[Cleonis]

Sure enough, the principles of Carnot's thermodynamics and the premises of statistical mechanisc look quite differently. The thing is: since both form the same thermodynamics there must be a connection.

I submit: the qualification 'completely independent axioms' is incorrect. There is the observation: temperature is transitive. This transitive property is a statement of conservation (In terms of Carnot's thermodynamic it used to be thought of as conservation of Caloric.) The concept of Conservation of a quantity correlates with information.

We have that statistical mechanics subsumed Carnot's thermodynamics.

The laws of Carnot's thermodynamics are theorems of statistical mechanics. (Those theorems weren't necessarily stated explicitly. I'm saying the principles of statistical mechanics are sufficient to imply the laws of Carnot's thermodynamics.)

[krastanov]

Statistical mechanics needs to assume the existence of atoms to give you the results of Carnot's. Carnot's thermodynamics does not. That makes it a bit less straightforward to say one strictly follows from the other. Otherwise the gist of what you say seems reasonable.