The (Black Hole Mass)-(Spheroid Stellar Density) Relations: $M_{\mathrm{B}\mathrm{H}}$--$\mu$ (and $M_{\mathrm{B}\mathrm{H}}$--$\mathrm{\Sigma }$) and $M_{\mathrm{B}\mathrm{H}}$--$\rho$

This paper is the fourth in a series presenting (galaxy morphology, and thus galaxy formation)-dependent black hole mass, $M_{\mathrm{B}\mathrm{H}}$, scaling relations. We have used a sample of 119 galaxies with directly-measured $M_{\mathrm{B}\mathrm{H}}$ and host spheroid parameters obtained from multi-component decomposition of, primarily, $3.6\mu$m Spitzer images. Here, we investigate the correlations between $M_{\mathrm{B}\mathrm{H}}$ and the projected luminosity density $\mu$, the projected stellar mass density $\mathrm{\Sigma }$, and the deprojected (internal) stellar mass density $\rho$, for various spheroid radii. We discover the predicted $M_{\rm BH}$--$\mu_{\rm 0,sph}$relationandpresentthefirst$M_{\rm BH}$--$\mu_{\rm e, sph}$and$M_{\rm BH}$--$\rho_{\rm e,int, sph}$ diagrams displaying slightly different (possibly curved) trends for early- and late-type galaxies (ETGs and LTGs) and an offset between ETGs with (fast-rotators, ES/S0) and without (slow-rotators, E) a disk. The scatter about various $M_{\rm BH}$--$\langle\Sigma\rangle_{\rm R,sph}$(and$\langle\rho\rangle_{\rm r,sph}$) relations is shown to systematically decrease as the enclosing aperture (and volume) increases, dropping from 0.69~dex when using the spheroid \enquote{compactness}, $⟨\mathrm{\Sigma }{⟩}_{1\mathrm{k}\mathrm{p}\mathrm{c},\mathrm{s}\mathrm{p}\mathrm{h}}$, to 0.59~dex when using $⟨\mathrm{\Sigma }{⟩}_{5\mathrm{k}\mathrm{p}\mathrm{c},\mathrm{s}\mathrm{p}\mathrm{h}}$. We also reveal that $M_{\mathrm{B}\mathrm{H}}$ correlates with the internal density, ${\rho }_{\mathrm{s}\mathrm{o}\mathrm{i},\mathrm{s}\mathrm{p}\mathrm{h}}$, at the BH’s sphere-of-influence radius, such that core-S'ersic (high S'ersic index, $n$) and (low-$n$) S'ersic galaxies define different relations with total rms scatters 0.21~dex and 0.77~dex, respectively.(Abridged)