Non-Commuting Covariance Geometry in Filter-Integrated Dynamical-Decoupling Spectroscopy

This paper studies the covariance geometry of filter-integrated observables in dynamical-decoupling spectroscopy. We show that the covariance admits a non-commuting additive decomposition,Ctotal=λCflat+CdevCᵗᵒᵗᵃˡ=λ Cᶠˡᵃᵗ+CᵈᵉᵛCtotal=λCflat+Cdev,where CflatCᶠˡᵃᵗCflat represents flat filter-overlap geometry and CdevCᵈᵉᵛCdev encodes spectral deformation induced by power-law noise weights. Numerical analysis of CPMG filters reveals persistent but diluted non-commutativity under protocol enlargement, selective intermediate-mode hybridization, and horizon-precursor growth of covariance ill-conditioning.