  The number of electrons in small metallic or semiconducting islands is
quantized. When tunnelling is enabled via opaque barriers this number can
change by an integer. In superconductors the addition is in units of two
electron charges (2e), reflecting that the Cooper pair condensate must have an
even parity. This ground state (GS) is foundational for all superconducting
qubit devices. Here, we study a hybrid superconducting-semiconducting island
and find three typical GS evolutions in a parallel magnetic field: a robust
2e-periodic even-parity GS, a transition to a 2e-periodic odd-parity GS,and a
transition from a 2e- to a 1e-periodic GS. The 2e-periodic odd-parity GS
persistent in gate-voltage occurs when a spin-resolved subgap state crosses
zero energy. For our 1e-periodic GSs we explicitly show the origin being a
single zero-energy state gapped from the continuum, i.e. compatible with an
Andreev bound states stabilized at zero energy or the presence of Majorana zero
modes.
