\section{\uppercase{#1}}}

\begin{document}

\title{Circumventing Heisenberg's uncertainty principle in atom interferometry tests
of the equivalence principle}

\author{Albert Roura}
\affiliation{Institut f\"ur Quantenphysik, Universit\"at Ulm, Albert-Einstein-Allee 11,
89081 Ulm, Germany}

\date{\today}
\begin{abstract}
Atom interferometry tests of universality of free fall based on the differential measurement of two different atomic species provide a useful complement to those based on macroscopic masses. However, when striving for the highest possible sensitivities, gravity gradients pose a serious challenge.
Indeed, the relative initial position and velocity for the two species need to be controlled with extremely high accuracy, which can be rather demanding in practice and whose verification may require rather long integration times.
Furthermore, in highly sensitive configurations gravity gradients lead to a drastic loss of contrast. These difficulties can be mitigated by employing wave packets with narrower position and momentum widths, but this is ultimately limited by Heisenberg's uncertainty principle.
We present a novel scheme that simultaneously overcomes the loss of contrast and the initial co-location problem. In doing so, it circumvents the fundamental limitations due to Heisenberg's uncertainty principle and eases the experimental realization by relaxing the requirements on initial co-location by several orders of magnitude.
\end{abstract}

\pacs{}

\maketitle

\section{Introduction}
\label{sec:introduction}

The equivalence principle is a cornerstone of general relativity and Einstein's key inspirational principle in his quest for a relativistic theory of gravitational phenomena. Experiments searching for small violations of the principle are being pursued in earnest \cite{will14} since they could provide evidence for violations of Loretnz invariance \cite{kostelecky11a} or for dilaton models inspired in string theory \cite{damour10}, and they could offer invaluable hints of a long sought underlying fundamental theory for gravitation and particle physics.
A central aspect which has been tested \comment{to high precision}
is the universality of free fall (UFF) for test masses. Indeed, torsion balance experiments have reached sensitivities at the $10^{-13}\, g$ level \cite{schlamminger08} and it is hoped that this can be improved two orders of magnitude in a forthcoming satellite mission \cite{touboul12}.

An interesting alternative that has been receiving increasing attention in recent years is to perform tests of UFF with quantum systems and, more specifically, using atom interferometry. Instead of macroscopic test masses this kind of experiments compare the gravitational acceleration experienced by atoms of different atomic species \cite{fray04,bonnin13,schlippert14,zhou15}. They offer a valuable complement to traditional tests with macroscopic objects because a wide range of new elements with rather different properties can be employed, so that better bounds on models parameterizing violations of the equivalence principle can be achieved even with lower sensitivities to differential accelerations \cite{hohensee13b,schlippert14}.
Furthermore, given the different kind of systematics involved, they could help to gain confidence in eventual evidence for violations of UFF.

By using neutral atoms prepared in magnetically insensitive states and an appropriate shielding of the interferometry region, one can greatly suppress the effect of spurious forces acting on the atoms, which constitute excellent inertial references \cite{borde89,kasevich91,peters01}.
State of the art gravimeters based on atom interferometry can reach a precision of the order of $10^{-9}\, g$ in one second \cite{hu13} and are mainly limited by the vibrations of the retro-reflecting mirror. When performing simultaneous differential interferometry measurements for both species and sharing the retro-reflecting mirror (as sketched in Fig.~\ref{fig:AI_sketch}), common-mode rejection techniques can be exploited to suppress the effects of vibration noise and enabling higher sensitivities for the measurement of differential accelerations \cite{fixler07,hogan08,varoquaux09,bonnin13,rosi14}.
Thus, although tests of UFF based on atom interferometry have reached sensitivities up to $10^{-8}\, g$ so far, there are already plans for future space missions that aim for sensitivities of $10^{-15}\, g$ \cite{aguilera14} by exploiting the longer interferometer times available in space and the fact that the sensitivity scales quadratically with the interferometer time.

\begin{figure}[h]
\begin{center}
\includegraphics[width=2.8cm]{figure1.pdf}
\end{center}
\caption{Sketch of an atom interferometry set-up for differential acceleration measurements of two different atomic species. The various laser beams driving the diffraction processes for both species share a common retro-reflection mirror so that vibration noise is highly suppressed in the differential phase-shift measurement.}
\label{fig:AI_sketch}
\end{figure}

As shown below, however, when targeting sufficiently high sensitivities, gravity gradients become a great challenge for this kind of experiments: they lead to a drastic loss of contrast in the interference signal and impose the need to control the initial co-location  of the two atomic species (i.e.\ their relative position and velocity) with very high accuracy.
In this paper we will present a novel scheme that simultaneously overcomes both difficulties.

\section{Challenges due to gravity gradients in tests of universality of free fall}
\label{sec:challenges}

In order to analyze the effects of gravity gradients, we will make use of a convenient description of the state evolution in a light-pulse AI developed in Ref.~\cite{roura14} and summarized in \comment{Appendix~\ref{sec:state_evolution}.}
Within this approach the evolution of the interfering wave packets along each branch of the interferometer is described in terms of centered states $|\psi_\text{c} (t) \rangle$ that characterize their expansion and shape evolution, as well as displacement operators that account for their motion and whose argument $\boldsymbol{\chi} (t) = \big(\boldsymbol{\mathcal{R}}(t), \boldsymbol{\mathcal{P}}(t) \big)^\text{T}$ corresponds to the central position and momentum of the wave packet, which are given by classical phase-space trajectories including the kicks from the laser pulses.
The state at the exit port~I (analogous results hold for the exit port~II) takes then the form
\begin{equation}
|\psi_\text{I} (t)\rangle =
\frac{1}{2} \Big[ e^{i \Phi_1} \hat{\mathcal{D}}(\boldsymbol{\chi}_1)  |\psi_\text{c} (t) \rangle
+ e^{i \Phi_2} \hat{\mathcal{D}}(\boldsymbol{\chi}_2)  |\psi_\text{c} (t) \rangle \Big]
\label{eq:port_I},
\end{equation}
and gives rise to the following oscillations in the fraction of atoms detected in that port as a function of the phase shift $\delta\phi$ between the interferometer branches:
\begin{equation}
\frac{N_\text{I}}{N_\text{I} + N_\text{II}}
= \big\langle \psi_\text{I} (t) \big| \psi_\text{I} (t) \big\rangle
= \frac{1}{2} \big(1 + C \cos \delta\phi \big)
\label{eq:fraction_I}.
\end{equation}
The positive quantity $C$, known as the contrast, characterizes the amplitude of these oscillations and is given by
\begin{equation}
C = \Big| \big\langle \psi_\text{c} (t) \big| \hat{\mathcal{D}}(\delta\boldsymbol{\chi})
\big| \psi_\text{c} (t) \big\rangle \Big| \leq 1
\label{eq:contrast},
\end{equation}
which takes its maximum value when the relative displacement $\delta\boldsymbol{\chi} = \boldsymbol{\chi}_2 - \boldsymbol{\chi}_1$ between the interfering wave packets vanishes.
The full result for the phase shift is provided in \comment{Appendix~\ref{sec:state_evolution}}, but for the present discussion it is sufficient to focus on the role of uniform forces (including inertial ones) and the dependence, caused by gravity gradients, on the central position and velocity ($\mathbf{r}_0$ and $\mathbf{v}_0$) of the initial wave packet:
\begin{equation}
\delta\phi = \mathbf{k}_\text{eff}^\text{T}\, (\mathbf{g} + \mathbf{a}')\, T^2
+ \mathbf{k}_\text{eff}^\text{T}\, \big(\Gamma\, T^2\big) \,
(\mathbf{r}_0 + \mathbf{v}_0 T)
+ \ldots
\label{eq:phase_shift0},
\end{equation}
where for simplicity we have assumed time-independent accelerations and gravity gradients and kept only the contributions to lowest order in $(\Gamma\, T^2)$. We have separated the acceleration $\mathbf{g}$ caused by the gravitational field from the acceleration $\mathbf{a}'$ caused by any other forces including inertial ones and accounting also for the vibrations of the retro-reflecting mirror. On the other hand, the gravity gradient tensor $\Gamma$, defined in Eq.~\eqref{eq:Gamma}, characterizes the deviations from a uniform gravitational field.

\subsection{Initial co-location}
\label{sec:challenges_co-location}

When performing a test of the UFF based on a simultaneous differential measurement with two different atomic species (labeled here $A$ and $B$) the relevant quantity is the phase-shift difference
\begin{align}
\delta\phi^A - \delta\phi^B \approx&\
\mathbf{k}_\text{eff}^\text{T}\, (\mathbf{g}_A - \mathbf{g}_B)\, T^2
+ \mathbf{k}_\text{eff}^\text{T}\, \big(\Gamma\, T^2\big) \, (\mathbf{r}_0^A - \mathbf{r}_0^B)
\nonumber \\
&+\mathbf{k}_\text{eff}^\text{T}\,\big(\Gamma\, T^2\big)\,(\mathbf{v}_0^A - \mathbf{v}_0^B)\,T
\label{eq:co-location}.
\end{align}
Here we have assumed that $\mathbf{k}_\text{eff}$ and $T$ are the same for both species and neglected any contributions to the residual accelerations $\mathbf{a}'$ which are not common to both species. We have made these simplifying assumptions to make the presentation here less cumbersome and to highlight the essential points, but such restrictions are lifted in the remaining sections, so that the strategies and results presented there are more generally applicable.

