= Solution
A <simultaneous transit in a resonant chain> requires both <conjunction> patterns to admit the observer's longitude at the same time. In the leading mean-conjunction geometry put $x=\Lambda-\varpi_c$. The necessary phase compatibility at symmetric centres is
$$
qx=\phi_{bc,0}\pmod{2\pi},\qquad nx=\phi_{cd,0}\pmod{2\pi}.
$$
Let $d=\gcd(q,n)$. Eliminating $x$, and using that $q\ell-nk$ ranges over multiples of $d$, gives the exact modular compatibility criterion
$$
\boxed{\frac{n\phi_{bc,0}-q\phi_{cd,0}}{d}\in2\pi\mathbb Z}.
$$
Here $\gcd$ denotes the <greatest common divisor>. For the orders $q,n\in\{1,2,3\}$ above, the two phase sets intersect only when
$$
\boxed{q=n=2}.
$$
The common directions are the two quadratures relative to <periapsis>. In reduced resonances this requires $p$ and $m$ odd; otherwise the apparent second-order ratios reduce to different resonance orders. There is no additional restriction on those odd values from angular compatibility alone. A shared direction and suitable temporal phase can be chosen, and the rational period ratios allow its recurrence.
This is a potential low-amplitude configuration, not a guarantee of an observed triple <exoplanet transit>. The observer must lie in the common <orbital plane> and near an allowed direction. Finite stellar radii, nonzero <libration amplitudes of a resonant argument>, apsidal motion and asymmetric centres broaden or change the possibilities. At finite $e_c$, true simultaneous alignment obeys the more precise equations $(p+q)y-px=\phi_{bc}$ and $(m+n)x-my=\phi_{cd}$, with $y=M_c(x)$. Thus the order-only conclusion is not a universal necessary condition for every eccentric resonant chain.
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