Pauli Exclusion Principle vs Periodic Table

In 1925 Wolfgang Pauli introduced a 4th quantum number into the budding new physics of Quantum Mechanics QM in order to explain the observed two-valued periodicity of the Periodic Table PT with $2\times n^2$ electrons in shell $n=1,2,3,...$ giving the sequence $2, 8, 18, 32,..$. 

Pauli was unhappy with his "two-valuedness" or as an ad hoc pick without physics. He was comforted a bit by Uhlenbeck and Goudsmit who named it spin with two values "spin-up" and "spin-down" still without physics. 

What emerged was Pauli's Exclusion Principle PEP stating that two electrons with different spin can occupy the same atomic orbital, but not with same spin, which quickly came to serve a fundamental role in QM. But Pauli was still unhappy with PEP when he for the “discovery of PEP” in 1945 received the Nobel Prize in Physics, because PEP lacked physics and so could be “discovered” as physics, just imagination. What in fact prevented two electrons with same spin to occupy the same orbital? It was like a law prohibiting same-sex marriage, unphysical and no longer valid.

Today PEP is enforced asking wave functions to be anti-symmetric motivated by a cocktail of Lorentz invariance, locality and stability taken from (relativistic) Quantum Field Theory QFT.

The argument chain is: 

• QFT implies anti-symmetry. 
• Anti-symmetry implies PEP.
• PEP implies two-valuedness/spin.
• Two-valuedness/spin is observed in PT. 
• Conclusion: QFT, antisymmetry, PEP and two-valuedness/spin is all confirmed. 

But the logic is the incorrect logic of confirming an assumption by observing a consequence, as noted by Aristotle. 

Back to the PT: The actual periodicity observed is 2, 8, 8, 18, 18, 32, 32,.. with repetition of periods, and this is not explained by PEP not really by QM either, as remarked by Eric Scerri as authority of PT.

How then to explain the actual periodicity? Let us take a look at Real Quantum Mechanics RealQM as an alternative to the Standard Quantum Mechanics StdQM of above with anti-symmetric wave functions.

In RealQM electrons appear as charge densities with non-overlapping supports and the arrangement of electrons in an atom becomes a packing problem. It starts with the two electrons of Helium packed to occupy two half-spheres meeting at a common separating plane. 

This arrangement serves as origin of two-valuedness with the next shell to be filled consisting of two half-shells each one allowing a natural division into $2\times 2$ subdomains, the next one into $3\times 3$ subdomains altogether forming the original sequence of periods $2\times n^2$. 

The period doubling can then be explained as the result of electron packing where the next shell to be filled after 2 and 8 is not wide enough to allow division into $3\times 3$ only $2\times 2$ et cet.

It appears thus that RealQM can give an explanation of the periodicity of PT based on solid physics of packing of electron charge densities. 

Pauli passed away in 1958, and since then there is nobody questioning PEP by asking for physics. Maybe there is still reason to do so? To explain the repeated periods of PT?

PS A $n\times n$ subdivision of a half-shell reflects eigenfunction configuration of a vibrating square membrane, which connects the the orbitals of StdQM given by the eigenfunctions of the Hydrogen atom.