PMO Polaris CO survey. II. Where is the dust?

Dust plays critical chemical and dynamical roles in the interstellar medium (ISM), but its specific association with molecular and atomic gas remains difficult to isolate. Combining the PMO Polaris CO Survey (PPCOS), EBHIS \ion{H}{I} data, and \textit{Planck} dust maps, this study investigates dust distributions across multiple gas components in the Polaris Flare. We employ multi-technique linear decomposition -- including full-spectrum fitting and a regularization approach -- to reconstruct the dust distribution from multi-component gas emissions. This framework quantifies dust contributions from CO-associated, \ion{H}{I}-associated, and CO-dark molecular gas phases. CO-associated dust accounts for 20--40\% of the total dust mass, whereas dust in the broad \ion{H}{I} (warm neutral medium, WNM) component is negligible. Instead, \ion{H}{I}-associated dust concentrates primarily within the narrow cold neutral medium (CNM) and a distinct, ultra-narrow component with a velocity width comparable to the \ion{H}{I} spectral resolution. Residual dust at atomic-to-molecular (\ion{H}{I}--CO) interfaces contributes 4--10\% to the global dust mass, but exceeds 25\% at molecular cloud boundaries, confirming a substantial presence of CO-dark molecular gas. Furthermore, the velocity fields of dust-associated \ion{H}{I} closely match those of CO, indicating active dynamical coupling between CO-emitting gas and the surrounding CNM. Guided by these results, we present a stepwise schematic cartoon illustrating the coupling between multi-phase gas structures, molecular formation, and dust growth.