Well, first of all, I do respect your authority on this subject, and I do appreciate you taking the time to teach me/us something about it. That's why I wanted to engage on this subject in the first place.

At the risk of infuriating you, let me just seek/make clarification on these points:

1.

T1 to T2 will be faster for the ball, not slower. It will be faster through sheer volume.

Are you saying it's cooling faster because the ball is bigger? If so, how would a bigger cube cool in comparison? Or, more precisely, a cube of the same size (mass) as the sphere? Or do you mean the distance of the surface of the sphere to the center where T3 is? If so, isn't the distance from the surface to the center of the cube sometimes further (depending on the surface point)?

2. I never said that the rate of cooling for spheres vs cubes should be the same. In fact, from the very beginning I've asserted that

obviously the ice with more surface area will cool faster than a sphere

The graph in your response shows that, yet you are insisting that I'm wrong. That chart does not show how much of the ice has melted, which is the key question I had posed.