11institutetext:  Department of Physics, Chungbuk National University,
Cheongju 28644, Republic of Korea
11email<EMAIL_ADDRESS>22institutetext: Astronomical
Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland
33institutetext: Korea Astronomy and Space Science Institute, Daejon 34055,
Republic of Korea 44institutetext: University of Canterbury, Department of
Physics and Astronomy, Private Bag 4800, Christchurch 8020, New Zealand
55institutetext: Korea University of Science and Technology, 217 Gajeong-ro,
Yuseong-gu, Daejeon, 34113, Republic of Korea 66institutetext: Max Planck
Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany
77institutetext: Department of Astronomy, The Ohio State University, 140 W.
18th Ave., Columbus, OH 43210, USA 88institutetext: Department of Particle
Physics and Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel
99institutetext: Center for Astrophysics $|$ Harvard & Smithsonian 60 Garden
St., Cambridge, MA 02138, USA 1010institutetext: Department of Astronomy and
Tsinghua Centre for Astrophysics, Tsinghua University, Beijing 100084, China
1111institutetext: School of Space Research, Kyung Hee University, Yongin,
Kyeonggi 17104, Republic of Korea 1212institutetext: Department of Astronomy &
Space Science, Chungbuk National University, Cheongju 28644, Republic of Korea
1313institutetext: Department of Physics & Astronomy, Seoul National
University, Seoul 08826, Republic of Korea 1414institutetext: Division of
Physics, Mathematics, and Astronomy, California Institute of Technology,
Pasadena, CA 91125, USA 1515institutetext: Department of Physics, University
of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK

# KMT-2018-BLG-1025Lb: microlensing super-Earth planet orbiting a low-mass
star

Cheongho Han 11 Andrzej Udalski 22 Chung-Uk Lee
(Leading authors)
Michael D. Albrow 3344 Sun-Ju Chung 3355 Andrew Gould 6677 Kyu-Ha Hwang 33
Youn Kil Jung 33 Doeon Kim 11 Hyoun-Woo Kim 33 Yoon-Hyun Ryu 33 In-Gu Shin 33
Yossi Shvartzvald 88 Jennifer C. Yee 99 Weicheng Zang 1010 Sang-Mok Cha 331111
Dong-Jin Kim 33 Seung-Lee Kim 3355 Dong-Joo Lee 33 Yongseok Lee 331111 Byeong-
Gon Park 3355 Richard W. Pogge 77 Chun-Hwey Kim 1212 Woong-Tae Kim
(The KMTNet Collaboration) 1313
Przemek Mróz 221414 Michał K. Szymański 22 Jan Skowron 22 Radosław Poleski 22
Igor Soszyński 22 Paweł Pietrukowicz 22 Szymon Kozłowski 22 Krzysztof Ulaczyk
1515 Krzysztof A. Rybicki 22 Patryk Iwanek 22 Marcin Wrona
(The OGLE Collaboration)
22

(Received ; accepted)

###### Abstract

Aims.  We aim to find missing microlensing planets hidden in the unanalyzed
lensing events of previous survey data.

Methods.  For this purpose, we conduct a systematic inspection of high-
magnification microlensing events, with peak magnifications $A_{\rm
peak}\gtrsim 30$, in the data collected from high-cadence surveys in and
before the 2018 season. From this investigation, we identify an anomaly in the
lensing light curve of the event KMT-2018-BLG-1025. The analysis of the light
curve indicates that the anomaly is caused by a very low mass-ratio companion
to the lens.

Results.  We identify three degenerate solutions, in which the ambiguity
between a pair of solutions (solutions B) is caused by the previously known
close–wide degeneracy, and the degeneracy between these and the other solution
(solution A) is a new type that has not been reported before. The estimated
mass ratio between the planet and host is $q\sim 0.8\times 10^{-4}$ for the
solution A and $q\sim 1.6\times 10^{-4}$ for the solutions B. From the
Bayesian analysis conducted with measured observables, we estimate that the
masses of the planet and host and the distance to the lens are $(M_{\rm
p},M_{\rm h},D_{\rm L})\sim(6.1\leavevmode\nobreak\
M_{\oplus},0.22\leavevmode\nobreak\ M_{\odot},6.7\leavevmode\nobreak\ {\rm
kpc})$ for the solution A and $\sim(4.4\leavevmode\nobreak\
M_{\oplus},0.08\leavevmode\nobreak\ M_{\odot},7.5\leavevmode\nobreak\ {\rm
kpc})$ for the solutions B. The planet mass is in the category of a super-
Earth regardless of the solutions, making the planet the eleventh super-Earth
planet, with masses lying between those of Earth and the Solar system’s ice
giants, discovered by microlensing.

###### Key Words.:

gravitational microlensing – planets and satellites: detection

##  1 Introduction

The microlensing method has unique advantages in detecting some specific
populations of planets. It enables one to detect planets orbiting very faint
low-mass stars, which are the most common populations of stars in the Galaxy,
because of the lensing characteristic that does not depend on the luminosity
of the planet host. Another very important advantage of the method is that its
detection efficiency extends to very low-mass planets because of the slow
decrease of the efficiency with the decrease of the planet/host mass ratio
$q$. The microlensing efficiency decreases as $\sqrt{q}$, while the efficiency
of other methods, for example, the radial-velocity method, decreases in direct
proportion to $q$. See Gaudi (2012) for a review on various advantages of the
microlensing method. With the sensitivity to planets that are difficult to be
detected by other methods, microlensing plays an important role to complement
other methods not only for the complete demographic census of planets but also
for the comprehensive understanding of the planet formation process.

However, these advantages of the microlensing method, especially the latter
one, i.e., the high sensitivity to low-mass planets, were difficult to be
fully realized during the early generation of microlensing experiments, for
example MACHO (Alcock et al, 1997) and OGLE (Udalski et al., 1994). A
planetary microlensing signal, in general, appears as a short-term anomaly to
the smooth and symmetric lensing light curve generated by the host of the
planet (Mao & Paczyński, 1991; Gould & Loeb, 1992). For this reason, a
microlensing planet search should be carried out in two steps: first by
detecting lensing events, and second by inspecting planet-induced anomalies in
the light curves of detected lensing events. The probability for a star to be
gravitationally lensed is very low, on the order of $10^{-6}$ for stars
located in the Galactic bulge field, toward which microlensing surveys have
been and are being carried out (Paczyński, 1991; Griest et al., 1991; Sumi &
Penny, 2016; Mróz et al., 2019), and thus a lensing survey should cover a
large area of sky to increase the number of lensing events by maximizing the
number of monitored stars. This requirement had limited the cadence of lensing
surveys, and subsequently the rate of planet detections, especially that of
very low-mass planets. Gould & Loeb (1992) proposed to overcome this problem
by conducting intensive follow-up observations of survey-detected events,
which led to the first detections of low mass planets (Beaulieu et al., 2006;
Gould et al., 2006). However, this approach is necessarily restricted by
telescope resources to a small number of events.

The planet detection rate has rapidly increased with the operation of high-
cadence lensing surveys including MOA II (Bond et al., 2001), OGLE-IV (Udalski
et al., 2015), and KMTNet (Kim et al., 2016). By employing multiple telescopes
equipped with large-format cameras, these surveys achieve an observation
cadence reaching down to 15 min for dense bulge fields. This cadence is
shorter than those of the first-generation MACHO and OGLE surveys, that had
been carried out with a $\sim 1$ day cadence, by about a factor 100.

Table 1: Microlensing super-Earth planets

Event |  $M_{\rm p}$ ($M_{\oplus}$) |  $M_{\rm host}$ ($M_{\odot}$) | Reference
---|---|---|---
OGLE-2005-BLG-390Lb | $\sim 5.5$ | $\sim 0.22$ | Beaulieu et al. (2006)
MOA-2007-BLG-192Lb | $\sim 3.3$ | $\sim 0.06$ | Bennett et al. (2008)
OGLE-2013-BLG-0341Lb | $\sim 2$ |  $\sim 0.1$–0.15 | Gould et al. (2014)
OGLE-2016-BLG-1195Lb | $\sim 1.25$ | $\sim 0.067$ |  Bond et al. (2017), Shvartzvald et al. (2017)
OGLE-2016-BLG-1928L | $\sim 0.3$ | – | Mróz et al. (2020)
OGLE-2017-BLG-1434Lb | $\sim 4.4$ | $\sim 0.23$ | Udalski et al. (2018)
KMT-2018-BLG-0029Lb | $\sim 7.6$ | $\sim 1.1$ | Gould et al. (2020)
OGLE-2018-BLG-0532Lb | $\sim 8$ | $\sim 0.25$ | Ryu et al. (2019)
OGLE-2018-BLG-0677Lb | $\sim 4.0$ | $\sim 0.12$ | Herrera-Martín et al. (2020)
KMT-2019-BLG-0842Lb | $\sim 10.2$ | $\sim 0.76$ | Jung et al. (2020)

111 The sample is selected with a planet mass limit of $M_{\rm p}\lesssim
10\leavevmode\nobreak\ M_{\oplus}$. OGLE-2016-BLG-1928L is a free-floating
planet, and thus the host mass is not included.