As seen from Eq.~\eqref{eq:co-location}, in the presence of gravity gradients small differences in the central position (and velocity) of the initial wave packets for the two atomic species can mimic the effects of a violation of UFF.
In principle preparing wave packets with very well defined central position and momentum does not suffer from limitations associated with Heisenberg's uncertainty principle, which only affects their position and momentum widths. However, the required degree of control on those quantities is rather demanding in practice: for example, testing the UFF at the level of
$10^{-15} g$
entails controlling the relative initial position and velocity with accuracies better than a few nm and several hundred pm/s respectively.
In fact, this systematic effect, often known in this context as the \emph{initial co-location} problem, is one of the biggest challenges faced by this kind of experiments.
Furthermore, verifying that the stringent requirements on initial co-location are fulfilled by measuring the relative position and velocity of the two species under the same experimental conditions as in the differential interferometry measurements (otherwise one could introduce additional systematic effects) is itself limited by Heisenberg's uncertainty principle, which implies
\begin{equation}
n\, N\, \sigma_x\, \sigma_p \geq \hbar / 2 \,
\label{eq:HUP},
\end{equation}
where $\sigma_x$ and $\sigma_p$ are the precisions for the measurement of the central position and momentum that can be achieved after a given integration time, $N$ is the number of atoms in each atom cloud and $n$ is the number of experimental runs, given by the integration time multiplied by the repetition rate.
Thus, unless the number of atoms in each atom cloud is rather high, the required integration time to achieve the desired accuracy in these checks of systematics may be comparable to or even exceed the entire lifetime of the mission, which has been raised as an objection against the use of atoms (rather than macroscopic test masses) as inertial references for high-precision tests of UFF \cite{nobili15}.

\subsection{Loss of contrast}
\label{sec:challenges_contrast}

In order to understand qualitatively the motion of the atomic wave packets along the branches of the interferometer in the presence of a gravity gradient, it is useful to consider a freely falling frame where the central trajectories for the two branches after the first beam-splitter pulse are symmetric with respect to the spatial origin of coordinates at $z=0$, as depicted in Fig.~\ref{fig:gg_trajectories}. It then becomes clear that the tidal forces associated with the gravity gradient tend to open up these trajectories and give rise to \comment{an \emph{open interferometer} with} non-vanishing relative position and momentum displacements at the exit ports. They are quantitatively obtained in \comment{Appendix~\ref{sec:state_evolution}} and are given by
\begin{equation}
\delta \boldsymbol{\mathcal{R}} = \left(\Gamma\, T^2 \right) \mathbf{v}_\text{rec} \, T \, ,
\quad
\delta \boldsymbol{\mathcal{P}} = (\Gamma\, T^2) \, m\, \mathbf{v}_\text{rec}
\label{eq:deltaR_deltaP} \, .
\end{equation}
The existence of a non-vanishing relative displacement $\delta\boldsymbol{\chi} \neq 0$ between the interfering wave packets, which no longer exhibit full quantum overlap, gives rise to a loss of contrast, as reflected by Eq.~\eqref{eq:contrast}.
\begin{figure}[h]
\begin{center}
\def8.5cm{8.5cm}
\input{figure2_pdf.tex}
\end{center}
\caption{Central trajectories for a Mach-Zehnder interferometer in the presence of gravity gradients as seen in a suitable freely falling frame. The tidal forces tend to open up the trajectories compared to the case without gravity gradients (dashed lines).
The momentum transfer from the laser pulses are also indicated for the different branches.
}
\label{fig:gg_trajectories}
\end{figure}

In terms of the Wigner function \cite{hillary84}, defined in Eq.~\eqref{eq:wigner_def}, the expression for the contrast takes the following suggestive form, which is also valid for mixed states~\cite{roura14}:
\begin{equation}
C = \left | \, \int d^3x \int d^3p \, W(\mathbf{x},\mathbf{p};t) \,
e^{-\frac{i}{\hbar} \delta \boldsymbol{\chi}^\text{T} J \, \boldsymbol{\xi}} \, \right |
\label{eq:contrast_wigner} ,
\end{equation}
where we introduced the phase-space vector $\boldsymbol{\xi} = (\mathbf{x}, \mathbf{p})^\text{T}$ and the symplectic form $J$ defined in Eq.~\eqref{eq:symplectic_form}.
The loss of contrast can be understood as a consequence of the oscillatory factor ``washing out'' the result of the integral. It otherwise tends to unity when the oscillations are negligible, as dictated by the normalization of the Wigner function.
The example of Gaussian states, whose Wigner function is given by Eq.~\eqref{eq:wigner_gaussian}, is particularly illustrative. Substituting into Eq.~\eqref{eq:contrast_wigner}, one obtains the following result for the contrast:
\begin{equation}
C = \exp \Big[ -\frac{1}{2 \hbar^2}\, \delta \boldsymbol{\chi}_0^\text{T} J^\text{T} \Sigma \,
J \, \delta \boldsymbol{\chi}_0 \Big]
\label{eq:contrast_gaussian},
\end{equation}
where $\Sigma$ is the covariance matrix of the initial state, detailed in Eq.~\eqref{eq:covariance}.
In turn, $\delta \boldsymbol{\chi}_0 \equiv \mathcal{T}^{-1} (t,t_0) \, \delta \boldsymbol{\chi}$ is related to $\delta \boldsymbol{\chi}$ through the transition matrix $\mathcal{T} (t,t_0)$ as explained in Appendix~\ref{sec:classical_trajectories} and well approximated by Eqs.~\eqref{eq:deltaR_free}-\eqref{eq:deltaP_free} for gravity gradients.
It is clear from Eq.~\eqref{eq:contrast_gaussian}
that the loss of contrast caused by gravity gradients can be reduced by simultaneously considering smaller position and momentum spreads, $\Sigma_{xx}$ and $\Sigma_{pp}$.
In fact, such a conclusion is not specific of Gaussian states and holds in general. Indeed, as the size of the main support of the Wigner function decreases, the wash-out effect from the oscillatory factor becomes less and less important.

Simultaneously decreasing $\Sigma_{xx}$ and $\Sigma_{pp}$ is, however, ultimately limited by Heisenberg's uncertainty principle. Moreover, achieving a sufficiently narrow momentum distribution can sometimes be rather demanding in practice even before the limit due to Heisenberg's uncertainty principle is reached. In order to alleviate these difficulties, an easily implementable mitigation strategy based on a small adjustment $\delta T / T \sim (\Gamma_{zz}\, T^2)$ of the timing for the last pulse was proposed in Ref.~\cite{roura14}. The key idea is that with a suitable choice of $\delta T$  one can change $\delta \boldsymbol{\mathcal{R}}$ (while keeping $\delta \boldsymbol{\mathcal{P}}$ essentially unchanged) so that the phase-space vector $(J\, \delta \boldsymbol{\chi})$ becomes aligned with the Wigner function and the wash-out effect of the oscillatory factor in Eq.~\eqref{eq:contrast_wigner} is minimized. (This is actually reminiscent of the use of squeezed states to beat the standard quantum limit in optical interferometers; rather than squeezing the state, here the observable is modified to achieve a similar effect.)
The strategy, which is briefly summarized in \comment{Appendix~\ref{sec:contrast_wigner_gaussian}}, was shown to be very effective for parameter ranges like those considered, for example, for the STE-QUEST mission \cite{aguilera14}. Nevertheless, it would eventually face significant limitations in future plans for further increasing by several orders of magnitude the sensitivity of UFF tests based on atom interferometry \cite{dimopoulos07}. Furthermore, it is somewhat less effective when applied to thermal clouds (even for ultracold atoms close to quantum degeneracy but with a negligible condensate fraction) rather than Bose-Einstein condensates%
\footnote{\comment{These limitations also apply to an alternative approach based on extracting the phase shift from the spatial location of the maxima within the fringe pattern that arises in the density profile at the exit ports of an open interferometer \cite{muentinga13,sugarbaker13}.}}.
In the next section we present a novel scheme which is not afflicted by these shortcomings and simultaneously overcomes both the initial co-location problem and the loss of contrast.

\section{Simultaneously overcoming loss of contrast and the initial co-location problem}
\label{sec:co-location}

As shown by Eq.~\eqref{eq:phase_shift3}, the contributions to the phase shift $\delta\phi$ that depend on the initial values of the central position and momentum, characterized by the phase-space vector $\boldsymbol{\chi}_0 = (\mathbf{r}_0, m \mathbf{v}_0)^\text{T}$, can be written in the following revealing form:
\begin{equation}
\delta\phi = - \frac{1}{\hbar}\, \delta \boldsymbol{\chi}^\text{T}(t)\, J \, \mathcal{T}(t,t_0) \, \boldsymbol{\chi}_0 + \ldots
\label{eq:phase_shift_initial},
\end{equation}
where $\mathcal{T}(t,t_0)$ is the transition matrix defined right after Eq.~\eqref{eq:full_solution}.
The next insight is then to realize that through a suitable adjustment of the laser wavelength for the second pulse, one can have a vanishing final displacement $\delta\boldsymbol{\chi} = 0$.
Indeed, this can be achieved by changing the effective momentum transfer associated with the second pulse to $\hbar\, \big( \mathbf{k}_\text{eff} + \Delta \mathbf{k}_\text{eff} \big)$ with
\begin{equation}
\Delta \mathbf{k}_\text{eff} = \big( \Gamma\, T^2 / 2 \big)\, \mathbf{k}_\text{eff}
\label{eq:Delta_k},
\end{equation}
where we neglected corrections of higher order in $\big( \Gamma\, T^2 \big)$ for simplicity [the exact result can be obtained using Eq.~\eqref{eq:transition_exact}]. This can be easily understood in a suitable freely falling frame where the central trajectories for the two branches of the interferometer are symmetric with respect to $z=0$, as shown in Fig.~\ref{fig:mitigation_trajectories}. The momentum transfer associated with the second pulse is chosen so that the trajectory after the pulse corresponds to the time reversal of the trajectory before it. Since the gravity gradient leads to the curvature of the trajectories in the space-time diagram as depicted in Fig.~\ref{fig:mitigation_trajectories}, an increase of the momentum transfer given by Eq.~\eqref{eq:Delta_k} is necessary.

\begin{figure}[h]
\begin{center}
\def8.5cm{8.5cm}
\input{figure3_pdf.tex}
\end{center}
\caption{Central trajectories for the same situation depicted in Fig.~\ref{fig:gg_trajectories} but with a suitable adjustment of the momentum transfer from the second laser pulse so that a closed interferometer, with vanishing relative displacement between the interfering wave packets in each port, is recovered.}
\label{fig:mitigation_trajectories}
\end{figure}

