An algebraic approach to certain cases of Thurston rigidity

In the parameter space of monic centered polynomials of degree $3$ with marked critical points $c_1$ and $c_2$, let $C_{1,n}$ be the locus of maps for which $c_1$ has period $n$ and let $C_{2,m}$ be the locus of maps for which $c_2$ has period $m$. A consequence of Thurston's rigidity theorem is that the curves $C_{1,n}$ and $C_{2,m}$ intersect transversally. We give a purely algebraic proof that the intersection points are $3$-adically integral and use this to prove transversality. We also prove an analogous result when $c_1$ or $c_2$ or both are taken to be preperiodic with tail length exactly $1$.