The great shortening of the observation cadence resulted in a rapid increase
of the planet detection rate. The population of planets with a remarkable
increase of the detection rate is super-Earth planets, which are defined as
planets having masses higher than the mass of Earth, but substantially lower
than those of the Solar System’s ice giants, Uranus and Neptune (Valencia et
al., 2007).222 We note that the term ”super-Earth” refers only to the mass of
the planet, and so does not imply anything about the atmosphere structure,
surface conditions, or size of the planet. In Table 1, we list the
microlensing super-Earth planets, along with the masses of the planets and
their hosts, that have been detected from the last 28 year operation of
lensing surveys since 1992. Among the total ten super-Earth planets, seven
were detected during the last four years since the full operation of the
KMTNet survey in 2016, and for all of these events, the KMTNet data played key
roles in detecting and characterizing the planets.

Figure 1:  Light curve of KMT-2018-BLG-1025. The upper and lower panels show
the whole view and the zoomed-in view of the peak region, respectively. The
colors of data points indicate the observatories, as given in the legend. The
curve plotted over the data points is the 1L1S model, for which the residuals
in the peak region are presented in the bottom panel. The dotted vertical
lines at $t_{1}\sim 8274.38$ and $t_{2}\sim 8274.66$ indicate the respective
times of the bump and dip in the residuals to the 1L1S model.

In this paper, we report the detection of a new microlensing super-Earth
planet. The planet was found from a project that has been conducted to search
for unrecognized planets in the previous KMTNet data collected in and before
the 2018 season. In the first part of this project, Han et al. (2020b)
investigated lensing events with faint source stars, considering the
possibility that planetary signals might be missed due to the noise or scatter
of data. From the investigation, they found four planetary events that had not
been reported before. The new planetary system that we report in this work is
found from the second part of the project that has been carried out by
inspecting subtle planetary signals in the light curves of high-magnification
lensing events with peak magnifications $A_{\rm peak}\gtrsim 30$. In this
project, high-magnification events are selected as targets for reinspection
because the sensitivity to planets for these events is high (Griest &
Safizadeh, 1998). Despite the high chance of planet perturbations, some
planetary signals produced by a non-caustic-crossing channel may not be
noticed due to their weak signals (Zhu et al., 2014).

For the presentation of the work, we organize the paper as follows. The
acquisition and reduction processes of the data used in the analysis are
discussed in Sect. 2. We describe the characteristics of the anomaly in the
lensing light curve in Sect. 3. We explain various models tested to explain
the observed anomaly, and show that the anomaly is of a planetary origin in
Sect. 4. The procedure to estimate the angular Einstein radius is discussed in
Sect. 5. We estimate the physical parameters of the planetary system,
including the mass and distance, in Sect. 6. We summarize the results and
conclude in Sect. 7.

Table 2: Data readjustment factors

Data set | $N_{\rm data}$ | $k$ | $\sigma_{\rm min}$
---|---|---|---
KMTA (BLG03) | 1423 | 1.562 | 0.005
KMTA (BLG43) | 1246 | 1.483 | 0.010
KMTC (BLG03) | 1563 | 1.183 | 0.010
KMTC (BLG43) | 1790 | 1.225 | 0.010
KMTS (BLG03) | 1357 | 1.208 | 0.005
KMTS (BLG43) | 1249 | 1.483 | 0.005
OGLE | 1457 | 1.575 | 0.005

333$N_{\rm data}$ indicates the number of each data set.

##  2 Observations and data

The planet is found from the analysis of the microlensing event
KMT-2018-BLG-1075. The source star of the event lies in the Galactic bulge
field with the equatorial coordinates of $({\rm R.A.},{\rm
decl.})=(17:59:27.94,-27:52:41.02)$, which correspond to the Galactic
coordinates of $(l,b)=(2^{\circ}\hskip-2.0pt.461,-2^{\circ}\hskip-2.0pt.082)$.
The flux from the source, which had been constant before the lensing-induced
magnification with an apparent baseline brightness of $I\sim 19.65$, was
highly magnified during about 10 days centered at ${\rm
HJD}^{\prime}\equiv{\rm HJD}-2450000\sim 8274.85$. The event was found from
the post-season inspection of the 2018 season data using the KMTNet Event
Finder System (Kim et al., 2018). At the time of finding, the event drew
little attention due to the similarity of the lensing light curve to that of a
regular single-source single-lens (1L1S) event. See more detailed discussion
in the following section.

Observations of the event by the KMTNet survey were conducted using three
telescopes that are located in Australia (KMTA), Chile (KMTC), and South
Africa (KMTS). Each telescope has a 1.6m aperture and is equipped with a
camera yielding 4 deg2 field of view. The event is located in the two
overlapping survey fields of “BLG03” and “BLG43”, which are displaced with a
slight offset to fill the gaps between the chips of the camera. Observations
in each field were conducted with a 30 min cadence, resulting in a combined
cadence of 15 min. Thanks to the high-cadence coverage of the event using the
globally distributed multiple telescopes, the peak region of the event was
continuously and densely covered.

Figure 2:  $\Delta\chi^{2}$ map in the $s$–$q$ plane. The color coding is set
to represent points with $<1n\sigma$ (red), $<2n\sigma$ (yellow), $<3n\sigma$
(green), $<4n\sigma$ (cyan), $<5n\sigma$ (blue), and $<6n\sigma$ (purple),
where $n=4$. The three encircled regions indicate the positions of the three
degenerate local solutions A, Bc, and Bw.

Images of the source were obtained mostly in the $I$ band, and a fraction of
images were acquired in the $V$ band for the measurement of the source color.
Reduction of the data was done using the KMTNet pipeline (Albrow et al., 2009)
based on the difference imaging method (Tomaney & Crotts, 1996; Alard &
Lupton, 1998), that is developed for the optimal photometry of stars lying in
very dense star fields. For a subset of the KMTC data, an additional
photometry was conducted using the pyDIA software (Albrow, 2017) to construct
a color-magnitude diagram (CMD) of stars and to measure the color of the
source star. The detailed procedure of determining the source color is
described in Sect. 5. Error bars of the data estimated from the automatized
photometric pipeline were readjusted using the method of Yee et al. (2012). In
this method, the error bars are renormalized by $\sigma=[\sigma_{\rm
min}^{2}+(k\sigma_{0})^{2}]^{1/2}$, where $\sigma_{0}$ denotes the error
estimated from the pipeline, $\sigma_{\rm min}$ is a scatter of data, and $k$
is a factor used to make $\chi^{2}$ per degree of freedom (dof) unity. In
Table 2, we list the numbers and the data readjustment factors for the
individual data sets.

Although not alerted at the time of the lensing magnification, the source star
of the event lies in the field covered by the OGLE survey. We, therefore,
checked the OGLE images containing the source and conducted photometry for the
source identified by the KMTNet survey. From this, we recover the OGLE
photometry data, among which seven data points cover the peak of the light
curve. OGLE observations were done using the 1.3m telescope of the Las
Campanas Observatory in Chile, and reduction is carried out using the OGLE
photometry pipeline (Udalski, 2003). We publish the photometry data to ensure
reproducibility of the analysis. The data are available at
http://astroph.chungbuk.ac.kr/$\sim$cheongho/data.html.

Figure 3:  Zoomed-in view of the light curve in the peak region and the
residuals from five tested models including 1L1S, 1L2S, and three 2L1S models
(solutions A, Bc, and Bw). Although three 2L1S model curves are drawn over the
data points in the top panel, it is difficult to distinguish them within the
line width due to the severity of the degeneracy among the solutions.

##  3 Characteristics of the anomaly

The light curve of KMT-2018-BLG-1025 is shown in Figure 1. At the first
glance, it appears to have the smooth and symmetric form of a 1L1S event. A
1L1S modeling yields an impact parameter (scaled to the angular Einstein
radius $\theta_{\rm E}$) of the lens-source approach and an event timescale of
$(u_{0},t_{\rm E})\sim(0.0064,10.1\leavevmode\nobreak\ {\rm days})$,
respectively, indicating that the event has a relatively short timescale with
a high peak magnification of $A_{\rm peak}\sim 1/u_{0}\sim 150$. The 1L1S
model curve is plotted over the data points in Figure 1. The full lensing
parameters and their uncertainties are listed in Table 3, where $t_{0}$
indicates the time of the closest lens-source approach. We note that finite-
source effects are considered in the 1L1S model, but the effects are
negligible, and thus the value of the normalized source radius $\rho$ is not
presented in the table. The normalized source radius is defined as the ratio
of the angular source radius $\theta_{*}$ to $\theta_{\rm E}$, that is,
$\rho=\theta_{*}/\theta_{\rm E}$.

The event was reanalyzed because it was selected in the list of high-
magnification events for close examinations among the KMTNet events detected
in and before the 2018 season in search for planetary signals that had not
been noticed previously. From this analysis, we find that the light curve
exhibits a subtle but noticeable deviation from a 1L1S model.

In the lower two panels of Figure 1, we present a zoomed-in view of the light
curve and residuals from the 1L1S model in the peak region, which shows a
slight bump in the residuals centered at $t_{1}\sim 8274.38$ and a dip
centered at $t_{2}\sim 8274.66$. Although minor, with $\Delta I\lesssim 0.05$
magnitude, the deviation drew our attention for two major reasons. The first
reason is that the deviation occurred in the central magnification region, in
which the chance of a planet-induced perturbation is high (Griest & Safizadeh,
1998). The second reason is that different data sets exhibit a consistent
pattern of deviation. The data sets obtained using the KMTC, located in Chile,
and KMTS, located in South Africa, telescopes show consistent deviations.
Considering that the two telescopes are remotely located, it is difficult to
explain the consistency with a coincidental systematics in the data such as
changes in transparency. Furthermore, the OGLE data in the deviation region
exhibit a consistent anomaly pattern with that of the KMTC data, although
their coverage is not very dense. Therefore, the deviation is very likely to
be real.

