Even a precessing clock is right twice per orbit -- The super-periods of eRO-QPE2 and challenges for quasi-periodic eruption orbital models

We present O$-$C (``observed minus calculated'') timing analysis of the quasi-periodic eruption (QPE) source eRO-QPE2 with a multi-mission X-ray campaign, which includes 32 observed eruptions spanning a month (i.e. 325 QPE cycles). In relation to accretion (e.g. disk instability) models, the O-C is consistent with a damped random walk of the QPE recurrence, albeit with highly uncertain parameters. If instead an underlying orbital clock is present, eRO-QPE2 is consistent with a period of $P \sim 2.24$\,h and two hierarchical super-periodic modulations, with periods of $\sim 4.4$\,d ($\sim47$\,P) and $\approx 95$\,d ($\approx 1000$\,P). We found no negative period derivative, with $|\dot{P}| \lesssim 2 \times 10^{-6}$\,s/s at $3\sigma$. This disfavors high-eccentricity WDs and high-mass/eccentricity IMBHs via GW decay. For disk-collision models, where the $\dot{P}$ from gas drag and the QPE integrated energy provide bounds on the local disk density, a main-sequence star is disfavored as EMRI secondary unless stellar debris streams are present, while stripped stars remain allowed. The correlated odd/even O-C disfavors both disk crossings per orbit being observed. Interpreting the data with one \emph{observed} event per orbit, the short modulation is consistent with apsidal precession for $a \sim 140\,R_g$, $e \approx 0.1$, and $M_{\rm BH} \approx 1.5 \times 10^{5}\,M_\odot$. The longer modulation (much less constrained) is inconsistent with EMRI nodal precession and disk precession is allowed for a limited parameter volume, while there is a solution with a stable hierarchical triple system with an outer massive black hole at $\sim 0.4\,\mathrm{mpc}$ and mass $\sim(0.1-1) \times M_{\rm BH}$. However, no reliable solution can be found with more robust EMRI trajectory models, possibly due to narrow likelihood peaks in a multi-dimensional parameter space with sparse data.