High to low temperature: $O(N)$ model at large $N$

We study the $O(N)$ vector model for scalars with quartic interaction at large $N$ on $S^1\times S^2$ without the singlet constraint. The non-trivial fixed point of the model is described by a thermal mass satisfying the gap equation at large $N$. We obtain the free energy and the energy density for the model as a series at low temperature in units of the radius of the sphere. We show these results agree with the Borel-Pad\'{e} extrapolations of the high temperature expansions of the free energy and energy density obtained in our previous work. This agreement validates both the expansions and demonstrates that low temperature expansions obtained here correspond to the same solution of the gap equation studied earlier at high temperature. We obtain the ratio of the free energy of the theory at the non-trivial fixed point to that of the Gaussian theory at all values of temperature. This ratio begins at $4/5$ when the temperature is infinity, decreases to a minimum value of $0.760753$, then increases and approaches unity as the temperature is decreased.