are the same too, it follows that for the so-obtained family $P_z$ of twisted polygons we have $P_1 = P$. Finally, we need to show that the projective equivalence classes of $P$ and $P_z$ are related by scaling $[P_z] = R_z[P]$. To that end, we use formulas expressing corner invariants in terms of coefficients of a recurrence relation satisfied by the lifts of vertices. Arguing as in the proof of [@ovsienko2010pentagram Lemma 4.5] one gets the following expressions for the corner invariants of $P$: $$\begin{aligned}
    x_{i+2}=\frac{a_ic_i}{b_i b_{i+1}} \qquad y_{i+2}=-\frac{b_{i+1} }{a_i a_{i+1}}. \end{aligned}$$ Accordingly, since equations and encoding $P$ and $P_z$ are connected by the transformation $b_i \mapsto z^{-1}b_i$, $c_i \mapsto z^{-1}c_i $, the corner invariants of $P_z$ are given by $$\begin{aligned}
    x_{i+2}(z)=\frac{a_i(z^{-1}c_i)}{(z^{-1}b_i) (z^{-1}b_{i+1})} = z x_{i+2} \qquad y_{i+2}(z)=-\frac{z^{-1}b_{i+1} }{a_i a_{i+1}} = z^{-1}y_{i+2}.\end{aligned}$$ Thus, the projective equivalence classes of the polygons $P$ and $P_z$ are indeed related by scaling, as desired.

We are now in a position to prove our main result. To that end, we will compute the monodromy of the polygon defined by , take its derivative at $z=1$, and hence find the infinitesimal monodromy.

We put the vectors $V_i(z)$ into columns of matrices as follows: define $$W_i(z):=\big[V_{i+2}(z) \quad 