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\end{tabular}  \caption{$\chi ^2$ and reduced $\chi ^2$ for various statistical samples of models described in text. Disk models (i.e. M31-like models, including the model based on M31 iteself) are strongly disfavored relative to our COP model or the simple isotropic case.}  \end{table}  In Table \ref{tab:chis} we give the $\chi ^2$ values describing the goodness of fit for several statistical samples of modelled systems. We first quote the $\chi ^2$ considering only $\alpha \lt 45^{\circ}$, corresponding to systems with satellites in close to oppositely aligned configurations. We then give the $\chi ^2$ for the full domain, $0^{/circ} \lt \alpha \lt 180^{\circ}$. The $\chi ^2$ are displayed this way in order to illustrate how the picture changes when full domain of $\alpha$ is considered, as in Figures \ref{fig:full} and Figure \ref{fig:zoom}. We will examine the statistical samples in more detail in the remainder of this section.  \subsubsection{Disk Model and M31 model}  In Figure \ref{fig:disks} we examine the kinematic SDSS data in comparison with our ``disk model." We plot the fraction of satellite pairs that are corotating as a function of the opening angle $\alpha$. The blue line corresponds to a statistical sample comprised only of satellite disks, the green line to a sample comprised 50\% of satellite disks and 50\% of isotropic satellites, and the red line 10\% of disks and 90\% of isotropic satellites. The dashed green line corresponds to a sample with 50\% M31 model and 50\% isotropic composition. The black line is a purely isotropic sample.  We find strong disagreement between the disk models with $\geq 50\%$ and the teal line denoting the SDSS data. Significantly, the presence of inclined, rotated and out-of-phase planes in the toy models results in a significant signal in the $\sim 20^{\circ} \, \lt \alpha \, \lt \sim 60^{\circ}$ regime which is not seen in the data. The data does agree reasonably well with models where 90\% of the hosts have satellites distributed isotropically in phase space; however, the 90\% isotropic model fails to reproduce the sharpness of the decline in corotation at $\alpha \sim 10^{\circ}$. The M31 sample is similar to the 50\% disk/50\% isotropic sample with the posiitoins of the satellites tailored to match the positions of M31 satellites. The Our results seem to strongly exclude the possibility of coherently rotating disks, the objection could be raised that the velocity selection criteria used to select the SDSS systems systematically removes all inclined, rotated and out-of-phase systems; we will address this possible objection in a later subsection.  \subsubsection{COP model}  Figure \ref{fig:bells} shows the same information as Figure \ref{fig:disks} with the disk model being replaced by the COP model. Due to the strong constraints of the model, the pure COP corotation fraction model is undefined over much of the regime (satellite pairs separated by e.g. 60 degrees simply do not exist no matter how the model is rotated), therefore we do not plot it. Instead, we plot only the 50\% isotropic and 90\% isotropic cases, both of which are in quite good agreement with the SDSS data. This agreement is particularly notable at small $\alpha$, where the sharp dropoff in corotating fraction at $\alpha \sim 10^{\circ}$ is captured by the model. Whether or not the COP model is physical will be discussed at length in \ref{sec:Discuss}; we conclude this subsection with the claim that the ``dumbbell model" is the better fit to the kinematic data than the disk model.  \subsection{Velocity Modeling and Cuts}  As mentioned previously, the argument that the disk model fails to adequetely describe the data depends on the extent to which the selection cuts adequately exclude edge-on and out-of-phase disks from our SDSS sample, if such things exist. This argument can be summarized by the claim that it would be difficult to observationally distinguish an edge-on rotating disk from an oppositely-aligned pair rotating in the line-of-sight direction. Previously, we had assigned a charactaristic velocity of 100 km/s to the satellites and only cut on satellites whos 1D velocity offset was less than $\sqrt{2} \times 25$ km/s. Since the toy model is essentially scale-free, this is equivalent to cutting on satellites that have a 1D velocity offset less than $\sqrt{2} \times 25\%$ of the charactaristic 3D velocity for satellites of the host, i.e. a velocity threshold of $0.25 \times V_0$ is applied, where $V_0$ is the charactaristic 3D velocity of the satellites.   Figure \ref{fig:vel_limit} shows how the shape of the 90\% isotropic disk sample changes wihen this velocity threshold chenges. The lines in the figure span from cutting on 10\% of the 3D velocity to cutting on 80\% of the 3D velocity. Overplotted is the SDSS data; it is apparent that reproducing the lack of signal at $\sim 20^{\circ} \, \lt \alpha \, \lt \sim 60^{\circ}$, or correspondingly the sharpness of the upturn at very small $\alpha$, requires a very high velocity threshold, cutting on 60-80\% of the characteristic velocity. Since we impose a cut of $\sqrt{2} \times 25$ km/s on our data, this would lead us to hypothesize a characteristic 3D velocity of $\sim 60$ km/s or lower for the SDSS satellite population. If this were to be the case, then we would expect the systems selected to be sufficiently edge-on to account for the lack of signal detected at $\alpha \sim 30^{\circ}$, however the figure indicates that this is not the case. Rather, the mean 1D velocity of the SDSS satellites is XXX $\kms$ and the mean 1D velocity of the 25 (?) pairs with $\alpha \lt 10^{\circ}$ is XXX $\kms$, consistent with the overall sample. This seems to indicate that a velocity cut of $\sqrt{2} \times 25$ km/s would be insufficient to isolate only edge-on planes, and if such systems existed, planes of moderate inclination would be seen in the data. Since they are not, we disfavor the idea that the data supports the ubiquity of corotating planes in SDSS in satellites brighter than r = -16.