Finite-Window Energy and Spectral Conservation

Finite-Window Energy and Spectral Conservation Auric Version 1.1 November 24, 2025 Abstract Establish energy and spectral conservation laws under finite-window measurement. Core result: for windowed readout with window wand spectral density ρ, windowed energy Ew=REw(E)ρ(E)dE satisfies: (i) Conservation: under unitary evolution ∂tEw= 0 when wtime-independent; (ii) Covariance: gauge-invariant under phase transformations; (iii) NPE closure: discrete approximation with alias+EM+tail error decomposition. 1 Definitions Windowed energy functional: Ew[ρ] = ZR E w(E)ρ(E)dE where w≥0 normalized window, ρspectral density. Spectral measure:dµρ(E) = ρ(E)dE for density, general dµρfor measure. 2 Main Theorems Theorem 2.1 (Windowed Energy Conservation).For unitary evolution ∂tρ=i[H, ρ]and time-independent window w, have ∂tEw= 0. Theorem 2.2 (Gauge Covariance).Under gauge transformation ρ7→ UρU†,H7→ UHU† with Uunitary, Ewinvariant. Theorem 2.3 (NPE Error for Windowed Energy).Discrete approximation b Ew= ∆ PnEnw(En)ρ(En) satisfies |Ew−b Ew|≤|εalias|+|εEM|+|εtail| with εalias = 0 under Nyquist. 3 Discussion Finite-window framework provides:  Energy conservation compatible with finite bandwidth 1  Gauge-invariant formulation  Non-asymptotic error control  Connection to thermodynamics and resource theories 2