We reveal properties of global modes of linear buoyancy instability in stars, characterized by the celebrated Schwarzschild criterion, using non-Hermitian topology. We identify a ring of exceptional points of order 4 that originates from the pseudo-Hermitian and pseudochiral symmetries of the system. The ring results from the merging of a dipole of degeneracy points in the Hermitian stably-stratified counterpart of the problem. Its existence is related to spherically symmetric unstable modes. We obtain the conditions for which convection grows over such radial modes. Those are met at early stages of low-mass stars formation. We finally show that a topological wave is robust to the presence of convective regions by reporting the presence of a mode transiting between the wavebands in the non-Hermitian problem, strengthening their relevance for asteroseismology. Published by the American Physical Society 2024