Circle actions on function field Bost-Connes systems: the compact group advantage is illusory, and the Weil zeta function is thermodynamic

The circle action coming from the degree grading on a function field Bost-Connes system is analyzed: the apparent advantage of a compact acting group is shown to be illusory for classification purposes, while the Weil zeta function of the underlying curve is exhibited as a thermodynamic quantity of the system. The algebra is stably finite, with a faithful scaling functional built from the gauge expectation, and the results are verified in full on an explicit curve, including point counts and Jacobian orders.