Figure 4:  Lens system configurations of the three degenerate 2L1S solutions:
A, Bc, and Bw. In each panel, the blue dots, marked by $M_{1}$ (host) and
$M_{2}$ (planet), are the lens positions, the line with an arrow is the source
trajectory, and the cuspy closed curves are the caustics. The dashed circle
centered at $M_{1}$ represents the Einstein ring. The enlarged views in the
central magnification regions for the individual solutions are presented in
Fig. 5.

##  4 Interpretation of the anomaly

The fact that the anomaly occurred in the peak region of a high-magnification
event suggests the possibility that the anomaly may be produced by a planetary
companion, $M_{2}$, to the primary lens, $M_{1}$. In order to check this
possibility, we conduct an additional modeling under a binary-lens (2L1S)
interpretation.

The modeling is carried out to find a set of lensing parameters that best
explain the observed anomaly in the light curve. In addition to the 1L1S
lensing parameters $(t_{0},u_{0},t_{\rm E},\rho)$, a 2L1S modeling requires
one to add three extra lensing parameters of $(s,q,\alpha)$, which represent
the projected separation (normalized to $\theta_{\rm E}$) and mass ratio
between the binary lens components, and the angle between the source
trajectory and the $M_{1}$–$M_{2}$ axis (source trajectory angle),
respectively. The parameter $\rho$ is included to account for potential
finite-source effects in the lensing curve caused by a source approach close
to or a crossing over lensing caustics induced by a lens companion. The 2L1S
modeling is done in two steps. In the first step, we conduct grid searches for
the binary lensing parameters $s$ and $q$, while the other parameters are
found using a downhill approach based on the Markov Chain Monte Carlo (MCMC)
algorithm. Considering that central anomalies can also be produced by a very
wide or a close binary companion with a mass roughly equal to the primary
(Han, 2009a), we set the ranges of $(s,q)$ wide enough to check the binary
origin of the anomaly: $-1.0\leq\log s\leq 1.0$ and $-5.0\leq\log q\leq 1.0$.
In the second step, the individual local solutions found from the first step
are refined by allowing all parameters (including $s$ and $q$) to vary.

Figure 5:  Lens system configurations in the central magnification region for
the three 2L1S solutions. Notations are same as those in Fig. 4. The two
orange circles represent the source positions at the times of the major
anomalies at $t_{1}$ and $t_{2}$ that are marked in Figs. 1 and 3. The size of
the circle is scaled to the source size. In the case of the solution A, for
which the source size cannot be securely measured, the radius of the circle is
set to that of the best-fit value.  Table 3: Lensing parameters of various
tested models

Parameter | 1L1S | 1L2S | 2L1S
---|---|---|---
|  |  | Solution A |  Solution Bc |  Solution Bw
$\chi^{2}$/dof | $10251.0/10082$ | $10127.9/10077$ | $10081.0/10078$ | $10089.4/10078$ | $10092.8/10078$
$t_{0}$ (HJD′) | $8274.845\pm 0.001$ | $8274.845\pm 0.001$ | $8274.843\pm 0.001$ | $8274.843\pm 0.001$ | $8274.843\pm 0.001$
$u_{0}$ (10-3) | $6.416\pm 0.177$ | $7.471\pm 0.532$ | $7.089\pm 0.237$ | $8.575\pm 0.319$ | $8.453\pm 0.334$
$t_{\rm E}$ (days) | $10.071\pm 0.2453$ | $9.893\pm 0.262$ | $9.571\pm 0.258$ | $8.889\pm 0.234$ | $8.986\pm 0.239$
$s$ | – | – | $0.937\pm 0.021$ | $0.883\pm 0.025$ | $1.097\pm 0.036$
$q$ (10-4) | – | – | $0.829\pm 0.270$ | $1.627\pm 0.499$ | $1.584\pm 0.564$
$\alpha$ (rad) | – | – | $6.188\pm 0.009$ | $2.697\pm 0.023$ | $2.693\pm 0.022$
$\rho$ (10-3) | – | – | $\lesssim 5.5$ | $7.657\pm 0.645$ | $7.351\pm 0.632$
$t_{0,2}$ (HJD′) | – | $8274.416\pm 0.017$ | – | – | –
$u_{0,2}$ (10-3) | – | $7.012\pm 1.793$ | – | – | –
$\rho_{2}$ (10-3) | – | – | – | – | –
$q_{F}$ | – | $0.012\pm 0.004$ | – | – | –

444 ${\rm HJD}^{\prime}\equiv{\rm HJD}-2450000$.

From the 2L1S modeling, we identify three degenerate local solutions. Figure 2
shows the locations of the local solutions in the $\Delta\chi^{2}$ map on the
$s$–$q$ plane obtained from the grid search. The individual locals lie at
$(s,q)\sim(0.94,0.8\times 10^{-4})$, solution “A”, $(0.89,1.6\times 10^{-4})$,
solution “Bc”, and $(1.10,1.6\times 10^{-4})$, solution “Bw”. Here the
subscripts “c”, standing for close, and “w”, standing for wide, imply that the
normalized binary separation is less ($s<1.0$, close solution) and greater
($s>1.0$, wide solution) than unity, respectively. The model curves of the
individual 2L1S solutions and the residuals from the models in the region
around the peak of the light curve are shown in Figure 3. In Table 3, we also
list the lensing parameters of the solutions along with the values of
$\chi^{2}$/dof for the individual models. The uncertainty of each lensing
parameter is estimated as the standard deviation of the distribution of points
in the MCMC chain under the assumption that the distribution is gaussian. The
$\Delta\chi^{2}=8.4$ difference between the A solution and the minimum of the
two B solutions is not big enough to confidently distinguish between them. To
be noted among the parameters is that the mass ratios, which are $q\sim
0.8\times 10^{-4}$ for the solution A and $\sim 1.6\times 10^{-4}$ for the
solutions B, are very low, indicating that the primary lens is accompanied by
a very low-mass planetary companion according to the models. From an
additional modeling considering microlens-parallax effects (Gould, 1992), we
find that it is difficult to securely constrain the microlens parallax
$\pi_{\rm E}$ due to the relatively short timescale, $\sim 9$ days, of the
event.

The identified local solutions are subject to two different types of
degeneracy. The ambiguity between the pair of the solutions Bc and Bw is
caused by the well-known close–wide degeneracy (Griest & Safizadeh, 1998;
Dominik, 1999; An, 2005). The solution A is not subject to this type of
degeneracy because the source trajectory of the corresponding wide solution
passes over the planetary caustic located at a position with a separation from
$M_{1}$ of $\sim s-1/s\sim 0.12$ on the planet side, and this causes a poor
fit of the wide solution to the observed data.

We note that the degeneracy between the A and B solutions is a new type that
has not been reported before. The degeneracy is accidental in the sense that
it is caused by the unexpected combination of multiple lens parameters instead
of being rooted in the lensing physics, for example, the close–wide degeneracy
that is originated in the invariance of the binary lens equations with $s$ and
$s^{-1}$. For such accidental degeneracies, it is difficult to identify them
from the exploration of the numerous combinations of lensing parameters and
the simulations of various observational conditions, and thus they are mostly
identified from the analyses of actual lensing events, as illustrated in the
cases of the events OGLE-2011-BLG-0526 and OGLE-2011-BLG-0950/MOA-2011-BLG-336
(Choi et al., 2012), OGLE-2012-BLG-0455/MOA-2012-BLG-206 (Park et al., 2014),
and MOA-2016-BLG-319 (Han et al., 2018). Although the offsets of the source
trajectory from the central caustic for both solutions, $\xi\sim
u_{0}/\cos\alpha\sim 7.1\times 10^{-3}$ for the solution A and $\xi\sim
7.7\times 10^{-3}$ for the solution B, are similar to each other, this
degeneracy is different from the caustic-chirality degeneracy reported by
Skowron et al. (2018) and Hwang et al. (2018) for two reasons. First, the
source stars of the two solutions A and B move in almost opposite directions,
while the source directions of the two solutions subject to the caustic-
chirality degeneracy are nearly identical. Second, while the caustic-chirality
degeneracy, in general, occurs when the source passes a planetary caustic,
around which the magnification pattern on the left and right sides are
approximately symmetric (Gaudi & Gould, 1997), the magnification pattern
around the central caustic inducing the observed anomaly is not symmetric
(Chung et al., 2005).

The lens system configurations of the individual 2L1S local solutions are
shown in Figure 4. In each panel of the figure, the blue dots marked by
$M_{1}$ and $M_{2}$ denote the positions of the lens components, the line with
an arrow represents the source trajectory, and the red closed curves are
caustics. The planet induces two sets of caustics, one lying near the position
of $M_{1}$ (central caustic) and the other lying at a position with a
separation from $M_{1}$ of $\sim s-1/s$ (planetary caustic). For all
solutions, the anomaly is explained by the passage of the source close to the
central caustic, but the source incidence angles of the solutions A and B
differ from one another: $\alpha\sim-5^{\circ}\hskip-2.0pt.4$ for the solution
A and $\alpha\sim 26^{\circ}$ for the solutions Bc and Bw.

Figure 6:  Cumulative distributions of $\Delta\chi^{2}$ for the three
degenerate 2L1S (solutions A, Bc, and Bw) models and 1L2S model with respect
to the 1L1S model. The dotted vertical lines denote the times of the major
anomalies at $t_{1}$ and $t_{2}$ that are marked in Fig. 1.