From Eqs.~\eqref{eq:contrast} and \eqref{eq:phase_shift_initial} it is clear that by achieving $\delta\boldsymbol{\chi} = 0$, this scheme simultaneously takes care of the loss of contrast caused by gravity gradients as well as the stringent requirements on the initial co-location of the two atomic species.
Indeed, thanks to the absence of a relative displacement between the interfering wave packets full contrast is recovered.
Furthermore, there is also an intuitive explanation for the solution of the initial co-location problem which is connected with the laser phases.
Because of the additional $\Delta \mathbf{k}_\text{eff}$ for the second pulse, in this new scheme the total momentum transfer to the two branches is unbalanced, i.e.\ one no longer has $\sum_j \delta \mathbf{k}_\text{eff}^{(j)} = 0$, where $\hbar\, \delta \mathbf{k}_\text{eff}^{(j)}$ corresponds to the difference between the momenta transferred to the two branches by the $j$-th pulse. This implies that the total contribution from the laser phases exhibits the following dependence on the initial values of the central position and velocity of the atomic wave packet:
\begin{align}
\delta\phi_\text{laser} \, \to \, &
\sum_j \delta \mathbf{k}_\text{eff}^{(j)} \cdot (\mathbf{r}_0 + \mathbf{v}_0 T) =
-2\, \Delta \mathbf{k}_\text{eff}^\text{T}\, (\mathbf{r}_0 + \mathbf{v}_0 T)  \nonumber \\
& \qquad\qquad\qquad = -\mathbf{k}_\text{eff}^\text{T}\, \big(\Gamma T^2\big) \,
(\mathbf{r}_0 + \mathbf{v}_0 T)
\label{eq:laser_phase} ,
\end{align}
where we made use of the choice of $\Delta \mathbf{k}_\text{eff}$ specified above in the second equality.
The key point is that the dependence on the initial position and velocity of the contribution from the laser phases compensates the effect of gravity gradients because the right-hand side of Eq.~\eqref{eq:laser_phase} exactly cancels the terms depending on the initial conditions in Eq.~\eqref{eq:phase_shift0}.

On the other hand, the fact that $\sum_j \delta \mathbf{k}_\text{eff}^{(j)} \neq 0$ also implies that the phase shift depends on the initial position (and velocity) of the retro-reflecting mirror, which is not known or controlled with very high precision.
More specifically, it gives rise to the following contribution to the phase shift:
\begin{equation}
\delta\phi_\text{laser} \, \to \, -\, 2\, \Delta\mathbf{k}_\text{eff} \cdot \mathbf{r}_\text{mirror}
= -2\, \mathbf{k}_\text{eff}^\text{T}\, \big(\Gamma\, \mathbf{r}_\text{mirror}\big) \, T^2
\label{eq:mirror_position} ,
\end{equation}
where $\mathbf{r}_\text{mirror}$ corresponds to the position of the retro-reflecting mirror at the time the second pulse is applied. The effect of this contribution can be post-corrected for a known mirror position $\mathbf{r}_\text{mirror}$. However, an uncertainty $\Delta\mathbf{r}_\text{mirror}$ on the position of the mirror leaves a residual contribution to the phase shift after post-correction corresponding to the expression in Eq.~\eqref{eq:mirror_position} but with $\mathbf{r}_\text{mirror}$ replaced by $\Delta\mathbf{r}_\text{mirror}$.
Fortunately, such dependence on $\Delta\mathbf{r}_\text{mirror}$ drops out (to sufficiently high degree) in the differential measurement. Indeed, when performing a differential measurement between species $A$ and $B$, the dependence on the position uncertainty $\Delta\mathbf{r}_\text{mirror}$ reduces to
\begin{equation}
\delta\phi_\text{laser}^A - \delta\phi_\text{laser}^B \, \to \, -2\, \big( \mathbf{k}_A\, T_A^2
- \mathbf{k}_B\, T_B^2 \big)^\text{T}
\big(\Gamma\, \Delta\mathbf{r}_\text{mirror}\big)
\label{eq:mirror_position_diff} ,
\end{equation}
which can be made small enough in exactly the same way as the contribution to the differential measurement of  residual accelerations common to both species.
For example, given the gravity gradient on Earth' surface, one has $\big| \Gamma\, \Delta\mathbf{r}_\text{mirror} \big| \sim 3\times10^{-13}\, g $ for an uncertainty $| \Delta \mathbf{r}_\text{mirror} | = 1\, \mu\text{m}$ in the position of the mirror. This means, for example, that if one can suppress residual accelerations $a' \lesssim 10^{-12}\, g$ through common-mode rejection, any contribution due to the unknown position of the mirror will be suppressed provided that this can be determined with an uncertainty $| \Delta \mathbf{r}_\text{mirror} | \lesssim 1\, \mu \text{m}$.
In fact, typical plans for high-precision tests of UFF with AIs are designed to reject much higher residual accelerations, so that the requirements on $| \Delta \mathbf{r}_\text{mirror} |$ can be further relaxed by several  orders of magnitude compared to this example.

So far we have assumed perfect knowledge of the gravity gradients, which would allow perfect compensation of their effects (up to the uncertainty in the mirror position) by using the scheme described above with $\Delta \mathbf{k}_\text{eff}$ given by Eq.~\eqref{eq:Delta_k}. In practice, however, the gravity gradient tensor $\Gamma$ employed in Eqs.~\eqref{eq:Delta_k}-\eqref{eq:mirror_position_diff} needs to be determined by simulations of the mass distribution, direct gradiometry measurements or a combination of both.
Since this can only be done with some finite accuracy $\Delta\Gamma$ characterizing the difference between the actual gravity gradients and the tensor $\Gamma$ determined by those means, the final displacement $\delta\boldsymbol{\chi}$ will take some non-vanishing residual value with $\Delta\Gamma$ appearing instead of $\Gamma$ in Eqs.~\eqref{eq:deltaR_deltaP}. Consequently, the dependence of the phase shift on the initial conditions appearing in Eq.~\eqref{eq:phase_shift_initial} will not be completely eliminated, but the co-location requirements for the differential measurement will be substantially relaxed by a factor of order $\|\Delta\Gamma\| / \|\Gamma\|$. Hence, determining for instance the gravity gradient tensor $\Gamma$ with a relative accuracy $\|\Delta\Gamma\| / \|\Gamma\| \lesssim 10^{-3}$ leads to a relaxation of the initial co-location requirements by 3 orders of magnitude.

We conclude this section by briefly discussing how feasible it is to implement the required momentum change for the second pulse, which is given by Eq.~\eqref{eq:Delta_k} and amounts to a relative frequency change $\Delta\nu / \nu = (\Gamma_{zz}\, T^2)$ if the gravity gradient is \comment{aligned} with $\mathbf{k}_\text{eff}$. Given Earth's gravity gradient $\Gamma_{zz} \approx 3 \times 10^{-6}\, \text{s}^{-2}$ and a moderate interferometer time $2\,T = 2\,\text{s}$, this corresponds to a frequency change $\Delta\nu \approx 1\, \text{GHz}$, which can be easily implemented with acousto-optical modulators (AOMs). This single-photon frequency change will be the same even if one has a large momentum transfer through higher-order Bragg diffraction, a sequence of multiple $\pi$ pulses or a combination of both. On the other hand, for a longer interferometer time $2\,T = 10\,\text{s}$ a single-photon frequency change $\Delta\nu \approx 25\, \text{GHz}$ would be necessary. Such a frequency change requires a more sophisticated set-up \comment{(e.g.\ two phase-locked lasers)}. Moreover, it gives rise to a substantially larger detuning of the single-photon transition, which requires a higher laser intensity in order to have a comparable Rabi frequency.
(The detuning could be approximately reduced in half by going from red- to blue-detuned transitions when changing from $\mathbf{k}_\text{eff}$ to $\mathbf{k}_\text{eff} + \Delta \mathbf{k}_\text{eff}$.)
Thus, the new scheme seems somewhat easier to implement directly in set-ups where higher sensitivity is achieved through large momentum transfer rather than very long interferometer times.

\section{Conclusion}
\label{sec:conclusion}

Gravity gradients pose a great challenge for high-precision tests of UFF based on atom interferometry. Indeed, for sufficiently large values of the effective momentum transfer or the interferometer time gravity gradients lead to a drastic loss of contrast and impose a serious limitation on the highest sensitivity that can be achieved.
Furthermore, they also imply stringent requirements on the initial co-location of the wave packets for the two species since the effects of a non-uniform gravitational field could otherwise mimic a violation of UFF.
Although there is in principle no fundamental limitation on the precision with which the central position and momentum of the wave packets for both species can be determined, in practice the requirements can be rather challenging. Moreover, the time needed to verify that such requirements are fulfilled can be comparable to the entire mission lifetime.

The situation concerning the loss of contrast and the time needed for verification of the systematics associated with initial co-location can be improved by considering smaller position and momentum widths for the initial state, but this is ultimately limited by Heisenberg's uncertainty principle.
In this paper we have presented a novel scheme that simultaneously overcomes the loss of contrast and the initial co-location problem. In doing so, it circumvents the limitations due to Heisenberg's uncertainty principle on the highest sensitivities that can be achieved and eases the experimental realization by relaxing the requirements on initial co-location by several orders of magnitude.
The key idea is that by slightly changing the wavelength of the second laser pulse, the momentum transfer to the two interferometer branches becomes unbalanced and this implies that the total phase-shift contribution from the laser phases depends on the initial values of the central position and momentum of the atomic wave packets. In fact, with a suitable choice of the change of wavelength this can exactly compensate the analogous contribution caused by the gravity gradients.
Furthermore, this choice automatically gives rise to a closed interferometer (vanishing relative displacement between the interfering wave packets) with no loss of contrast.

The results and discussions presented here can also be applied directly to experiments performed under microgravity conditions (this would actually correspond to the freely falling frame considered in Figs.~\ref{fig:gg_trajectories} and \ref{fig:mitigation_trajectories}). In that case employing a retro-reflection set-up naturally leads to double diffraction \cite{leveque09,giese13} and \comment{the associated} symmetric interferometers, which have a number of advantages, such as immunity to a number of systematic effects and noise sources (including laser phase noise). Only one detail needs then to be changed in Figs.~\ref{fig:gg_trajectories} and \ref{fig:mitigation_trajectories}: the central velocity of the initial wave packet vanishes and there is an additional third exit port with vanishing central velocity.
Interestingly, the double diffraction scheme can be generalized to experiments performed in a laboratory under normal gravity conditions by adding a third laser frequency per species \cite{malossi10,zhou15}, so that one can still benefit from many of the advantages associated with double diffraction (including also the cancellation of the photon-recoil term quadratic in $\mathbf{k}_\text{eff}$ mentioned in Appendix~\ref{sec:phase_shift}).

