The Flat Sphere... can *anyone* solve this one?

First, we must consider a hypothetical sphere of unspecified and irrelevant physical attributes.  Only a few basic properties matter:

1) It is a physical object.

2) It exists somewhere in the universe.

3) It is solid, or atleast has a solid surface.

4) It is perfectly spherical.

5) Its size is dynamic (explanation below).

So, a completely round solid spherical object... a cue ball from a pool table suffices perfectly for this hypothetical object.  Let us also assume that physical forces such as gravity, electromagnetism, inertia, and angular momentum are not at work, or atleast that this object is immune to them and they are irrelevant for our purposes; our cue ball is not going to be collapsing under its own gravity or breaking apart due to shearing forces.  Our sphere, for that matter, is not even comprised of atoms or other particles, but is constructed merely from a solid material.  While it exists somewhere in our universe, let us consider it to be indestructible.

And, finally, to the crucial element.  We can take this sphere and increase or decrease its diameter, and thus its radius, circumference, surface area, volume, etc., at will.  Although it really makes no difference, let's assume it begins at the size of a normal cue ball, a few inches in diameter.  From here, we can manipulate it and alter its size down to the diameter of a neutrino or smaller, or up to the diameter of the Milky Way or larger.  We can make it as large or as small as we possibly want.  Now, here is the riddle: As the diameter of a sphere increases, its curvature decreases, and vice versa.  So, what size does our object have to be in order for the surface of the sphere to become *completely flat*?

There is, in fact, despite first appearances and protests to the contrary that I have received in the past, a correct answer, and only *one* correct answer, to this riddle.  In addition, *all* of the information necessary to arrive at the correct answer has been provided here, so I’m afraid no further help or details can be given.  Best of luck!