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.
Categories: math.AT math.KT