Rotations, which were not explicitly considered here, also lead to open interferometers, loss of contrast and dependence of the phase shift on the central velocity of the initial wave packet. These effects can be mitigated by employing satellites with sufficiently small angular velocity, the use of a tip-tilt mirror that \comment{compensates rotations} \cite{hogan08,lan12,dickerson13} or a combination of both. Moreover, by combining it with  the use of such a tip-tilt mirror, the scheme presented here can be extended to the case of non-aligned gravity gradients, where the direction of $\mathbf{k}_\text{eff}$ does not coincide with a principle axis of the tensor $\Gamma$.

Finally, it is worth pointing out that \comment{our method} can also be applied to differential measurements involving two (or more) spatially separated atom interferometers interrogated by common laser beams, a configuration employed for gradiometry measurements \cite{snadden98,rosi15}. Indeed, a scheme completely analogous to that presented here could be  exploited to mitigate the loss of contrast for each single interferometer (for sufficiently high $\mathbf{k}_\text{eff}$ or interferometer time) as well as the additional loss of contrast in the differential measurement due to the coupling of static gravity gradients to initial position and velocity jitter from shot to shot, which can become particularly important when considering long baselines between the interferometers. This would be relevant for gravitational antennas  capable of monitoring very precisely changes in the local gravitational field, such as the MIGA facility \cite{geiger15} currently under construction, and with interesting applications to geophysics and hydrology. It may also be relevant for gravitational antennas with very long baselines which have been proposed for the detection of low-frequency gravitational waves \cite{dimopoulos08b,graham13,hogan15}.

\begin{acknowledgments}
This work was supported by the German Space Agency (DLR) with funds provided by the Federal Ministry of Economics and Technology (BMWi) under Grant No.~50WM1556 (QUANTUS IV).
It is a pleasure to thank Wolfgang Zeller, Stephan Kleinert and Wolfgang Schleich for collaboration in related earlier work.
\end{acknowledgments}

\bibliographystyle{apsrev4-1}


\section{Feature}
\subsection{Detector \& Descriptor}
In our method, we implemented three kinds of features, pure DISK feature, pure SuperPoint\cite{superpoint_paper} feature, and a mix of SuperPoint and DISK\cite{tyszkiewicz2020disk} feature.

For SuperPoint, we additionally applied test-time homographic adaptation with 100 iterations to detect keypoints, and in some cases, we added a trained convolutional autoencoder to halve the feature dimension.

The Non-Maximum Suppression (NMS) window size and keypoints threshold was estimated theoretically based on the size of the image, to achieve a rather equally keypoints distribution across the image. Those values were then fine tuned around their initial estimates, to obtain the best performance on the validation set for each feature for each dataset.

\subsection{Pyramid Extraction}
We observed large-scale variation between images in some sets, such as the lizard in PragueParks. It troubled our matcher since the same feature under different scales may look very different. To solve this, we extracted the features in three scales and then concatenate those associated with the same keypoint.

Other than scale, sometimes orientation is also a problem. Similar to the approach taken for the scale problem, we extracted the feature in seven orientations by affine transformations (i.e. rotate 90 degrees left and right, respectively) and perspective transformations (i.e. horizontally left/right or vertically up/down by 45 degrees, respectively), and then concatenate them.

In order to save some computational resources and make the task more efficient, we only apply scale and/or orientation in some conditions, which will be elaborated in the Appendix.

\subsection{Pre-Process \& Post-Process}
\textbf{Masking}
For the Phototoursim and GoogleUrban datasets, some dynamic objects occur frequently, which introduce unreliable and unrepeatable keypoints to our solution. To overcome this, we segmented the scene and masked out some classes, including Person, Sky, Car, Bus, and Bicycle. Except for the class Person, which was segmented by Mask-RCNN\cite{Maskrcnn} trained on COCO, all other classes were removed by the pspnet\cite{zhao2017pspnet} trained on ade20k.

While those object segmentation nets worked well under most of the circumstances, they performed poorly when distinguishing the sculpture from humans; thus we additionally trained a binary classifier to take pedestrians apart from sculptures.

Moreover, in order to preserve the details of buildings, when the masking was enabled, we eroded the area masked by a 5x5 kernel for sky and 3x3 for the other classes.
\\

\textbf{Refinement}
We applied an argsoftmax function for keypoints refinement with a radius of 2.
\\

\section{Matching}
We trained the SuperGlue\cite{sarlin20superglue} together with its official feature extractor SuperPoint\cite{superpoint_paper} in an end-to-end manner on the MegaDepth dataset with IMW2021 competition testset removed. More specifically, we splited the original SuperPoint\cite{superpoint_paper} into two networks, the first one was fixed with the official weights to extract the keypoints from the image, while the other one was fine-tuned to provide descriptions. However, we found this adjustment advanced the model performance slightly, since the SuperGlue\cite{sarlin20superglue} can already match the given points well.

Furthermore, we take the advantage of the latest feature DISK\cite{tyszkiewicz2020disk} instead of SuperPoint\cite{superpoint_paper}, which built upon a simple CNN backbone. For the SuperGlue\cite{sarlin20superglue} matcher compatible with DISK\cite{tyszkiewicz2020disk}, we only trained the matcher part, while directly used the official weights of DISK\cite{tyszkiewicz2020disk} and did not conduct any fine-tuning on it. We leveraged DISK\cite{tyszkiewicz2020disk} to extract $2,048$ points for each image in the training phase, and $8,000$ points in the testing phase. In this way, as shown in Table.\ref{table:1 superglue}, the AUC could be improved significantly. The evaluation was conducted following the methodology in the SuperGlue\cite{sarlin20superglue} paper.

We have trained an indoor and an outdoor version for the SuperGlue matcher compatible with either SuperPoint or DISK, but we did not find the indoor one help with performance advancement; thus we stuck with the outdoor weights for this competition.

\begin{table}[ht!]
\resizebox{\columnwidth}{!}{%
\begin{tabular}{|l|c|c|c|}
\hline
                                              & \multicolumn{3}{c|}{Exact AUC} \\ \hline
                                              & 5°       & 10°      & 20°      \\ \hline
SuperPoint + SuperGlue (Official)             & 38.72    & 59.13    & 75.81    \\ \hline
SuperPoint + SuperGlue (Our trained, outdoor) &  38.88    &   59.27   &  75.71  \\ \hline
DISK + SuperGlue (Outdoor)                    & 42.72    & 62.54    & 97.68    \\ \hline
\end{tabular}}
\caption{SuperGlue Weights Evaluation}
\label{table:1 superglue}
\end{table}

\textbf{Guided Pyramid Matching} Only when the number of matches found by SuperGlue was less than 100, SuperGlue matching would be applied on the pyramid extraction results, i.e. multiple scales and/or multiple orientations, we might combine the matches in different scales (\textit{ALL}) or trust the one with the most number of matches (\textit{MAX}).
\section{Outlier Rejection \& Adaptive F/H}
Based on our experiments, DegenSAC\cite{Mishkin2015MODS} outperformed other outlier rejection methods in most cases. So we applied DegenSAC to find the fundamental matrix and under certain circumstances, the homography simultaneously.

Inspired by ORB-SLAM\cite{ORBSLAM3_2020}, different transformation matrix should be selected for different scenes and then be decomposed to get pose. Although the process is fixed in the offical bankend, considering different transformation matrix still helps us to filter matching outliers. In the GoogleUrban dataset, for example, many correct matches are on the flat buildings along the street, but many wrong matches are on the ground or on the isolation belt of the road. Suffering from the weak constraint of the F matrix, some wrong matches also could pass the filter (the distance to the epipolar line is less than the threshold). While if the H matrix is selected, only the correct matches on the street-side flat building is retained. Even if the pose is decomposed by calculating the F matrix later in the offical process, the accuracy can still be improved
since some wrong matches could be removed by adaptive F/H strategy. For the method implementation, we refer to the ORB-SLAM\cite{ORBSLAM3_2020}, respectively calculating the F matrix and the H matrix, as well as their scores (SF and SH), which are related to the symmetric transfer error less than the threshold. After obtaining SF and SH, calculate RH=SH/(SH+SF), and then we select H matrix if RH is greater than 0.45. Otherwise, we select the F matrix, which s called adaptive F-H policy. In the Pragueparks dataset, there are not many these kind of cases, so th
e performance improvement was not obvious by applying adaptive F/H. In the Phototourism dataset, most of the non-planar matches are correct matches, so applying adaptive F/H led to the correct non-planar matches being removed and thus worsen the accuracy.

\section{Conclusion}
This report provided Megvii-3D team's strategies towards CVPR2021 IMW competition for both the unlimited keypoints category and the restricted one. \\

\textbf{Limitations of our method}

From our analysis on the corner cases, we realized that a simple feature-matcher-filter solution could not suffice all conditions, without multiple scales and/or multiple orientations matching augmentation. We observed there are image pairs with scale difference more than a factor of 3, more than 45 degree of rotation, or large perspective transformation. A single level solution can not work them out by any means.

While those cases are rare but still possible in some applications for real life, for example, when you are using an AR map guider and conducting a sharp turn-around or you suddenly fail down, visual localizer is likely to fail. However, we argue that those extreme cases could be leveraged by compensations from other type of data or sensors, such as the IMU, GPS, and QR code positioning system. On the other hand, the performance of current feature-matcher might be saturated already since the convolutional neural network block has its own limitations, including the limited receptive field, hard to break the spatial relationships between pixels, cannot model shape deformations (though we have DCNs now), etc.

