Shell Structure: StdQM vs RealQM

A fundamental conception of atom physics is that the electrons surrounding an atomic kernel are arranged in a sequence of shells $S_n$ for $n=1,2,3,...$ with $S_n$ containing $2n^2$ electrons when filled, which gives the Periodic Table with periods 2, 8, 8, 18, 18, 32,,,  including repetitions.

A fundamental question in Standard Quantum Mechanics StdQM is if the shell structure of the Periodic Table is carried by solutions of the Schrödinger equation for the atom? Can an answer be given when such solutions are uncomputable because they involve $3N$ spatial dimensions for an atom with $N$ electrons? 

• Does the shell structure of an atom come out from StdQM? 
• Is the Periodic Table well explained by StdQM? 

The view of Eric Scerri as world leading expert on the subject is summarised as follows by chatGPT:

• In short, Scerri agrees that quantum mechanics supplies the essential skeleton of the periodic system, but he rejects the stronger claim that Schrödinger’s equation alone “explains” the periodic table in a purely deductive sense. The full story, in his view, requires a blend of quantum theory, empirical ordering principles, and chemical reasoning.

OK, so the answer is No rather then Yes. 

On the other hand, in RealQM as an alternative to StdQM, the shell structure comes out in a deductive sense as solution to a non-overlapping electron packing problem resulting in the shell structure of the Periodic Table. Details are given in the RealQM book.