Overall, we find that our measurements are generally consistent with the
previous total SFRD measurements within the errors. At $z>2$, although we find
that the central values in our measurements are higher than the previous
measurements in both Madau & Dickinson (2014) and Bouwens et al. (2020), the
optical–NIR dark objects that have been recently reported even up to $z\simeq
7$ (e.g., Fujimoto et al. 2016; Franco et al. 2018; Wang et al. 2019b;
Yamaguchi et al. 2019; Williams et al. 2019; Casey et al. 2019; Romano et al.
2020; Fudamoto et al. 2021; Gómez-Guijarro et al. 2021; Talia et al. 2021;
Fujimoto et al. 2022; Xiao et al. 2022; Manning et al. 2022; Barrufet et al.
2022; Giulietti et al. 2023) are not included in the optical–NIR surveys used
in these previous measurements. Our blind survey for the mm sources includes
the optical–NIR dark objects and thus provides a complete estimate of the
total SFRD measurement, which is likely the cause of the slight excess in our
estimate. Similarly at $z\lesssim 2$, we also find that the central values in
our measurements are higher than the previous studies. Although Madau &
Dickinson (2014) include those measurements obtained from the IR surveys that
unlikely miss the optical–NIR dark objects, those measurements are obtained
with a flat faint-end slope assumption in the IR LFs that has not been well
constrained in the previous Herschel studies at $z>0.3$. Therefore, the
potential difference at $z\lesssim 2$ is attributed to the faint-end slope in
the IR LFs.

Note that these cosmic SFRD estimates can vary with the integration range. In
Section 5.4, we find that the faint-end slope in the 1.2-mm number counts may
have a turnover or flattening ($S_{\rm turn}$) above $\sim 0.2\mu$Jy not to
make the integrated 1.2-mm flux density exceed the upper limit of the CIB.
Given that $S_{\rm turn}=0.2\mu$Jy is $\sim 2$-dex deeper than our detection
limit in the direct counts of $S_{\rm 1.2mm}\sim 7\mu$Jy, we assume the
turnover or flattening is also taking place in the IR LFs at $\sim 2$-dex
deeper than the detection limit of $L_{\rm IR}=10^{10}\,L_{\odot}$. We thus
regard $L_{\rm IR}=10^{8}\,L_{\odot}$ as the turnover or flattening IR
luminosity $L_{\rm IR,turn}$ and compute the integration of IR LFs down to
$L_{\rm IR,turn}$, below which the integrated values are thought to be
negligible. In Figure 18, the open circles present our SFRD measurements with
$L_{\rm IR,turn}$. We find that the integrated values are not increased more
than a factor of $\sim 2$ in the entire redshift range, and at $z>4$, the
increase is almost negligible. At $z>4$, the total cosmic SFRD measurements
with $L_{\rm IR,turn}$ is estimated to be $161^{+25}_{-21}\%$ of the previous
measurement of Madau & Dickinson (2014). Our results indicate that the general
understanding of the cosmic SFRD is unchanged by a factor of $\sim$2, while
there may be an additional ($\approx 60$%) SFRD component contributed by the
faint-mm population, including NIR dark objects.

###  6.4 Tension with Cosmic SMD

We caution that the integrated value of the previous cosmic SFRD measurement
has been argued to be $\sim 2$ times greater than the observed cosmic stellar
mass density (SMD) at $0\leq z\leq 3$ (e.g., Madau & Dickinson 2014; Leja et
al. 2015). Thus, the potential excess of our total SFRD measurement than the
previous measurement across the cosmic time (Section 6.3) may enhance the
tension to be incorporated with the cosmic SMD. As discussed in Madau &
Dickinson (2014), the possible reasons for the discrepancy are a) overestimate
in SFR, b) underestimate in stellar mass, c) uncertainty in the faint-end of
both UV luminosity function and stellar mass function, and subsequently, their
integrated values of SFRD and SMD, and d) IMF.

For a), the dust-obscured component dominates the total SFRD estimate at
around the peak of cosmic SFRD ($z\simeq$1–2), suggesting the importance of
the contribution from the IR-based SFR. In our $L_{\rm IR}$ measurements
(Section 6.1), we include the Herschel photometry in the IR SED fitting, while
$\sim 20$% of our ALCS sources are not detected ($2\sigma$) in any Herschel
bands (Sun et al. 2022), leaving a systematic uncertainty in the $L_{\rm IR}$
estimate. Even with an accurate $L_{\rm IR}$ estimate, the systematic
uncertainty in the conversion between $L_{\rm IR}$ to SFR cannot be ruled out,
including the potential contribution of AGNs in $L_{\rm IR}$. For b), young
stellar populations are more luminous than older stellar populations, where
the light from older ones is lost in the glare of more recent star formation,
so-called “outshining”, resulting in the stellar mass underestimated (e.g.,
Papovich et al. 2001; Maraston et al. 2010; Pforr et al. 2012). This effect is
more significant at high redshifts with limited broad-band photometry in the
rest-frame optical wavelengths ($\sim$0.3–1.0 dex at $z>4$; Giménez-Arteaga et
al. 2022; Papovich et al. 2022). For c), recent JWST NIRCam and MIRI images
unveil the presence of dusty young ($\sim$1–4 Myr), massive ($\sim
10^{6}\,M_{\odot}$) star clusters that have been invisible in previous deep
HST images even in local galaxies (Linden et al. 2022). Long et al. (2022)
also discuss the potential contribution of the massive end of stellar mass
function due to heavily dust-obscured massive galaxies. These heavily dust-
obscured populations would increase the cosmic SMD estimate, while these
populations would also increase the cosmic SFRD estimate. Therefore, the gap
between SFRD and SMD unlikely disappears by accounting for the dusty-obscured
populations alone. The presence of hidden quiescent galaxies would help (cf.
Leja et al. 2015). For d), we adopt the Chabrier IMF in both our and previous
measurements by applying the correction factor when necessary. Generally, a
more top-heavy or bottom-light IMF leads to a smaller stellar-mass per UV
luminosity. It has long been predicted that the first galaxies would have a
top-heavy IMF (e.g., Larson 1998; Bromm et al. 2001; Chon et al. 2021; Sharda
et al. 2022), which would suggest the potential redshift evolution of IMF.
Interestingly, recent JWST studies report a high abundance of UV-bright
($M_{\rm UV}\lesssim-20$) galaxies at $z\gtrsim 9$ (e.g., Adams et al. 2022;
Atek et al. 2022; Bouwens et al. 2022a; Donnan et al. 2022; Labbe et al. 2022;
Morishita & Stiavelli 2022; Naidu et al. 2022) than theoretical predictions
before (e.g., Dayal et al. 2017; Yung et al. 2019; Behroozi et al. 2019; Davé
et al. 2019), which may be explained by a top-heavy IMF in these early
galaxies (e.g., Finkelstein et al. 2022). Not only such high redshift regimes,
the redshift evolution of IMF is also suggested by incorporating a
temperature-dependency of IMF in the photometric template fitting to galaxy
SEDs (Sneppen et al. 2022; Steinhardt et al. 2022). In this evolution, the
typical IMF becomes progressively top-heavier with increasing redshift, which
helps reconcile the cosmic SMD estimate with the time-integrated SFRD.

Overall, although our SFRD measurement, potentially higher than the previous
measurement of Madau & Dickinson (2014), produces more tension between the
measures of the SMD and the time-integrated SFRD, this requires further
improvements in the systematic uncertainties relevant to both measures,
including the factors of a)–d).

##  7 Summary

In this paper, we identify 180 faint 1.2 mm continuum sources with a flux
density of $\simeq$ 0.007–3.8 mJy in the 33 lensing clusters homogeneously
observed in the large ALMA project of ALCS. The ALCS explores the faintest
layer of mm wavelengths most comprehensively and efficiently with
gravitational lensing support, resulting in a large, blind sample of faint mm
sources out to $z\sim 6$. This allows for the identification of unique
sources, such as a strongly lensed and multiply imaged ($\mu\sim 20$ and
$\sim$160) galaxy at $z=6.07$, as well as NIR dark populations with no
counterparts in deep HST and even Spitzer images. Using this large sample, we
derive the 1.2 mm number counts and discuss the contributions of these faint
sources to the Cosmic Infrared Background light (CIB), including potential
uncertainties in the faint-end shape. We also derive the infrared (IR)
luminosity functions (LFs) and the total (= obscured and unobscured) cosmic
star-formation rate density (SFRD) at $z=0.6$–7.5, in conjunction with the
recent serendipitous detection of mm sources at $z>7$ in the literature. The
main findings of this paper are summarized as follows:

  1. 1.

The target clusters were selected from the best-studied 33 massive galaxy
clusters drawn from HST treasury programs of HFF, CLASH, and RELICS, where the
deep HST and Spitzer maps are available. The observations were carried out
with two frequency setups in ALMA Band 6, spanning a total of 15 GHz width
with the center at 261.25 GHz ($\sim$1.15 mm) to enlarge the blind
identifications of the line emitters, in addition to the continuum sources.
The total survey area above a relative sensitivity of 30% in the mosaic map
covers 133 arcmin2 in the observed frame. After the lens correction at $z=6$,
this is decreased down to 20 arcmin2 and 4 arcmin2 at detection limits of 0.1
mJy and 0.04 mJy, which are larger than those in ASPECS at the same detection
limits by factors of $\sim$4.8 and $\sim$2.0, respectively.

  2. 2.

we produce natural-weighted and $uv$-tapered maps ($2\farcs 0\times 2\farcs
0$) for each cluster not to miss strongly elongated mm sources due to the
lensing effect. Based on the positive and negative source histogram as a
function of peak SNR in the maps, we adopt thresholds of SNR $=$ 5.0 and 4.5
for the natural-weight and $uv$-tapered maps, respectively, to maintain a high
purity $>0.99$. This yields 141 ALMA sources, referred to as the blind sample.
From cross-matching with counterparts in the Spitzer/IRAC maps, we also
identify additional 39 ALMA sources with SNR down to 4.0 in the natural-
weighted maps. Among the blind sample, we find 19 ALMA sources with no HST
counterparts, some showing no counterparts even in Spitzer/IRAC ch1 and ch2
maps.

  3. 3.

After the lens correction and removing the dust emission from the cluster
member galaxies, we obtain 146 ALCS sources which span in the intrinsic flux
density range of $\simeq 0.007$–3.8 mJy and $z\simeq 0$–6 with the median
values of 0.35 mJy and $z=2.03$ and are the most widely distributed in both
parameter spaces among the sources identified in recent ALMA surveys. These
median values are consistent with the general trend of the increasing median
flux density at submm/mm wavelengths in the increasing median redshift. The
median magnification factor in our ALCS sources is estimated to be 2.7.

  4. 4.

With careful corrections for the flux measurements and completeness via
dedicated Monte Carlo (MC) simulations based on a constant FIR size
assumption, we derive the 1.2 mm number counts by using the blind sample,
exploring its faint end down to $\sim 7\mu$Jy. We find that the number counts
are well represented by both Schechter and double power law (DPL) functions
and generally consistent with previous measurements in a wide flux range over
a $\sim$3 dex scale. We estimate a faint-end slope of
$\alpha=-2.05^{+0.12}_{-0.10}$ for our fiducial model. These results suggest
little flattening in the counts below $\sim 0.1$ mJy as previously claimed by
earlier ALMA studies.

  5. 5.

To assess the sensitivity of the completeness correction on the intrinsic
source size, we derive 1.2 mm number counts by assuming a positive or negative
correlation of FIR size with IR luminosity $L_{\rm IR}$ as reported in the
literature and evaluate the impact of this on the form of the faint number
counts. We find that the faint-end slope varies depending on the FIR size
assumptions, potentially changing the CIB resolved fraction by factors of
$\sim$ 0.6–1.1.

  6. 6.

We also test the potential impact on the derived number counts of the varying
precision of the lens model among the clusters by separately estimating the
1.2 mm counts for the HFF, CLASH, and RELICS clusters. The highest quality
lens models, those for HFF, yield the shallowest faint-end slope,
$\alpha=-1.71^{+0.34}_{-0.29}$ (but with a larger cosmic variance
uncertainty). This might indicate that the high magnification areas in the
CLASH and RELICS clusters are underestimated due to their less detailed mass
models with fewer multiple images, which decreases the granularity and small-
scale structures included in the mass models. Combining all the clusters
weighted by the quality of the lens models yields a slightly shallower value
of $\alpha=-1.96^{+0.14}_{-0.13}$. We also note the estimate of the faint-end
slope excluding the highly magnified sources ($\mu\gtrsim$3–10) also yields
$\alpha\sim-2.0$, similar to our fiducial model. Hence, despite the potential
remaining uncertainties, all faint-end slope estimates agree within
$\sim$1–2$\sigma$, suggesting a conservative $\alpha$ range of
$-2.0^{+0.2}_{-0.3}$, regardless of the clusters used or the different FIR
size assumptions.

  7. 7.

We integrate our fiducial Schechter function fit (which assumes a constant FIR
size and no weights among clusters) to the 1.2 mm counts and obtain an
integrated intensity of $20.7^{+8.5}_{-6.5}$ Jy deg-2 down to a flux limit of
7 $\mu$Jy. This integrated flux density corresponds to $\sim 80$% of the CIB
measured by COBE/FIRAS assuming the model for Galactic foreground emission of
Odegard et al. (2019). Extrapolating our fiducial Schechter function to
$\lesssim 7$ $\mu$Jy will exceed the CIB measurements unless there is a
turnover or flattening in the counts at fainter flux densities or a shallower
slope of $\alpha\gtrsim-1.8$ whose integration down to zero does not diverge.
However, we note that there is also significant systematic uncertainty in the
CIB constraint due to the subtraction of the foreground emission.

  8. 8.

We calculate $L_{\rm IR}$ of our ALCS sources by modeling the IR–mm SED with
the Spitzer, Herschel, and ALMA photometry and also derive the IR luminosity
functions (LFs) out to $z=7.5$, combining our ALCS sources with the faint-mm
sources serendipitously detected in recent ALMA studies at $z>7$. We obtain
the faint-end slope of $\simeq$0.9–1.0 in the Double Power Law (DPL) function,
which is higher than previous IR LF measurements of $\simeq 0.4$–0.6 at
$z\lesssim 0.3$, but close to reports in the UVLF studies of $\simeq 1.5$–2.0.
The difference may be explained by the relatively large luminosity range
($\sim 2$–3 dex) covered by our IR LFs owing to the lensing support, compared
to the previous studies, and/or by the different triggering mechanisms of
dusty starbursts in the local universe due to the significant decrease of the
gas fraction. Our ALCS IR LF results are generally consistent with the
predictions from simulations within $\sim 1$–2$\sigma$.

  9. 9.

We integrate the IR LFs and derive the redshift evolution of the obscured and
total (= obscured + unobscured) star-formation rate density (SFRD). We find
that our total cosmic SFRD measurement at $z=4-8$ is estimated to be
$161^{+25}_{-21}$% of the previous measurements of Madau & Dickinson (2014),
where the measurements at $z>4$ relied only on optical–NIR surveys. This
indicates that our current understanding of the cosmic SFRD might be
underestimated by a factor of $\approx 1.6$ due to missing the SFRD component
contributed by both the very faint mm populations and NIR-dark galaxies.

We thank Carlos Gómez-Guijarro and Aswin Vijayan for sharing their
measurements and the compilation of the literature. We are also grateful to
Caitlin Casey for useful discussions on the evolution of the IR LFs, Steven
Finkelstein, and Peter Behroozi for the useful comments on the cosmic star-
formation rate and stellar mass density evolutions, and Soh Ikarashi for the
helpful comments on the impact of the size assumptions to the faint-end of the
1.2-mm number counts. This paper makes use of the ALMA data: ADS/JAO. ALMA
#2018.1.00035.L, #2019.1.00237.S, #2021.1.01246.S, and #2021.1.00407.S. ALMA
is a partnership of the ESO (representing its member states), NSF (USA) and
NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI
(Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by the ESO, AUI/NRAO, and NAOJ. This work is based on
observations and archival data made with the Spitzer Space Telescope, which is
operated by the Jet Propulsion Laboratory, California Institute of Technology,
under a contract with NASA, along with archival data from the NASA/ESA Hubble
Space Telescope. This research also made use of the NASA/IPAC Infrared Science
Archive (IRSA), which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics and
Space Administration. This work was supported in part by World Premier
International Research Center Initiative (WPI Initiative), MEXT, Japan, and
JSPS KAKENHI Grant Numbers 22H01260, 20H05856, 20H00181, 17H06130, 22K21349,
20H00180, and 21H04467. The Cosmic Dawn Center is funded by the Danish
National Research Foundation under grant No. 140. This work was supported by
the joint research program of the Institute for Cosmic Ray Research (ICRR),
University of Tokyo. This project has received funding from the European
Union’s Horizon 2020 research and innovation program under the Marie
Sklodowska-Curie grant agreement No. 847523 ‘INTERACTIONS’ and from NASA
through the NASA Hubble Fellowship grant #HST-HF2-51505.001-A awarded by the
Space Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Incorporated, under NASA contract
NAS5-26555. S.F. acknowledges support from the European Research Council (ERC)
Consolidator Grant funding scheme (project ConTExt, grant No. 648179). K.
Kohno acknowledges support from the NAOJ ALMA Scientific Research Grant Number
2017-06B. G.E.M. acknowledges the Villum Fonden research grants 13160 and
37440. I.R.S acknowledges support from dSTFC (ST/T000244/1). FEB acknowledges
support from ANID-Chile BASAL CATA FB210003, FONDECYT Regular 1200495 and
1190818, and Millennium Science Initiative Program–ICN12_009. D.E.
acknowledges support from a Beatriz Galindo senior fellowship (BG20/00224)
from the Spanish Ministry of Science and Innovation, projects
PID2020-114414GB-100 and PID2020-113689GB-I00 financed by
MCIN/AEI/10.13039/501100011033, project P20-00334 financed by the Junta de
Andalucía, and project A-FQM-510-UGR20 of the FEDER/Junta de Andalucía-
Consejería de Transformación Económica, Industria, Consejerá, Universidades.