Despite that many excellent researchers will continue on improving the accuracy, robustness together with the generalization of feature extraction and matching, it might be doubted that stereo matching plays a dominant role in optimizing the performance of the whole visual localization system.
\section*{Appendix: Details about each Submission}

\input{table}

\end{landscape}
\section{Instructions}

\TODO{Remove this section for the final version.}
This document should provide just enough information for readers to reproduce
your outcomes.
It also should list all datasets and pre-trained models for this purpose. This document should be shorter than 2 pages in length.

\section{Method and Technical Details}

This section should discuss the technical details regarding your method.
You should reference the method that was used, and provide any modifications that you performed to the method to achieve your performance.

In case you use your own RANSAC, please provide all RANSAC parameters as well. For example, the number of iterations would be important to make a fair comparison with other RANSAC methods.

SuperPoint 2k + SuperGlue

SuperPoint 4k + SuperGlue

\section{Dataset and Pre-trained Models}
We used the MegaDepth dataset with validation and test set removed to train our model from scratch.
\section{Pipeline}

\begin{figure}[ht!]
    \centering
    \includegraphics[width=8.5cm]{pipeline.png}
    \caption{Method Pipeline}
    \label{fig:pipeline}
\end{figure}

\input{1_feature}
\input{2_matching}
\input{3_ransac}
\input{4_conclusion}

{\small
\bibliographystyle{IEEEtran}


\section{Introduction}

Graphs in this paper are simple and undirected.
We use $V(G)$, $E(G)$, and $F(G)$, respectively, to represent the vertices, edges, and faces of a graph $G$.
A \emph{coloring} of a graph $G$ is a function $c$ that assigns an element $c(v)$ to each vertex $v \in V(G)$.
A \emph{proper coloring} is a coloring such that $c(u) \neq c(v)$ whenever $uv \in E(G)$.

Vizing~\cite{V76}, and independently Erd\H{o}s, Rubin, and Taylor~\cite{ERT76} introduced list coloring, a generalization of proper coloring.
A \emph{list assignment} $L$ gives each vertex $v$ a set of available colors $L(v)$.
A graph is \emph{$L$-colorable} if it has a proper coloring $c$ with $c(v) \in L(v)$ for every vertex $v$.
A graph is \emph{$k$-choosable} (or \emph{$k$-list-colorable}) if it is $L$-colorable whenever $|L(v)| \ge k$ for each $v \in V(G)$.
The \emph{choosability} (or \emph{list chromatic number}) $\chi_\ell(G)$ of a graph $G$ is the least $k$ such that $G$ is $k$-choosable; the analogue for coloring is the \emph{chromatic number} $\chi(G)$.
In the case that $L(V) = [k]$ for each $v \in V(G)$, any $L$-coloring of $G$ is also a \emph{proper $k$-coloring}, where $[k]$ denotes the set of integers $\{1, 2, \dots, k\}$.
Thus we always have $\chi_\ell(G) \ge \chi(G)$.

More recently, \Dv{} and Postle~\cite{DP18} introduced the following idea of correspondence coloring, which has since become known as DP-coloring. This notion generalizes choosability.

\begin{definition} Let $G$ be a simple graph with $n$ vertices and let $L$ be a list assignment for $G$.
For each $v \in V(G)$, let $L_v = \{v\} \times L(v)$.
For each edge $uv \in E(G)$, let $M_{uv}$ be a matching (possibly empty) between the sets $L_u$ and $L_v$, and let $\M_L = \{M_{uv} : uv \in E(G)\}$, called the \emph{matching assignment}.
Let $G_L$ be a graph that satisfies the following conditions:
\begin{itemize}
\item $V\left(G_L\right) = \cup_{v \in V(G)} L_v$,
\item for each $v \in V(G)$, the set $L_v$ is a clique in $G_L$,
\item if $uv \in E(G)$, then the edges between $L_u$ and $L_v$ are exactly those of $M_{uv}$, and
\item if $uv \notin E(G)$, then there are no edges between $L_u$ and $L_v$.
\end{itemize}
We say that $G$ has an $\M_L$-coloring if $G_L$ contains an independent set of size $n$.
The graph $G$ is \emph{DP-$k$-colorable} if, for any list assignment $L$ with $|L(v)| = k$ for each $v \in V(G)$, the graph is $\M_L$-colorable for every matching assignment $\M_L$.
The least $k$ such that $G$ is DP-$k$-colorable is the \emph{DP-chromatic number} of $G$, denoted $\chi_{DP}(G)$.
\end{definition}

We generally identify the elements of $L_v$ with those of $L(v)$ and refer to the elements as colors.
We will often assume without loss of generality that $L(v) = [k]$ for all $v \in V(G)$, as the existence of an independent set in $G_L$ depends only on the matching assignment $\M_L$.
Suppose $G$ is $\M_L$-colorable and $I$ is an independent set of size $n$ in $G_L$.
Then $|I \cap L_v| = 1$ and we refer to the element $i \in I \cap L_v$ as the color given to $v$.

If $L(v) = [k]$ for each $v \in V(G)$ and $M_{uv} = \{(u, i) (v, i) : i \in [k]\}$ for each $uv \in E(G)$, then an $\M_L$-coloring is exactly a proper $k$-coloring.
Additionally, DP-coloring also generalizes $k$-choosability, even with the restriction that $L(v) = [k]$ for each $v \in V(G)$.
To see this, consider a list assignment $L'$ with $|L'(v)| = k$ for all $v \in V(G)$.
For each vertex $v \in V(G)$, there exists a bijection from the elements of $L'(v)$ to $[k]$, and we simply let $M_{uv}$ be the matching between the colors of $u$ and $v$ that correspond to equal elements of $L'(u)$ and $L'(v)$.
Accounting for relabeling, an $\M_L$-coloring is equivalent to an $L'$-coloring.
Thus, any DP-$k$-colorable graph must be $k$-choosable, and so $\chi_{DP}(G) \ge \chi_\ell(G)$ for all graphs $G$.

One difficulty in the study of list coloring is that some techniques useful in solving coloring problems, such as identifation of vertices, are not feasible in the list coloring setting.
DP-coloring can be used to apply these coloring techniques in some situations.
In the paper introducing DP-coloring, \Dv{} and Postle~\cite{DP18} use identification to prove that planar graphs without cycles of length 4 to 8 are 3-choosable.
However, they impose conditions on the matching assignment $\M_L$, and their proof does not give the analogous result that such graphs are DP-3-colorable.
In their paper, \Dv{} and Postle note that DP-coloring is strictly more difficult than list coloring, in the sense that it is possible for $\chi_{DP}(G) > \chi_\ell(G)$ for some graphs $G$.
In particular, they showed that cycles of even length are 2-choosable, but they are not DP-2-colorable.
In addition, while Alon and Tarsi~\cite{AT92} showed that planar bipartite graphs are 3-choosable, Bernshteyn and Kostochka~\cite{BK18+} provide a bipartite planar graph $G$ with $\chi_{DP}(G) = 4$.

These differences, particularly for even cycles, result in difficulties in extending results from list-coloring to DP-coloring.
However, some proofs for list-coloring do extend to DP-coloring.
For example, \Dv{} and Postle note that Tommassen's proofs~\cite{T94,T95} that $\chi_\ell(G) \le 5$ for planar graphs and $\chi_\ell(G) \le 3$ for planar graphs with no 3-cycles or 4-cycles immediately extend to DP-coloring.

There has been considerable recent interest in extending results for choosability of planar graphs to DP-coloring.
Liu and Li~\cite{LL18+}, Sittitrai and Nakprasit~\cite{SN18+}, Kim and Yu~\cite{KY18+}, and Kim and Ozeki~\cite{KO18+} all extend results on 4-choosability of planar graphs to DP-4-coloring.
Yin and Yu~\cite{YY18+} extend results for 3-choosability to DP-3-coloring, in some cases for a larger class of graphs than the analogous choosability result.
Among other results extending that conditions for 3-choosability hold for DP-3-coloring, Liu, Loeb, Yin, and Yu~\cite{LLYY18+} show that planar graphs with no $\{4, 5, 6, 9\}$-cycles or with no $\{4, 5, 7, 9\}$-cycles are DP-3-colorable.
In this paper, we extend the three previous results for 3-choosability of planar graphs stated in Theorem~\ref{thm1}.

\begin{thm}\label{thm1}
A planar graph $G$ is $3$-choosable if one of the following conditions holds
\begin{itemize}

\item $G$ contains no $\{4,6,7,9\}$-cycles. (Wang, Lu, and Chen~\cite{WLC08})
\item $G$ contains no $\{4,6,8,9\}$-cycles. (Shen and Wang~\cite{SW07})
\item $G$ contains no $\{4,7,8,9\}$-cycles. (Wang and Shen~\cite{WS11})
\end{itemize}
\end{thm}

Our main result is the following.

\begin{thm}\label{awesome}
If $a$ and $b$ are distinct values from $\{6, 7, 8\}$, then every planar graph without cycles of lengths $\{4, a, b, 9\}$ is DP-3-colorable.
\end{thm}

Our proofs use the discharging method, which uses strong induction.  We say a structure is \emph{reducible} if it cannot appear in a minimal counterexample $G$.  The proofs of the results in Theorem~\ref{thm1} rely on the fact that an even cycle with all vertices of degree 3 is reducible.  Such a structure is not necessarily reducible in the setting of DP-coloring.  In Section~\ref{lemma}, we use the lemma about ``near-$(k-1)$-degenerate'' subgraphs from~\cite{LLYY18+} which fills a similar role in our reducible structures.
In Section~\ref{lemma}, we also provide our reducible structures and a lemma about how much charge can be given by large faces in our subsequent discharging arguments.
Section~\ref{sec:threeproofs} provides the proofs for Theorem~\ref{awesome}.
We use different initial charges from the ones in~\cite{LLYY18+}, and provide a new unified set of discharging rules for all three cases.