Figure 5 shows the enlarged views of the configuration in the central
magnification region for the individual solutions. In each panel, we mark the
source positions corresponding to the times $t_{1}$ and $t_{2}$ (two orange
circles), and draw equi-magnification contours (grey curves around the
caustic). Around a central caustic, the magnification excess, defined by
$\epsilon=(A_{\rm 2L1S}-A_{\rm 1L1S})/A_{\rm 1L1S}$, varies depending on the
region. Here $A_{\rm 2L1S}$ and $A_{\rm 1L1S}$ denote the 2L1S and 1L1S
lensing magnifications, respectively. Positive anomalies occur in the regions
around the cusps of the caustic, and negative anomalies arise in the outer
region of the fold caustic and the back end region of the wedge-shaped
caustic. See example maps of magnification excess around central caustics
presented in Han (2009a) and Han (2009b). According to the solution A, the
bump at $t_{1}$ is produced when the source passes through the positive excess
region extending from the protrudent cusp of the central caustic, and the dip
at $t_{2}$ is produced when the source moves through the negative excess
region formed along the caustic fold. According to the solutions B, on the
other hand, the bump and dip are produced by the successive passage of the
positive and negative excess regions formed in the back end region of the
caustic, respectively.

Models with the addition of a planetary companion to the lens improves the fit
by $\Delta\chi^{2}\sim 158$ – 170 with respect to the 1L1S solution. To show
the region of the fit improvement, we present the cumulative distributions of
$\Delta\chi^{2}$ for the three 2L1S solutions in Figure 6. The distributions
show that the major fit improvement occurs at around $t_{1}$ and $t_{2}$, that
are the times of the major anomalies from the 1L1S model. This can also be
seen in the residuals of the 2L1S solutions, shown in Figure 3, which shows
that the major residuals from the 1L1S model at around $t_{1}$ and $t_{2}$
disappear in the residuals of the 2L1S solutions.

Figure 7:  Distributions of points in the MCMC chains for the three degenerate
2L1S solutions. The red, yellow, green, cyan, and blue colors represent points
within 1$\sigma$, 2$\sigma$, 3$\sigma$, 4$\sigma$, and 5$\sigma$,
respectively. The significance level is determined so that $n\sigma$
corresponds to $\Delta\chi^{2}=n^{2}$ with rescaled uncertainties.

We also check the possibility that the anomaly was produced by a companion to
the source: 1L2S model. Similar to the case of a 2L1S modeling, extra
parameters in addition to those of the 1L1S modeling are needed for a 1L2S
modeling. Following the parameterization of Hwang et al. (2013), these extra
parameters are $(t_{0,2},u_{0,2},\rho_{2},q_{F})$, which denote the time of
the closest approach of the second source to the lens, the companion-lens
separation at that time, the normalized radius of the source companion, and
the flux ratio between the source stars, respectively. Considering the
possibility that source stars approach very close to the lens, we consider
finite-source effects in the 1L2S modeling by including two parameters
($\rho$, $\rho_{2}$), which denote the normalized source radii of the first
and second source stars, respectively. In the 1L2S modeling, we use the
parameters of the 1L1S model as initial parameters, and set the other
parameters considering the anomaly features of the light curve. We list the
best-fit lensing parameters of the 1L2S model in Table 3, present the
residuals from the model in Figure 3, and show the cumulative $\Delta\chi^{2}$
distribution with respect to the 1L1S model in Figure 6. We note that the
normalized source radii of both source stars are not measurable due to very
week finite-source effects, and thus the values of $\rho$ and $\rho_{2}$ are
not listed in Table 3. It is found that the 1L2S model reduces the residuals
at around $t_{1}$, but the model still leaves noticeable residuals near the
peak of the lightcurve. The fit of the 1L2S model is better than the 1L1S
model by $\Delta\chi^{2}\sim 123$, but it is worse than the 2L1S models by
$\Delta\chi^{2}\sim 35$ – 46. We, therefore, conclude that the anomaly in the
lensing light curve was generated by a companion to the lens rather than a
companion to the source.

##  5 Angular Einstein radius

In general cases of lensing events, the angular Einstein radius is estimated
from the combination of the angular source radius $\theta_{*}$ and the
normalized source radius $\rho$ by

$\theta_{\rm E}={\theta_{*}\over\rho}.$ (1)

The value of $\theta_{*}$ can be derived from the color and brightness of the
source, and the value of $\rho$ is decided from the analysis of the light
curve affected by finite-source effects. Then, the prerequisite for the
measurement of $\theta_{\rm E}$ is that a lensing light curve should be
affected by finite-source effects to yield the normalized source radius
$\rho$.555The angular Einstein radius can also be measured by separately
imaging the lens and source. By resolving the images, one can measure the
vector separation $\Delta\mbox{\boldmath$\theta$}$ between the lens and source
and hence their heliocentric relative proper motion by
$\mbox{\boldmath$\mu$}_{\rm hel}=\Delta\mbox{\boldmath$\theta$}/\Delta t$,
where $\Delta t$ represents the time elapsed since the event. Then, the
Einstein radius is determined by $\theta_{\rm E}=\mu_{\rm geo}t_{\rm E}$. Here
the geocentric relative proper motion is related to
$\mbox{\boldmath$\mu$}_{\rm hel}$ by $\mbox{\boldmath$\mu$}_{\rm
geo}=\mbox{\boldmath$\mu$}_{\rm hel}-{\bf v}_{\oplus,\perp}\pi_{\rm rel}/{\rm
au}$, where ${\bf v}_{\oplus,\perp}$ is Earth’s velocity projected on the
plane of the sky at $t_{0}$. Due to the long time span $\Delta t$ required for
the lens–source resolution together with the limited access to high-resolution
instrument, there exist just five cases of planetary lens events for which the
values of $\theta_{\rm E}$ are measured by this method: OGLE-2005-BLG-071
(Bennett et al., 2020), OGLE-2005-BLG-169 (Batista et al., 2015; Bennett et
al., 2015), OGLE-2012-BLG-0950 (Bhattacharya et al., 2018), MOA 2013 BLG-220
(Vandorou et al., 2020), and MOA-2007-BLG-400 (Bhattacharya et al., 2020).

Figure 8:  The source (cross mark) position with respect to the centroid of
the red giant clump (RGC, red dot) in the instrumental color-magnitude diagram
of stars lying in the vicinity of the source. We mark source positions
corresponding to the three degenerate solutions, which yield very similar
source locations. Also marked is the position of the blend (green dot).

In the case of KMT-2018-BLG-1025, the feasibility of measuring $\rho$ varies
depending on the solution. We find that finite-source effects are securely
detected according to the solutions Bc and Bw, but according to the solution
A, the effects are not firmly detected and the model is consistent with a
point-source model within 3$\sigma$. This is shown in Figure 7, in which we
present the $\Delta\chi^{2}$ distributions of points in the MCMC chain
obtained from the modeling runs of the three degenerate 2L1S solutions. These
scatter plots show that the normalized source radii of the B solutions,
$\rho\sim 7$ – $8\times 10^{-3}$, are well determined, but only a upper limit,
$\rho_{\rm max}\sim 5.5\times 10^{-3}$, can be placed for the solution A. As a
result, the angular Einstein radius is determined for the solutions Bc and Bw,
but only a lower limit can be placed for the solution A. Below, we describe
the procedure of $\theta_{\rm E}$ estimation for the individual solutions.

Table 4: Source color, brightness, Einstein radius, and proper motion

Value | Solution A |  Solution Bc |  Solution Bw
---|---|---|---
$(V-I,I)$ | $(2.585\pm 0.029,21.465\pm 0.003)$ | $(2.584\pm 0.027,21.384\pm 0.002)$ | $(2.582\pm 0.027,21.386\pm 0.002)$
$(V-I,I)_{\rm RGC}$ | $(2.535,16.461)$ | $\leftarrow$ | $\leftarrow$
$(V-I,I)_{{\rm RGC},0}$ | $(1.060,14.324)$ | $\leftarrow$ | $\leftarrow$
$(V-I,I)_{0}$ | $(1.110\pm 0.029,19.328\pm 0.003)$ | $(1.109\pm 0.027,19.247\pm 0.002)$ | $(1.107\pm 0.027,19.249\pm 0.002)$
$\theta_{*}$ (uas) | $0.67\pm 0.05$ | $0.70\pm 0.05$ | $0.69\pm 0.05$
$\theta_{\rm E}$ (mas) | $\geq 0.12$ | $0.091\pm 0.007$ | $0.094\pm 0.007$
$\mu$ (mas yr-1) | $\geq 4.3$ | $3.73\pm 0.28$ | $3.83\pm 0.29$

666 The notation “$\leftarrow$” indicates that the value is same as that
presented in the left column.

The angular source radius and the resulting Einstein radius for each solution
is estimated following the routine procedure outlined in Yoo et al. (2004). In
the first step of the procedure, we specify the source type by placing the
positions of the source and the centroid of the red giant clump (RGC) in the
CMD of stars lying in the vicinity of the source. Figure 8 shows the positions
of the source, marked by a black cross at $(V-I,I)=(2.585\pm 0.029,21.465\pm
0.003)$, estimated from the solution A, and the RGC centroid, red dot at
$(V-I,I)_{\rm RGC}=(2.535,16.461)$, in the instrumental CMD constructed using
the pyDIA photometry data of the KMTC $I$\- and $V$-band images. We note that
the source color and brightness estimated from the other solutions, marked by
grey crosses and listed in Table 4, result in similar values. Also marked is
the position of the blend, green dot. As we will show in the following
section, the lens is a very low-mass M dwarf, while the color and brightness
of the blend correspond to an early main-sequence star or a subgiant. This
implies that the contribution of the lens flux to the blended flux is
negligible. We calibrate the source color and brightness using the known de-
reddened values of the RGC centroid, $(V-I,I)_{0}=(1.060,14.324)$ (Bensby et
al., 2013; Nataf et al., 2013), as references. From the measured offsets in
the color $\Delta(V-I)$ and brightness $\Delta I$ between the source and RGC
centroid, the reddening and extinction corrected values of the source color
and brightness are estimated by

