Free-energy formula for the one-dimensional 3-permaspin model

Let $c=e^{-2\beta H}$ and $d=e^{-2\beta J}$, and consider the one-dimensional 3-permaspin model with couplings $J$ and $H$. Let $z$ be the unique root of the cubic

$$\lambda^3-(cd^2+c^2+2cd+c+1)\lambda^2-c(cd^3+3c^2d+cd^2+c^2+cd+c+3d+1)(d-1)\lambda+(d^2+4d+1)c^3(d+1)(d-1)^3=0$$

whose argument satisfies $\mathbf{\arg}(z)\in[2\pi/3,\pi]$. Free-energy formula. The free energy is

$$f^{(\mathrm{cddes})}(\beta)=-(J+H)-\frac{1}{\beta}\ln\left(c^2d+c^2+2cd+c+1-2\mathbf{Re}(z)\right).$$

This summarizes the preceding experimental observations and spectral analysis of the transfer matrix. The parser provides no evidence that the formula has been proved, so its status is left open.