you will find Center of Mass is indeed correct to such an extent that unless you are in a quantum world you will not doubt it.
Let's check that:
Consider a sphere. The center of mass is located at the Geometrical center. It is a fixed point. Let us place a test-particle a few diameters away from a solid sphere on the right along the x-axis. (make the radius of the sphere 1 unit, and put the sphere at the origin). According to the CMT, the mass acts as if it were concentrated at the centre and the distance will be measured centre to centre for purposes of using Newton's Gmm/d^2 formula.
(1) divide the sphere logically into two halves vertically. Each half will have its own centre of mass, located 3/8 radians down the axis of symmetry. The actual position is n't important, but because it is an average mean of the atoms in the solid, it is fixed relative to the geometric skin of the half-sphere.
(2) Naturally, the CM for the other half will be the same distance away from the geometric centre of the sphere in the opposite direction away from the test-particle. The total force will be the vector addition of the two halves. But by inspection this is impossible. The increase in force for one half the mass now located closer cannot balance the decrease for the other half, because while the distances are equal, the forces have changed by an unequal amount. Gravity is an inverse exponential force.
(3) The actual force calculated by summing the halves separately will be larger than the one calculated treating the sphere as a whole. Conversely, the force calculated by dividing the sphere horizontally will be weaker than the 'whole' force, since vertical components from each half will cancel.
(4) The Centre of Mass Theorem contradicts itself, and also the Sphere Theorem as well, which is a special case of the CM.
So as it turns out, the failure of the theorem is not directly connected to absolute sizes, or the discrete localization of mass or charge, but simply the error is directly connected to the ratio of the effective radius of the objects versus the distance between them.
Congratulations, you have discovered that center of gravity is not the same as center of mass!
No, I have discovered what I suspected:
That people often confuse and interchange the names of three related but different theorems:
Centre of Mass (CM), Centre of Gravity (CG) and the Sphere Theorem (ST).
We are not discussing the Centre of Gravity concept at all, which only applies in a uniform gravitational field. I could start another thread for that if you wish.