##  Appendix A Serendipitous and follow-up line detection

Figure 19:  ALCS Band 6 continuum-subtracted spectra for the 10 serendipitous
line detection in the ALCS continuum sources. The S/N, central frequency, and
line width are summarized in Table 9.  Figure 20:  Zoom-in spectra of the
27-GHz wide ($\sim$85–112 GHz) follow-up ALMA Band 3 line cube (left) and the
ALCS 15-GHz wide Band 6 line cube (right) for a NIR-dark lensed galaxy of
M0553-C67. The Band 3 follow-up line scan observations yield a robust single
line detection (S/N=12.1) at $\sim 97.1$ GHz. Based on the photometric
redshift of $z_{\rm phot}\gtrsim 4$ and the non-detection of neighboring
lines, we regard the line as CO(5-4) at $z=4.93$ in this paper (see Appendix
A). The blue and green dashed vertical lines indicate the line frequency of [C
i](1-0) at $z=3.74$ and [C ii]158$\mu$m at $z=6.12$ in case the 97.1 GHz line
corresponds to CO(4-3) and CO(6-5), respectively.

We extract the ALMA Band 6 spectra with a $1\farcs 0$-radius aperture for the
180 ALCS sources with the 15-GHz ALCS data cube and identify the serendipitous
line emission from 13 ALCS continuum sources (Section 3.4). Three lines are
the CO emission line from the BCGs in the clusters, and the remaining ten
lines are interpreted as [C ii] and CO emission lines from the sources behind
the clusters. We summarize these 13 source properties in Table 9. The ALMA
Band 6 line spectra for three sources are presented in Figure 4, and Figure 19
presents the remaining 10 sources.

Several ALMA CO line scan follow-up programs have been ongoing, and we
additionally detect emission lines from 6 ALCS sources, which is summarized in
Table 10. Figure 20 shows an ALMA Band 3 line spectrum for a NIR-dark lensed
object of M0553-C67 that has been observed in a follow-up line scan program.
We provide further details for each line emission as follows.

M1931-C41 (serendipitous)

– The emission line is serendipitously identified in the ALCS data cube at
255.69 GHz with SNR = 46.0. The source position corresponds to one of the BCGs
in M1931, and the emission line is determined as CO(3-2) at $z=0.352$, which
is in excellent agreement with the cluster redshift of M1931.

A370-C31 (serendipitous)

– The emission line is serendipitously identified in the ALCS data cube at
251.77 GHz with SNR = 18.9 in the velocity integrated line flux density over
210 km s-1. The source position corresponds to one of the BCGs in A370, and
the emission line is determined as CO(3-2) at $z=0.375$ which is in excellent
agreement with the cluster redshift of A370.

M0035-C94 (serendipitous)

– The emission line is serendipitously identified in the ALCS data cube at
254.43 GHz with SNR = 9.5 in the velocity integrated line flux density over
420 km s-1. The source position corresponds to one of the BCGs in M0035, and
the emission line is determined as CO(3-2) at $z=0.359$, which is in excellent
agreement with the cluster redshift of M0035.

R0600-C13 (serendipitous)

– The emission line is serendipitously identified in the ALCS data cube at
253.77 GHz with SNR = 7.8 in the velocity integrated line flux density over
420 km s-1. The source falls outside of the HST/WFC3 images, but the template
fitting with other multi-wavelength suggests $z_{\rm
phot}=1.00^{+0.05}_{-0.05}$ (Appendix B. An independent SED analysis based on
the FIR photometry with magphys suggests $z_{\rm phot}=1.23^{+0.40}_{-0.45}$.
We thus regard this line as CO(5-4) at $z=1.271$, while this interpretation
might be updated with a second-line detection in future follow-up
observations.

R0600-C67 (follow-up)

– Although the ALMA continuum is very brightly and securely detected
($\sim$2.3 mJy, SNR=62), M0600-C67 shows no counterparts in HST/WFC3 and even
IRAC maps (F160W $\gtrsim$ 27 mag, IRAC $\gtrsim$ 25 mag, 3$\sigma$; Coe et
al. 2019). Because of its uniquely NIR-faint nature even in the IRAC bands,
despite the brightness in the ALMA Band 6, a follow-up ALMA Band 3+4 line scan
observation was carried out (#2021.1.00407.S, PI: F. Bauer). Only marginal
line features are detected (SNR $=$ 4.6–5.0), where no convincing redshift
solutions are found among the combinations of these marginal line features. We
thus regard the marginal line feature in the Band3+4 spectrum with the highest
SNR of 5.0 detected at 138.979 GHz as CO(7-6) at $z=4.80$ and use this
redshift interpretation in our analysis throughout the paper. However, future
follow-up observations might update this interpretation with a secure second-
line detection. Further details of the follow-up line scan observation results
will be presented in Bauer et al. (in prep.).

R0949-C10 & C19 (follo-up)

– Similar to R0600-C13, R0949-C10 and R0949-C19 are characterized as very
bright in ALMA Band 6 ($\sim$5.5 mJy & $\sim$1.6 mJy, SNR=62 & 13), but
relatively very faint in NIR with no counterparts in HST/WFC3 and even IRAC
maps (F160W $\gtrsim$ 27 mag, IRAC $\gtrsim$ 25 mag, 3$\sigma$; Coe et al.
2019). The upper limits of the flux ratio at 4.5 $\mu$m (IRAC ch2) and 1.2 mm
(ALMA) indicate their source redshifts at $z>5.5$ and $z>4.5$ in R0949-C10 and
R0949-C19, respectively, based on the composite SED of the high-$z$ dusty
galaxies (Dudzevičiūtė et al. 2020). A follow-up ALMA Band 3+4 line scan
observation was conducted (#2021.1.00407.S, PI: F. Bauer). The multiple lines
are successfully detected in R0949-C10, determining the source redshift of
$z=5.753$, which meets the above lower limit of the redshift. On the other
hand, only marginal line features are detected in R0949-C19 (SNR $=$ 4.3–4.7),
where no convincing redshift solutions are found among the combinations of
these marginal line features. We thus regard the marginal line feature in the
Band3+4 spectrum with the highest SNR of 4.7 detected at 88.266 GHz as CO(5-4)
at $z=5.53$ and use this redshift interpretation for R0949-C19 in our analysis
throughout the paper. However, future follow-up observations might update this
interpretation with a secure second-line detection. Further details of the
follow-up line scan observation results will be presented in Bauer et al. (in
prep.).

M0416-C120 (serendipitous)

– The line is serendipitously identified in the ALCS data cube at 267.34 GHz
with SNR = 5.3 in the velocity integrated line flux density over 240 km s-1.
The source has a counterpart in the HST/F160W map and IRAC/ch1 and ch2, while
it is not detected in any Herschel bands. The SED fitting with eazy suggests
$z_{\rm phot}=9.62^{+0.07}_{-0.06}$ (Kokorev et al. 2022). Although our ALCS
15-GHz data cube has the potential to detect bright FIR emission lines of [C
ii] 158$\mu$m at $z=$ 5.97–6.17 and 6.38–6.60 as well as [O iii] 88$\mu$m at
$z=$ 11.45–11.80 and 12.18–12.57, the redshift solution falls outside of all
of these redshift ranges. Moreover, the morphology in the HST/F160W map
implies a large system (Figure 26) which deviates from typical galaxy sizes at
$z>9$ (e.g., effective radius of $<0\farcs 1$; e.g., Bouwens et al. 2017b;
Kawamata et al. 2018) even with the lensing effect ($\mu=1.6$ at $z=9.6$)
taken into account. We thus interpret that the Lyman-break and the
4000${\rm\AA}$-beak features are degenerated in the SED fitting, and $z_{\rm
phot}\sim$2.2, instead of beyond 9. A public HST+IRAC catalog in the same
field also shows a consistent estimate of $z_{\rm phot}=2.34\pm 0.20$ for this
source (Shipley et al. 2018). We thus regard this line as CO(8-7) at $2.448$,
while this interpretation might be updated with a second-line detection in
future follow-up observations.

M0553-C67 (follow-up)

Although the ALMA continuum is brightly and securely detected ($\sim$0.8 mJy,
SNR=10), M0553-C67 shows no HST/WFC3 counterparts, but with a faint
counterpart in IRAC ch1 and ch2 (F160W $\gtrsim$27 mag, IRAC $\sim$24 mag;
Kokorev et al. 2022). The template fitting with the IRAC and ALMA photometry
(Section 3.5) suggests its photometric redshift of $z_{\rm phot}=4.2\pm 0.3$,
while it could be higher because the IRAC photometry might be overestimated
due to a nearby bright object. A follow-up $\sim$27-GHz wide ALMA Band 3 line
scan observation was carried out (#2019.1.00237.S, PI: S. Fujimoto), and we
reduced the data in the same manner as the ALCS data cubes (Section 3). A
robust single line is detected at $97.140\pm 0.003$ GHz with a line width of
FWHM = $224\pm 25$ km s-1 and SNR = 12.1 in the velocity-integrated map. In
Figure 20, we show the robust single-line detection from M0553-C67 in the
follow-up Band 3 observations. Given the $z_{\rm phot}$ estimate and non-
detection of the secondary lines in the $\sim$27-GHz wide Band 3 data cube, we
interpret the line at 97.1 GHz as CO(4-3) at $z=3.74$, CO(5-4) at $z=4.93$, or
CO(6-5) at $z=6.12$. In the case of CO(4-3) at $z=3.74$, [C i](1-0) should
also be detected with SNR $>3$ in the same Band 3 data cube, given the typical
line ratio of $\log$([C i](1-0)/CO(4-3)) ranging in $[-0.6:0.5]$ among
high-$z$ star-forming galaxies (Valentino et al. 2020). In the case of CO(6-5)
at $z=6.12$, the [C ii] line should be detected with SNR $>5$ in the ALCS Band
6 data cube, given the typical ratio of $L_{\rm[CII]}/L_{\rm IR}$. Since we do
not identify such secondary line features in the Band 3 and Band 6 data cubes
(Figure 20), we regard the single line at 97.1 GHz as CO(5-4) and determine
its redshift of $z=4.9323$. This interpretation might be updated with a
second-line detection in future follow-up observations.

R0600-C164 (serendipitous)

– The line is serendipitously identified in the ALCS data cube at 268.68 GHz
with SNR = 7.1 in the velocity integrated line flux density over 150 km s-1.
The source has an HST counterpart with an arc shape elongated over $\sim
6^{\prime\prime}$ scale, and the SNR is increased out to 9.1 with an optimized
aperture along with the arc shape (Fujimoto et al. 2021). The arc shows a
clear Lyman-break feature at $z\sim 6$ in the HST bands. The follow-up
Gemini/GMOS spectroscopy detects the continuum break and confirms the source
redshift at $z=6.19^{+0.06}_{-0.16}$ (Laporte et al. 2021), and the line is
determined as [C ii] 158 $\mu$m at $z=6.072$. The two multiple images of the
arc are identified with the line detection at the same frequency and the same
SED properties, while the dust continuum is not detected ($<4\sigma$) from
these multiple images because of different magnifications among multiple
images. Three independent lens models consistently suggest that an outskirt
region of the galaxy behind the cluster is crossing the caustic line, which is
strongly lensed into the arc shape in the image plane with a magnification
factor of 163${}^{+27}_{-13}$ (29${}^{+4}_{-7}$) in the local (global) scale
(Fujimoto et al. 2021). In this paper, we adopt the magnification factor of
29${}^{+4}_{-7}$ in our analyses to estimate the intrinsic flux density of the
host galaxy on the global scale, given the scopes in the number count and IRLF
measurements.

M0553-C133/190/249 (serendipitous)

– The lines are serendipitously and consistently identified in the ALCS data
cube from nearby three ALCS sources at 269.12 GHz with SNR$\sim$15–30 in the
velocity integrated line flux density over 660 km s-1 (Figure 4). Their HST
counterparts are known to be a multiply imaged system spectroscopically
confirmed at $z=1.14$ (Ebeling et al. 2017), and thus these lines are
determined as CO(5-4) at $z=1.14$. M0553-C113 falls in the FoV of follow-up
ALMA Band 3 line scan observations for M0553-C67 (#2019.1.00237.S, PI: S.
Fujimoto), and we also detect the CO(2-1) line from M0553-C113 in the Band 3
data cube.

M0553-C303 (serendipitous)

– The line is serendipitously identified in the ALCS data cube at 250.95 GHz
with SNR=9.5 in the velocity integrated line flux density over 270 km s-1. The
source falls outside of the HST/WFC3 footprint, while it is detected in the
IRAC ch1 and ch2 maps with the forced-aperture photometry (Appendix B) and the
Herschel/SPIRE maps with the debelended technique (Sun et al. 2022). The
template fitting with these IRAC, Herschel, and ALMA photometry suggests
$z_{\rm phot}=0.65^{+0.05}_{-0.05}$ (Appendix B), which is consistent with the
independent FIR SED fitting results of $0.98^{+0.39}_{-0.39}$ (Sun et al.
2022) within the errors. We thus regard this line as CO(4-3) at $z=0.836$,
while this interpretation might be updated with a second-line detection in
future follow-up observations. Note that the cluster redshift of M0553 is
0.430, and M0553-C303 is a lensed galaxy behind the cluster.

R0032-C127/131/198 (serendipitous)

– The lines are serendipitously consistently identified in the ALCS data cube
at $\sim$272 GHz from nearby three ALCS sources with SNR $\sim$ 5.0–9.2 in the
velocity integrated line flux density over 800 km s-1. The counterparts are
very faint, but identified in HST/WFC3 at their source positions, and
secondary emission lines are securely detected at 101.97 GHz in a follow-up
ALMA Band 3 & 4 line scan program in this field (#2021.1.01246.S, PI: K.
Kohno). Although the central frequencies of the line features observed in the
ALCS data cube are slightly different, this is likely explained either by the
sensitivity or the differential magnification effect. Together with the
similar NIR dark properties, we interpret these three sources as the multiply
imaged system. Based on the frequencies of these two lines, we regard them as
CO(8-7) in Band 6 and CO(3-2) in Band 3 at $z=$2.391. Our fiducial lens model
also confirms the interpretation of the multiple images. Further details of
the follow-up line scan observation results will be presented in Tsujita et
al. (in prep.).

R0032-C208/281/304 (follow-up)

– Although the ALMA continuum is brightly and securely detected ($>$1.0 mJy,
SNR$\gtrsim$10), no HST/WFC3 counterparts are identified at their source
positions (F160W $\gtrsim$ 27 mag, 3$\sigma$; Coe et al. 2019). The multiple
lines are securely detected in follow-up ALMA Band 3 & 4 line scan
observations (#2021.1.01246.S, PI: K. Kohno) exactly at the same frequencies
at 86.80 GHz and 144.60 GHz. We conclude that these lines are CO(3-2) and
CO(5-4) at $z=2.985$ and that these three sources are the multiply imaged
system, which is consistent with the predictions from our fiducial lens model.
Further details of the follow-up line scan observation results will be
presented in Tsujita et al. (in prep.).

M0417-C46/58/121 (follow-up)

– Similar to the multiple images of R0032-C208/281/304, no HST/WFC3
counterparts are identified at their source positions (F160W $\gtrsim$ 27 mag,
3$\sigma$; Coe et al. 2019), although the ALMA continuum is brightly and
securely detected ($>$2–4 mJy, SNR$>$20). A single line is robustly detected
consistently at 99.11 GHz from all three sources in follow-up ALMA Band 3 line
scan observations (#2021.1.01246.S, PI: K. Kohno). The secondary line is also
securely detected at 148.65 GHz from M0417-C46 in the Band 4 line scan
observations in the same program, while the same observations were not carried
out for the other two sources. Our fiducial lens model suggests that these
three sources are multiple images, and given that the line at 99.1 GHz is
detected exactly at the same frequency among these three sources, we conclude
that these are multiple images. Based on the two emission lines, we conclude
that their redshift at $z=3.652$. Further details of the follow-up line scan
observation results will be presented in Tsujita et al. (in prep.).

