Simultaneous transit in a resonant chain

In leading mean-conjunction geometry a common observer phase $x$ for two resonant arguments requires $qx={\varphi }_1$ and $nx={\varphi }_2$ modulo $2\pi$. With $d=\gcd (q,n)$, the exact modular criterion isNecessity follows by subtracting the two congruences after multiplication; sufficiency follows from the integer linear combinations of $q$ and $n$, whose set is $d\mathbb{Z}$. For exterior centres $\pi$ and interior centres $0,\pi ,0$ at orders one to three, only $q=n=2$ satisfies it. This is phase compatibility, not a guarantee of an observed exoplanet transit: timing, observer orientation, resonant-argument libration widths and true-versus-mean longitude must also be considered.