$(V-I,I)_{0}=(V-I,I)_{\rm RGC,0}+\Delta(V-I,I).$ (2)

The values of $(V-I,I)_{0}$ corresponding to the individual solutions are
listed in Table 4. The estimated color and brightness are
$(V-I,I)_{0}\sim(1.1,19.3)$, indicating that the source is an early K-type
main sequence star. We then convert $V-I$ color into $V-K$ color using the
color-color relation of Bessell & Brett (1988), and estimate the angular
source radius using the $(V-K)$ – $\theta_{*}$ relation of Kervella et al.
(2004). The measured source radii are in the range of
$0.67\lesssim\theta_{*}/\mu{\rm as}\lesssim 0.70$. Finally, the angular
Einstein radius and the relative lens-source proper motion are estimated by
the relations $\theta_{\rm E}=\theta_{*}/\rho$ and $\mu=\theta_{\rm E}/t_{\rm
E}$, respectively.

In Table 4, we list the values of $\theta_{*}$, $\theta_{\rm E}$, and $\mu$
corresponding to the individual solutions. We note that the lower limits of
$\theta_{\rm E}$ and $\mu$ are presented for the solution A, for which only
the upper limit of $\rho$ is constrained. We note that the Einstein radius
estimated from the solutions B, $\theta_{\rm E}=0.091$ for the solution Bc and
0.094 mas Bw, is substantially smaller than $\sim 0.5$ mas of a typical
lensing event produced by an M dwarf with a mass $\sim 0.3\leavevmode\nobreak\
M_{\odot}$ located roughly halfway between the lens and source. The angular
Einstein radius is related to the lens mass and distance by

$\theta_{\rm E}=(\kappa M\pi_{\rm rel})^{1/2};\qquad\pi_{\rm rel}={\rm
au}\left({1\over D_{\rm L}}-{1\over D_{\rm S}}\right),$ (3)

where $\kappa=4G/(c^{2}{\rm au})$ and $D_{\rm S}$ is the distance to the
source. Then, the small value of $\theta_{\rm E}$ for the solutions B suggests
that the lens has a low mass or it is located close to the source.

##  6 Physical lens parameters

The lens mass and distance are unambiguously determined by simultaneously
measuring $\theta_{\rm E}$ and $\pi_{\rm E}$, which are related to the
physical lens parameters by

$M={\theta_{\rm E}\over\kappa\pi_{\rm E}},\qquad D_{\rm L}={{\rm
au}\over\pi_{\rm E}\theta_{\rm E}+\pi_{\rm S}}.$ (4)

Here $\pi_{\rm S}={\rm au}/D_{\rm S}$ denotes the parallax of the source. In
the case of KMT-2018-BLG-1025, only $\theta_{\rm E}$ is measured for the
solutions Bc and Bw, and neither of $\theta_{\rm E}$ and $\pi_{\rm E}$ is
measured for the solution A. Although this makes it difficult to uniquely
determine $M$ and $D_{\rm L}$, these parameters can be statistically
constrained from a Bayesian analysis with the priors of a lens mass function
and a Galactic model.

Table 5: Physical lens parameters Parameter | Solution A |  Solution Bc |  Solution Bw
---|---|---|---
$M_{\rm h}$ ($M_{\odot}$) | $0.219^{+0.297}_{-0.120}$ | $0.082^{+0.126}_{-0.044}$ | $0.084^{+0.131}_{-0.045}$
$M_{\rm p}$ ($M_{\oplus}$) | $6.059^{+8.210}_{-3.321}$ | $4.444^{+6.808}_{-2.401}$ | $4.447^{+6.932}_{-2.359}$
$D_{\rm L}$ (kpc) | $6.705^{+0.994}_{-1.166}$ | $7.470^{+0.908}_{-0.920}$ | $7.450^{+0.904}_{-0.916}$
$a_{\perp}$ (au) | $1.305^{+0.194}_{-0.227}$ | $0.508^{+0.062}_{-0.063}$ | $0.650_{-0.079}^{+0.080}$

In the Bayesian analysis, we conduct a Monte Carlo simulation to produce
artificial lensing events. For the production of events, we use priors of a
mass function, to assign lens masses, and a Galactic model, to assign lens
locations and relative lens-source transverse velocities. For the mass
function, we use a model constructed by combining those of Zhang et al. (2019)
and Gould (2000) mass functions, and the model considers not only stellar
lenses but also substellar brown dwarfs and stellar remnants. For the physical
lens distribution, we use the modified version of Han & Gould (2003) model, in
which the original double-exponential disk model is replaced with the Bennett
et al. (2014) model. We note that the distance to the source, $D_{\rm S}$, is
allowed to vary by choosing $D_{\rm S}$ from the physical distribution model
of the bulge instead of using a fixed value. For the dynamical distribution of
the lens and source motion, we adopt the Han & Gould (1995) model. A detailed
description of the adopted priors is given in Han et al. (2020a). The number
of events produced by the simulation for each solution is $10^{7}$. With the
events produced by the simulation, posteriors of $M$ and $D_{\rm L}$ are
obtained by constructing the probability distributions of events that are
consistent with the measured observables. Although the $\rho$ value is not
tightly constrained for the solution A, we use its distribution obtained based
on the MCMC links to weight the posteriors of the $M$ and $D_{\rm L}$.

Figure 9:  Bayesian posteriors of the host mass for the three degenerate 2L1S
solutions. The three curves in each panel represent contributions by the disk
(blue), bulge (red), and total (black) lens populations. The solid vertical
line represents the median, and the two dotted vertical lines indicate
1$\sigma$ range of the distribution.

The posteriors for the host mass, $M_{\rm h}$, and distance to the lens are
shown in Figures 9 and 10, respectively. For each posterior, we present three
distributions, in which the red and blue distributions are contributions by
the bulge and disk lens populations, respectively, and the black distribution
is the sum of contributions by the both lens populations. In Table 5, we list
the estimated physical parameters of the host and planet ($M_{\rm p}$) masses,
distance, and projected physical separation ($a_{\perp}$) of the planet from
its host. For each physical parameter, we choose a median of the probability
distribution as a representative value, and the upper and lower limits are
estimated as the 16% and 84% ranges of the distributions. The estimated masses
of the planet and host are

$(M_{\rm p},M_{\rm h})\sim\begin{cases}(6.1\leavevmode\nobreak\
M_{\oplus},0.22\leavevmode\nobreak\ M_{\odot})&\text{for solution A},\\\
(4.4\leavevmode\nobreak\ M_{\oplus},0.08\leavevmode\nobreak\
M_{\odot})&\text{for solution B.}\\\ \end{cases}$ (5)

The planet mass is in the category of a super-Earth regardless of the
solutions, and thus the planet is the eleventh super-Earth planet discovered
by microlensing. The host mass varies depending on the solutions: a mid-M
dwarf for the solution A and a very late M dwarf or possibly a substellar
brown dwarf for the solutions B. The estimated distance to the lens is

$D_{\rm L}\sim\begin{cases}6.7\leavevmode\nobreak\ {\rm kpc}&\text{for
solution A},\\\ 7.5\leavevmode\nobreak\ {\rm kpc}&\text{for solution B.}\\\
\end{cases}$ (6)

The host mass estimated from the solutions B is substantially smaller than the
corresponding value of the solution A. This is because the host mass of the
solution A is estimated mostly based on the single constraint of the event
timescale, $t_{\rm E}\sim 9.6$ days, but the mass of the solutions B is
estimated with the additional constraint of the small Einstein radius,
$\theta_{\rm E}\sim 0.09$ mas. For the same reason, the distance to the lens
predicted by the solutions B, $\sim 7.5\leavevmode\nobreak\ {\rm kpc}$, is
greater than the distance expected from the solution A, $\sim
6.7\leavevmode\nobreak\ {\rm kpc}$.

Figure 10:  Bayesian posteriors of the lens distance for the three 2L1S
solutions. Notations are same as those of in Fig. 9.

