Logarithmic-derivative conjecture for torus and cylinder generating functions

For an $L_v\times L_h$ lattice at $z=-1$, let $G^{CC}_{L_h}$ and $G^{CF}_{L_h}$ be the generating functions along the periodic direction for the torus and cylinder, respectively, and let $P^{CC}_{L_h}$ and $P^{CF}_{L_h}$ be the corresponding transfer-matrix characteristic polynomials. Logarithmic-derivative conjecture. The generating functions are the negative logarithmic derivatives of these characteristic polynomials:

$$G^{CC}_{L_h}=-\frac{d}{dx}\ln\left(P^{CC}_{L_h}\right),\qquad G^{CF}_{L_h}=-\frac{d}{dx}\ln\left(P^{CF}_{L_h}\right).$$

The relation is specific to the torus and cylinder at $z=-1$ and the periodic direction; the source explicitly notes that it does not hold for general $z$, Möbius bands, or Klein bottles. It is used to derive further conjectural period formulas.