Table 9: Serendipitous Line Detection from ALCS Continuum Sources in ALMA Band 6 ID | S/N | $\nu$ | FWHM | $I_{\rm line}$ | $f_{\rm cont}$ | Line | $z$ | Note
---|---|---|---|---|---|---|---|---
|  | GHz | km s-1 | Jy km s-1 | % |  |  |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
M1931-C41 | 46.0 | $255.67\pm 0.00$ | $224\pm 11$ | $2.05\pm 0.14$ | 1.7 | CO(3-2) | 0.352 | BCG
A370-C31 | 18.9 | $251.75\pm 0.00$ | $168\pm 10$ | $0.63\pm 0.05$ | 12.0 | CO(3-2) | 0.375 | BCG
M0035-C94 | 9.5 | $254.40\pm 0.02$ | $332\pm 50$ | $0.48\pm 0.1$ | 10.9 | CO(3-2) | 0.359 | BCG
R0600-C164 | 9.1 | $268.68\pm 0.01$ | $162\pm 30$ | $0.32\pm 0.08$ | 4.9 | [C ii] | 6.072 |  HST-detected (F21,L21)
M0553-C133 | 15.2 | $269.07\pm 0.02$ | $557\pm 62$ | $1.53\pm 0.22$ | 3.2 | CO(5-4) | 1.142 |  HST-detected, multiple images (E17)
M0553-C190 | 32.8 | $269.08\pm 0.02$ | $562\pm 55$ | $3.36\pm 0.43$ | 4.1 | CO(5-4) | 1.142 |  HST-detected, multiple images (E17)
M0553-C249 | 24.2 | $269.06\pm 0.02$ | $592\pm 62$ | $2.14\pm 0.29$ | 4.0 | CO(5-4) | 1.142 |  HST-detected, multiple images (E17)
M0416-C120 | 5.4 | $267.30\pm 0.02$ | $157\pm 55$ | $0.25\pm 0.12$ | 6.7 | CO(8-7) | 2.448 |  HST-detected, $z_{\rm phot}$ \+ single line
R0600-C13 | 7.8 | $253.76\pm 0.02$ | $325\pm 45$ | $0.49\pm 0.09$ | 4.5 | CO(5-4) | 1.271 |  HST-detected, $z_{\rm phot}$ \+ single line
M0553-C303 | 6.2 | $250.92\pm 0.02$ | $260\pm 56$ | $0.34\pm 0.1$ | 3.4 | CO(5-4) | 0.836 |  HST-detected, $z_{\rm phot}$ \+ single line
R0032-C127 | 9.2 | $271.74\pm 0.04$ | $674\pm 119$ | $0.83\pm 0.19$ | 1.6 | CO(8-7) | 2.391 |  HST-detected, two lines$\dagger$, multiple images
R0032-C131 | 6.3 | $272.04\pm 0.02$ | $210\pm 54$ | $0.28\pm 0.1$ | 0.9 | CO(8-7) | 2.391 |  HST-detected, two lines$\dagger$, multiple images
R0032-C198 | 5.0 | $271.93\pm 0.08$ | $551\pm 219$ | $0.33\pm 0.17$ | 2.1 | CO(8-7) | 2.391 |  HST-detected, two lines$\dagger$, multiple images
Table 10: Line Detection in Follow-up Line Scan in ALMA Band 3, 4 for NIR dark Galaxies in ALCS ID | $z$ | Band | Line | Note | Ref.
---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6)
M0553-C67 | 4.932 | 3 | CO(5-4) | No counterparts in HST/WFC3, $z_{\rm phot}$ \+ single line | App. A
R0032-C127 | 2.391 | 3, 4 | CO(3-2), CO(4-3), [C i](1-0) | two lines, multiple images | T23
R0032-C131 | 2.391 | 3, 4 | CO(3-2), CO(4-3), [C i](1-0) | two lines, multiple images | T23
R0032-C198 | 2.391 | 3, 4 | CO(3-2), CO(4-3), [C i](1-0) | two lines, multiple images | T23
R0032-C208 | 2.985 | 3, 4 | CO(3-2), CO(5-4) | No counterparts in HST/WFC3, two lines, multiple images | T23
R0032-C281 | 2.985 | 3, 4 | CO(3-2), CO(5-4) | No counterparts in HST/WFC3, two lines, multiple images | T23
R0032-C304 | 2.985 | 3, 4 | CO(3-2), CO(5-4) | No counterparts in HST/WFC3, two lines, multiple images | T23
M0417-C46 | 3.652 | 3, 4 | CO(4-3), CO(6-5) | No counterparts in HST/WFC3, two lines, multiple images | T23
M0417-C58 | 3.652 | 3, 4 | CO(4-3), CO(6-5) | No counterparts in HST/WFC3, two lines, multiple images | T23
M0417-C121 | 3.652 | 3, 4 | CO(4-3), CO(6-5) | No counterparts in HST/WFC3, two lines, multiple images | T23
R0600-C67 | (4.80) | 3, 4 | CO(7-6)$?$, CO(6-5)$?$ | No counterparts in HST, IRAC , two tentative line + $z_{\rm phot}^{\dagger}$ | B23
R0949-C10 | 5.753 | 3, 4 | CO(6-5), CO(5-4) | No counterparts in HST, IRAC, two lines | B23
R0949-C19 | (5.53) | 3, 4 | CO(5-4)$?$ | No counterparts in HST, IRAC, single tentative line + $z_{\rm phot}^{\dagger}$ | B23

##  Appendix B $z_{\rm phot}$ estimate by template fitting

In Section 3.5, we identify 26 ALCS sources lacking the HST counterparts due
to their faintness in the HST bands ($N=18$) and their sky positions outside
of the HST footprints ($N=8$), respectively. We obtain the forced-aperture
photometry in IRAC ch1 and ch2 at the ALCS source position with a diameter of
$3\farcs 0$ on the residual IRAC maps after subtracting the models for the
HST-detected sources as well as the debelended Herschel photometry if detected
(Sun et al. 2022). There remain 4 ALCS sources that are not detected in any
bands in IRAC and Herschel (ACT0102-C11, ACT0102-C251, R1347-C51, and
M2129-C24). These 4 ALCS sources are all included in the primary catalog. With
our purity cut for the primary catalog ($>$99%) based on the positive and
negative source distribution (Figure 2), one of these 4 ALCS sources might be
spurious. Still, the majority of them are the most likely real.

Given the limited number of photometry data points, we simply perform the
$\chi^{2}$ minimization with a fixed template to evaluate $P(z)$ for these 23
(=27-4) ALCS sources. We use the SED fitting code of stardust (Kokorev et al.
2021) by implementing the SED template of Arp220 and a composite SED model
from 707 dusty galaxies identified in the AS2UDS survey (e.g., Dudzevičiūtė et
al. 2020). We find that the AS2UDS template suggests a lower redshift solution
than Arp220 in general. This is because the AS2UDS template is fainter in the
rest-frame optical than Arp220 due to a lower typical stellar mass or much
higher obscuration (Dudzevičiūtė et al. 2020).

Figure 21:  Example of our template fitting for the ALCS sources with no HST
counterparts. Left: The best-fit SEDs for M0417-C121 based on the templates of
Arp220 (black line) and the composite of 707 dusty galaxies in AS2UDS
(Dudzevičiūtė et al. 2020). Right: $P(z)$ from the $\chi^{2}$ minimazation.
The red dashed line indicates $z_{\rm spec}=3.652$ of M0417-C121 determined
from follow-up ALMA Band 3 & 4 line scan observations with multiple line
detection (Tsujita et al. in prep.).

Figure 21 shows an example of the NIR–mm photometry and the best-fit SED
(left) and $P(z)$ (right) with the Arp220 and AS2UDS templates for M0417-C121.
M0417-C121 is robustly detected with ALMA (SNRnat=20.7) without any HST
counterparts, and follow-up ALMA Band 3 & 4 line scan observations determine
its redshift at $z_{\rm spec}=3.652$ with two secure CO line detection
(Tsujita et al. in prep.). We find that these SED templates suggest the source
redshift at $z\sim$ 3–4, where the peak of $P(z)$ with the AS2UDS template
shows an excellent agreement with $z_{\rm spec}$ of M0417-C121. Given the
intrinsically faint ($\sim$ less massive) and potentially high dust
obscuration aspects without any HST counterparts, the SEDs of these ALCS
sources may be closer to the AS2UDS template. We thus use $P(z)$ obtained with
the AS2UDS template for the 23 ALCS sources.

##  Appendix C Lens Models

We construct lens models using cluster member galaxies as well as multiple
images behind clusters based on the photometric redshift, colors, and
morphology of galaxies in the HST images with independent algorithms including
glafic (Oguri 2010; Kawamata et al. 2016, 2018; Okabe et al. 2020), Lenstool
(Jullo et al. 2007; Caminha et al. 2016, 2017b, 2017a, 2019; Richard et al.
2014; Niemiec et al. 2020), Light-Traces-Mass (Zitrin-LTM; Zitrin et al. 2013,
2015), and Pseudo-Isothermal Elliptical Mass Distribution plus elliptical
Navarro–Frenk–White (Zitrin-dPIEeNFW; Zitrin et al. 2013, 2015). We use the
lens model of glafic as a fiducial model for our analyses throughout this
paper, while other available models are also used to evaluate uncertainties in
magnification factors (Appendix D). Note that we do not use the median
magnification estimate among different lens models, because the process of
choosing the median estimate systematically reduces the effective survey area
at high magnifications of $\mu\gtrsim 10$ (Appendix D). In Table 11, we
summarize the number of the multiple images used for constructing the glafic
models and the r.m.s of positional differences between the observed and model-
predicted multiple images.

Table 11: Summary of our fiducial and available lens models in the 33 ALCS clusters Cluster | $z$ | $N_{\rm src,all}$ | $N_{\rm src,spec}$ | $N_{\rm img}$ | rmsimg | Ref. | Caminha | CATS | Zitrin
---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
HFF
A2744 | 0.308 | 45 | 24 | 132 | 0.42 | K18 | N | Y | Y
A370 | 0.375 | 49 | 19 | 135 | 0.50 | K18 | N | Y | Y
M0416 | 0.396 | 75 | 34 | 202 | 0.44 | K18 | Y | Y | Y
M1149 | 0.543 | 36 | 16 | 108 | 0.31 | K16 | N | Y | Y
AS1063 | 0.348 | 53 | 19 | 141 | 0.38 | K18 | Y | Y | Y
CLASH
A209 | 0.206 | 3 | 0 | 7 | 0.52 | Z15 | N | Y | Y
A383 | 0.187 | 8 | 6 | 23 | 0.41 | Z15 | N | Y | Y
M0329 | 0.45 | 11 | 10 | 28 | 0.62 | Ca19, R21 | Y | Y | Y
M0429 | 0.399 | 3 | 2 | 11 | 0.32 | Ca19 | Y | N | Y
M1115 | 0.355 | 3 | 1 | 9 | 0.44 | Ca19 | Y | Y | Y
M1206 | 0.439 | 37 | 37 | 109 | 0.47 | Ca17, R21 | Y | Y | Y
M1311 | 0.494 | 3 | 1 | 8 | 0.57 | Ca19 | Y | N | Y
R1347 | 0.451 | 40 | 38 | 129 | 0.65 | U18, Ca19, R21 | Y | Y | Y
M1423 | 0.545 | 3 | 2 | 12 | 0.65 | Z15 | N | Y | Y
M1931 | 0.352 | 7 | 7 | 19 | 0.39 | Ca19 | Y | Y | Y
M2129 | 0.57 | 11 | 11 | 38 | 0.73 | Ca19 | Y | Y | Y
R2129 | 0.234 | 7 | 7 | 22 | 0.43 | Ca19 | Y | Y | Y
RELICS
R0032 | 0.396 | 12 | 2 | 36 | 0.49 | A20 | Y | Y | N
M0035 | 0.352 | 5 | 0 | 13 | 0.24 |  $\cdots$ | Y | N | N
AC0102 | 0.87 | 10 | 0 | 28 | 0.52 | Ce18 | Y | N | N
M0159 | 0.405 | 4 | 0 | 10 | 0.47 | Ci18 | Y | Y | N
A295 | 0.3 | 6 | 4 | 18 | 0.22 | Ci18 | N | N | N
M0257 | 0.505 | 4 | 0 | 12 | 0.36 | Z11 | N | Y | N
P171 | 0.27 | 5 | 0 | 16 | 0.34 | A18 | N | N | Y
A3192 | 0.425 | 5 | 2 | 16 | 0.51 | H13 | N | N | N
M0417 | 0.443 | 21 | 8 | 57 | 0.45 | M19 & ALCS† | Y | Y | N
M0553 | 0.43 | 10 | 2 | 30 | 0.73 | E17 | N | Y | N
R0600 | 0.46 | 8 | 5 | 26 | 0.66 | F21 | Y | Y | N
SM0723 | 0.39 | 4 | 0 | 13 | 0.27 |  $\cdots$ | N | Y | N
R0949 | 0.383 | 4 | 0 | 11 | 0.26 |  $\cdots$ | N | Y | N
A2163 | 0.203 | 4 | 3 | 15 | 0.3 | Ce18, Re20 | Y | N | N
RJ2211 | 0.397 | 3 | 1 | 11 | 0.14 | Ce18 | N | N | N
A2537 | 0.297 | 8 | 1 | 29 | 0.44 | Ce18 | N | Y | N

##  Appendix D Lens Models Uncertainty

Figure 22:  Difference of the magnification factor at random positions among
different lens models. We evaluate the difference with $\Delta\mu$/$\mu$ and
$\Delta\mu=|\mu-\mu_{\rm other}|$, where $\mu$ and $\mu_{\rm other}$ are the
magnification factors with our fiducial and other lens models, respectively.
The different color plots indicate different lens models for $z=2$, where
there are 8, 3, and 4 publically available lens models in A2744 (HFF, left),
M1115 (CLASH, middle), and R0032 (RELICS, right), respectively, including our
fiducial lens model constructed with glafic (Oguri 2010). The black squares
denote the median of $\Delta\mu$/$\mu$ in each $\mu$ bin.

We compare the magnification factors between our fiducial and other lens
models at random pixel positions of the maps and evaluate the systematic
uncertainty from the choice of the lens model. To evaluate the systematic
uncertainty, we calculate $\Delta\mu/\mu$ ($\equiv|\mu_{\rm other}-\mu|$). The
accuracy of the lens model generally depends on the richness of the multi-
wavelength data in the cluster field for identifying the multiple images and
the cluster member galaxies. Thus we separately calculate $\Delta\mu/\mu$ for
HFF, CLASH, and RELICS. The clusters of A2744, M1115, and R0032 are used for
the calculation of $\Delta\mu/\mu$ as the representatives of HFF, CLASH, and
RELICS, respectively, and we fix the redshift at $z=2$ for the lens model.

Figure 23:  Comparison of the effective survey area among 7 different lens
models publically available in AS1063. The red line is drawn from a median
magnification map created from these lens models. We find that the median
magnification map underestimates the effective survey, especially at $0.01$
mJy which corresponds to the area with $\mu\gtrsim 10$. This is because the
highly magnified regions are smeared out by sampling the median values due to
slight differences in the critical curve positions among different lens
models.

In Figure 22, we show $\Delta\mu/\mu$ at random positions in the cluster
fields. The median values as a function of $\mu$ are also plotted with black
squares. We find an increasing trend in $\Delta\mu/\mu$ as a function of
$\mu$. We also find that $\Delta\mu/\mu$ is the smallest in HFF at $\mu\leq
5$, while there is no large difference in $\Delta\mu/\mu$ among HFF, CLASH,
and RELICS at $\mu>5$. Based on these results, we adopt 20% (40%), 50%, 60%,
and 80% for the systematic uncertainty of the different lens models at
$\mu\leq 5$, $5<\mu\leq 10$, $10<\mu\leq 30$, and $\mu>30$ in HFF (CLASH &
RELICS).

We also evaluate the difference in the effective survey area among different
lens models. In Figure 23, we show the survey area after correcting the
lensing effect with 7 different lens models in AS1063. The redshift is fixed
again at $z=2$ for all these lens models. We find that the survey area is
unchanged by more than $\sim$10% in the wide magnification ranges. This
indicates that the overall shape of the magnification distribution is not
significantly changed among the different lens models, although the small
variations should remain. On the other hand, however, if we create the median
magnification map from these 7 different lens models, the survey area is
underestimated at high magnifications ($\mu\gtrsim 10$). This is because such
highly magnified regions are smeared out even by the small variation of the
critical curve positions among the different models.

Figure 24 shows the comparison of the effective survey area among the HFF,
CLASH, and RELICS. As discussed in Section 5.6, the less detailed models with
fewer multiple images in CLASH and RELICS than HFF may lead to underestimating
the survey area at high magnification ($\mu\gtrsim 3$) by factors of $\sim
1.4$–1.8 (Jauzac et al. 2015). The sum of the survey areas from CLASH and
RELICS at the high magnification regime contribute to almost constantly
$\sim$80% of the total survey area, suggesting that our survey area estimate
can be increased by factors of $\sim$1.1–1.5 in future deep follow-up
observations in the CLASH and RELICS clusters. On the other hand, the faint-
end estimates in the number counts and LFs may not be changed much, because
the increase of the high magnification area also increases the sources with
high magnification estimates.

Figure 24:  Comparison of the effective survey area among the HFF, CLASH, and
RELICS clusters based on our fiducial lens model at $z=2$.