\section{Lemmas and a brief discussion of the discharging.}\label{lemma}

Graphs mentioned in this paper are all simple.  A $k$-vertex (resp., $k^+$-vertex, $k^-$-vertex) is a vertex of degree $k$ (resp., at least $k$, at most $k$). The \emph{length} of a face is the number of vertices on its boundary, with repetition included.
A face with length $k$ (resp., at least $k$, at most $k$) is a $k$-face (resp., $k^+$-face, $k^-$-face).
We may also refer to an $(\ell_1, \ell_2, \ldots, \ell_k)$-face, which is a $k$-face $f = v_1 v_2 \ldots v_k$ with facial walk $v_1, v_2, \ldots, v_k$ such that $d(v_i) = \ell_i$.
An $(\ell_1, \ell_2, \ldots, \ell_k)$-path and $(\ell_1, \ell_2)$-edge are defined similarly, and we may replace  $\ell_i$ with $\ell_i^+$ to indicate $d(v_i) \ge \ell_i$.
A $3$-vertex is {\em{triangular}} if it is incident to a $3$-face.

\begin{lemma}\label{minimum}
Let $G$ be a smallest graph (with respect to the number of vertices) that is not DP-$k$-colorable. Then $\delta(G)\ge k$.
\end{lemma}

\begin{proof}
Suppose there is a vertex $v$ with $d(v) < k$.
Any $\mathcal{M}_L$-coloring of $G - v$ can be extended to $G$ since $v$ has at most $d(v)$ elements of $L(v)$ forbidden by the colors selected for the neighbors of $v$, while $|L(v)| = k$.
\end{proof}

Let $H$ be a subgraph of $G$.
For each vertex $v \in H$, let $A(v)$ be the set of vertices (of $G_L$) in $L_v$ that are not matched with vertices in $\cup_{u \in G - H} L_u$.
One may think of $A(v)$ as the colors available at $v$ after coloring $G - H$.

\begin{lemma}\label{near-2-degenerate}\cite{LLYY18+}
Let $k \ge 3$ and $H$ be a subgraph of $G$. If the vertices of $H$ can be ordered as $v_1, v_2, \ldots, v_{\ell}$ such that the following hold
\begin{itemize}
\item[(1)] $v_1v_{\ell}\in E(G)$, and $|A(v_1)|>|A(v_{\ell})|\ge 1$,
\item[(2)] $d(v_{\ell})\le k$ and $v_{\ell}$ has at least one neighbor in $G-H$,
\item[(3)] for each $2\le i\le \ell-1$, $v_i$ has at most $k-1$ neighbors in $G[\{v_1, \ldots, v_{i-1}\}]\cup (G-H)$,
\end{itemize}
then a DP-$k$-coloring of $G-H$ can be extended to a DP-$k$-coloring of $G$.
\end{lemma}

For the remainder of this paper, we will let $G$ denote a minimal counterexample to Theorem~\ref{awesome}.
That is $G$ is a planar graph with no $\{4, a, b, 9\}$-cycles, where $a, b \in \{6, 7, 8\}$ are distinct, such that $G$ is not DP-3-colorable, but any planar graph on fewer than $|V(G)|$ vertices with no $\{4, a, b, 9\}$-cycles is DP-3-colorable.
We now use Lemma~\ref{near-2-degenerate} to provide some reducible configurations we will need in Section~\ref{sec:threeproofs}.

\begin{lemma}\label{lem:reducible}
The graph $G$ does not contain any of the following subgraphs:
\end{lemma}
\begin{figure}[h]
\includegraphics[scale=0.18]{2Bad5Faces.pdf}
\includegraphics[scale=0.17]{5Face3Vertices.pdf}\\
\includegraphics[scale=0.20]{Good10FaceNoSR4Vertex.pdf}\quad\quad
\includegraphics[scale=0.18]{10Face5Face3Vertices.pdf}  \quad\quad\includegraphics[scale=0.18]{10Face3Face3Vertices.pdf}
\end{figure}

\begin{proof}
Let $H$ be the subgraph of $G$ consisting of the labeled vertices, and order the vertices according to their labels.
It is straightforward to verify that all labeled vertices must be distinct, since otherwise cycles of forbidden lengths are created.
From Lemma~\ref{near-2-degenerate}, it follows that a DP-$3$-coloring of $G - H$ can be extended to a DP-$3$-coloring of $G$.
\end{proof}

We use balanced discharging and assign an initial charge of $\mu(x) = d(x) - 4$ to each $x \in V(G) \cup F(G)$.
Let $\mu^*(x)$ be the final charge after the discharging procedure.
From Euler's formula, we have
\begin{equation}\label{eq}
\sum_{x \in V(G) \cup F(G)}(d(x) - 4) = -8.
\end{equation}
We will move charge around and argue in Section~\ref{sec:threeproofs} that each vertex and face ends with non-negative final charge.
This contradiction to (\ref{eq}) will prove our conclusion.

\section{Proof of Theorem~\ref{awesome}} \label{sec:threeproofs}

We say a $10^+$-face $f$ is \emph{good} to a $10$-face $f'$ (and $f'$ is \emph{poor} to $f$) if $f'$ is incident to ten $3$-vertices, and $f$ and $f'$ share a $(3, 3)$-edge such that each end of the shared edge is an end of a $(3, 4^+, 3^+, \dots)$-path of $f$, where the second vertex of the path is incident to either a $(3, 3, 4^+)$-face or a $(3, 3, 3, 3, 4^+)$-face adjacent to both $f$ and $f'$.
A $4^+$-vertex $v$ is {\em{poor}}, {\em{semi-rich}}, or {\em{rich}} to a $10^+$-face $f$ if $v$ is incident to two $3$-faces, one $3$-face, or no $3$-faces adjacent to $f$, respectively.
Moreover, we call a semi-rich $4^+$-vertex $v$ {\em{special}} if $v$ is on a $10^+$-face $f$ such that $v$ is rich to $f$, and we call a $5$-face {\em{bad}} if it is incident to five $3$-vertices and adjacent to two $5$-faces.

The following are our discharging rules:

\begin{enumerate}[(R1)]
\item Each $3$-face gets $\frac{1}{3}$ from each adjacent $5^+$-face, and each $5^+$-face gets $\frac{1}{5}$ from each incident $5^+$-vertex.

\item Each $10^+$-face gets $\frac{1}{6}$ from each incident special semi-rich $4^+$-vertex, and gives $\frac{1}{3}$ to each incident rich $4$-vertex which is on a $3$-face.

\item If $f$ is good to $f'$, then $f$ gives $\frac{1}{6}$ to $f'$.

\item[(R4a)] If $G$ contains no $\{4, 7,8,9\}$-cycles, then each $5$-face that shares a $(3,3)$-edge with a $3$-face gets $\frac{1}{3}$ from each adjacent $10^+$-face, each $5$-face that shares a $(3,4^+)$-edge with a $3$-face gets $\frac{1}{3}$ from the $10^+$-face incident to the $3$-vertex of the $(3,4^+)$-edge, and each $5$-face sends $1$ to any incident triangular $3$-vertex and $\frac{1}{3}$ to any adjacent $3$-face, then distributes its remaining charge evenly to its adjacent $10$-faces. Each $3$-vertex gets its remaining needed charge evenly from its incident $6^+$-faces.

\item[(R4b)] If $G$ contains no $\{4, 6, a, 9\}$-cycles for $a \in \{7, 8\}$, then each 3-vertex gets $1$ evenly from its incident $5^+$-faces, each $5$-face gets $\frac{1}{6}$ from each adjacent $7^+$-face, and each bad $5$-face gets an additional $\frac{1}{12}$ from each adjacent $5$-face.
Following this, if a $5$-face has positive charge, then  it distributes its surplus charge to its adjacent $5$-faces.
\end{enumerate}

\begin{lemma}
Each vertex and each $8^-$-face have non-negative final charge.
\end{lemma}

\begin{proof}

Note that if a graph $G$ contains no $\{4, 7, 8, 9\}$-cycles, then each $3$-vertex must be incident to at least one $6^+$-face.
By (R4a) and (R4b), the final charge of each $3$-vertex is $0$.
Note that by (R2), a $4$-vertex $v$ sends out charge only if it is special, in which case $v$ is rich to a $10^+$-face and semi-rich to at most two $10^+$-faces.
So if $d(v) = 4$, then $\mu^*(v) \ge (4 - 4) - \frac{1}{6} \cdot 2 + \frac{1}{3} = 0$.
Now let $v$ be a $5^+$-vertex.
Then by (R1) and (R2), $v$ sends out at most $\frac{1}{5}$ to each of its incident faces, so $\mu^*(v) \ge (d(v) - 4) - \frac{1}{5}d(v) \ge 0$.
Hence all vertices end with non-negative charge.

Let $f$ be an $8^-$-face in $G$.
If $d(f) = 3$, then $f$ is adjacent to three $5^+$-faces.
So by (R1), $\mu^*(f) \ge 0$.
For the remaining $8^-$-faces, we consider the consequences of rules (R4a) and (R4b) as separate cases.

{\bf Case 1: $G$ has no $\{4,7,8,9\}$-cycles.}

Then the only $8^-$-faces of $G$ left to consider are $5$- and $6$-faces.
Note that a $5$-face cannot be adjacent to a $6$-face.
If $d(f) = 6$, then $f$ is adjacent to no $3$-face and by (R4a) sends at most $\frac{1}{3}$ to each incident vertex.
So $\mu^*(f) \ge (6 - 4) - \frac{1}{3} \cdot 6 = 0$.
If $d(f) = 5$, then $f$ is adjacent to at most one $3$-face and at least four $10^+$-faces.
If $f$ is not adjacent to any $3$-face, then it only needs to send $1$ evenly to its adjacent 10-faces by (R4a), so $\mu^*(f) \ge 0$.
If $f$ shares a $(3,3)$-edge with a $3$-face, then by (R1) and (R4a), $f$ sends $\frac{1}{3}$ to its adjacent $3$-face and $1$ to each incident triangular $3$-vertex, and $f$ gets $\frac{1}{3}$ from each adjacent $10^+$-face.
So $\mu^*(f) \ge (5 - 4) - \frac{1}{3} - 1 \cdot 2 + \frac{1}{3} \cdot 4 = 0$.
If $f$ shares a $(3,4^+)$-edge with a $3$-face, then by (R1) and (R4a), $f$ sends $\frac{1}{3}$ to its adjacent $3$-face and $1$ to its incident triangular $3$-vertex, and $f$ gets $\frac{1}{3}$ from one $10^+$-face.
So $\mu^*(f) \ge (5 - 4) - \frac{1}{3} - 1 + \frac{1}{3} = 0$.
If $f$ shares a $(4^+, 4^+)$-edge with a $3$-face, then by (R1) and (R4a), $f$ sends $\frac{1}{3}$ to its adjacent $3$-face and sends the remaining charge evenly to adjacent $10$-faces.
Thus, $\mu^*(f)\ge0$.

