> You still have an arbitrary threshold: at high resolution you see two disordered atoms off-axis and at low resolution you see one ordered atom on-axis. However, somewhere in between you or the program has to decide whether you still see two atoms or if the data (resolution) does not warrant such a statement and you switch to the one-atom model.

Switching between interpretations happens all the time as higher
resolution data is obtained!  Let's say at low resolution you see
density apparently for one copy of a side-chain (i.e. the density is
not of sufficient resolution to warrant interpreting it as disordered
two half side-chains) and you fit that.  To keep it simple I'm
assuming it's on a general, not a special position.  Then you collect
high-resolution data and now you see that the same side-chain is
disordered.  Rephrasing your statement: "somewhere in between you or
the program has to decide whether you still see one (ordered, with
occupancy=1) side-chain or if the data (resolution) warrants such a
statement and you switch to the two side-chain (disordered, now with
sum occupancy=1) model".

> As George Sheldrick confirmed, there is a discontinuous transition between the two, which does not correspond to the physical reality. There is no "quantum transition" or something when the atom get closer than a certain limit to a crystallographic symmetry element. The atom does not care, its position is just determined by the local force fields and if those force fields have two local minima close together, the atom will be disordered.

I'm sorry I don't see this discontinuity that you are referring to at
all (I think you have forgotten to include the symmetry copy), and I'm
certainly not claiming there is any "quantum transition".  Let's start
with a disordered (1/2 occupancy) atom off a 2-fold axis and see what
happens to the electron density as it approaches and finally sits on
the 2-fold.  Here are the electron densities (this would obviously
look at lot better graphed - my apologies!):

     1   6  10  6  1   *   1  6  10  6  1

Now move the atom closer in steps to the axis so it overlaps more and
more with its symmetry copy:

                           *

         1   6  10  6  2  6  10  6  1

           1   6  10  7 7  10  6  1

            1  6  11  12  11  6  1

               1  7  16  16  7  1

                2  12  20  12  2

                          *

On the final step the fully overlapped atom has twice the occupancy
(i.e. 1 instead of 1/2) as the original as evidenced by a peak height
of 20 units, compared with 10.  In which step did the discontinuity
occur?  Clearly we could make the steps as small as we like, and you
would see a smooth transition from 2 1/2 atoms to 1 whole one.


> The decision to switch from a model where the atom is added once with full occupancy to the fourier transform calculation, or whether the atom is added twice with half occupancy is an arbitrary decision, made by the programmer or the user of the program.

I completely agree, both ways of doing it work equally well and it's
all down to convention.  As I pointed out to Dale, the way I'm
describing does work in practice, as evidenced by the fact that
CRYSTALS which does it the way I describe, has been doing it this way
for the last 40 years.  So I can't accept that it can't work in
practice when plainly it does!

This issue here is purely one of divergence of agreed convention (CIF,
mmCIF & PDB) and practice.

Cheers

-- Ian