂Earg det S(Friedel phase derivative), standard Wigner– Smith delay theory. 1.5 Paley–Wiener Bandlimited Even Window and Fourier Convention PWeven Ω={w: supp bw⊂[−Ω,Ω],w(E) = w(−E)}, w(0) = 1. This paper adopts non-angular frequency convention, for energy variable Etake Fourier transform (frequency denoted ξ): b f(ξ) = ZR e−iEξf(E)dE, f(E) = 1 2πZR eiEξ b f(ξ)dξ. Scaled window definition: Let bandlimited even window w∈PWeven Ω,scaled window defined as wR(E) := w(E/R). Then cwR(ξ) = Rbw(Rξ), support located in [−Ω/R, Ω/R]. 2 Born = Information Projection (If and Only If) Theorem 2.1 (Born Probability as I-Projection).Set PVM elements Pi, denote ϕi:= Piψ |Piψ|(|Piψ|>0), wi:= ⟨ψ, Piψ⟩=|Piψ|2. For linear constraint family C={p:Pipiai=b}and reference distribution q, if reference support contains Born support (supp w⊆supp q), then I-projection 2 p⋆= arg min p∈C DKL(p∥q) has exponential family form p⋆ i∝qieλai. Alignment condition (necessary and sufficient):p⋆=w(Born) if and only if on {i:wi>0}exists λsuch that log(wi/qi)affinely expressible in constraint space spanned by {ai}. Proof. Strict convexity of KL and Lagrange multipliers give exponential family and uniqueness. Alignment condition ensures Born weights match I-projection solution. POVM case by Naimark dilation. 3 Pointer = Spectral Minimum (If and Only If) Theorem 3.1 (Pointer Basis as Ky Fan Minimum).For self-adjoint observable Aand distinguishable window family {w, h}, define windowed trace operators Kw,h. Let {ek}m k=1 orthonormal system. Then m X k=1 ⟨ek, Kw,hek⟩ ≥ m X k=1 λ↑ k(Kw,h), equality if and only if {ek}spans minimal eigensubspace of Kw,h (Ky Fan minimum sum). Under window distinguishability, Kw,h commute with spectral projection of A, thus minimal eigensubspace equals (modulo degeneracy) spectral eigenbasis of A. Proof. Standard Ky Fan variational principle. Window distinguishability ensures commutativity via Stone–Weierstrass and direct integral decomposition. 4 Windows = Minimax Optimal Theorem 4.1 (Window Minimax Optimality with KKT Condition).On PWeven Ωwith normalization w(0) = 1, consider strongly convex proxy J(w) = M−1 X j=1 γj∥w(2j) R∥2 L2+λ∥1{|E|>T }wR∥2 L2. Exists unique minimizer w⋆satisfying frequency-domain bandlimited projection–KKT equation: P(ξ) BM−1 X j=1 γjξ4jc w⋆ R+λc w⋆ R−λ π(T·)∗c w⋆ R=ηc w⋆ R where P(ξ) Bfrequency projection to B= [−Ω/R, Ω/R],ηnormalization multiplier. Proof. Strong convexity ensures unique minimizer. Frechet derivative with constraint gives KKT condition in frequency domain. 3 5 Error Closure: Nyquist–Poisson–EM Decomposition Theorem 5.1 (NPE Three-Term Error Decomposition).For windowed readout discretization with step ∆and truncation N, have Error =εalias |{z} Poisson +R2M |{z} EM +εtail |{z} truncation . When Fbandlimited with supp b F⊂[−ΩF,ΩF]and ∆≤π/ΩF(Nyquist), alias term εalias = 0. EM remainder satisfies |R2M| ≤ 2ζ(2M) (2π)2MR|F(2M)|. Tail controlled by function decay: |εtail| ≤ R|E|>N∆|F|. Proof. Poisson summation + Euler–Maclaurin expansion + truncation analysis. 6 Unified Scale Chain The trinity theorem unified by scale chain holding a.e. on absolutely continuous spectrum: φ′(E) π=ρrel(E) = 1 2πtr Q(E) connecting:  Born: Information projection optimal under alignment  Pointer: Ky Fan minimum of windowed operators  Windows: Minimax optimal under NPE error  Phase–Density: Birman–Kre˘ın–Wigner–Smith bridge 7 Discussion and Outlook This work establishes trinity of windowed measurement: 1. Born probability as information-geometric optimum 2. Pointer basis as spectral-geometric minimum 3. Window design as minimax-optimal under finite-sample error Key contributions:  Rigorous if-and-only-if characterizations  Unified via Birman–Kre˘ın–Wigner–Smith scale chain  Non-asymptotic error bounds via NPE decomposition  DBK canonical system theoretical foundation Future directions: 4  Extension to continuous POVM and general observables  Numerical optimization of window families  Applications to quantum metrology and sensing  Connections to quantum thermodynamics 5