##  Appendix E Predictions from the EAGLE simulation

For EAGLE, we retrieved the publicly available spectral energy distributions
of all galaxies from $z=10$ to $z=0$ (see Camps et al. 2018 for details of how
these fluxes are computed). We compute the total IR luminosity using all four
(rest-frame) IRAC bands available in the database and the conversion between
those fluxes and the total IR luminosity of Sanders & Mirabel (1996). From
those luminosities, we construct the IR luminosity functions at all redshifts.
To compute the number counts at 1.2mm, we retrieve the observer-frame ALMA
band 6 fluxes and use all the snapshots of the simulation to construct a
lightcone. The lightcone is constructed as follows: for each redshift, we
compute the implied projected sky area of a 100 Mpc2 assuming the EAGLE
cosmology (Planck Collaboration et al. 2014a) and the number of galaxies per
unit area at each flux bin contributed by that redshift $N(z,S)$. We then
integrate under the $N(z,S)$ vs $z$ curve to obtain $N(S)$. We note that the
fluxes available in the EAGLE database are computed only for galaxies that
have a number of particles representing the galaxy’s body of dust $>250$. The
latter mostly removes quenched galaxies that are more prevalent at low
redshift. Hence, we do not think that affects the 1.2mm number counts in the
flux range investigated here.

##  Appendix F Footprints of 33 target clusters in ALCS

In Figure 25, we present the footpritns of ALMA Band 6, HST/F160W, and
IRAC/ch1 for the 33 target lensing clusters in the same format as Figure 1.

