Coherence and cross-phase of density and field magnitude. (a) Density–field-magnitude coherence γ²(δn, δ∣B∣) against spacecraft-frame frequency f, formed from gradient-removed fluctuations; the LP-OB band (0.003–0.03 Hz) is shaded, the dotted line is the noise floor of a phase-randomized control, the dashed line marks the ion-kinetic onset, and the arrow points toward the precursor scale L_EP, which lies below the plotted frequency range. (b) The signed density–field cross-phase Φ(δn, δ∣B∣); the shaded band marks ±30^∘ about zero.