The degeneracy between solutions A and B can very likely be lifted if the lens
and source can be resolved from future follow-up observations using high-
resolution adaptive optics (AO) instrument. KMT-2018-BLG-1025 presents an
unusual case for which the degeneracy can be lifted purely by the proper
motion measurement, which is the most robust result from AO follow-up
observations. From Figure 7, $\rho_{\rm Sol\ A}<5.5\times 10^{-3}$ and
$\rho_{\rm Sol\ B}>5.9\times 10^{-3}$, both at $3\,\sigma$, while from Tables
3 and 4, the quantity $\theta_{*}/t_{\rm E}$ is about 10% larger for solution
A than solutions Bc and Bw. Therefore, the $3\sigma$ limits for
$\mu=\theta_{*}/(\rho t_{\rm E})$ barely overlap. Most likely, the actual
proper motion measurement will be well away from this boundary, e.g., near the
best fits $\mu\simeq 3.8\leavevmode\nobreak\ {\rm mas}\leavevmode\nobreak\
{\rm yr}^{-1}$ (for solutions B) or $\mu\simeq 5.7\leavevmode\nobreak\ {\rm
mas}\leavevmode\nobreak\ {\rm yr}^{-1}$ (for solution A). Only if the measured
value is about half way between will the correct solution remain undetermined.
Note that the long tail in the solution A distribution, which prevented a
precise estimate of $\theta_{\rm E}$ and $\mu$ for this case, does not affect
the resolution of the degeneracy: if the true value of $\rho$ is in this tail,
then the proper motion will be high, and solution A will be unambiguously
favored. To be confident of detecting the lens, one should allow for proper
motions as low as $\mu\sim 3\leavevmode\nobreak\ {\rm mas}\leavevmode\nobreak\
{\rm yr}^{-1}$, which are permitted by solutions B. However, even at this slow
pace, the source and lens will be separated by $30$ mas in 2028, the earliest
possible date for first AO light 30m class telescopes. At that point the
source and lens can be easily resolved. By contrast, the close–wide degeneracy
between the solutions Bc and Bw cannot be resolved because the relative proper
motions expected from the degenerate solutions are similar to one another.

##  7 Conclusion

We reported the discovery of a super-Earth planet that was found from the
analysis of the lensing event KMT-2018-BLG-1025. The planetary signal in the
lensing light curve had not been noticed during the season of the event
discovery, and was found from the systematic inspection of high-magnification
events in the KMTNet data collected in and before the 2018 season. We
identified three degenerate solutions, in which the ambiguity between a pair
of solutions was caused by the previously known close–wide degeneracy, and the
degeneracy between these and the other solution was a new type that had not
been reported before. The estimated mass ratio between the planet and host was
$q\sim 0.8\times 10^{-4}$ for one solution and $\sim 1.6\times 10^{-4}$ for
the other pair of solutions. From the Bayesian analysis carried out with the
measured observables, we estimated that the masses of the planet and host and
the distance to the lens were $(M_{\rm p},M_{\rm h},D_{\rm
L})\sim(6.1\leavevmode\nobreak\ M_{\oplus},0.22\leavevmode\nobreak\
M_{\odot},6.7\leavevmode\nobreak\ {\rm kpc})$ for one solution and
$\sim(4.4\leavevmode\nobreak\ M_{\oplus},0.08\leavevmode\nobreak\
M_{\odot},7.5\leavevmode\nobreak\ {\rm kpc})$ for the other solutions. The
planet mass was in the category of a super-Earth regardless of the solutions,
making the planet the eleventh super-Earth planet discovered by microlensing.
Due to the substantial difference between the relative lens-source proper
motions expected from the two sets of solutions, the degeneracy between the
solutions can be lifted by resolving the lens and source from future high
resolution imaging observations. These observations will also yield the mass
and distance of the lens, and so the mass of the planet.

###### Acknowledgements.

Work by CH was supported by the grants of National Research Foundation of
Korea (2020R1A4A2002885 and 2019R1A2C2085965). Work by AG was supported by JPL
grant 1500811. This research has made use of the KMTNet system operated by the
Korea Astronomy and Space Science Institute (KASI) and the data were obtained
at three host sites of CTIO in Chile, SAAO in South Africa, and SSO in
Australia. The OGLE project has received funding from the National Science
Centre, Poland, grant MAESTRO 2014/14/A/ST9/00121 to AU.

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${\Delta^{{\mathrm{sand}}}}_{i}^{\prime}-{\Delta^{{\mathrm{sand}}}}_{i}=\mathsf{PNL}_{i}$.

##  Appendix H Proof of Proposition 4

Using the bound from Proposition 4 we have for a permutation $\pi$:

$\displaystyle\mathsf{PNL}_{\pi(i)}-\mathsf{PNL}_{i}$ $\displaystyle\leq
a\Delta_{\pi(i)}+b+c\left(\sum_{\ell=1}^{\pi(i)-1}d^{\pi(i)-\ell-1}\Delta_{\ell}\right)-\Delta_{i}-e^{i}$

and correspondingly, we have a lower bound:

$\displaystyle\mathsf{PNL}_{\pi(i)}-\mathsf{PNL}_{i}$
$\displaystyle\geq\Delta_{\pi(i)}+e^{\pi(i)}-a\Delta_{i}-b-c\sum_{\ell=1}^{i-1}d^{i-\ell-1}\Delta_{\ell}$

Applying $\max$ and taking expectations over $\pi\sim S_{k}$:

$\displaystyle\mathop{\bf E{}}_{\pi\sim
S_{k}}\left[\max_{i\in[k]}\,\lvert\mathsf{PNL}_{\pi(i)}-\mathsf{PNL}_{i}\rvert\right]$
$\displaystyle\leq\mathop{\bf E{}}_{\pi\sim
S_{k}}\left[\max_{i\in[k]}a\Delta_{\pi(i)}+b+c\left(\sum_{\ell=1}^{\pi(i)-1}d^{\pi(i)-\ell-1}\Delta_{\ell}\right)-\Delta_{i}-e^{i}\right]$

We now adapt the methodology used by [CAE22] to get our final bound. First,
define the partial sums
$R_{i}(T_{k},\pi)=a\Delta_{\pi(i)}+b+c\left(\sum_{\ell=1}^{\pi(i)-1}d^{\pi(i)-\ell-1}\Delta_{\ell}\right)-\Delta_{i}-e^{i}$
and consider the binary search tree $\mathsf{BST}(\mathbf{R}(T_{k},\pi))$
whose root is $R_{1}(T_{k},\pi)$. The elements $R_{j}(T_{k},\pi)$ are added
sequentially to this tree. Then, following [CAE22, §3], we have the following
bounds:

$\displaystyle\max_{i}|\mathsf{PNL}_{\pi(i)}-\mathsf{PNL}_{i}|$
$\displaystyle\leq|R_{1}(T_{k},\pi)|+\max_{j}\left|R_{j}(T_{k},\pi)\right|\mathsf{height}(\mathsf{BST}(\mathbf{R}(T_{k},\pi)))$

Now, recalling from [Ree03] that for equiprobable permutations $\mathop{\bf
E{}}_{\pi\sim
S_{k}}[\mathsf{height}(\mathsf{BST}(\mathbf{R}(T_{k},\pi)))]=\alpha\log
k-\beta\log\log k$, we have:

$\displaystyle\mathop{\bf E{}}_{\pi\sim
S_{k}}\left[\max_{i}|\mathsf{PNL}_{\pi(i)}-\mathsf{PNL}_{i}|\right]$
$\displaystyle\leq\mathop{\bf E{}}_{\pi\sim
S_{k}}\left[|R_{1}(T_{k},\pi)|+\max_{j}\left|R_{j}(T_{k},\pi)\right|\mathsf{height}(\mathsf{BST}(\mathbf{R}(T_{k},\pi)))\right]$
$\displaystyle=\mathop{\bf E{}}_{\pi\sim
S_{k}}|R_{1}(T_{k},\pi)|]+\left(\max_{j}|R_{j}(T_{k},\pi)|\right)\mathop{\bf
E{}}_{\pi\sim S_{k}}[\mathsf{height}(\mathsf{BST}(\mathbf{R}(T_{k},\pi)))]$
$\displaystyle\leq\mathop{\bf E{}}_{\pi\sim S_{k}}|R_{1}(T_{k},\pi)|]$
$\displaystyle+\max_{i,j}\left|a\Delta_{i}+b+c\sum_{\ell=1}^{i-1}d^{i-\ell-1}\Delta_{\ell}-\Delta_{j}-e^{j}\right|(\alpha\log
k-\beta\log\log k)$

where the last inequality uses the following identity

$\max_{j}\left|a\Delta_{\pi(j)}+b+c\sum_{\ell=1}^{\pi(j)-1}d^{\pi(j)-\ell-1}\Delta_{\ell}-\Delta_{j}-e^{j}\right|\leq\max_{i,j}\left|a\Delta_{i}+b+c\sum_{\ell=1}^{i-1}d^{i-\ell-1}\Delta_{\ell}-\Delta_{j}-e^{j}\right|$

Now, note that we can bound:

$\displaystyle\mathop{\bf E{}}_{\pi\sim
S_{k}}|R_{1}(T_{k},\pi)|=\frac{1}{k}\sum_{j=1}^{k}\left|a\Delta_{j}+b+c\sum_{\ell=1}^{j-1}d^{j-\ell-1}\Delta_{\ell}-\Delta_{1}-e^{1}\right|\leq\max_{i,j}\left|a\Delta_{i}+b+c\sum_{\ell=1}^{i-1}d^{i-\ell-1}\Delta_{\ell}-\Delta_{j}-e^{j}\right|$

which gives us the bound:

$\displaystyle\mathop{\bf E{}}_{\pi\sim
S_{k}}\left[\max_{i}|\mathsf{PNL}_{\pi(i)}-\mathsf{PNL}_{i}|\right]\leq\max_{i,j}\left|a\Delta_{i}+b+c\sum_{\ell=1}^{i-1}d^{i-\ell-1}\Delta_{\ell}-\Delta_{j}-e^{j}\right|(\alpha\log
k-\beta\log\log k)$

which allows us to conclude that $\mathop{\bf E{}}_{\pi\sim
S_{k}}[\max_{i}|\mathsf{PNL}_{\pi(i)}-\mathsf{PNL}_{i}|]=O(\log k)$.

