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In The Feynman Lectures on Physics, Vol. 3, Section 15-5, there's a discussion of the stability of various benzene ions. It turns out that the ions for which the all the energy levels up to a certain energy are completely filled, are more stable than other ions.

This is supposed to be explained by figure 15-14, where a smoothed energy curve is plotted and we see that the saturated ions' actual energy lies below the smoothed energy curve.

I do not understand this argument. What's the justification for looking at this smoothed out curve? Is there some statistical mechanical meaning to the average energy of the two neighbouring energy levels? Can one make a precise mathematical statement, assuming the independent particle approximation holds exactly?

For added motivation let me note that, as much as I understand, this stability of saturated states also explains the inertness of noble gasses, and begins to explain the "magic numbers" determining the stability of nuclei.

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