David Springer assumes [not quite rightly] “Maximum entropy is when there is no mechanical energy gradient.”
Well *THERE’S* yer problem!
Fact The well-validated Sackur-Tetrode entropy function depends upon *BOTH* mechanical energy *AND* particle density.
Consequence The distribution of density and temperature, in a gravitational column at maximal entropy, is stationary under variations of *BOTH* mechanical energy *AND* particle density.
Result When density and energy *BOTH* are varied, subject to conservation of total energy and total mass, and the total entropy is required to be stationary, then the gravito-thermal effect disappears, and all the results of standard thermodynamics (like Boyle’s Law) are obtained.
It is a pleasure to concretely assist your thermodynamical understanding, David Springer!