##  Appendix I Proof of Proposition 5

Recall that we have the following lower bound on
$\mathsf{PNL}_{\pi(i)}-\mathsf{PNL}_{i}$ from Section G:

$\displaystyle\mathsf{PNL}_{\pi(i)}-\mathsf{PNL}_{i}$
$\displaystyle\geq\Delta_{\pi(i)}+e^{\pi(i)}-a\Delta_{i}-b-c\sum_{\ell=1}^{i-1}d^{i-\ell-1}\Delta_{\ell}$

Now, taking absolute values and averages, we have:

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\lvert\mathsf{PNL}_{\pi(i)}-\mathsf{PNL}_{i}\rvert$
$\displaystyle\geq\frac{1}{n}\sum_{i=1}^{n}\lvert\Delta_{\pi(i)}+e^{\pi(i)}-a\Delta_{i}-b-c\sum_{\ell=1}^{i-1}d^{i-\ell-1}\Delta_{\ell}\rvert$
$\displaystyle\geq\frac{1}{n}n\min_{i}\left[\Delta_{\pi(i)}+e^{\pi(i)}-a\Delta_{i}-b-c\sum_{\ell=1}^{i-1}d^{i-\ell-1}\Delta_{\ell}\right]$
$\displaystyle\geq\min_{i}\left[\Delta_{\pi(i)}+e^{\pi(i)}-a\Delta_{i}-b-c\sum_{\ell=1}^{i-1}d^{i-\ell-1}\Delta_{\ell}\right]$

which allows us to conclude that
$\frac{1}{n}\sum_{i=1}^{n}\lvert\mathsf{PNL}_{\pi(i)}-\mathsf{PNL}_{i}\rvert=\Omega(1)$.

##  Appendix J Proof of Theorem 1

We combine Propositions 5 and 6 to get the main result:

$\displaystyle\mathsf{CoF}(T_{n})$ $\displaystyle=\frac{\mathop{\bf
E{}}_{\pi\sim
S_{n}}\left[\max_{i\in[n]}\lvert\mathsf{PNL}_{\pi(i)}-\mathsf{PNL}_{i}\rvert\right]}{\mathop{\bf
E{}}_{\pi\sim
S_{n}}\left[\frac{1}{n}\sum_{i=1}^{n}|\mathsf{PNL}_{\pi(i)}-\mathsf{PNL}_{i}|\right]}$
$\displaystyle\leq\frac{\max_{i,j}\left|a\Delta_{i}+b+c\sum_{\ell=1}^{i-1}d^{i-\ell-1}\Delta_{\ell}-\Delta_{j}-e^{j}\right|(\alpha\log
n-\beta\log\log n)}{\mathop{\bf E{}}_{\pi\sim
S_{n}}\left[\frac{1}{n}\sum_{i=1}^{n}|\mathsf{PNL}_{\pi(i)}-\mathsf{PNL}_{i}|\right]}$
$\displaystyle\leq\frac{\max_{i,j}\left|a\Delta_{i}+b+c\sum_{\ell=1}^{i-1}d^{i-\ell-1}\Delta_{\ell}-\Delta_{j}-e^{j}\right|(\alpha\log
n-\beta\log\log
n)}{\min_{i}\left[\Delta_{\pi(i)}+e^{\pi(i)}-a\Delta_{i}-b-c\sum_{\ell=1}^{i-1}d^{i-\ell-1}\Delta_{\ell}\right]}$
$\displaystyle=O(\log n)$ (34)

##  Appendix K Routing MEV

In this section, we provid proofs of the optimality conditions for the CFMM
Pigou example and Proposition 2. The latter establishes the locality of
sandwich attacks on CFMM networks.

###  K.1 CFMM Pigou Example

We now derive the optimality condition for the optimal routing problem with no
sandwiching:

$\displaystyle\mathrm{maximize}$ $\displaystyle
G_{1}(\Delta_{1})+G_{2}(\Delta_{2})$ $\displaystyle\mathrm{subject\ to}$
$\displaystyle\Delta=\Delta_{1}+\Delta_{2}$
$\displaystyle\Delta_{1},\Delta_{2}\geq 0.$

Setting up the Lagrangian, we have:

$\displaystyle\mathcal{L}(\Delta_{1},\Delta_{2},\nu,\mu_{1},\mu_{2})=G_{1}(\Delta_{1})+G_{2}(\Delta_{2})+\nu(\Delta-\Delta_{1}-\Delta_{2})-\mu_{1}\Delta_{1}-\mu_{2}\Delta_{2}$

Recall that any optimal $\Delta_{1}^{*},\Delta_{2}^{*}$ must satisfy:

$\displaystyle\frac{\partial\mathcal{L}}{\partial\Delta_{1}}=\frac{\partial\mathcal{L}}{\partial\Delta_{2}}=0$

and the complementary slackness conditions:

$\displaystyle\mu_{1}\Delta_{1}^{*}=0$ $\displaystyle\mu_{2}\Delta_{2}^{*}=0$

The first (stationarity) conditions give us:

$\displaystyle G_{1}^{\prime}(\Delta_{1}^{*})=\nu+\mu_{1}$ $\displaystyle
G_{2}^{\prime}(\Delta_{2}^{*})=\nu+\mu_{2}$

Therefore, for any $\Delta_{1}^{*},\Delta_{2}^{*}>0$, we must have:
$G_{1}^{\prime}(\Delta_{1}^{*})=g_{1}(\Delta_{1}^{*})=\nu^{*}=g_{2}(\Delta_{2}^{*})=G_{2}^{\prime}(\Delta_{2}^{*})$,
as desired.

###  K.2 Proof of Proposition 2

#### Defining Sandwich Profit on a Graph.

In order to prove the bounds of Proposition 2, we first need to define what
the sandwich profit will be. This profit function can then be used to
implicitly define sandwich sizes and we can use $(\mu,\kappa)$-smoothness to
construct the bounds mentioned.

######  Definition 4.

Define the _cumulative output without sandwiches_ as

$\displaystyle\bar{G}_{p_{i}}(\alpha_{p}\Delta)=G_{p_{i}^{k_{i}}}(G_{p_{i}^{k_{i}-1}}(\dots(G_{p_{i}^{2}}(G_{p_{i}^{1}}(\alpha_{p}\Delta)))))$

The interpretation of this quantity is as the amount the user is expecting to
receive from the path under no sandwiching. We can now write a defining
equation for ${\Delta^{{\mathrm{sand}}}}_{e}$ on every edge $e\in 1,\dots,|p|$
as a function of the flow entering that edge. We write this equation in words
first and then incorporate symbols: At the terminal node of path $p$, we know
we need to receive $(1-\eta)\bar{G}_{p}(\Delta)=(1-\eta)G_{T}(\Delta)$ units
of output token out. Now, look at the node immediately preceding it. Call this
node $e$. We can write an equation:

$\displaystyle
G_{e}(\Delta_{e}+{\Delta^{{\mathrm{sand}}}}_{e})-G_{e}({\Delta^{{\mathrm{sand}}}}_{e})=(1-\eta)G_{T}(\Delta)-G_{e}(\hat{G}_{e-1}(\Delta))$

where $\hat{G}_{e-1}(\Delta)$ is the profit up to node $e-1$.

#### Implied Slippage Limits over a Path.

We seek an explicit representation of ${\Delta^{{\mathrm{sand}}}}_{e}$ in
terms of the flow entering that edge, $\Delta_{e}$ and the global slippage
limit $\eta$. Suppose that we have a path $\mathcal{P}=(e_{1},\ldots,e_{T})$
where $e_{T}$ is the terminal edge (e.g. returns desired output token when
traversed). To do this, we write a Bellman-type equation that writes
${\Delta^{{\mathrm{sand}}}}_{e_{i}}$ on every edge as a function of the
terminal slippage $\eta$ and the slippages that occurred before. For the final
node, we have the following equivalence.

$\displaystyle
G_{e_{T-1}}(\Delta_{e_{T-1}}+{\Delta^{{\mathrm{sand}}}}_{e_{T-1}})-G_{e_{T-1}}({\Delta^{{\mathrm{sand}}}}_{e_{T-1}})=(1-\eta)G_{T}(\Delta_{T})-G_{e_{T-2}}({\Delta^{{\mathrm{sand}}}}_{e_{T-2}})$

The left hand side of this equation represents the excess price impact that
occurs at edge $e_{i}$ when the path $\mathcal{P}$ is traversed. The right
hand side is contribution to the terminal impact (a boundary term) from the
$e_{i-1}$th edge. This effectively says the flow into $e_{T-1}$ needs to be
routed such that it exactly compensates for the excess price impact plus the
output quantity. Another way of framing this condition is as a divergence-free
condition for the flow (e.g. input flow and output flows have to be equal in
terms of their net price impact). Similarly, we can recursively construct
slippage limits for each $e_{i}$ as

$\displaystyle
G_{e_{i-1}}(\Delta_{e_{i-1}}+{\Delta^{{\mathrm{sand}}}}_{e_{i-1}})-G_{e_{i-1}}({\Delta^{{\mathrm{sand}}}}_{e_{i-1}})=(1-\eta_{i})G_{e_{i}}(\Delta_{e_{i}})-G_{e_{i-2}}({\Delta^{{\mathrm{sand}}}}_{e_{i-2}})$
(35)

From this we have a sequence of $T-1$ equations for solving for $T-1$ unknown
variables $\eta_{e_{1}},\ldots,\eta_{e_{T-1}}$. This can be solved via dynamic
programming, as this is an analogue of the Kolmogorov backward equation,
albeit for slippage limits. Therefore, there exists a unique way to solve for
implied slippage limits along a route $\eta$.

