The BKM-SLD Metric Defect: Commutator Geometry, Channel Generation, and Lindblad Onset

For faithful finite-dimensional quantum states, we study the positive quadratic-form difference Δ_K(ρ;X)=g_BKM,ρ(X,X)-g_SLD,ρ(X,X) between the Bogoliubov-Kubo-Mori (BKM) and symmetric-logarithmic-derivative (SLD) quantum Fisher metrics. Its spectral kernel is the difference between the reciprocals of the logarithmic and arithmetic means. We prove that Δ_K(ρ;X)=0 exactly when [ρ,X]=0 and rewrite the defect as a positive spectrally weighted Hilbert-Schmidt norm of the commutator, with explicit dimension-free bounds in terms of the extremal eigenvalues of ρ. Although both Petz metrics are individually contractive under quantum channels, their difference is not: amplitude damping generates a strictly positive defect from a zero-defect pair. We define the channel defect-generation strength and show, for channels mapping faithful inputs to faithful outputs, that zero generation is equivalent to commutativity preservation. Finally, for GKSL dynamics starting from [ρ₀,X₀]=0, we derive the short-time law Δ_K(ρ_t;X_t)=A_L t²+O(t³). The Hamiltonian derivation cancels identically from A_L, so the leading creation of the defect is determined by the non-derivation part of the generator.