  In the seminal proposal by Fu and Kane, the superconducting proximity effect
was predicted to transform the surface state of a topological insulator (TI)
into a topological superconductor, forming a nonchiral 1D Majorana state within
a linear Josephson junction on the TI surface. The hallmark of this 1D Majorana
state is a robust gap closing as a function of the superconducting phase
difference $\varphi$ across the junction, which alternates in and out of the
topological phase. These topological phase-transitions occur at $\varphi =
(2n+1)\pi$ with integer $n$, leading to a $4\pi$-periodicity of the ground
state. While the $4\pi$-periodicity has been indirectly inferred in the AC
Josephson effect, the direct observation of the 1D Majorana state in a TI
Josephson junction has remained contentious. Here, we report the direct
observation of topological phase-transitions in a TI Josephson junction, where
the local density of states is probed via tunnel contacts and $\varphi$ is
controlled by a flux loop. The observed transitions are independent of the
chemical potential, reinforcing their topological origin. Under an applied
perpendicular magnetic field, Josephson vortices form, making $\varphi$
position-dependent. In this case, the gap closing occurs locally at the
Josephson vortex cores where $\varphi = (2n+1)\pi$, which we also observe. Our
findings provide direct confirmation of the Fu-Kane proposal and robust
evidence for the emergence of topological superconductivity in a TI Josephson
junction.