This means that the net amount of output token the user receives from the CFMM
network under sandwiching must be no more than $1-\eta$ times the amount the
user would have received under no sandwiching. The optimal sandwich attacks
${\Delta^{{\mathrm{sand}}}}_{e}$ solve the above equation
(LABEL:eq:_sandwichConditionRouting). As there is just one equation for the
network, but $|E|$ sandwiches to be solved for, we provide a heuristic that
can be used to solve for each individual sandwich
${\Delta^{{\mathrm{sand}}}}_{e}$ using a fixed point iteration, and use the
solution that results to provide price of anarchy bounds for the network.

#### Proof of (9) and (10).

We use the equations (35) to construct the bounds on
${\Delta^{{\mathrm{sand}}}}_{e}$ described in Proposition 2. We assume that we
have uniform upper and lower bounds on all $G_{e}(\cdot)$. That is,
$\kappa\Delta\leq G_{e}(\Delta)\leq\mu\Delta$ for all $e\in E$. Recall the
equations:

$\displaystyle
G_{T-1}(\Delta_{T-1}+{\Delta^{{\mathrm{sand}}}}_{T-1})-G_{e_{T-1}}({\Delta^{{\mathrm{sand}}}}_{e_{T-1}})=(1-\eta)G_{T-1}(G_{T-2}(\dots(\Delta)))$

for the terminal sandwich ${\Delta^{{\mathrm{sand}}}}_{T-1}$ and:

$\displaystyle
G_{i}(\Delta_{i}+{\Delta^{{\mathrm{sand}}}}_{i})-G_{i}({\Delta^{{\mathrm{sand}}}}_{i})=(1-\eta)G_{T-1}(G_{T-2}(\dots(\Delta)))$
$\displaystyle-
G_{i}(G_{i-1}(G_{i-2}(\dots(G_{1}(\Delta)+{\Delta^{{\mathrm{sand}}}}_{1})+\dots\Delta^{{\mathrm{sand}}}_{i-2})+{\Delta^{{\mathrm{sand}}}}_{i-1}))$

for the intermediate sandwiches ${\Delta^{{\mathrm{sand}}}}_{i}$ for
$i=1,\dots,T-2$. Let
$\Delta_{T-1}=G_{T-2}(G_{T-3}(\dots(G_{1}(\Delta)+{\Delta^{{\mathrm{sand}}}}_{1})+\dots\Delta^{{\mathrm{sand}}}_{T-3})+{\Delta^{{\mathrm{sand}}}}_{T-2})$.
We now apply $\mu$ and $\kappa$ bounds to the above equation to get:

$\displaystyle\Delta_{T-1}\leq\mu^{T-2}(\Delta+{\Delta^{{\mathrm{sand}}}}_{1})+\mu^{T-3}{\Delta^{{\mathrm{sand}}}}_{2}+\dots+\mu{\Delta^{{\mathrm{sand}}}}_{T-2}$

Which gives us:

$\displaystyle
G_{T-1}(\Delta_{T-1}+{\Delta^{{\mathrm{sand}}}}_{T-1})-G_{T-1}({\Delta^{{\mathrm{sand}}}}_{T-1})\leq\mu^{T-1}(\Delta+{\Delta^{{\mathrm{sand}}}}_{1})+\mu^{T-2}{\Delta^{{\mathrm{sand}}}}_{2}$
$\displaystyle+\dots+\mu^{2}{\Delta^{{\mathrm{sand}}}}_{T-2}+\mu{\Delta^{{\mathrm{sand}}}}_{T-1}-\kappa{\Delta^{{\mathrm{sand}}}}_{T-1}$

and

$\displaystyle(1-\eta)G_{T}(G_{T-1}(G_{T-2}(\dots(\Delta))))$ $\displaystyle-
G_{T-1}(G_{T-2}(G_{T-3}(\dots(G_{1}(\Delta)+{\Delta^{{\mathrm{sand}}}}_{1})+\dots\Delta^{{\mathrm{sand}}}_{T-3})+{\Delta^{{\mathrm{sand}}}}_{T-2}))$
$\displaystyle\leq(1-\eta)\mu^{T-1}\Delta-\kappa^{T-1}(\Delta+{\Delta^{{\mathrm{sand}}}}_{1})+\kappa^{T-2}{\Delta^{{\mathrm{sand}}}}_{2}+\dots+\kappa{\Delta^{{\mathrm{sand}}}}_{T-2}$

Forcing the bound on the RHS to be greater than the bound on the LHS, we have:

$\displaystyle\mu^{T-1}(\Delta+{\Delta^{{\mathrm{sand}}}}_{1})+\mu^{T-2}{\Delta^{{\mathrm{sand}}}}_{2}+\dots+\mu^{2}{\Delta^{{\mathrm{sand}}}}_{T-2}+\mu{\Delta^{{\mathrm{sand}}}}_{T-1}-\kappa{\Delta^{{\mathrm{sand}}}}_{T-1}$
$\displaystyle\geq(1-\eta)\mu^{T-1}\Delta-\kappa^{T-1}(\Delta+{\Delta^{{\mathrm{sand}}}}_{1})+\kappa^{T-2}{\Delta^{{\mathrm{sand}}}}_{2}+\dots+\kappa{\Delta^{{\mathrm{sand}}}}_{T-2}$

Moving all the ${\Delta^{{\mathrm{sand}}}}_{T-j}$ terms for $j>1$ to the RHS,
we have:

$\displaystyle(\mu-\kappa){\Delta^{{\mathrm{sand}}}}_{T-1}$
$\displaystyle\leq(\mu^{T-1}-\kappa^{T-1})\Delta-(\eta+1)\mu^{T-1}\Delta$
$\displaystyle-(\mu^{T-1}+\kappa^{T-1}){\Delta^{{\mathrm{sand}}}}_{1}-\dots-(\mu^{2}+\kappa^{2}){\Delta^{{\mathrm{sand}}}}_{T-2}$

and dividing by $\mu-\kappa$:

$\displaystyle{\Delta^{{\mathrm{sand}}}}_{T-1}$
$\displaystyle\leq\frac{1}{\mu-\kappa}\left((\mu^{T-1}-\kappa^{T-1})\Delta-(\eta+1)\mu^{T-1}\Delta\right.$
$\displaystyle\left.-(\mu^{T-1}+\kappa^{T-1}){\Delta^{{\mathrm{sand}}}}_{1}-\dots-(\mu^{2}+\kappa^{2}){\Delta^{{\mathrm{sand}}}}_{T-2}\right)$
(36)

Recall the defining recursion for a sandwich:

$\displaystyle
G_{e_{i-1}}(\Delta_{e_{i-1}}+{\Delta^{{\mathrm{sand}}}}_{e_{i-1}})-G_{e_{i-1}}({\Delta^{{\mathrm{sand}}}}_{e_{i-1}})=(1-\eta_{i})G_{e_{i}}(\Delta_{e_{i}})-G_{e_{i-2}}({\Delta^{{\mathrm{sand}}}}_{e_{i-2}})$
(37)

Using the $(\mu,\kappa)$-smoothness of $G_{e_{i-1}}$ we can upper bound the
left hand side and lower bound the right hand side (which is the defining
relation for $\eta_{i}$) as:

$\displaystyle\mu(\Delta_{e_{i-1}}+{\Delta^{{\mathrm{sand}}}}_{e_{i-1}})-\kappa{\Delta^{{\mathrm{sand}}}}_{e_{i-1}}\leq(1-\eta_{i})\kappa\Delta_{e_{i}}-\mu{\Delta^{{\mathrm{sand}}}}_{e_{i-2}}$

Rearranging and collecting terms:

$\displaystyle{\Delta^{{\mathrm{sand}}}}_{e_{i-1}}\leq-\frac{\mu}{\mu-\kappa}{\Delta^{{\mathrm{sand}}}}_{e_{i-2}}+\left(\frac{(1-\eta_{i})\kappa-\mu}{\mu-\kappa}\right)\Delta_{e_{i}}\leq-\Delta_{e_{i}}-\frac{\mu}{\mu-\kappa}{\Delta^{{\mathrm{sand}}}}_{e_{i-2}}$

If combining this equation with (K.2) gives:

$\displaystyle{\Delta^{{\mathrm{sand}}}}_{T-1}$
$\displaystyle\leq\frac{1}{\mu-\kappa}\left((\mu^{T-1}-\kappa^{T-1})\Delta-(\eta+1)\mu^{T-1}\Delta\right.$
$\displaystyle\left.+(\mu^{T-1}+\kappa^{T-1}){\Delta^{{\mathrm{sand}}}}_{e_{1}}-\dots-(\mu^{2}+\kappa^{2})\left(\Delta_{T-2}+\frac{\mu}{\mu-\kappa}{\Delta^{{\mathrm{sand}}}}_{T-2}\right)\right)$
(38)

Note that this gives an upper bound on the terminal sandwich, which implies a
bound on the total path sandwich attack ${\Delta^{{\mathrm{sand}}}}_{p}$.
Solving the recursions using Propositions 7 and 8 for this bound yields Eq.
(9). We can compute a similar bound for $\Delta_{e_{i}}$ using the other bound
for (37) and arrive at a similar bounded recursion, yielding (10).