{\bf Case 2: $G$ has no $\{4, 6, a, 9\}$-cycles for $a \in \{7, 8\}$.}

Let $f$ be an $8$-face.
Then $G$ contains no $\{4, 6, 7, 9\}$-cycles, so $f$ is not adjacent to any 3-face.
Thus, by (R4b), $f$ gives $\frac{1}{3}$ to each incident 3-vertex and $\frac{1}{6}$ to each adjacent 5-face.
So $\mu^*(f) \ge (8 - 4) - \frac{1}{3} \cdot 8 - \frac{1}{6} \cdot 8 = 0$.

Let $f$ be a $7$-face.
Then $G$ contains no $\{4, 6, 8, 9\}$-cycles, so $f$ is again not adjacent to any 3-face.
By (R1) and (R4b), $f$ gives $\frac{1}{3}$ to each incident 3-vertex and $\frac{1}{6}$ to each adjacent 5-face.  Note that no two $5$-faces are adjacent.
So $\mu^*(f) \ge (7 - 4) - \max\big\{\frac{1}{3} \cdot 7 + \frac{1}{6} \cdot 3, \frac{1}{3} \cdot 6 + \frac{1}{6} \cdot 4, \frac{1}{3} \cdot 5 + \frac{1}{6} \cdot 7 \big\} > 0$.

It remains to consider $5$-faces.
So let $f$ be a $5$-face.
Let $r_5(f)$ and $s_3(f)$ be the number of adjacent 5-faces and incident 3-vertices of $f$, respectively.
Since $G$ contains no $6$-cycles, $f$ cannot be adjacent to a $3$-face.

If $f$ is a bad $5$-face, then $s_3(f) = 5$ and $r_5(f) = 2$.
By Lemma~\ref{lem:reducible}$(i)$, $f$ is not adjacent to another bad $5$-face.
Then by (R4b), $f$ gives $\frac{1}{3}$ to each incident $3$-vertex, and $f$ gets $\frac{1}{6}$ from each adjacent $7^+$-face and $\frac{1}{12}$ from each adjacent $5$-face.
So $\mu^*(f) \ge (5 - 4) - \frac{1}{3} \cdot 5 + \frac{1}{12} \cdot 2 + \frac{1}{6} \cdot 3 = 0$.
Thus we may assume that $f$ is not bad.
By (R4b),
\[ \mu^*(f) \ge (5 - 4) - \frac{1}{3} s_3(f) + \frac{1}{6} (5 - r_5(f)) - \frac{1}{12} b_5(f) = \frac{1}{6} \left(11 - 2s_3(f) - r_5(f) - \frac{1}{2}b_5(f)\right), \]
where $b_5(f)$ is the number of bad $5$-faces adjacent to $f$.
Clearly, $b_5(f)\le r_5(f)$.

Note that no $3$-vertex can be incident to three 5-faces since $G$ contains no $9$-cycles.
If $s_3(f) \le 2$, then $r_5(f) \le 5$ and $b_5(f) \le 1$.
So $\mu^*(f) \ge \frac{1}{6} \left(11 - 2 \cdot 2 - 5 - \frac{1}{2}\right) > 0$.
If $s_3(f) = 3$, then $r_5(f) \le 3$ and $b_5(f) \le 1$.
Thus $\mu^*(f) \ge \frac{1}{6} \left(11 - 2 \cdot 3 - 3 - \frac{1}{2}\right) > 0$.
If $s_3(f) = 5$, then $r_5(f) \le 1$ since $f$ is not bad.
By Lemma~\ref{lem:reducible}$(i)$, $b_5(f) = 0$, so $\mu^*(f) \ge \frac{1}{6}(11 - 2 \cdot 5 - 1) = 0$.
Lastly, if $s_3(f) = 4$, then $r_5(f) \le 3$.
So $\mu^*(f) \ge \frac{1}{6}\left(11 - 2 \cdot 4 - r_5(f) - \frac{1}{2}b_5(f)\right)$, and $\mu^*(f) < 0$ only if $r_5(f) = 3$ and $b_5(f) = 1$, in which case $\mu^*(f) \ge -\frac{1}{12}$.
Let $v$ be the $4^+$-vertex incident to $f$.
If $d(v) \ge 5$, then by (R1), $f$ gets $\frac{1}{5} > \frac{1}{12}$ from $v$ and ends with non-negative charge.
If $d(v) = 4$, then by Lemma~\ref{lem:reducible}$(ii)$, at least one of the $5$-faces adjacent to $f$ and incident to $v$, say $f'$, has at least two $4^+$-vertices.
Note that $r_5(f') \le 4$.
By (R4b), $f'$ can send
\begin{align*}
&\frac{1}{r_5(f')} \cdot \frac{11 - 2s_3(f') - r_5(f') - \frac{1}{2} b_5(f')}{6} \\
&\ge \max\left\{\frac{1}{4} \cdot \frac{11 - 2 \cdot 2 - 4}{6},\ \frac{1}{3} \cdot \frac{11 - 2 \cdot 3 - 3 - \frac{1}{2}}{6},\ \frac{1}{2} \cdot \frac{11 - 2 \cdot 3 - 2 - \frac{1}{2}}{6},\ \frac{11 - 2 \cdot 3 - 1}{6}\right\}\\
&> \frac{1}{12}
\end{align*}
to each of its adjacent $5$-faces, which includes $f$.
Hence $\mu^*(f) \ge 0$.
\end{proof}

We now only need to verify that $10^+$-faces end with non-negative charge.
Let $f$ be a $10^+$-face.
Let $P$ be a maximal path (or possibly a cycle) along $f$ such that every edge of $P$ is adjacent to a $5^-$-face.
Let $\mathcal{P}$ be a collection of all such paths $P$ along $f$.
By construction, the paths of $\mathcal{P}$ are disjoint.
Let $t_i$ denote the number of paths of $\mathcal{P}$ with $i$ vertices for $i \ge 2$, and let $t_1$ denote the number of vertices incident to $f$ not contained in any path $P \in \mathcal{P}$.
Then $\sum_{i \ge 1} i \cdot t_i = d(f)$.
We will use the following two lemmas to simplify our final analysis.

\begin{lemma}\label{12+faces}
A $10^+$-face $f$ can afford to give out at least \begin{equation}\label{eq-general}
\frac{1}{3} \sum_{i \ge 1} i \cdot t_i + \frac{2}{3} \sum_{i \ge 2} t_i + \frac{1}{3}t_1 + \frac{1}{3} \sum_{i \ge 3} (i - 2)t_i - \frac{x}{3},
\end{equation}
where $x = 0$ if $d(f) \ge 12$, $x = 1$ if $d(f) = 11$, and $x = 2$ if $d(f) = 10$.
\end{lemma}

\begin{proof}
From $\sum_{i \ge 1} i \cdot t_i = t_1 + 2 \sum_{i \ge 2} t_i + \sum_{i \ge 3} (i - 2)t_i$, we have
\[ 2 \sum_{i \ge 2} t_i = \sum_{i \ge 1} i \cdot t_i - t_1 - \sum_{i \ge 3}(i - 2)t_i. \]
Therefore,
\begin{align*}
&\frac{1}{3} \sum_{i \ge 1} i \cdot t_i + \frac{2}{3} \sum_{i \ge 2} t_i + \frac{1}{3} t_1 + \frac{1}{3} \sum_{i \ge 3} (i - 2)t_i - \frac{x}{3} \\
&= \frac{1}{3} \sum_{i \ge 1} i \cdot t_i + \frac{1}{3} \left(\sum_{i \ge 1} i \cdot t_i - t_1 - \sum_{i \ge 3}(i - 2)t_i\right) + \frac{1}{3} t_1 + \frac{1}{3} \sum_{i \ge 3} (i - 2)t_i - \frac{x}{3} \\
&= \frac{2}{3} \sum_{i \ge 1} i \cdot t_i - \frac{x}{3} \\
&= \frac{2}{3}d(f) - \frac{x}{3}.
\end{align*}
So $\mu^*(f) \ge (d(f) - 4) - \frac{2}{3}d(f) + \frac{x}{3} = \frac{1}{3}(d(f) - 12 + x) \ge 0$ when $d(f) \ge 10$.
\end{proof}

\begin{lemma}\label{lem:eq-10+}
Each $10^+$-face $f$ needs to send out at most
\begin{equation}\label{eq-10+}
\frac{1}{3} \sum_{i \ge 1} i \cdot t_i + \frac{2}{3} \sum_{i \ge 2} t_i + \frac{1}{6} \sum_{i \ge 6}(i - 5)t_i + \frac{1}{6} \sum_{i \ge 3}t_i,
\end{equation}
\end{lemma}

\begin{proof}
Note that $f$ gives at most $\frac{1}{3}$ to each vertex not on any path of $\mathcal{P}$ by (R2), (R4a) or (R4b).
Now we show that $f$ needs to send out at most $\sum_{i \ge 2} \frac{1}{3}(i + 2)t_i + \frac{1}{6} \sum_{i \ge 6} (i - 5)t_i$ to the $5^-$-faces and vertices along the paths of $\mathcal{P}$.
In the case of forbidding $\{4, a, 8, 9\}$-cycles with $a \in \{6, 7\}$, $f$ gives at most $\frac{1}{2}$ to each endpoint and $\frac{1}{3}$ to each adjacent $5^-$-face along an $i$-path.
So $f$ gives $\frac{1}{2} \cdot 2 + \frac{1}{3}(i - 1) = \frac{1}{3}(i + 2)$ to each $i$-path.
In the case of forbidding $\{4, 6, 7, 9\}$-cycles, $f$ may instead need to give at most $\frac{1}{3}$ to each vertex and $\frac{1}{6}$ to each adjacent $5$-face along an $i$-path when all vertices of the path are $3$-vertices and all adjacent faces are $5$-faces.
Then $f$ gives $\frac{1}{3} i + \frac{1}{6}(i - 1) = \frac{1}{2} i - \frac{1}{6}$ to each $i$-path of this form.
In addition, $\frac{1}{2} i - \frac{1}{6} > \frac{1}{3}(i + 2)$ only if $i \ge 6$.
So in any case, $f$ needs to send to each $i$-path at most $\sum_{i \ge 2} \frac{1}{3}(i + 2)t_i + \sum_{i \ge 6} \frac{1}{6}(i - 5)t_i$.
By (R3), $f$ may need to send out an additional $\frac{1}{6} \sum_{i \ge 3}t_i$ to poor $10$-faces.
Note that if $f$ sends charge over a $2$-path to a poor face, then the $2$-path has a $4^+$-vertex as an endpoint, and $f$ sends at most $\frac{1}{2} + \frac{1}{3} + \frac{1}{6} \le \frac{1}{3}(2 + 2)$ across this path, so the $\frac{1}{6}$ sent to the poor face is already accounted for in the above formula.
Therefore $f$ sends at most
\[ \frac{1}{3} t_1 + \sum_{i \ge 2} \frac{1}{3}(i + 2)t_i + \sum_{i \ge 6} \frac{1}{6}(i - 5)t_i + \frac{1}{6} \sum_{i \ge 3} t_i \]
to its incident vertices and adjacent faces, from which ~(\ref{eq-10+}) follows.
\end{proof}