##  Appendix G Full list of ALCS 180 sources

Table 12: ALCS Continuum Source Catalog

ID | R.A. | Dec | SNR | PB | $S_{\rm nat}$ | $S_{\rm tap}$ | $S_{\rm ap}$ | $S_{\rm imf}$ | flag
---|---|---|---|---|---|---|---|---|---
| deg | deg |  |  | mJy/beam | mJy/beam | mJy | mJy |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
Blind Catalog ($N=141$)
AC0102-C11 | 15.7477863 | $-49.2904001$ | 5.4 (3.1) | 0.96 | 0.39 $\pm$ 0.07 | 0.34 $\pm$ 0.11 | 0.28 $\pm$ 0.12 | 0.35 $\pm$ 0.1 | $0$
AC0102-C22 | 15.7626821 | $-49.2864566$ | 7.9 (7.4) | 0.79 | 0.71 $\pm$ 0.09 | 0.97 $\pm$ 0.13 | 1.23 $\pm$ 0.14 | 0.96 $\pm$ 0.19 | $2$
AC0102-C50 | 15.7475095 | $-49.2821233$ | 5.8 (5.6) | 0.99 | 0.41 $\pm$ 0.07 | 0.58 $\pm$ 0.1 | 0.66 $\pm$ 0.11 | 0.67 $\pm$ 0.16 | $2$
AC0102-C52 | 15.7561196 | $-49.2823856$ | 5.8 (3.2) | 0.99 | 0.41 $\pm$ 0.07 | 0.33 $\pm$ 0.11 | 0.04 $\pm$ 0.11 | 0.4 $\pm$ 0.12 | $0$
AC0102-C118 | 15.7128798 | $-49.2607919$ | 34.3 (26.5) | 0.92 | 2.63 $\pm$ 0.08 | 2.99 $\pm$ 0.11 | 3.29 $\pm$ 0.12 | 2.93 $\pm$ 0.14 | $2$
AC0102-C160 | 15.7508527 | $-49.2677044$ | 4.7 (4.9) | 0.98 | 0.34 $\pm$ 0.07 | 0.52 $\pm$ 0.11 | 0.72 $\pm$ 0.12 | 0.87 $\pm$ 0.23 | $2$
AC0102-C215 | 15.7288441 | $-49.2540865$ | 42.4 (32.7) | 0.99 | 3.0 $\pm$ 0.07 | 3.4 $\pm$ 0.1 | 3.83 $\pm$ 0.11 | 3.6 $\pm$ 0.18 | $2$
AC0102-C223 | 15.7051647 | $-49.2524655$ | 7.9 (6.7) | 0.96 | 0.58 $\pm$ 0.07 | 0.72 $\pm$ 0.11 | 1.01 $\pm$ 0.12 | 0.78 $\pm$ 0.26 | $2$
AC0102-C224 | 15.7320137 | $-49.2525191$ | 91.8 (78.4) | 0.98 | 6.57 $\pm$ 0.07 | 8.24 $\pm$ 0.11 | 9.5 $\pm$ 0.12 | 8.99 $\pm$ 0.26 | $2$
AC0102-C241 | 15.742206 | $-49.2489458$ | 5.4 (4.5) | 0.45 | 0.85 $\pm$ 0.16 | 1.05 $\pm$ 0.23 | 1.08 $\pm$ 0.25 | 0.53 $\pm$ 0.16 | $1$
AC0102-C251 | 15.7294397 | $-49.2386615$ | 5.1 (4.4) | 0.63 | 0.56 $\pm$ 0.11 | 0.72 $\pm$ 0.16 | 0.97 $\pm$ 0.18 | 0.75 $\pm$ 0.24 | $2$
AC0102-C276 | 15.7053793 | $-49.2439317$ | 9.9 (8.6) | 0.81 | 0.86 $\pm$ 0.09 | 1.09 $\pm$ 0.13 | 1.41 $\pm$ 0.14 | 1.16 $\pm$ 0.18 | $2$
AC0102-C294 | 15.7056994 | $-49.2514717$ | 14.6 (19.1) | 0.97 | 1.06 $\pm$ 0.07 | 2.03 $\pm$ 0.11 | 3.1 $\pm$ 0.12 | 3.16 $\pm$ 0.27 | $2$
M0417-C46 | 64.3888354 | $-11.9175841$ | 48.0 (36.7) | 0.97 | 4.04 $\pm$ 0.08 | 4.26 $\pm$ 0.12 | 4.29 $\pm$ 0.12 | 4.3 $\pm$ 0.13 | $1$
M0417-C49 | 64.4172807 | $-11.9168995$ | 36.1 (28.6) | 0.71 | 4.18 $\pm$ 0.12 | 4.57 $\pm$ 0.16 | 4.61 $\pm$ 0.16 | 3.32 $\pm$ 0.17 | $2$
M0417-C58 | 64.3985559 | $-11.9147976$ | 30.4 (24.3) | 0.95 | 2.63 $\pm$ 0.09 | 2.9 $\pm$ 0.12 | 3.18 $\pm$ 0.12 | 2.89 $\pm$ 0.14 | $2$
M0417-C121 | 64.4042424 | $-11.9055111$ | 20.7 (16.3) | 0.89 | 1.91 $\pm$ 0.09 | 2.07 $\pm$ 0.13 | 2.14 $\pm$ 0.13 | 1.93 $\pm$ 0.14 | $2$
M0417-C204 | 64.3843029 | $-11.877534$ | 4.1 (4.5) | 0.82 | 0.41 $\pm$ 0.1 | 0.62 $\pm$ 0.14 | 1.05 $\pm$ 0.14 | 1.18 $\pm$ 0.46 | $2$
M0417-C218 | 64.3833817 | $-11.8826035$ | 14.5 (11.2) | 0.91 | 1.32 $\pm$ 0.09 | 1.4 $\pm$ 0.13 | 1.61 $\pm$ 0.13 | 1.29 $\pm$ 0.16 | $2$
M0417-C221 | 64.3853268 | $-11.884119$ | 7.7 (6.4) | 0.9 | 0.7 $\pm$ 0.09 | 0.8 $\pm$ 0.13 | 1.04 $\pm$ 0.13 | 0.75 $\pm$ 0.22 | $2$
M0417-C223 | 64.3856046 | $-11.8847084$ | 6.0 (4.3) | 0.9 | 0.54 $\pm$ 0.09 | 0.54 $\pm$ 0.13 | 0.98 $\pm$ 0.13 | 1.93 $\pm$ 0.51 | $2$
P171-C69 | 48.2198449 | $8.3791528$ | 19.7 (15.2) | 0.62 | 2.3 $\pm$ 0.12 | 2.56 $\pm$ 0.17 | 3.0 $\pm$ 0.17 | 1.69 $\pm$ 0.14 | $2$
P171-C162 | 48.2446035 | $8.3914414$ | 9.0 (6.8) | 0.3 | 2.16 $\pm$ 0.24 | 2.32 $\pm$ 0.34 | 2.69 $\pm$ 0.35 |  $\cdots$ | $-2$
P171-C177 | 48.2311509 | $8.3955503$ | 6.9 (7.3) | 0.86 | 0.57 $\pm$ 0.08 | 0.87 $\pm$ 0.12 | 1.19 $\pm$ 0.12 | 1.31 $\pm$ 0.29 | $2$
M0329-C11 | 52.4239873 | $-2.1824407$ | 11.1 (6.3) | 0.52 | 1.5 $\pm$ 0.14 | 1.16 $\pm$ 0.18 | 0.79 $\pm$ 0.18 | 0.58 $\pm$ 0.09 | $0$
M0159-C5 | 29.9653751 | $-8.8194848$ | 16.0 (14.3) | 0.83 | 1.2 $\pm$ 0.08 | 1.45 $\pm$ 0.1 | 1.59 $\pm$ 0.1 | 1.38 $\pm$ 0.13 | $2$
M0159-C24 | 29.9472208 | $-8.8258977$ | 6.3 (5.1) | 0.54 | 0.72 $\pm$ 0.11 | 0.79 $\pm$ 0.16 | 0.89 $\pm$ 0.16 | 0.2 $\pm$ 0.09 | $0$
M0159-C46 | 29.9556285 | $-8.8330682$ | 32.8 (25.5) | 0.99 | 2.07 $\pm$ 0.06 | 2.17 $\pm$ 0.09 | 2.22 $\pm$ 0.09 | 2.22 $\pm$ 0.1 | $2$
M0159-C61 | 29.9648314 | $-8.8370309$ | 5.8 (5.3) | 0.9 | 0.4 $\pm$ 0.07 | 0.49 $\pm$ 0.09 | 0.57 $\pm$ 0.09 | 0.59 $\pm$ 0.16 | $1$
R1347-C41 | 206.8796443 | $-11.7441022$ | 7.2 (6.0) | 0.98 | 0.38 $\pm$ 0.05 | 0.47 $\pm$ 0.08 | 0.55 $\pm$ 0.08 | 0.56 $\pm$ 0.12 | $2$
R1347-C51 | 206.8905708 | $-11.7469165$ | 5.1 (3.3) | 0.99 | 0.27 $\pm$ 0.05 | 0.25 $\pm$ 0.08 | 0.12 $\pm$ 0.08 | 0.2 $\pm$ 0.06 | $0$
R1347-C75 | 206.8776098 | $-11.7526462$ | 57.0 (39.9) | 0.99 | 3.02 $\pm$ 0.05 | 3.09 $\pm$ 0.08 | 3.12 $\pm$ 0.08 | 3.06 $\pm$ 0.1 | $0$
R1347-C145 | 206.8651991 | $-11.7641845$ | 22.0 (19.0) | 0.93 | 1.24 $\pm$ 0.06 | 1.56 $\pm$ 0.08 | 1.91 $\pm$ 0.09 | 1.62 $\pm$ 0.15 | $2$
R1347-C148 | 206.8660282 | $-11.765497$ | 18.9 (21.4) | 0.89 | 1.11 $\pm$ 0.06 | 1.83 $\pm$ 0.09 | 2.63 $\pm$ 0.09 | 2.45 $\pm$ 0.2 | $2$
R1347-C166 | 206.8666684 | $-11.7700462$ | 5.1 (4.2) | 0.41 | 0.65 $\pm$ 0.13 | 0.79 $\pm$ 0.19 | 0.99 $\pm$ 0.2 | 0.52 $\pm$ 0.14 | $1$
M1115-C2 | 168.9612161 | $1.5098552$ | 11.3 (10.5) | 0.32 | 2.16 $\pm$ 0.19 | 2.9 $\pm$ 0.28 | 3.34 $\pm$ 0.29 |  $\cdots$ | $-2$
M1115-C4 | 168.9668036 | $1.5078324$ | 7.6 (6.9) | 0.83 | 0.57 $\pm$ 0.08 | 0.75 $\pm$ 0.11 | 0.91 $\pm$ 0.11 | 0.74 $\pm$ 0.16 | $2$
M1115-C34 | 168.9756027 | $1.4989589$ | 16.1 (13.7) | 0.68 | 1.47 $\pm$ 0.09 | 1.81 $\pm$ 0.13 | 1.83 $\pm$ 0.14 | 1.34 $\pm$ 0.12 | $1$
M1115-C36 | 168.96623 | $1.4985808$ | 7.5 (5.5) | 1.0 | 0.46 $\pm$ 0.06 | 0.49 $\pm$ 0.09 | 0.54 $\pm$ 0.09 | 0.47 $\pm$ 0.1 | $1$
M0257-C13 | 44.2968806 | $-23.4287678$ | 11.4 (12.2) | 0.91 | 1.02 $\pm$ 0.09 | 1.52 $\pm$ 0.13 | 1.96 $\pm$ 0.13 | 1.83 $\pm$ 0.21 | $2$
M0553-C17 | 88.314182 | $-33.6931782$ | 8.5 (8.0) | 0.53 | 1.03 $\pm$ 0.12 | 1.4 $\pm$ 0.18 | 1.91 $\pm$ 0.19 | 1.03 $\pm$ 0.23 | $2$
M0553-C18 | 88.319805 | $-33.6932357$ | 4.0 (4.5) | 0.54 | 0.48 $\pm$ 0.12 | 0.78 $\pm$ 0.17 | 1.42 $\pm$ 0.19 | 0.91 $\pm$ 0.36 | $2$
M0553-C58 | 88.3757309 | $-33.6978091$ | 7.5 (5.6) | 0.97 | 0.5 $\pm$ 0.07 | 0.54 $\pm$ 0.1 | 0.64 $\pm$ 0.11 | 0.54 $\pm$ 0.1 | $1$
M0553-C61 | 88.3395244 | $-33.6985393$ | 10.7 (9.5) | 0.96 | 0.71 $\pm$ 0.07 | 0.92 $\pm$ 0.1 | 1.22 $\pm$ 0.11 | 1.1 $\pm$ 0.18 | $2$
M0553-C67 | 88.3720848 | $-33.698982$ | 10.0 (7.8) | 0.99 | 0.65 $\pm$ 0.07 | 0.74 $\pm$ 0.1 | 0.91 $\pm$ 0.1 | 0.79 $\pm$ 0.12 | $2$
M0553-C133 | 88.3657754 | $-33.7044538$ | 39.0 (29.4) | 0.96 | 2.6 $\pm$ 0.07 | 2.84 $\pm$ 0.1 | 3.13 $\pm$ 0.11 | 2.8 $\pm$ 0.12 | $2$
M0553-C190 | 88.3660559 | $-33.7084931$ | 67.5 (49.9) | 0.99 | 4.37 $\pm$ 0.07 | 4.69 $\pm$ 0.09 | 4.6 $\pm$ 0.1 | 4.74 $\pm$ 0.12 | $1$
M0553-C200 | 88.3891584 | $-33.7088061$ | 4.7 (5.1) | 0.98 | 0.31 $\pm$ 0.07 | 0.48 $\pm$ 0.1 | 0.76 $\pm$ 0.1 | 0.86 $\pm$ 0.27 | $2$
M0553-C249 | 88.3651136 | $-33.7121852$ | 44.0 (32.8) | 0.97 | 2.93 $\pm$ 0.07 | 3.16 $\pm$ 0.1 | 3.3 $\pm$ 0.11 | 3.11 $\pm$ 0.13 | $2$
M0553-C275 | 88.3585617 | $-33.714338$ | 6.6 (5.5) | 0.96 | 0.44 $\pm$ 0.07 | 0.54 $\pm$ 0.1 | 0.71 $\pm$ 0.11 | 0.6 $\pm$ 0.18 | $2$
M0553-C303 | 88.3832348 | $-33.7167949$ | 7.6 (8.2) | 1.0 | 0.49 $\pm$ 0.07 | 0.77 $\pm$ 0.09 | 1.23 $\pm$ 0.1 | 1.56 $\pm$ 0.39 | $2$
M0553-C355 | 88.3469942 | $-33.7210992$ | 12.4 (10.4) | 0.94 | 0.85 $\pm$ 0.07 | 1.04 $\pm$ 0.1 | 1.06 $\pm$ 0.11 | 1.07 $\pm$ 0.13 | $2$
M0553-C375 | 88.3053831 | $-33.7225917$ | 8.3 (8.1) | 0.74 | 0.72 $\pm$ 0.09 | 1.02 $\pm$ 0.13 | 1.49 $\pm$ 0.14 | 1.06 $\pm$ 0.22 | $2$
M0553-C398 | 88.3744572 | $-33.7238804$ | 13.5 (10.8) | 0.49 | 1.79 $\pm$ 0.13 | 2.07 $\pm$ 0.19 | 2.39 $\pm$ 0.21 | 1.09 $\pm$ 0.16 | $0$
SM0723-C61 | 110.7660662 | $-73.4517267$ | 5.8 (5.3) | 0.57 | 0.66 $\pm$ 0.11 | 0.97 $\pm$ 0.19 | 0.99 $\pm$ 0.21 | 0.8 $\pm$ 0.2 | $2$
SM0723-C93 | 110.7631337 | $-73.4558425$ | 4.4 (4.7) | 0.49 | 0.58 $\pm$ 0.13 | 1.01 $\pm$ 0.22 | 1.66 $\pm$ 0.25 | 0.87 $\pm$ 0.21 | $2$
SM0723-C124 | 110.8545175 | $-73.4608018$ | 5.7 (3.7) | 0.9 | 0.41 $\pm$ 0.07 | 0.43 $\pm$ 0.12 | 0.33 $\pm$ 0.14 | 0.44 $\pm$ 0.13 | $0$
M1931-C41 | 292.956802 | $-26.5758904$ | 83.3 (67.3) | 1.0 | 4.58 $\pm$ 0.06 | 5.21 $\pm$ 0.08 | 5.92 $\pm$ 0.08 | 5.65 $\pm$ 0.16 | $2$
M1931-C47 | 292.9669811 | $-26.5767779$ | 7.7 (6.9) | 0.61 | 0.69 $\pm$ 0.09 | 0.88 $\pm$ 0.13 | 0.89 $\pm$ 0.13 | 0.65 $\pm$ 0.15 | $0$
M1931-C55 | 292.9504041 | $-26.5787259$ | 10.4 (11.6) | 0.97 | 0.59 $\pm$ 0.06 | 0.93 $\pm$ 0.08 | 1.33 $\pm$ 0.08 | 1.34 $\pm$ 0.2 | $2$
M1931-C61 | 292.9553168 | $-26.5808187$ | 9.2 (10.5) | 1.0 | 0.51 $\pm$ 0.06 | 0.82 $\pm$ 0.08 | 1.16 $\pm$ 0.08 | 1.18 $\pm$ 0.21 | $2$
R0949-C10 | 147.4672003 | $17.0964321$ | 61.9 (56.3) | 0.99 | 3.83 $\pm$ 0.06 | 4.83 $\pm$ 0.09 | 5.57 $\pm$ 0.09 | 5.51 $\pm$ 0.17 | $2$
R0949-C19 | 147.4743782 | $17.0993478$ | 13.1 (10.7) | 0.62 | 1.31 $\pm$ 0.1 | 1.47 $\pm$ 0.14 | 1.66 $\pm$ 0.14 | 1.01 $\pm$ 0.16 | $2$
R0949-C122 | 147.4631762 | $17.1368889$ | 5.3 (4.4) | 0.67 | 0.49 $\pm$ 0.09 | 0.56 $\pm$ 0.13 | 0.76 $\pm$ 0.13 | 0.61 $\pm$ 0.23 | $0$
R0949-C124 | 147.4775828 | $17.1362404$ | 12.1 (8.6) | 0.32 | 2.3 $\pm$ 0.19 | 2.27 $\pm$ 0.26 | 2.19 $\pm$ 0.26 |  $\cdots$ | $-2$
A209-C38 | 22.9724037 | $-13.6214995$ | 5.3 (3.9) | 0.47 | 0.7 $\pm$ 0.13 | 0.71 $\pm$ 0.18 | 0.64 $\pm$ 0.19 | 0.35 $\pm$ 0.11 | $0$
M0416-C51 | 64.0290222 | $-24.0666569$ | 7.1 (6.3) | 0.95 | 0.4 $\pm$ 0.06 | 0.51 $\pm$ 0.08 | 0.62 $\pm$ 0.08 | 0.59 $\pm$ 0.14 | $1$
M0416-C79 | 64.0486442 | $-24.0720833$ | 5.3 (4.0) | 0.87 | 0.33 $\pm$ 0.06 | 0.36 $\pm$ 0.09 | 0.35 $\pm$ 0.09 | 0.35 $\pm$ 0.11 | $1$
M0416-C117 | 64.0449843 | $-24.0798771$ | 15.7 (13.4) | 0.84 | 1.01 $\pm$ 0.06 | 1.23 $\pm$ 0.09 | 1.14 $\pm$ 0.1 | 1.08 $\pm$ 0.11 | $0$
M0416-C160 | 64.0367835 | $-24.0895812$ | 6.0 (3.8) | 0.82 | 0.4 $\pm$ 0.07 | 0.36 $\pm$ 0.1 | 0.25 $\pm$ 0.1 | 0.36 $\pm$ 0.12 | $1$
R0600-C12 | 90.0263215 | $-20.1679383$ | 5.3 (4.6) | 0.95 | 0.31 $\pm$ 0.06 | 0.39 $\pm$ 0.09 | 0.42 $\pm$ 0.09 | 0.38 $\pm$ 0.13 | $2$
R0600-C13 | 90.0210292 | $-20.1642048$ | 9.9 (9.0) | 0.98 | 0.56 $\pm$ 0.06 | 0.73 $\pm$ 0.08 | 0.98 $\pm$ 0.09 | 0.88 $\pm$ 0.15 | $2$
R0600-C67 | 90.0208748 | $-20.1567574$ | 38.1 (28.9) | 0.99 | 2.16 $\pm$ 0.06 | 2.34 $\pm$ 0.08 | 2.55 $\pm$ 0.09 | 2.37 $\pm$ 0.11 | $2$
R0600-C111 | 90.0371425 | $-20.1488705$ | 5.2 (3.7) | 0.96 | 0.3 $\pm$ 0.06 | 0.31 $\pm$ 0.08 | 0.36 $\pm$ 0.09 | 0.54 $\pm$ 0.14 | $0$
R0600-C164 | 90.0379199 | $-20.1407069$ | 4.6 (4.8) | 0.98 | 0.26 $\pm$ 0.06 | 0.4 $\pm$ 0.08 | 0.49 $\pm$ 0.09 | 0.76 $\pm$ 0.29 | $2$
M1149-C77 | 177.400434 | $22.406799$ | 5.3 (5.5) | 0.96 | 0.34 $\pm$ 0.07 | 0.48 $\pm$ 0.09 | 0.58 $\pm$ 0.09 | 0.62 $\pm$ 0.16 | $2$
M1149-C229 | 177.3918136 | $22.3879592$ | 4.0 (4.7) | 0.76 | 0.33 $\pm$ 0.08 | 0.52 $\pm$ 0.11 | 0.63 $\pm$ 0.12 | 0.59 $\pm$ 0.27 | $2$
A2744-C7 | 3.5796823 | $-30.3783961$ | 8.0 (4.5) | 0.48 | 0.83 $\pm$ 0.1 | 0.77 $\pm$ 0.17 | 0.73 $\pm$ 0.22 | 0.48 $\pm$ 0.1 | $1$
A2744-C21 | 3.5920469 | $-30.3804818$ | 6.8 (4.2) | 0.65 | 0.53 $\pm$ 0.08 | 0.54 $\pm$ 0.13 | 0.71 $\pm$ 0.17 | 0.42 $\pm$ 0.12 | $2$
A2744-C33 | 3.5850158 | $-30.3818137$ | 15.0 (13.6) | 0.91 | 0.83 $\pm$ 0.06 | 1.24 $\pm$ 0.09 | 1.52 $\pm$ 0.12 | 1.41 $\pm$ 0.15 | $2$
A2744-C56 | 3.5732619 | $-30.383509$ | 11.9 (6.5) | 0.61 | 0.98 $\pm$ 0.08 | 0.88 $\pm$ 0.14 | 1.05 $\pm$ 0.18 | 0.63 $\pm$ 0.11 | $0$
A2744-C81 | 3.5825059 | $-30.3854775$ | 26.7 (16.2) | 0.98 | 1.38 $\pm$ 0.05 | 1.37 $\pm$ 0.09 | 1.62 $\pm$ 0.11 | 1.31 $\pm$ 0.08 | $1$
A2744-C319 | 3.5760949 | $-30.4131819$ | 14.2 (11.4) | 0.46 | 1.55 $\pm$ 0.11 | 2.05 $\pm$ 0.18 | 3.28 $\pm$ 0.23 | 1.31 $\pm$ 0.14 | $2$
M1423-C38 | 215.94587 | $24.0817102$ | 5.5 (4.8) | 0.99 | 0.35 $\pm$ 0.06 | 0.42 $\pm$ 0.09 | 0.51 $\pm$ 0.09 | 0.64 $\pm$ 0.16 | $2$
M1423-C50 | 215.9494919 | $24.0784418$ | 23.1 (18.3) | 1.0 | 1.47 $\pm$ 0.06 | 1.59 $\pm$ 0.09 | 1.64 $\pm$ 0.09 | 1.81 $\pm$ 0.19 | $0$
R2129-C20 | 322.4164909 | $0.0891782$ | 17.8 (12.3) | 0.99 | 0.71 $\pm$ 0.04 | 0.72 $\pm$ 0.06 | 0.77 $\pm$ 0.06 | 0.75 $\pm$ 0.08 | $1$
R2129-C37 | 322.4093493 | $0.0812868$ | 5.8 (5.7) | 0.59 | 0.39 $\pm$ 0.07 | 0.57 $\pm$ 0.1 | 0.74 $\pm$ 0.1 | 0.51 $\pm$ 0.12 | $2$
M2129-C24 | 322.3553503 | $-7.6854665$ | 4.9 (5.2) | 0.92 | 0.25 $\pm$ 0.05 | 0.38 $\pm$ 0.07 | 0.51 $\pm$ 0.08 | 0.5 $\pm$ 0.14 | $2$
