 Solution

For constant mean motions, consecutive mean conjunctions are separated by $P_{\mathrm{syn}}=P_b{P}_c/(P_c-P_b)$ and advance in longitude by $\Delta \Lambda =2\pi /(P_c/P_b-1)$. To follow one of the $q$ branches, compare conjunctions $q$ steps apart. At exact resonance that advance is $2\pi p$; subtract it to measure the slow drift. Hence conjunction-pattern precession near a mean-motion resonance isThe printed expression is its leading small-offset term:Positive $\delta$ gives retrograde drift and negative $\delta$ prograde drift. The tracked-branch rotation time is about $q{P}_b/(p|\delta |)$; the full set of $q$ interchangeable directions repeats after a rotation $2\pi /q$, giving the usual mean-pattern super-period $P_b(1+\delta )/(p|\delta |)$. The slow, continuously tracked branch interpretation assumes the resonance neighbourhood $|\delta |(p+q)\ll q$; merely $|\delta |\ll 1$ need not give a slow pattern for an arbitrarily large integer ratio $p/q$. At $\delta =0$ there is no offset-driven drift.

The allowed directions can consequently move through the observer's line of sight. Triple exoplanet transits require the $bc$ and $cd$ patterns, together with their timing phases, to overlap that line of sight within the finite exoplanet transit windows; differing drift rates can open and close such windows. A small offset neither guarantees a triple event nor prohibits it permanently.

Moreover, in a true mean-motion resonance, bounded ${\varphi }_{bc}$ can coexist with a period offset because $(p+q)n_c-p{n}_b=q{\dot{\varpi }}_c+{\dot{\varphi }}_{bc}$. The mean conjunction pattern then follows apsidal precession plus the resonant-argument libration correction. For appreciable orbital eccentricity, the true-conjunction geometry has the periodic corrections identified above; the drift calculation is a mean-phase result.