The filling of the 4s sub-shell occurs before the 3d sub-shell due to its lower energy level, as dictated by the Aufbau principle.

The Aufbau principle is a core concept in quantum mechanics that describes the sequence in which electrons occupy atomic orbitals. It asserts that electrons will fill the lowest available energy levels first before progressing to higher levels. Consequently, the filling order starts with the 1s sub-shell, followed by 2s, 2p, 3s, and 3p, and continues in a similar manner.

However, the filling order is not strictly determined by the principal quantum number ($n$). This is primarily because the varying shapes and orientations of different types of orbitals (s, p, d, f) can impact their energy levels. For example, the 4s orbital is spherical, while the 3d orbital has a more complex cloverleaf shape. This difference in geometry results in the 4s orbital having a lower energy than the 3d orbital.

Moreover, the energy levels of orbitals can be affected by electron-electron interactions. When an electron occupies an orbital, it may increase the energy of that orbital due to repulsion between electrons. This effect is particularly pronounced in the d and f orbitals, which can accommodate more electrons and thus experience greater electron-electron interactions compared to the s and p orbitals. As a result, the 4s orbital, which can hold only two electrons, maintains a lower energy than the 3d orbital, which can hold up to ten electrons.

In summary, according to the Aufbau principle, the 4s sub-shell is filled before the 3d sub-shell because it possesses a lower energy level. This illustrates the intricate and often counterintuitive nature of quantum mechanics in determining atomic structure.