The descent spectral sequence for topological modular forms

We prove the Gap Theorem for the spectrum of topological modular forms $\mathrm{T}\mathrm{m}\mathrm{f}$. This removes a longstanding circularity in the literature, thereby confirming the computation of ${\pi }_{\ast }\mathrm{t}\mathrm{m}\mathrm{f}$ from over two decades ago by Hopkins and Mahowald. Our approach is crucially a modern one, developing and refining many techniques in synthetic spectra.