In cases when the \(4s\) shell are filled up first, it is because the last solution overlaps with the last but one.
In which case \(4s\) is physically closer to the center than the four solutions on the last but one set of curves. Such overlaps results in the higher shell being filled up first.
For the diagram above, when fully filled, it is the element Zinc (Zn). The following diagram shows the furthest curve overlapping by one pair of solution with the last but one curve. When filled up to the \(4s\) shell, this is titanium (Ti).
Although \(4s\) is a solution closer to the center than the last pair of solutions for the \(3p\) shells, it is a solution based on all particles before it, taken as a whole, around which the next particle orbits with velocity \(v^2\). Each set of curves provides for the addition of four possible particles. This means for the case of electrons around a positive nucleus a possible charge discrepancy from \(+7\) (after losing seven electrons) to \(-7\) (after gaining seven electrons) due to a single set of curves. (\(+3\) to \(-3\) due to a single loop pair of the curve.) Higher discrepancy are possible when the curves overlap.
The first set of curves may provide 2, 4, 6 or 8 solutions. Beyond the first set of curves, the next solutions come in 4 pairs (total 8). These curves unfortunately does not fit well with the boundaries of electronic shells (1s, 2s, 2p, 3s...). Grouping the electron into shells based on their relative distance from the center does not fit the solutions provide for by considering \(\psi\), energy density.
The last pair of solutions is still the furthest. And we can make the correspondent that a closed shell is a single solution with a set of curves of eight solutions.
