We are obviously talking about two different problems.
OK, I think I get it now.
I think I was solving the wrong problem, or rather, solving only half of
the problem.
I think John M. had the right idea, or rather, a more complete
analysis. Let me explain it in my words and see if it makes sense.
Consider two thin sheets. Let's use rectangular sheets for
simplicity; disks would be harder to describe but the result would be the
same. Label the parcels as follows:
1 2 3 4 5 6 7 8
A B C D E F G H
Now, keeping the gap the same, fold the top sheet:
1 8 2 7 3 6 4 5
A B C D E F G H
At this stage, the force on the top sheet is the same as it was
originally. The force per unit area is greater, but the area is less.
Now fold the other sheet:
1 8 2 7 3 6 4 5
A H B G C F D E
Now something interesting has happened: The force on the top sheet has
doubled. The useless EFGH parcels have been brought over and made useful.
Of course in all cases the total force on one sheet is equal-and-opposite
to the total force on the other sheet. Newton's third law, conservation of
momentum, and all that.....
========================
The foregoing is half of the required analysis. The rest of the story goes
like this:
0) Start with two sheets, very large (size on the order of R*N), very thin
and dilute, very close together (gap on the order of R/N).
1) Pull the sheets apart, to get to the specified distance R. The force is
(to a good approximation) unchanged by this operation, according to the
Olber's argument as discussed in my previous notes.
2) Now fold the sheets up, to concentrate the mass in a region of lateral
extent comparable to R. This operation increases the force by a factor of
roughly N.