M2129-C46 | 322.3727048 | $-7.691988$ | 6.2 (4.5) | 0.98 | 0.3 $\pm$ 0.05 | 0.31 $\pm$ 0.07 | 0.25 $\pm$ 0.07 | 0.35 $\pm$ 0.09 | $2$
A3192-C31 | 59.7084164 | $-29.9418886$ | 32.7 (27.5) | 0.99 | 2.36 $\pm$ 0.07 | 2.78 $\pm$ 0.1 | 2.9 $\pm$ 0.1 | 2.94 $\pm$ 0.15 | $2$
A3192-C40 | 59.7130272 | $-29.9196827$ | 26.0 (20.7) | 0.66 | 2.82 $\pm$ 0.11 | 3.15 $\pm$ 0.15 | 3.36 $\pm$ 0.15 | 2.2 $\pm$ 0.15 | $2$
A3192-C131 | 59.6984479 | $-29.942652$ | 11.5 (8.5) | 0.75 | 1.1 $\pm$ 0.1 | 1.13 $\pm$ 0.13 | 1.13 $\pm$ 0.13 | 0.79 $\pm$ 0.12 | $0$
A3192-C138 | 59.7380434 | $-29.9148732$ | 8.1 (6.7) | 0.95 | 0.61 $\pm$ 0.08 | 0.71 $\pm$ 0.11 | 0.75 $\pm$ 0.11 | 0.87 $\pm$ 0.14 | $1$
M1206-C27 | 181.5469134 | $-8.7954315$ | 7.3 (7.7) | 1.0 | 0.38 $\pm$ 0.05 | 0.57 $\pm$ 0.07 | 0.87 $\pm$ 0.08 | 1.04 $\pm$ 0.23 | $2$
M1206-C54 | 181.5664156 | $-8.7998352$ | 6.1 (7.4) | 0.97 | 0.32 $\pm$ 0.05 | 0.57 $\pm$ 0.08 | 0.92 $\pm$ 0.08 | 1.13 $\pm$ 0.24 | $2$
M1206-C55 | 181.5447563 | $-8.800161$ | 8.2 (9.3) | 0.99 | 0.43 $\pm$ 0.05 | 0.69 $\pm$ 0.08 | 0.92 $\pm$ 0.08 | 1.15 $\pm$ 0.18 | $2$
M1206-C58 | 181.550609 | $-8.800985$ | 6.4 (6.5) | 0.99 | 0.34 $\pm$ 0.05 | 0.48 $\pm$ 0.08 | 0.54 $\pm$ 0.08 | 0.52 $\pm$ 0.13 | $1$
M1206-C60 | 181.5447729 | $-8.8014626$ | 9.8 (10.4) | 0.99 | 0.51 $\pm$ 0.05 | 0.78 $\pm$ 0.08 | 1.1 $\pm$ 0.08 | 1.28 $\pm$ 0.21 | $2$
M1206-C61 | 181.5449871 | $-8.8024698$ | 6.5 (8.0) | 0.99 | 0.34 $\pm$ 0.05 | 0.6 $\pm$ 0.08 | 0.95 $\pm$ 0.08 | 1.5 $\pm$ 0.23 | $2$
M0429-C4 | 67.3958701 | $-2.8775789$ | 8.4 (7.7) | 0.88 | 0.85 $\pm$ 0.1 | 1.08 $\pm$ 0.14 | 1.4 $\pm$ 0.14 | 1.37 $\pm$ 0.29 | $2$
M0429-C19 | 67.4001178 | $-2.8852218$ | 27.6 (21.5) | 1.0 | 2.48 $\pm$ 0.09 | 2.69 $\pm$ 0.13 | 3.01 $\pm$ 0.12 | 2.45 $\pm$ 0.38 | $0$
M0429-C27 | 67.4046007 | $-2.8931148$ | 6.9 (6.7) | 0.92 | 0.67 $\pm$ 0.1 | 0.9 $\pm$ 0.14 | 1.15 $\pm$ 0.13 | 1.14 $\pm$ 0.29 | $2$
M0429-C19 | 332.9487419 | $-3.8509998$ | 6.5 (5.6) | 0.98 | 0.51 $\pm$ 0.08 | 0.59 $\pm$ 0.11 | 0.6 $\pm$ 0.11 | 0.67 $\pm$ 0.14 | $2$
M0429-C35 | 332.9287744 | $-3.8449034$ | 16.5 (17.8) | 0.93 | 1.35 $\pm$ 0.08 | 1.98 $\pm$ 0.11 | 2.7 $\pm$ 0.11 | 2.62 $\pm$ 0.23 | $2$
M0429-C171 | 332.9258442 | $-3.8399459$ | 10.1 (9.1) | 0.64 | 1.22 $\pm$ 0.12 | 1.48 $\pm$ 0.16 | 1.89 $\pm$ 0.17 | 1.14 $\pm$ 0.2 | $2$
M1311-C27 | 197.7554922 | $-3.1759898$ | 8.9 (9.2) | 1.0 | 0.54 $\pm$ 0.06 | 0.83 $\pm$ 0.09 | 1.09 $\pm$ 0.1 | 1.15 $\pm$ 0.26 | $2$
M1311-C33 | 197.7505853 | $-3.1784355$ | 8.3 (6.8) | 0.68 | 0.75 $\pm$ 0.09 | 0.91 $\pm$ 0.13 | 0.88 $\pm$ 0.14 | 0.67 $\pm$ 0.13 | $1$
R0032-C32 | 8.0406932 | $18.1067957$ | 9.6 (7.2) | 0.97 | 0.69 $\pm$ 0.07 | 0.73 $\pm$ 0.1 | 0.74 $\pm$ 0.1 | 0.75 $\pm$ 0.12 | $1$
R0032-C53 | 8.034269 | $18.1112068$ | 11.3 (11.3) | 0.99 | 0.8 $\pm$ 0.07 | 1.12 $\pm$ 0.1 | 1.39 $\pm$ 0.1 | 1.61 $\pm$ 0.23 | $2$
R0032-C55 | 8.0326614 | $18.1128138$ | 17.7 (18.9) | 1.0 | 1.24 $\pm$ 0.07 | 1.88 $\pm$ 0.1 | 2.66 $\pm$ 0.1 | 3.11 $\pm$ 0.43 | $2$
R0032-C57 | 8.0321383 | $18.1135584$ | 7.5 (4.8) | 1.0 | 0.53 $\pm$ 0.07 | 0.48 $\pm$ 0.1 | 1.32 $\pm$ 0.1 |  $\cdots$ | $-1$
R0032-C58 | 8.0315196 | $18.1142688$ | 12.2 (11.8) | 1.0 | 0.86 $\pm$ 0.07 | 1.17 $\pm$ 0.1 | 1.45 $\pm$ 0.1 | 1.54 $\pm$ 0.32 | $2$
R0032-C81 | 8.0535714 | $18.1172188$ | 5.2 (4.1) | 0.94 | 0.38 $\pm$ 0.07 | 0.43 $\pm$ 0.11 | 0.41 $\pm$ 0.11 | 0.55 $\pm$ 0.19 | $1$
R0032-C127 | 8.0548414 | $18.1372963$ | 53.6 (41.8) | 0.95 | 3.95 $\pm$ 0.07 | 4.36 $\pm$ 0.1 | 4.47 $\pm$ 0.11 | 4.25 $\pm$ 0.13 | $2$
R0032-C131 | 8.0507777 | $18.1370193$ | 46.8 (36.7) | 0.96 | 3.41 $\pm$ 0.07 | 3.78 $\pm$ 0.1 | 3.84 $\pm$ 0.11 | 4.21 $\pm$ 0.28 | $2$
R0032-C198 | 8.0355827 | $18.1407431$ | 18.3 (13.3) | 1.0 | 1.29 $\pm$ 0.07 | 1.33 $\pm$ 0.1 | 1.37 $\pm$ 0.1 | 1.35 $\pm$ 0.14 | $1$
R0032-C208 | 8.05821 | $18.1426942$ | 11.2 (10.4) | 0.95 | 0.83 $\pm$ 0.07 | 1.08 $\pm$ 0.11 | 1.25 $\pm$ 0.11 | 1.42 $\pm$ 0.19 | $1$
R0032-C220 | 8.0358851 | $18.1524648$ | 6.5 (6.8) | 0.97 | 0.47 $\pm$ 0.07 | 0.69 $\pm$ 0.1 | 0.96 $\pm$ 0.1 | 0.91 $\pm$ 0.23 | $2$
R0032-C238 | 8.0326402 | $18.1593962$ | 6.6 (5.9) | 0.43 | 1.09 $\pm$ 0.16 | 1.36 $\pm$ 0.23 | 2.05 $\pm$ 0.24 | 0.86 $\pm$ 0.23 | $2$
R0032-C250 | 8.0364996 | $18.1557851$ | 8.0 (7.3) | 0.97 | 0.58 $\pm$ 0.07 | 0.74 $\pm$ 0.1 | 0.93 $\pm$ 0.1 | 0.86 $\pm$ 0.15 | $2$
R0032-C276 | 8.0358657 | $18.1497658$ | 14.9 (10.5) | 0.98 | 1.07 $\pm$ 0.07 | 1.06 $\pm$ 0.1 | 0.95 $\pm$ 0.1 | 1.07 $\pm$ 0.14 | $0$
R0032-C281 | 8.0510423 | $18.1469397$ | 9.7 (9.0) | 0.97 | 0.7 $\pm$ 0.07 | 0.92 $\pm$ 0.1 | 1.2 $\pm$ 0.1 | 1.14 $\pm$ 0.22 | $2$
R0032-C287 | 8.032096 | $18.1480371$ | 11.4 (8.9) | 0.72 | 1.1 $\pm$ 0.1 | 1.22 $\pm$ 0.14 | 1.39 $\pm$ 0.14 | 0.98 $\pm$ 0.13 | $1$
R0032-C304 | 8.0373105 | $18.1449397$ | 11.2 (9.4) | 1.0 | 0.79 $\pm$ 0.07 | 0.93 $\pm$ 0.1 | 1.09 $\pm$ 0.1 | 1.03 $\pm$ 0.12 | $2$
A383-C40 | 42.01413 | $-3.5291164$ | 34.2 (24.2) | 1.0 | 2.05 $\pm$ 0.06 | 2.16 $\pm$ 0.09 | 2.21 $\pm$ 0.1 | 2.07 $\pm$ 0.13 | $0$
A370-C18 | 39.9591641 | $-1.5950176$ | 12.9 (8.9) | 0.59 | 1.01 $\pm$ 0.08 | 1.04 $\pm$ 0.12 | 1.38 $\pm$ 0.13 | 0.67 $\pm$ 0.09 | $2$
A370-C31 | 39.976781 | $-1.5601292$ | 5.9 (5.6) | 0.68 | 0.4 $\pm$ 0.07 | 0.57 $\pm$ 0.1 | 0.88 $\pm$ 0.11 | 0.58 $\pm$ 0.16 | $2$
A370-C103 | 39.9923102 | $-1.5734715$ | 6.8 (6.4) | 0.5 | 0.63 $\pm$ 0.09 | 0.89 $\pm$ 0.14 | 1.14 $\pm$ 0.15 | 0.57 $\pm$ 0.12 | $2$
A370-C110 | 39.985689 | $-1.5739784$ | 24.9 (21.3) | 0.78 | 1.48 $\pm$ 0.06 | 1.9 $\pm$ 0.09 | 2.19 $\pm$ 0.1 | 1.59 $\pm$ 0.12 | $2$
A370-C146 | 39.9643502 | $-1.5782083$ | 5.4 (4.2) | 1.0 | 0.25 $\pm$ 0.05 | 0.29 $\pm$ 0.07 | 0.31 $\pm$ 0.08 | 0.35 $\pm$ 0.11 | $2$
A2163-C11 | 243.9543963 | $-6.1372017$ | 4.0 (4.6) | 0.85 | 0.23 $\pm$ 0.06 | 0.37 $\pm$ 0.08 | 0.55 $\pm$ 0.08 | 0.75 $\pm$ 0.2 | $2$
A2537-C42 | 347.1017953 | $-2.1848524$ | 6.8 (6.3) | 0.63 | 0.74 $\pm$ 0.11 | 0.93 $\pm$ 0.15 | 1.11 $\pm$ 0.15 | 0.84 $\pm$ 0.22 | $2$
A2537-C49 | 347.0746776 | $-2.1868689$ | 5.2 (4.4) | 0.41 | 0.86 $\pm$ 0.17 | 0.99 $\pm$ 0.23 | 0.88 $\pm$ 0.23 | 0.45 $\pm$ 0.12 | $1$
A2537-C66 | 347.0820153 | $-2.1903513$ | 4.9 (4.5) | 0.98 | 0.34 $\pm$ 0.07 | 0.42 $\pm$ 0.09 | 0.45 $\pm$ 0.1 | 0.58 $\pm$ 0.15 | $1$
M0035-C41 | 8.8352511 | $-20.2597643$ | 7.2 (6.8) | 0.43 | 0.85 $\pm$ 0.12 | 1.11 $\pm$ 0.16 | 0.98 $\pm$ 0.16 | 0.52 $\pm$ 0.1 | $2$
M0035-C68 | 8.8781846 | $-20.2652598$ | 11.1 (10.4) | 0.87 | 0.66 $\pm$ 0.06 | 0.84 $\pm$ 0.08 | 0.88 $\pm$ 0.08 | 0.86 $\pm$ 0.13 | $1$
M0035-C94 | 8.8635169 | $-20.2715042$ | 4.5 (4.6) | 0.91 | 0.26 $\pm$ 0.06 | 0.36 $\pm$ 0.08 | 0.46 $\pm$ 0.08 | 0.43 $\pm$ 0.13 | $2$
AS1063-C15 | 342.1940949 | $-44.513018$ | 7.4 (7.7) | 0.55 | 0.7 $\pm$ 0.09 | 1.03 $\pm$ 0.13 | 1.58 $\pm$ 0.15 | 0.84 $\pm$ 0.16 | $2$
AS1063-C17 | 342.1970228 | $-44.5133754$ | 15.5 (11.9) | 0.66 | 1.22 $\pm$ 0.08 | 1.32 $\pm$ 0.11 | 1.42 $\pm$ 0.12 | 0.91 $\pm$ 0.09 | $1$
AS1063-C147 | 342.174236 | $-44.5326395$ | 4.9 (6.1) | 0.96 | 0.27 $\pm$ 0.05 | 0.47 $\pm$ 0.08 | 0.87 $\pm$ 0.09 | 1.23 $\pm$ 0.31 | $2$
AS1063-C222 | 342.2044085 | $-44.5403889$ | 85.7 (60.0) | 0.7 | 6.37 $\pm$ 0.07 | 6.31 $\pm$ 0.11 | 7.78 $\pm$ 0.12 | 5.42 $\pm$ 0.09 | $2$
IRAC Prior Catalog ($N=39$)
A2537-C6 | 347.0866914 | $-2.1762725$ | 4.2 (3.1) | 0.84 | 0.34 $\pm$ 0.08 | 0.34 $\pm$ 0.11 | 0.01 $\pm$ 0.11 | 0.19 $\pm$ 0.07 | $0$
A2537-C24 | 347.0881901 | $-2.1817937$ | 4.5 (4.1) | 0.99 | 0.31 $\pm$ 0.07 | 0.39 $\pm$ 0.09 | 0.47 $\pm$ 0.09 | 0.51 $\pm$ 0.17 | $1$
A2744-C17 | 3.5813087 | $-30.380165$ | 4.4 (3.0) | 0.67 | 0.33 $\pm$ 0.08 | 0.37 $\pm$ 0.12 | 0.61 $\pm$ 0.16 | 0.34 $\pm$ 0.15 | $2$
A2744-C47 | 3.5720304 | $-30.3829544$ | 4.7 (4.1) | 0.47 | 0.51 $\pm$ 0.11 | 0.72 $\pm$ 0.18 | 0.67 $\pm$ 0.23 | 0.43 $\pm$ 0.15 | $2$
A2744-C176 | 3.5723343 | $-30.3959509$ | 4.2 (4.2) | 0.87 | 0.24 $\pm$ 0.06 | 0.4 $\pm$ 0.1 | 0.72 $\pm$ 0.12 | 0.59 $\pm$ 0.21 | $2$
A2744-C178 | 3.6005008 | $-30.3961199$ | 4.7 (4.1) | 0.98 | 0.24 $\pm$ 0.05 | 0.35 $\pm$ 0.08 | 0.63 $\pm$ 0.11 | 0.58 $\pm$ 0.18 | $2$
A2744-C227 | 3.5690078 | $-30.4027867$ | 4.6 (2.5) | 0.5 | 0.46 $\pm$ 0.1 | 0.41 $\pm$ 0.17 | -0.15 $\pm$ 0.22 | 0.26 $\pm$ 0.12 | $1$
A3192-C83 | 59.7246463 | $-29.9320537$ | 4.2 (3.2) | 0.99 | 0.3 $\pm$ 0.07 | 0.32 $\pm$ 0.1 | 0.39 $\pm$ 0.1 | 0.33 $\pm$ 0.22 | $2$
A3192-C154 | 59.7362229 | $-29.9055828$ | 4.2 (4.1) | 0.46 | 0.66 $\pm$ 0.16 | 0.89 $\pm$ 0.22 | 0.97 $\pm$ 0.22 | 0.57 $\pm$ 0.15 | $1$
A370-C27 | 39.9730709 | $-1.5597934$ | 4.3 (4.0) | 0.73 | 0.27 $\pm$ 0.06 | 0.38 $\pm$ 0.1 | 0.66 $\pm$ 0.1 | 0.94 $\pm$ 0.22 | $2$
A370-C172 | 39.9473985 | $-1.5814055$ | 4.3 (3.0) | 0.31 | 0.64 $\pm$ 0.15 | 0.67 $\pm$ 0.22 | 0.18 $\pm$ 0.24 |  $\cdots$ | $-2$
A383-C24 | 42.0117763 | $-3.5240168$ | 4.0 (3.7) | 0.99 | 0.24 $\pm$ 0.06 | 0.33 $\pm$ 0.09 | 0.39 $\pm$ 0.1 | 0.61 $\pm$ 0.22 | $2$
A383-C50 | 42.0117548 | $-3.5326342$ | 4.2 (2.0) | 1.0 | 0.25 $\pm$ 0.06 | 0.18 $\pm$ 0.09 | 0.41 $\pm$ 0.1 | 0.54 $\pm$ 0.22 | $1$
AC0102-C128 | 15.7103067 | $-49.2615949$ | 4.5 (2.8) | 0.67 | 0.48 $\pm$ 0.11 | 0.43 $\pm$ 0.15 | 0.61 $\pm$ 0.17 | 0.25 $\pm$ 0.09 | $0$
AS295-C9 | 41.4026213 | $-53.0576036$ | 4.3 (3.3) | 0.92 | 0.35 $\pm$ 0.08 | 0.43 $\pm$ 0.13 | 0.66 $\pm$ 0.15 | 0.63 $\pm$ 0.23 | $0$
AS295-C269 | 41.3430354 | $-53.0135883$ | 4.3 (3.8) | 0.37 | 0.84 $\pm$ 0.2 | 1.21 $\pm$ 0.32 | 2.28 $\pm$ 0.38 | 1.24 $\pm$ 0.47 | $2$
M0035-C33 | 8.8562578 | $-20.2581929$ | 4.1 (2.9) | 0.95 | 0.22 $\pm$ 0.05 | 0.21 $\pm$ 0.07 | 0.09 $\pm$ 0.08 |  $\cdots$ | $-1$
M0416-C120 | 64.0437953 | $-24.0802677$ | 4.4 (2.9) | 0.84 | 0.28 $\pm$ 0.06 | 0.27 $\pm$ 0.09 | 0.22 $\pm$ 0.09 | 0.23 $\pm$ 0.1 | $0$
M0416-C138 | 64.0438489 | $-24.0846586$ | 4.9 (3.9) | 0.83 | 0.32 $\pm$ 0.07 | 0.36 $\pm$ 0.09 | 0.28 $\pm$ 0.1 | 0.34 $\pm$ 0.09 | $2$
M0416-C156 | 64.0366134 | $-24.0882085$ | 4.6 (3.2) | 0.85 | 0.29 $\pm$ 0.06 | 0.29 $\pm$ 0.09 | 0.2 $\pm$ 0.09 | 0.22 $\pm$ 0.07 | $0$
M1115-C33 | 168.9764012 | $1.4992074$ | 4.3 (2.9) | 0.54 | 0.5 $\pm$ 0.12 | 0.48 $\pm$ 0.17 | 0.86 $\pm$ 0.17 | 0.59 $\pm$ 0.56 | $2$
M1149-C27 | 177.4166485 | $22.4158657$ | 4.6 (3.4) | 0.46 | 0.64 $\pm$ 0.14 | 0.64 $\pm$ 0.19 | 0.17 $\pm$ 0.2 | 0.27 $\pm$ 0.11 | $0$
M1149-C95 | 177.4210623 | $22.4046942$ | 4.1 (3.9) | 0.46 | 0.55 $\pm$ 0.14 | 0.71 $\pm$ 0.18 | 0.87 $\pm$ 0.19 | 0.71 $\pm$ 0.35 | $2$
M1206-C38 | 181.5343916 | $-8.7975222$ | 4.4 (3.1) | 0.74 | 0.31 $\pm$ 0.07 | 0.31 $\pm$ 0.1 | 0.07 $\pm$ 0.11 | 0.27 $\pm$ 0.11 | $0$
M1206-C84 | 181.5547108 | $-8.8075633$ | 4.4 (4.3) | 0.96 | 0.24 $\pm$ 0.05 | 0.33 $\pm$ 0.08 | 0.47 $\pm$ 0.09 | 0.43 $\pm$ 0.14 | $2$
M1423-C52 | 215.9481902 | $24.0771052$ | 4.4 (3.1) | 1.0 | 0.28 $\pm$ 0.06 | 0.27 $\pm$ 0.09 | 0.25 $\pm$ 0.09 | 0.17 $\pm$ 0.09 | $0$
M1423-C76 | 215.9501077 | $24.0700621$ | 5.0 (4.3) | 0.92 | 0.34 $\pm$ 0.07 | 0.41 $\pm$ 0.09 | 0.52 $\pm$ 0.1 | 0.46 $\pm$ 0.13 | $2$
M1931-C42 | 292.9561655 | $-26.5753642$ | 4.2 (2.5) | 1.0 | 0.23 $\pm$ 0.06 | 0.19 $\pm$ 0.08 | 0.46 $\pm$ 0.08 | 5.37 $\pm$ 0.27 | $0$
M1931-C69 | 292.9484625 | $-26.5839529$ | 4.3 (3.0) | 0.83 | 0.29 $\pm$ 0.07 | 0.28 $\pm$ 0.09 | 0.23 $\pm$ 0.09 | 0.16 $\pm$ 0.05 | $0$
M2129-C62 | 322.3734434 | $-7.693796$ | 4.3 (3.3) | 0.96 | 0.21 $\pm$ 0.05 | 0.23 $\pm$ 0.07 | 0.32 $\pm$ 0.07 | 0.32 $\pm$ 0.13 | $2$
P171-C161 | 48.2306214 | $8.3954731$ | 4.3 (3.0) | 0.86 | 0.36 $\pm$ 0.08 | 0.36 $\pm$ 0.12 | 0.93 $\pm$ 0.12 | 2.21 $\pm$ 0.35 | $2$
R0032-C63 | 8.0474404 | $18.1145036$ | 4.5 (3.4) | 0.95 | 0.33 $\pm$ 0.07 | 0.36 $\pm$ 0.1 | 0.39 $\pm$ 0.11 | 0.47 $\pm$ 0.17 | $1$
R0032-C162 | 8.0480361 | $18.1311965$ | 4.8 (4.0) | 0.96 | 0.35 $\pm$ 0.07 | 0.41 $\pm$ 0.1 | 0.46 $\pm$ 0.11 | 0.44 $\pm$ 0.13 | $1$
R0032-C245 | 8.0394333 | $18.1580407$ | 4.6 (3.6) | 0.99 | 0.33 $\pm$ 0.07 | 0.36 $\pm$ 0.1 | 0.41 $\pm$ 0.1 | 0.46 $\pm$ 0.19 | $2$
R0949-C14 | 147.4681688 | $17.0962861$ | 4.7 (3.4) | 0.99 | 0.29 $\pm$ 0.06 | 0.29 $\pm$ 0.09 | 0.73 $\pm$ 0.09 |  $\cdots$ | $-1$
R0949-C113 | 147.4744436 | $17.1380636$ | 4.0 (2.8) | 0.36 | 0.69 $\pm$ 0.17 | 0.66 $\pm$ 0.24 | 0.71 $\pm$ 0.24 |  $\cdots$ | $-2$
R0949-C119 | 147.4671533 | $17.1360239$ | 4.2 (3.9) | 0.8 | 0.32 $\pm$ 0.08 | 0.41 $\pm$ 0.11 | 0.58 $\pm$ 0.11 | 0.42 $\pm$ 0.12 | $2$
R0949-C164 | 332.9363309 | $-3.8371329$ | 4.6 (3.5) | 0.98 | 0.36 $\pm$ 0.08 | 0.37 $\pm$ 0.11 | 0.44 $\pm$ 0.11 | 0.38 $\pm$ 0.12 | $1$
SM0723-C98 | 110.7609477 | $-73.4569162$ | 4.1 (2.3) | 0.44 | 0.61 $\pm$ 0.15 | 0.56 $\pm$ 0.24 | 0.04 $\pm$ 0.28 | 0.45 $\pm$ 0.22 | $1$

