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Unified Measurement via Windowed Readout: Born Probability = Minimal KL, Pointer Basis = Minimal Energy Eigenbasis (With Non-Asymptotic Error Closure and Window/Kernel Optimization) Auric Date: October 25, 2025 November 24, 2025 Abstract Within unified framework of mirror kernel–de Branges–Kre˘ın 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 wRand 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(E+i0) = πρ(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 Λ(ρ) = logPjwje⟨βj,ρ⟩,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 Γ-limit to hard projection (Hilbert orthogonal) as softening parameter τ↓0 (equivalently inverse temperature κ= 1/τ ↑∞). Equivalently using Fenchel–Legendre duality / Bregman–KL identity / Csisz´ar I-projection. 3. Pointer Basis = Eigenbasis of Minimal Energy/Information Projection: Under finite dictionary, coefficient vector of minimal energy mollifier β⋆=G−1c c∗G−1c; in Gram spectral decomposition G=UΛU∗,β⋆expanded along {uk}weighted by λ−1 k, thus direction contributing strongest to β⋆realized by arg max k |⟨uk, c⟩|2 λk ; small eigenvalue trend amplifies that direction, but whether dominates depends on simultaneously having sufficiently large projection |⟨uk, c⟩|. Soft version information Hessian ∇2Λ spectral basis isomorphic to this. On scattering side, via Birman–Kre˘ın and Wigner–Smith standard construction, single-channel phase derivative and (relative) spectral density satisfy 1
φ′(E) = π ρrel(E) = π ξ′(E),Q(E) = −i S(E)†dS dE , hence 1 2πtr Q(E) = ρrel(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. 1 Notation and Background 1.1 Basic Conventions Energy and upper half-plane:E∈R,C+={z:ℑz > 0}. Fourier convention: Uniformly adopt b f(ξ) = Rf(t)e−itξ dt, where ξis angular frequency (rad/energy). Unit convention: Fix ℏ= 1. 1.2 Spectral Function and Boundary Values Weyl–Titchmarsh and LDOS: If m:C+→C+is Herglotz–Nevanlinna function, has non-tangential boundary value at Lebesgue-a.e. energy points (Fatou boundary theory). When its Herglotz representation measure absolutely continuous part has density ρm(E) at E, a.e. ℑm(E+i0) = πρm(E). Here πfrom Stieltjes inversion standard constant, independent of Fourier transform convention. 1.3 Scattering and Spectral Shift Notation convention: This paper fixes Birman–Kre˘ın “positive sign” convention det S(E) = e+2πi ξ(E), ξ′(E) = ρrel(E). Define relative (spectral shift) density ρrel(E) := ξ′(E) (a.e.). Single-channel S(E) = e2iφ(E)gives φ′(E) = π ξ′(E) = π ρrel(E) (a.e.). 2 Main Theorem I: Windowed Readout and Non-Asymptotic Error Closure Theorem 2.1 (Windowed Readout; Nyquist–Poisson–EM Three-Term Decomposition).Assumption: Sampled function F(E) = wR(E) [h⋆ρ⋆](E)belongs to L1(R)or tempered distribution S′satisfying Poisson summation interchange condition; h∈L1∩L2;wReven window; ρ⋆absolute or relative LDOS. Take even window wR(x) = w(x/R)and frontend kernel h∈L1∩L2. For absolute or relative LDOS ρ⋆∈ {ρm, ρrel}define readout 2
Obs∆,T := ∆ X |n|≤M wR(En) [h⋆ρ⋆](En), En=n∆, T =M∆. Then Obs∆,T =ZR wR(E) [h⋆ρ⋆](E)dE +εalias +εEM +εtail, where (i) εalias: spectral aliasing from Poisson summation; (ii) εEM:finite-order Euler–Maclaurin sum formula remainder; (iii) εtail: out-of-window truncation tail. Alias zero necessary and sufficient condition: By Poisson summation formula, εalias = 0 necessary and sufficient condition:Fbandlimited with b F⊂[−ΩF,ΩF]and ∆≤π/ΩF(Nyquist). Proof. Apply Poisson summation to connect discrete sum with continuous integral. Euler– Maclaurin gives Bernoulli corrections. Truncation produces tail term. Nyquist condition ensures alias cancellation. 3 Main Theorem II: Born Probability = Minimal KL Theorem 3.1 (Born as I-Projection).For constraint family C={p:Pipiai=b}and reference q, minimal KL-divergence p⋆= arg min p∈C DKL(p∥q) has exponential family form p⋆ i∝qieλai. Alignment condition:p⋆equals Born weights wi=⟨ψ, Eiψ⟩if and only if log(wi/qi) affinely expressible in constraint space. Softmax probability pj(ρ;τ) = wje⟨βj,ρ⟩/τ Pℓwℓe⟨βℓ,ρ⟩/τ converges to Born via Γ-limit as τ↓0. Proof. Strict convexity of KL and Lagrange multipliers yield exponential family. Alignment condition ensures match with Born. Γ-limit follows from log-sum-exp concentration. 4 Main Theorem III: Pointer Basis = Minimal Energy Eigenbasis Theorem 4.1 (Pointer Basis Characterization).Under finite dictionary with Gram matrix G=Pjwjβjβ∗ j, minimal energy mollifier coefficient β⋆=G−1c c∗G−1c, where cconstraint vector. In spectral decomposition G=UΛU∗, direction maximizing contribution k⋆= arg max k |⟨uk, c⟩|2 λk . Small eigenvalues amplify corresponding eigendirections, but dominance requires sufficient projection |⟨uk, c⟩|. 3
Information Hessian ∇2Λ = Covp(ρ)(β)has spectral basis isomorphic to Gram decomposition, thus pointer basis corresponds to minimal curvature directions of log-partition function. Proof. Minimization with quadratic constraint yields β⋆=G−1c/norm. Spectral decomposition G=UΛU∗gives β⋆=Pk ⟨uk,c⟩ λkc∗G−1cuk. Contribution of direction kproportional to |⟨uk, c⟩|2/λk. 5 Phase–Density Unification Core scale chain holding a.e. on absolutely continuous spectrum: φ′(E) π=ρrel(E) = 1 2πtr Q(E) connecting: Scattering phase derivative φ′ Relative spectral density ρrel Wigner–Smith delay trace tr Q via Birman–Kre˘ın formula det S=e2πiξ and Q=−iS†∂ES. 6 Discussion and Outlook This work unifies: 1. Windowed readout with non-asymptotic NPE error closure 2. Born probability as information-geometric I-projection 3. Pointer basis as minimal energy eigenbasis 4. Phase–density correspondence via Birman–Kre˘ın–Wigner–Smith Key formulas: Error: εtotal =εalias +εEM +εtail Born: p⋆ i∝qieλaiwith alignment condition Pointer: k⋆= arg maxk|⟨uk, c⟩|2/λk Scale: φ′/π =ρrel = (2π)−1tr Q Future directions: Extension to continuous POVM and general observables Numerical optimization algorithms for window design Applications to quantum metrology and thermometry Connections to resource theories and entanglement 4
