Thermal winding dominance of a free rotor

For the quantum rotor, low temperature makes the angular-momentum sum converge fastest: n=0 gives $Z\to 1$. High temperature makes the temporal winding sum converge fastest: m=0 gives $Z\sim \sqrt{2\pi I/(\beta {\hslash }^2)}$. At low temperature many winding sectors contribute and cancel the vanishing fixed-endpoint prefactor. A zero-winding approximation valid at high temperature cannot be extended to low temperature merely because winding paths look geometrically longer.