We develop an abstract theory of spectral windows for self-adjoint operators on Hilbert spaces. The theory is organised in four layers (PRE-PURE, PURE, CORE, EXEC) and is self-contained: all definitions, lemmas, and theorems are proved locally, without external assumptions on geometry or operator construction. The PRE-PURE layer introduces the structural apparatus: windows, window families, band partitions, refinement chains, and the ACC condition. The PURE layer establishes the window contract — an abstract mass on windows together with distributional consistency under refinement. The CORE layer realises the contract via functional calculus for an axiomatically given non-negative self-adjoint operator: it introduces the Paley–Wiener class as a canonical realisation (kernels, traces, regularity), proves that the Nowak–Stachowiak reference family belongs to PW_Ω ∩ S, and establishes perturbation stability (Kato–Rellich, KLMN, Mosco convergence). The EXEC layer provides benchmarks (Fejér, Gaussian), a reference realisation on the q³D torus, and engineering diagnostics. The paper provides a self-contained theoretical foundation for the spectral window contract and its realisations; it is designed to be cited independently as upstream input.
Nowak et al. (Thu,) studied this question.