Note. — (1) Flux range used in the differential number counts. (2)
Differential number counts ($dN/dS$) in the logarithm scale. The 1$\sigma$
uncertainties are estimated from the combination of the number count Poisson
statistical errors and the uncertainty of the intrinsic flux density estimate
associated with the measurement, redshift, and magnification errors (see the
text). (3) Average source number in each flux bin among the 1,000 MC
realizations.

Note. — (1)–(3): Best-fit parameter set for the Schechter function. (4):
$\chi^{2}$ over the degree of freedom. (5)– (8): Best-fit parameter set for
the DPL function.

Note. — (1) Various data sets for the $\alpha$ measurement. (2) FIR size
assumption in the completeness correction estimate. (3) Derived faint-end
slope of $\alpha$ in the Schechter function.

$\dagger$ Assuming positive and negative correlations with $L_{\rm IR}$ (see
Sec. 5.5).

Note. — (1): Redshift bin range of ALCS IR LFs. (2): Faint-end slope in DPL
function. It is fixed in the fitting in the redshift bins of $z=4.0$–6.0 and
$z=6.0$–7.5. (3): Bright-end slope in DPL function. It is fixed in the fitting
in the redshift bins, except for $z=1.0$–2.0. (4): Characteristic volume
density in DPL function. (5): Characteristic IR luminosity in DPL function.

Note. — (1) ALCS continuum source ID. (2) SNR of the $1\farcs 0$-diameter
aperture photometry in the velocity-integrated map for the line emission. (3)
Central frequency of the line emission. (4) FWHM of the line width. (5)
Velocity-integrated line flux. (6) Contribution of the line flux to the 1.2-mm
continuum flux density estimate, which is corrected in Section 3.3. (7) Line
classification. (8) Spectroscopic redshift from the line. (9) Reference: E17
(Ebeling et al. 2017), F21 (Fujimoto et al. 2021), and L21 (Laporte et al.
2021). The values in (3–5) are measured from a single Gaussian fitting in the
spectrum.

$\dagger$ The second line is securely detected in follow-up Band 3 line scan
observations (Table 10) at a consistent frequency from these three sources.

Note. — (1) ALCS continuum source ID. (2) Redshift determined by the emission
lines. (3) ALMA Band used in the follow-up line scan observations. (4) Lines
detected in the follow-up observations. (4) Notes. (5) Reference: T23 (Tsujita
et al. in prep.) and B23 (Bauer et al. in prep).

$\dagger$ The upper limit of the flux ratio at 4.5 $\mu$m (IRAC ch2) and 1.2
mm (ALMA) suggests the source redshift at $z>4.5$–5.0 based on the SED
template of the high-$z$ dusty galaxy. We regard the line feature observed
with the highest SNR ($\sim 5$) in the Band 3+4 spectrum as the CO line, which
satisfies the lowest-$z$ solution at $z>4.5$ (Appendix A).

Note. — (1) Cluster Name. (2) Cluster redshift. (3) Number of the multiple
system identified in the cluster. (4) Number of multiple system
spectroscopically confirmed identified in the cluster. (5) Number of multiple
images identified in the image plane used for constructing our fiducial mass
model with glafic (Oguri 2010). (6) RMS of positional differences between
observed and model-predicted multiple images, in units of arcseconds. (7)
References for the multiple images: K18 (Kawamata et al. 2018), Z15 (Zitrin et
al. 2015), Ca19 (Caminha et al. 2019), R21 (Richard et al. 2021), Ce18 (Cerny
et al. 2018), Ci18 (Cibirka et al. 2018), Z11 (Zitrin et al. 2011), M19
(Mahler et al. 2019), E17 (Ebeling et al. 2017), R20 (Rescigno et al. 2020),
U18 (Ueda et al. 2018), H13 (Hsu et al. 2013), F21 (Fujimoto et al. 2021)
(8–10) Availability (Y: Yes, No: No) of other mass models so far constructed
by G. Caminha with Lenstool (Caminha et al. 2016, 2017b, 2017a, 2019), CATS
team with Lenstool (Mahler et al. 2019; Lagattuta et al. 2019; Richard et al.
2010, 2011, 2021; Jauzac et al. 2016, 2021; Rexroth et al. 2017; Dessauges-
Zavadsky et al. 2017; Livermore et al. 2012; Repp et al. 2016; Ebeling et al.
2017; Newman et al. 2013), and A. Zitrin with LTM and dPIEeNFW algorithms
(Zitrin et al. 2013, 2015) in the ALCS 33 clusters. All model links will be
available on the ALCS website. 777 http://www.ioa.s.u-tokyo.ac.jp/ALCS/
Interested readers may find more lens models on HFF888
https://archive.stsci.edu/prepds/frontier/ , CLASH999
https://www.stsci.edu/~postman/CLASH/ , and RELICS101010
https://relics.stsci.edu/ websites.

$\dagger$ The multiple images of the ALCS sources without HST counterparts,
M0417-C46/58/121, are also used for refining our fiducial lens model.

Note. — (1) Source ID, including the prefix of the cluster name + “AC” (ALCS
continuum). (2) Source coordinate of the continuum peak in the natural-
weighted map. (3) Signal-to-noise ratio of the peak pixel in the natural-
weighted map before primary beam correction. (4) Primary beam sensitivity in
the mosaic map. (5) Source flux density at 1.2 mm measured by the peak pixel
count in the natural-weighted map. (6) Source flux density at 1.2 mm measured
by the peak pixel count in the $uv$-tapered ($2^{\prime\prime}\times
2^{\prime\prime}$) map. (7) Source flux density at 1.2 mm measured by a
$2.0^{\prime\prime}$-radius aperture in the natural-weighted map. (8) Source
flux density at 1.2 mm measured by 2D elliptical Gaussian fitting with CASA
imfit. (9) Flag of the imfit output (0: point source, 1: mergry resovled, 2:
fully resolved, $-$1: fitting error, $-2$: near the edge of the map).

