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The energy released from the crystalization goes towards increasing the potential energy of the top of the water.

Freezing water releases energy to the surroundings, usually in the form of heat (the water gets colder, while the surroundings get warmer). It's the same effect as a handwarmer.... when the pack crystalizes, the energy released is absorbed by your hands, which get warmer.

In your hypothetical, (if the ball actually does raise up) I'd expect the water to freeze at an infinitesimally lower temperature than typical at your conditions.

I'd more expect the water to rise around the ball, but certainly can't know until doing it haha.

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The increased gravitational potential energy comes from the work done by the pressure exerted by the ice on the ball as the water volume expands during freezing. The pressure causes the ball to move upward, thus doing work on the ball and increasing its potential energy with respect to gravity. The work done is equal to the change in potential energy, so the increased potential energy comes from the work done by the pressure.

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The ball will raise the water level in the container, which increases the average pressure, so the freezing point decreases by an extremely tiny amount.

We are talking about really small effects here. Freezing one liter of water releases 334,000 J. Raising a 10 gram ball by a centimeter needs 0.001 J.

The more measurable effect is if you have a closed column of water with a piston at the top and freeze it, the increased pressure can do work by moving the piston up. Conversely if you increase the pressure on the piston it will melt the ice.

This is how freeze-thaw erosion works when water enters cracks in rocks, freezes and breaks the rock as it freezes.

So that thermal energy "came from" the liquid water, and the water was liquid in the first place due to energy from the sun.

id describe it as follows: the movement (heat) of the water molecules cause them to be liquid. as heat is lost to the surroundings the molecules move slower, and eventually can "snap" to eachother due to the difference in electrical charge that is a property of the water molecule. the solid structure of ice will form and, being of lower density, in effect convert some heat into height, including the ball on top

Maybe not the Sun itself, any source of energy would work (some chemical reaction such as fire, nuclear, even mechanical energy from stirring the water would convert to some thermal energy), but yes. It was already in the water when the ball was floating.

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I honestly haven't seen an answer here that addresses the real question:

How can you remove energy from a system (lowering it's temperature) and yet end up with more potential energy on the ball?

I don't have the answer but I realize that while the entire system will have less energy overall, some of the energy will get rearranged by the freezing process and the ball will end up with a bigger percentage of the total energy in the system after freezing. I just don't know exactly how it happens.

It’s been too long since I sat in a physics (or thermodynamics) class to say with certainty, but my guess is that the answer has to do with “types” of energy and sounds something like “some of the thermal potential energy translated to gravitational potential energy”. They alluded to this in the above comments too with the notes on the energy required to move the ball - if the ball wasn’t there, then additional (thermal kinetic?) energy would have been released into the environment.

To say it another way - the gravitational potential energy isn’t the only energy in the system. The total energy of the system after freezing will still be less then before despite the increase in gravitational potential energy due to the decrease in thermal energy.

Because water is really weird! It's unusual for a compound to expand in volume as it goes from liquid to solid. Water freezing is an exothermic process: heat must be given off. While there is net loss, the amount of kinetic energy that water needed in it's liquid, kinetic form is still less than the energy needed that was converted into potential energy with the raising of the ice.

You're correct, significant kinetic energy is present in the liquid form and lost during the transition, some of that is converted to potential energy by way of the 9% decrease in the density of the ice compared to liquid water.

Latent heat. At higher pressures water freezes at lower temperatures. At higher pressures the latent heat of fusion increases.

As a material passes from liquid phase down to solid energy is released. Same as how steam releases a lot of heat as it condenses on your hand when you delid a boiling pot of water. This is called latent heat of fusion, and latent heat of vaporization.

Water is weird. It is most dense at 4°C then begins to expand toward 0°C as it forms transient nano scale clusters of molecules. At high pressures these nano clusters are less able to form.

Ice has less thermal energy than water at the same temperature, which more than offsets the slight increase in gravitational potential energy due to ice Ih (the stable form at STP) being less dense.

On a molecular level, when the water molecules have slowed enough with cooling they can form permanent (ish) hydrogen bonds with each other and the shape of the molecules creates the ice crystal structure, even though this means pushing the molecules apart a bit more.

Oh, so that's why water heats up slightly when it freezes... I could never quite understand that, thanks kind stranger