  We provide a simple proof that the unit map from the sphere spectrum to the
connective image-of-$J$ spectrum $\mathrm{j}$ is surjective on homotopy groups.
This is achieved using a novel $t$-structure on the category of $E$-synthetic
spectra and a specific construction of $\mathbf{F}_p$- and BP-synthetic lifts
of $\mathrm{j}$. These synthetic lifts then easily produce modified Adams and
Adams--Novikov spectral sequences for $\mathrm{j}$ which we use the prove the
above detection statement, all without ever calculating $\mathbf{F}_p$- or
BP-homology nor the associated Ext groups.
