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In filtration 1 of the Adams spectral sequence, using secondary cohomologyoperations, Adams computed the differentials on the classes $h_j$, resolvingthe Hopf invariant one problem. In Adams filtration 2, using equivariant andchromatic homotopy theory, Hill--Hopkins--Ravenel proved that the classes$h_j^2$ support non-trivial differentials for $j \geq 7$, resolving thecelebrated Kervaire invariant one problem. The precise differentials on theclasses $h_j^2$ for $j \geq 7$ and the fate of $h_6^2$ remains unknown.  In this paper, in Adams filtration 3, we prove an infinite family ofnon-trivial $d_4$-differentials on the classes $h_j^3$ for $j \geq 6$,confirming a conjecture of Mahowald. Our proof uses two different deformationsof stable homotopy theory -- $\mathbb{C}$-motivic stable homotopy theory and$\mathbb{F}_2$-synthetic homotopy theory -- both in an essential way. Along theway, we also show that $h_j^2$ survives to the Adams $E_5$-page and that$h_6^2$ survives to the Adams $E_9$-page.

The Adams differentials on the classes $h_j^3$