This question prompts me to ask something more specific about the periodic table. As far as I know, the main significance of the periodic table is that

This question prompts me to ask something more specific about the periodic table. As far as I know, the main significance of the periodic table is that

Added 22, November:

I've succeeded in making the question entirely unintelligible with all my additions. So I thought I would summarize it in the simplest form I could manage and add it to the title. The question is thereby somewhat narrower in scope than my original query, but I would be happy to have this focused version answered.

The main point is that the loose answer

(A) because the outer shell is filled

frequently heard is definitely wrong. If we take the usual definition of a shell, all the noble gases but Helium have just the s and p subshells (corresponding to the representations $V_0\otimes S$ and $V_1\otimes S$) filled in the outermost shell, and this is not a full shell once the atom is bigger than Argon. The wikipedia article on noble gases offers an amusing formulation whereby, for a noble gas, 'the outer shell of valence electrons is considered to be "full"' (my emphasis).

All this led to my initial confusion: I thought that the term 'shell' must mean something else for multi-electron systems in a manner adapted to the answer (A). Such an alternative definition does not seem to exist, and it is not at all obvious how to come up with one in a non-tautological way. So I believe the essence of what I am asking is whether or not there is

a natural mathematical explanation for the stability of noble gases

that is more or less independent of experiments confirmed by difficult computations using approximation schemes.

Anyone interested in the background and details is invited to read the incoherent paragraphs below, or to look up a proper reference like Atkin's book on physical chemistry.

L-shell $2s\oplus 2p$

and so on. (In case you're wondering, the representations after $V_3\otimes S=f$ are labeled consecutively in the alphabet.) In this form, a shell does not necessarily consist of a grouping of subshells of similar energies. Rather, the conventional description says that more than one shell can be incomplete in an atom. For example, in the fourth row of the periodic table, starting with potassium (K) and up to copper (Cu), they say both the $M$-shell and the $N$-shell are incomplete. If this is confusing to you, I suggest you don't worry about it. I just wanted to point out that my question `what is a shell?' has a clear-cut answer in this usage. If we don't want to go against this convention, we need to stick to the version

L-shell: $2s\oplus 2p$

and so on. (In case you're wondering, the representations after $V_3\otimes S=f$ are labeled consecutively in the alphabet.) The key point is that, in this form, a shell does not consist of a grouping of subshells of similar energies when more than a few electrons are present. Rather, the conventional description of the phenomena says that more than one shell can be incomplete in an atom. For example, in the fourth row of the periodic table, starting with potassium (K) and up to copper (Cu), they say both the $M$-shell and the $N$-shell are incomplete. If this is confusing to you, I suggest you don't worry about it. I just wanted to point out that my question `what is a shell?' has a clear-cut answer in this usage. If we don't want to go against this convention, we need to stick to the version