Assume that $\mu^*(f) < 0$.
Let $s(f)$ be the number of semi-rich $4$-vertices and $5^+$-vertices on $f$.
Note that each semi-rich $4$-vertex on $f$ saves at least $\frac{1}{3} - \frac{1}{6} = \frac{1}{6}$ and by (R2) each $5^+$-vertex on $f$ gives $\frac{1}{5}$ to $f$.
Then by Lemmas~\ref{12+faces} and~\ref{lem:eq-10+},
\[ \frac{1}{6}\left(2t_1 + 2\sum_{i \ge 3} (i - 2)t_i - 2x\right) < \frac{1}{6} \left(\sum_{i \ge 6} (i - 5)t_i + \sum_{i \ge 3} t_i - s(f)\right), \]
which implies that
\[ 2x > s(f) + 2t_1 + \sum_{i \ge 3} (2i - 5)t_i - \sum_{i \ge 6} (i - 5)t_i \ge s(f) + 2t_1 + t_3 + 3t_4 + 5 \sum_{i \ge 5} t_i. \]

Clearly, $x > 0$, so $d(f) \le 11$.
Recall that $f$ gives at most $\frac{1}{3}(i + 2)$ across any $i$-path with $i \le 5$, and note that this is only possible when the ends of the path are $3$-vertices requiring $\frac{1}{2}$ from $f$, and each face adjacent to $f$ along the path is a $5^-$-face requiring $\frac{1}{3}$ from $f$.
Let $d(f) = 11$.
Then $x = 1$.
So $t_1 = 0$, $t_3 \le 1$ and $t_i = 0$ for $i \ge 4$.
By parity, $t_3 = 1$ and $t_2 = 4$, so $s(f) = 0$.
In this case, $f$ is not good to any $10$-face, so we have $\mu^*(f) \ge (11 - 4) - \frac{5}{3} - 4 \cdot \frac{4}{3} = 0$.

Let $d(f) = 10$.
Then $x = 2$, and $t_1 \le 1$ and $t_i = 0$ for $i \ge 5$.
Let $t_1 = 1$.
Then $t_3 \le 1$ and $t_4 = 0$.
By parity, $t_3 = 1$, and $t_2 = 3$.
It follows that $s(f) = 0$.
In this case, $f$ is not good to any $10$-face, so $\mu^*(f) \ge (10 - 4) - \frac{5}{3} - 3 \cdot \frac{4}{3} - \frac{1}{3} = 0$.
Thus we may assume $t_1 = 0$.
Then $s(f) + t_3 + 3t_4 \le 3$.
By parity, we have three primary cases: $t_4 = 1$ and $t_2 = 3$, or $t_3 = t_2 = 2$, or $t_2 = 5$.

In the first case, $s(f) = 0$, so $f$ is not good to any $10$-face, and $\mu^*(f) \ge 6 - 2 - 3 \cdot \frac{4}{3} = 0$.

In the second case, $s(f) \le 1$, and $f$ is good to at most two $10$-faces.
If $s(f) = 0$, then by Lemma \ref{lem:reducible}$(iii)$, $f$ cannot be good to a $10$-face, so $\mu^*(f) \ge 6 - 2 \cdot \frac{5}{3} - 2 \cdot \frac{4}{3} = 0$.
So let $s(f) = 1$.
If $f$ is good to at most one $10$-face, then $\mu^*(f) \ge 6 - 2 \cdot \frac{5}{3} - 2 \cdot \frac{4}{3} - \frac{1}{6} + \frac{1}{6} = 0$, where the final $\frac{1}{6}$ is the minimum $f$ saves or receives from the vertex counted by $s(f)$.
If $f$ is good to two $10$-faces, then the $4^+$-vertex counted by $s(f)$ must be the end of a $2$-path, so $f$ saves at least $\frac{1}{3}$ from this vertex and $\mu^*(f) \ge 6 - 2 \cdot \frac{5}{3} - 2 \cdot \frac{4}{3} - 2 \cdot \frac{1}{6} + \frac{1}{3} = 0$.

In the last case, $\mu^*(f) \ge 6 - \frac{4}{3} \cdot 5 = -\frac{2}{3}$.
We first assume that $G$ contains no $\{4, 7, 8, 9\}$-cycles.
Note that by (R4a), $f$ gives no charge to adjacent special $5$-faces, where a $5$-face is special if it does not share a $(3,3)$-edge with a $3$-face.
Thus we may assume that $f$ is adjacent to at most one special $5$-face, for otherwise, $f$ saves at least $\frac{2}{3}$ and ends with non-negative charge.
If $f$ is incident to at least two $4^+$-vertices, then $f$ saves at least $2\left(\frac{1}{2} - \frac{1}{6}\right) = \frac{2}{3}$, where the $\frac{1}{6}$ is because $f$ may now be good to adjacent $10$-faces.
If $f$ is incident to one $4^+$-vertex, then $f$ saves at least $\frac{1}{2} + \frac{1}{6} = \frac{2}{3}$ since the $4^+$-vertex must be rich to a $10^+$-face adjacent to $f$, and $f$ cannot be good to any $10$-face.
We may thus assume that $f$ is incident to ten $3$-vertices.
By Lemma~\ref{lem:reducible}$(iv)$ and $(v)$, each $5^-$-face adjacent to $f$ must contain a $4^+$-vertex.
This implies that all $3$-faces adjacent to $f$ are $(3,3,4^+)$-faces, and each adjacent $5$-face other than at most one special $5$-face contains a $4^+$-vertex and shares a $(3,3)$-edge with a $3$-face.
It follows that $f$ is poor to at least three $10^+$-faces if $f$ contains a special $5$-face, and is poor to five $10^+$-faces if $f$ contains no special $5$-faces.
Therefore, by (R2) and (R4a), $f$ receives $\min\left\{\frac{1}{3} + \frac{1}{6} \cdot 3, \frac{1}{6} \cdot 5\right\} > \frac{2}{3}$ from adjacent $10^+$-faces it is poor to, so $f$ ends with non-negative charge.
Now we assume that $G$ contains no $\{4, 6, a, 9\}$-cycles for $a \in \{7, 8\}$.
If $f$ is adjacent to a $5$-face, then $f$ gives at most $\frac{1}{3} \cdot 2 + \frac{1}{6} = \frac{5}{6}$ across this $2$-path.
Thus we may assume that $f$ is adjacent to at most one $5$-face, for otherwise, $f$ saves $2\left(\frac{4}{3} - \frac{5}{6}\right) > \frac{2}{3}$ and ends with non-negative charge.
If $f$ is adjacent to exactly one $5$-face and at least one $4^+$-vertex, then $f$ saves at least $\left(\frac{4}{3} - \frac{5}{6}\right) + \left(\frac{1}{3} - \frac{1}{6}\right) = \frac{2}{3}$.
If $f$ is not adjacent to any $5$-faces, then $f$ saves at least $2\left(\frac{1}{2} - \frac{1}{6}\right) = \frac{2}{3}$ if $f$ is incident to at least two $4^+$-vertices, and at least $\frac{1}{2} + \frac{1}{6} = \frac{2}{3}$ if $f$ is incident to exactly one $4^+$-vertex, where in the latter case the $4^+$-vertex must be a special semi-rich $4$-vertex which gives $\frac{1}{6}$ to $f$ by (R2).
We may therefore assume that all vertices incident to $f$ are $3$-vertices.
By Lemma~\ref{lem:reducible}$(v)$, all $3$-faces adjacent to $f$ are $(3,3,4^+)$-faces.
Thus $f$ is poor to five $10^+$-faces if $f$ is not adjacent to a $5$-face, and $f$ is poor to three $10^+$-faces otherwise.
Therefore $f$ gets $\frac{1}{6}$ from each $10^+$-face good to $f$ and saves $\frac{4}{3} - \frac{5}{6}$ if it is adjacent to a $5$-face, for a total of at least $\min\left\{\frac{1}{6} \cdot 5, \left(\frac{4}{3} - \frac{5}{6}\right) + \frac{1}{6} \cdot 3 \right\} > \frac{2}{3}$.
Therefore in all cases, $f$ ends with non-negative charge.

\section{Final remarks.}

We remark that we are yet unable to prove that planar graphs without $\{4, 5, 8, 9\}$-cycles are DP-3-colorable.  While some of our lemmas may be useful in such a proof (in particular, Lemmas~\ref{near-2-degenerate} and~\ref{12+faces}), this case is considerably more difficult than any of the three cases of Theorem~\ref{awesome}, and a unified proof of all four cases does not seem possible. Another remark is that \Dv{} and Postle showed that planar graphs without cycles of lengths from $4$ to $8$ are ``weakly'' DP-3-colorable. It remains open to know if such planar graphs are DP-$3$-colorable.