In the set-up you propose, the puck would stay static. Water is (pretty much) incompressible, so if the puck perfectly touches the walls of the container, there is nowhere for the water below it go. In order for the puck to sink, the volume of water below it would have to decrease, and that can only happen if it flows upwards, between the puck and the wall of the container.

Are isometries a special class of coordinate transformation, or are they completely distinct?

Perhaps you could manually calculate the second derivative of E wrt k by obtaining numerical values from the diagram. This might be of use: math.stackexchange.com/questions/654507/…

Thanks for your response. An expansion on your thought process seems to be found here: aapt.scitation.org/doi/10.1119/1.17844 ... thanks also to @Jon Custer for pointing me in the direction of this resource.