Table 13: Physical Properties of ALCS Continuum Sources

ID | $z_{\rm spec}$ | $z_{\rm phot}$ | $S_{\rm obs}$ | $L_{\rm IR}$ | $\mu$ |  $z$ type | Ref. | Note
---|---|---|---|---|---|---|---|---
|  |  | (mJy) | ($L_{\odot}$) |  |  |  |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
Blind Catalog ($N=$125)
AC0102-C11 |  $\cdots$ | $(3.0\pm 2.0)$ | $0.3\pm 0.10$ | $12.00\pm 0.19$ | 1.6${}_{-0.6}^{+0.7}$ | 5 |  $\cdots$ | No counterparts in HST, IRAC, Herschel
AC0102-C22 |  $\cdots$ | $3.45^{+0.17}_{-0.17}$ | $1.11\pm 0.13$ | $12.53\pm 0.19$ | 2.1${}_{-0.8}^{+0.8}$ | 4 | App.B | No counterparts in HST
AC0102-C50 |  $\cdots$ | $2.38^{+0.10}_{-0.11}$ | $0.60\pm 0.10$ | $12.85\pm 0.07$ | 2.4${}_{-1.0}^{+1.0}$ | 3 | K22 |
AC0102-C52 |  $\cdots$ | $2.02^{+0.35}_{-0.16}$ | $0.30\pm 0.09$ | $12.36\pm 0.11$ | 2.3${}_{-0.9}^{+1.0}$ | 3 | K22 |
AC0102-C160 |  $\cdots$ | $1.99^{+0.18}_{-0.14}$ | $0.65\pm 0.10$ | $12.61\pm 0.13$ | 2.3${}_{-0.9}^{+0.9}$ | 3 | K22 |
AC0102-C224 | 4.320 |  $\cdots$ | $3.19\pm 0.15$ | $13.07\pm 0.14$ | 5.1${}_{-3.6}^{+3.6}$ | 1 | C21 | multiple images (AC0102-C118/215/224)
AC0102-C241 |  $\cdots$ | $2.82^{+0.05}_{-0.67}$ | $0.97\pm 0.23$ | $12.70\pm 0.14$ | 2.0${}_{-0.8}^{+0.8}$ | 3 | K22 |
AC0102-C251 |  $\cdots$ | $(3.0\pm 2.0)$ | $0.9\pm 0.16$ | $11.84\pm 0.51$ | 1.0${}_{-0.0}^{+0.4}$ | 5 |  $\cdots$ | No counterparts in HST, IRAC, Herschel
AC0102-C276 |  $\cdots$ | $3.46^{+0.30}_{-0.43}$ | $1.41\pm 0.14$ | $12.95\pm 0.10$ | 5.2${}_{-2.7}^{+2.6}$ | 3 | K22 |
AC0102-C294 |  $\cdots$ | $4.0^{+0.5}_{-0.5}$ | $2.12\pm 0.28$ | $12.54\pm 0.13$ | 3.7${}_{-2.1}^{+2.1}$ | 2 |  $\cdots$ | multiple images (AC0102-C223/294)
M0417-C49 |  $\cdots$ | $3.16^{+0.10}_{-0.10}$ | $4.69\pm 0.24$ | $13.19\pm 0.25$ | 2.0${}_{-0.8}^{+0.8}$ | 3 | K22 |
M0417-C121 | 3.652 |  $\cdots$ | $3.05\pm 0.15$ | $13.09\pm 0.14$ | 4.0${}_{-2.3}^{+2.3}$ | 1 | App.A | multiple images (M0417-C46/58/121)
M0417-C204 |  $\cdots$ | $1.80^{+0.22}_{-0.22}$ | $0.94\pm 0.13$ | $12.57\pm 0.16$ | 2.8${}_{-1.2}^{+1.1}$ | 4 | App.B | Outside of HST/WFC3
M0417-C218 |  $\cdots$ | $2.00^{+0.07}_{-0.07}$ | $1.42\pm 0.18$ | $12.76\pm 0.29$ | 4.8${}_{-1.9}^{+1.9}$ | 4 | App.B | No counterparts in HST
M0417-C221 |  $\cdots$ | $1.85^{+0.09}_{-0.29}$ | $0.72\pm 0.11$ | $12.86\pm 0.13$ | 4.3${}_{-1.8}^{+1.7}$ | 3 | K22 |
M0417-C223 |  $\cdots$ | $1.07^{+0.05}_{-0.10}$ | $0.48\pm 0.11$ | $12.65\pm 0.08$ | 2.7${}_{-1.1}^{+1.1}$ | 3 | K22 |
P171-C69 |  $\cdots$ | $3.40^{+0.15}_{-0.15}$ | $2.75\pm 0.23$ | $12.94\pm 0.18$ | 2.6${}_{-1.1}^{+1.1}$ | 4 | App.B | No counterparts in HST
P171-C162 |  $\cdots$ | $3.95^{+0.19}_{-0.19}$ | $2.69\pm 0.35$ | $12.22\pm 0.54$ | 2.1${}_{-0.9}^{+0.9}$ | 4 | App.B | No counterparts in HST
P171-C177 |  $\cdots$ | $0.69^{+2.38}_{-0.01}$ | $1.07\pm 0.11$ | $11.71\pm 0.31$ | 1.8${}_{-0.7}^{+1.1}$ | 3 | K22 |
M0329-C11 |  $\cdots$ | $2.84^{+0.01}_{-0.01}$ | $1.12\pm 0.17$ | $12.30\pm 0.09$ | 2.5${}_{-1.0}^{+1.0}$ | 3 | K22 |
M0159-C5 |  $\cdots$ | $1.30^{+0.10}_{-0.10}$ | $1.66\pm 0.16$ | $12.05\pm 0.09$ | 2.1${}_{-0.8}^{+0.8}$ | 3 | K22 |
M0159-C24 |  $\cdots$ | $0.66^{+0.03}_{-0.15}$ | $0.80\pm 0.14$ | $11.80\pm 0.13$ | 1.9${}_{-0.8}^{+2.2}$ | 3 | K22 |
M0159-C46 | 0.405 | $0.89^{+0.07}_{-0.07}$ | $2.25\pm 0.10$ | $11.88\pm 0.02$ | 1.0 | 1 |  $\cdots$ | BCG
M0159-C61 |  $\cdots$ | $0.88^{+0.08}_{-0.04}$ | $0.51\pm 0.08$ | $11.55\pm 0.49$ | 2.3${}_{-0.9}^{+1.0}$ | 3 | K22 |
R1347-C41 | 0.848 | $0.84^{+0.06}_{-0.05}$ | $0.50\pm 0.07$ | $11.59\pm 0.04$ | 3.5${}_{-1.4}^{+1.4}$ | 1 | Ca19 |
R1347-C51 |  $\cdots$ | $(3.0\pm 2.0)$ | $0.2\pm 0.07$ | $11.87\pm 0.15$ | 7.1${}_{-5.9}^{+5.8}$ | 5 |  $\cdots$ | No counterparts in HST, IRAC, Herschel
R1347-C75 | 0.451 | $0.64^{+0.02}_{-0.04}$ | $3.10\pm 0.10$ | $10.86\pm 0.13$ | 1.0 | 1 |  $\cdots$ | BCG
R1347-C145 | 1.768 | $1.50^{+0.34}_{-0.07}$ | $1.74\pm 0.16$ | $12.77\pm 0.03$ | 3.1${}_{-1.2}^{+1.2}$ | 1 | R21 |
R1347-C148 | 1.768 | $1.64^{+0.23}_{-0.02}$ | $2.75\pm 0.22$ | $12.92\pm 0.03$ | 5.7${}_{-2.8}^{+2.8}$ | 1 | R21 |
R1347-C166 |  $\cdots$ | $2.65^{+0.09}_{-1.94}$ | $0.89\pm 0.18$ | $12.32\pm 0.18$ | 3.0${}_{-2.0}^{+1.2}$ | 3 | K22 |
M1115-C2 |  $\cdots$ | $3.17^{+0.11}_{-0.20}$ | $3.34\pm 0.29$ | $13.65\pm 0.00$ | 6.0${}_{-3.0}^{+3.0}$ | 3 | K22 |
M1115-C4 | 1.596 | $2.41^{+0.02}_{-0.01}$ | $0.82\pm 0.10$ | $12.71\pm 0.18$ | 2.2${}_{-0.9}^{+0.9}$ | 1 | Ca19 |
M1115-C34 |  $\cdots$ | $4.25^{+0.22}_{-0.22}$ | $1.98\pm 0.18$ | $12.68\pm 0.20$ | 2.0${}_{-0.8}^{+0.8}$ | 4 | App.B | No counterparts in HST
M1115-C36 | 0.352 | $0.53^{+0.05}_{-0.03}$ | $0.48\pm 0.08$ | $10.30\pm 0.23$ | 1.0 | 1 |  $\cdots$ | BCG
M0257-C13 |  $\cdots$ | $2.73^{+0.91}_{-0.03}$ | $2.00\pm 0.23$ | $12.63\pm 0.26$ | 1.1${}_{-0.1}^{+0.4}$ | 3 | K22 |
M0553-C17 |  $\cdots$ | $2.45^{+0.12}_{-0.12}$ | $1.91\pm 0.19$ | $12.91\pm 0.16$ | 1.6${}_{-0.6}^{+0.6}$ | 4 | App.B | Outside of HST/WFC3
M0553-C18 |  $\cdots$ | $2.20^{+0.17}_{-0.17}$ | $1.28\pm 0.17$ | $11.96\pm 0.59$ | 1.6${}_{-0.6}^{+0.6}$ | 4 | App.B | Outside of HST/WFC3
M0553-C58 |  $\cdots$ | $1.74^{+2.73}_{-0.03}$ | $0.58\pm 0.09$ | $12.64\pm 0.12$ | 1.9${}_{-0.8}^{+0.9}$ | 3 | K22 |
M0553-C61 |  $\cdots$ | $1.58^{+0.03}_{-0.18}$ | $1.14\pm 0.19$ | $12.49\pm 0.17$ | 2.1${}_{-0.8}^{+0.8}$ | 3 | K22 |
M0553-C67 | 4.930 |  $\cdots$ | $0.80\pm 0.13$ | $12.69\pm 0.09$ | 2.6${}_{-1.0}^{+1.0}$ | 1 | App.A | No counterparts in HST
M0553-C200 |  $\cdots$ | $1.10^{+0.14}_{-0.14}$ | $0.68\pm 0.09$ | $11.95\pm 0.42$ | 1.8${}_{-0.7}^{+0.7}$ | 4 | App.B | Outside of HST/WFC3
M0553-C249 | 1.140 |  $\cdots$ | $2.71\pm 0.12$ | $12.73\pm 0.09$ | 4.4${}_{-2.5}^{+2.5}$ | 1 | E17 | multiple images (M0553-C133/190/249)
M0553-C275 |  $\cdots$ | $0.80^{+0.05}_{-0.06}$ | $0.64\pm 0.09$ | $12.11\pm 0.20$ | 2.1${}_{-0.9}^{+0.9}$ | 3 | K22 |
M0553-C303 | 0.836 | $0.84^{+0.02}_{-0.09}$ | $1.07\pm 0.09$ | $12.35\pm 0.37$ | 1.5${}_{-0.5}^{+0.6}$ | 1 | App.A |
M0553-C355 |  $\cdots$ | $1.81^{+0.20}_{-0.05}$ | $1.14\pm 0.14$ | $12.29\pm 0.24$ | 1.8${}_{-0.7}^{+0.7}$ | 3 | K22 |
M0553-C375 |  $\cdots$ | $2.05^{+0.17}_{-0.17}$ | $1.49\pm 0.14$ | $12.72\pm 0.16$ | 1.0${}_{-0.0}^{+0.4}$ | 4 | App.B | Outside of HST/WFC3
M0553-C398 |  $\cdots$ | $4.41^{+0.07}_{-0.49}$ | $2.25\pm 0.33$ | $13.05\pm 0.11$ | 1.8${}_{-0.7}^{+0.7}$ | 3 | K22 |
SM0723-C61 |  $\cdots$ | $2.12^{+0.69}_{-0.39}$ | $0.89\pm 0.19$ | $11.80\pm 0.58$ | 2.9${}_{-1.2}^{+1.2}$ | 3 | K22 |
SM0723-C93 |  $\cdots$ | $0.46^{+0.03}_{-0.11}$ | $1.49\pm 0.22$ | $11.63\pm 0.10$ | 1.2${}_{-0.2}^{+0.5}$ | 3 | K22 |
SM0723-C124 |  $\cdots$ | $1.05^{+0.05}_{-0.09}$ | $0.39\pm 0.11$ | $11.45\pm 0.34$ | 2.3${}_{-0.9}^{+0.9}$ | 3 | K22 |
M1931-C41 | 0.352 | $0.32^{+0.02}_{-0.00}$ | $5.56\pm 0.16$ | $12.35\pm 0.07$ | 1.0 | 1 | App.A | BCG
M1931-C47 |  $\cdots$ | $1.79^{+0.41}_{-0.27}$ | $0.80\pm 0.11$ | $11.69\pm 0.03$ | 2.6${}_{-1.1}^{+1.1}$ | 3 | K22 |
M1931-C55 |  $\cdots$ | $3.59^{+0.50}_{-0.06}$ | $1.38\pm 0.21$ | $13.21\pm 0.15$ | 6.7${}_{-3.4}^{+3.4}$ | 3 | K22 |
M1931-C61 |  $\cdots$ | $3.57^{+0.10}_{-0.10}$ | $1.16\pm 0.08$ | $12.59\pm 0.12$ | 4.1${}_{-1.7}^{+1.7}$ | 3 | K22 |
R0949-C10 | 5.753 | $>5.5$ | $5.57\pm 0.17$ | $13.16\pm 0.19$ | 3.2${}_{-1.3}^{+1.3}$ | 1 | App.A | No counterparts in HST, IRAC
R0949-C19 | 5.530 | $>4.5$ | $1.64\pm 0.26$ | $12.01\pm 0.51$ | 2.6${}_{-1.0}^{+1.0}$ | 1 | App.A | No counterparts in HST, IRAC, Herschel
R0949-C122 |  $\cdots$ | $1.77^{+0.70}_{-0.70}$ | $0.68\pm 0.11$ | $12.29\pm 0.26$ | 2.8${}_{-1.3}^{+1.2}$ | 4 | App.B | No counterparts in HST
R0949-C124 |  $\cdots$ | $1.24^{+0.03}_{-0.75}$ | $2.27\pm 0.26$ | $12.07\pm 0.04$ | 2.4${}_{-1.4}^{+0.9}$ | 3 | K22 |
A209-C38 |  $\cdots$ | $2.06^{+0.36}_{-0.07}$ | $0.64\pm 0.16$ | $11.87\pm 0.29$ | 1.4${}_{-0.4}^{+0.6}$ | 3 | K22 |
M0416-C51 | 1.960 | $2.07^{+0.12}_{-0.03}$ | $0.56\pm 0.08$ | $12.12\pm 0.05$ | 2.3${}_{-0.5}^{+0.5}$ | 1 | T15,S14 |
M0416-C79 |  $\cdots$ | $2.03^{+0.10}_{-0.14}$ | $0.32\pm 0.08$ | $11.87\pm 0.21$ | 1.9${}_{-0.4}^{+0.4}$ | 3 | K22 |
M0416-C117 | 2.087 | $1.91^{+0.01}_{-0.02}$ | $1.29\pm 0.13$ | $12.56\pm 0.03$ | 1.6${}_{-0.3}^{+0.3}$ | 1 | T15,S14 |
M0416-C160 |  $\cdots$ | $1.75^{+0.40}_{-0.04}$ | $0.32\pm 0.09$ | $12.12\pm 0.18$ | 1.5${}_{-0.3}^{+0.3}$ | 3 | K22 |
R0600-C12 |  $\cdots$ | $0.45^{+0.13}_{-0.13}$ | $0.38\pm 0.08$ | $11.15\pm 0.54$ | 1.0${}_{-0.0}^{+0.2}$ | 4 | App.B | Outside of HST/WFC3
R0600-C13 | 0.817 | $0.82^{+0.23}_{-0.50}$ | $0.94\pm 0.08$ | $12.07\pm 0.22$ | 1.4${}_{-0.4}^{+0.6}$ | 1 | App.A |
R0600-C67 | 4.800 | $>4.5$ | $2.39\pm 0.11$ | $12.66\pm 0.24$ | 4.5${}_{-1.8}^{+1.8}$ | 1 | App.A | No counterparts in HST, IRAC, Herschel
R0600-C111 |  $\cdots$ | $1.56^{+0.10}_{-0.06}$ | $0.33\pm 0.08$ | $12.58\pm 0.09$ | 3.5${}_{-1.4}^{+1.4}$ | 3 | K22 |
R0600-C164 | 6.072 | $4.38^{+0.07}_{-0.60}$ | $0.42\pm 0.08$ | $11.43\pm 0.50$ | 29.0${}_{-7.0}^{+4.0}$ | 1 | F21,L21 | multiple images
M1149-C77 | 1.460 | $1.56^{+0.13}_{-0.03}$ | $0.52\pm 0.08$ | $11.95\pm 0.10$ | 3.1${}_{-0.6}^{+0.6}$ | 1 | M22 |
M1149-C229 |  $\cdots$ | $2.05^{+0.70}_{-0.70}$ | $0.56\pm 0.11$ | $12.30\pm 0.16$ | 1.9${}_{-0.5}^{+0.4}$ | 4 | App.B | No counterparts in HST, IRAC
A2744-C7 | 2.409 | $2.14^{+0.17}_{-0.04}$ | $0.69\pm 0.15$ | $12.06\pm 0.11$ | 1.8${}_{-0.4}^{+0.4}$ | 1 | M22 |
A2744-C21 | 2.644 | $7.87^{+0.17}_{-0.14}$ | $0.64\pm 0.15$ | $12.87\pm 0.16$ | 1.8${}_{-0.4}^{+0.4}$ | 1 | M22 |
A2744-C33 | 3.058 | $3.12^{+0.21}_{-0.10}$ | $1.55\pm 0.17$ | $12.45\pm 0.10$ | 2.4${}_{-0.5}^{+0.5}$ | 1 | M22 |
A2744-C56 | 1.498 | $1.42^{+0.10}_{-0.14}$ | $1.03\pm 0.17$ | $12.76\pm 0.00$ | 2.4${}_{-0.5}^{+0.5}$ | 1 | M22 |
A2744-C81 | 2.900 | $2.99^{+0.24}_{-0.08}$ | $1.35\pm 0.08$ | $12.57\pm 0.16$ | 3.6${}_{-0.7}^{+0.7}$ | 1 | M22 |
A2744-C319 | 2.482 | $3.21^{+0.05}_{-0.42}$ | $2.84\pm 0.30$ | $12.70\pm 0.03$ | 1.7${}_{-0.3}^{+0.3}$ | 1 | M22 |
M1423-C38 |  $\cdots$ | $1.95^{+0.16}_{-0.11}$ | $0.46\pm 0.08$ | $12.44\pm 0.17$ | 8.1${}_{-4.1}^{+4.2}$ | 3 | K22 |
M1423-C50 | 0.545 | $0.81^{+0.01}_{-0.04}$ | $1.81\pm 0.19$ | $11.47\pm 0.03$ | 1.0 | 1 |  $\cdots$ | BCG
R2129-C20 | 0.234 | $0.71^{+0.11}_{-0.03}$ | $0.76\pm 0.08$ | $10.77\pm 0.11$ | 1.0 | 1 |  $\cdots$ | BCG
R2129-C37 |  $\cdots$ | $2.73^{+0.32}_{-0.32}$ | $0.67\pm 0.09$ | $12.76\pm 0.16$ | 2.4${}_{-1.0}^{+1.0}$ | 3 | K22 |
M2129-C24 |  $\cdots$ | $(3.0\pm 2.0)$ | $0.5\pm 0.07$ | $11.48\pm 0.57$ | 41.3${}_{-40.3}^{+53.9}$ | 5 |  $\cdots$ | No counterparts in HST, IRAC, Herschel
M2129-C46 | 1.480 | $1.47^{+0.08}_{-0.03}$ | $0.28\pm 0.06$ | $12.01\pm 0.18$ | 14.9${}_{-8.9}^{+8.9}$ | 1 | T15,S14 |
A3192-C31 |  $\cdots$ | $1.76^{+0.03}_{-0.08}$ | $2.99\pm 0.15$ | $13.56\pm 0.16$ | 2.9${}_{-1.2}^{+1.2}$ | 3 | K22 |
A3192-C40 |  $\cdots$ | $3.43^{+0.16}_{-0.04}$ | $3.36\pm 0.23$ | $13.15\pm 0.13$ | 2.8${}_{-1.1}^{+1.1}$ | 3 | K22 |
A3192-C131 |  $\cdots$ | $1.90^{+0.14}_{-0.14}$ | $1.06\pm 0.16$ | $12.53\pm 0.18$ | 2.6${}_{-1.1}^{+1.1}$ | 4 | App.B | No counterparts in HST
A3192-C138 |  $\cdots$ | $3.46^{+0.37}_{-0.36}$ | $0.75\pm 0.11$ | $13.11\pm 0.08$ | 15.3${}_{-9.5}^{+10.5}$ | 3 | K22 |
M1206-C54 |  $\cdots$ | $2.06^{+0.10}_{-0.26}$ | $0.83\pm 0.08$ | $11.83\pm 0.30$ | 4.7${}_{-1.9}^{+1.9}$ | 3 | K22 |
M1206-C58 | 0.440 | $0.56^{+0.03}_{-0.04}$ | $0.48\pm 0.07$ | $11.19\pm 0.32$ | 1.0 | 1 | BCG |
M1206-C61 | 1.037 |  $\cdots$ | $1.23\pm 0.08$ | $12.45\pm 0.01$ | 69.8${}_{-68.8}^{+78.9}$ | 1 | E09 | multiple images (M1206-C27/55/60/61)
M0429-C4 |  $\cdots$ | $2.51^{+0.43}_{-0.17}$ | $1.40\pm 0.14$ | $11.99\pm 0.55$ | 4.4${}_{-1.8}^{+1.8}$ | 3 | K22 |
M0429-C19 | 0.399 | $0.79^{+0.03}_{-0.01}$ | $2.45\pm 0.38$ | $11.52\pm 0.01$ | 1.0 | 1 | BCG |
M0429-C27 |  $\cdots$ | $3.21^{+0.19}_{-0.87}$ | $1.03\pm 0.12$ | $12.91\pm 0.16$ | 4.1${}_{-1.7}^{+1.7}$ | 3 | K22 |
R2211-C19 |  $\cdots$ | $2.85^{+0.10}_{-0.10}$ | $0.54\pm 0.10$ | $13.03\pm 0.04$ | 2.4${}_{-1.0}^{+1.0}$ | 3 | K22 |
R2211-C35 |  $\cdots$ | $1.89^{+0.99}_{-0.16}$ | $2.81\pm 0.25$ | $13.58\pm 0.19$ | 3.4${}_{-1.4}^{+1.6}$ | 3 | K22 |
R2211-C171 |  $\cdots$ | $1.75^{+0.11}_{-0.07}$ | $1.79\pm 0.31$ | $12.64\pm 0.15$ | 2.3${}_{-0.9}^{+0.9}$ | 3 | K22 |
M1311-C27 | 2.187 | $2.42^{+0.03}_{-0.02}$ | $1.09\pm 0.10$ | $12.77\pm 0.00$ | 36.3${}_{-29.0}^{+29.0}$ | 1 | Ca19 |
M1311-C33 |  $\cdots$ | $1.81^{+0.82}_{-0.01}$ | $0.88\pm 0.15$ | $12.31\pm 0.19$ | 2.6${}_{-1.0}^{+1.2}$ | 3 | K22 |
R0032-C58 | 3.631 |  $\cdots$ | $1.62\pm 0.23$ | $13.34\pm 0.14$ | 9.3${}_{-6.6}^{+6.6}$ | 1 | D17 | multiple images (R0032-C32/53/55/57/58)
R0032-C198 | 2.391 |  $\cdots$ | $1.33\pm 0.14$ | $12.36\pm 0.07$ | 2.2${}_{-1.2}^{+1.3}$ | 1 | App.A | multiple images (R0032-C127/131/198)
R0032-C220 |  $\cdots$ | $1.53^{+0.82}_{-0.16}$ | $0.86\pm 0.09$ | $12.51\pm 0.16$ | 2.2${}_{-0.9}^{+0.9}$ | 3 | K22 |
R0032-C238 |  $\cdots$ | $2.80^{+0.12}_{-0.12}$ | $1.84\pm 0.21$ | $12.62\pm 0.26$ | 1.7${}_{-0.7}^{+0.7}$ | 4 | App.B | Outside of HST/WFC3
R0032-C250 |  $\cdots$ | $2.60^{+0.14}_{-0.14}$ | $0.93\pm 0.10$ | $12.52\pm 0.19$ | 2.2${}_{-0.9}^{+0.9}$ | 4 | App.B | No counterparts in HST
R0032-C276 |  $\cdots$ | $3.05^{+0.15}_{-0.15}$ | $1.09\pm 0.14$ | $13.31\pm 0.04$ | 2.9${}_{-1.2}^{+1.2}$ | 4 | App.B | No counterparts in HST
R0032-C287 |  $\cdots$ | $2.70^{+0.28}_{-0.32}$ | $1.36\pm 0.17$ | $13.43\pm 0.10$ | 1.7${}_{-0.7}^{+0.7}$ | 3 | K22 |
R0032-C304 | 2.984 |  $\cdots$ | $1.04\pm 0.12$ | $12.06\pm 0.11$ | 2.5${}_{-1.4}^{+1.4}$ | 1 | App.A | multiple images (R0032-C127/131/198)
A383-C40 | 0.187 | $0.12^{+0.06}_{-0.01}$ | $2.07\pm 0.13$ | $10.30\pm 0.03$ | 1.0 | 1 | BCG |
A370-C18 |  $\cdots$ | $2.49^{+0.24}_{-0.10}$ | $1.14\pm 0.15$ | $12.53\pm 0.17$ | 2.6${}_{-0.5}^{+0.5}$ | 3 | K22 |
A370-C31 | 0.375 | $0.22^{+0.02}_{-0.01}$ | $0.70\pm 0.09$ | $11.43\pm 0.01$ | 1.0 | 1 | W12, App.A | BCG
A370-C103 |  $\cdots$ | $1.23^{+0.04}_{-0.06}$ | $1.02\pm 0.14$ | $12.24\pm 0.03$ | 1.8${}_{-0.4}^{+0.4}$ | 3 | K22 |
A370-C110 | 1.062 | $1.17^{+0.05}_{-0.03}$ | $2.05\pm 0.16$ | $12.73\pm 0.02$ | 1.2${}_{-0.2}^{+0.2}$ | 1 | S99 |
A370-C146 | 1.070 | $0.96^{+0.09}_{-0.05}$ | $0.28\pm 0.07$ | $11.36\pm 0.08$ | 5.1${}_{-2.6}^{+2.6}$ | 1 | T15,S14 |
A2163-C11 |  $\cdots$ | $1.92^{+0.26}_{-0.42}$ | $0.34\pm 0.07$ | $13.07\pm 0.10$ | 2.8${}_{-1.1}^{+1.1}$ | 3 | K22 |
A2537-C42 |  $\cdots$ | $1.00^{+0.07}_{-0.05}$ | $1.00\pm 0.13$ | $12.29\pm 0.04$ | 3.3${}_{-1.3}^{+1.3}$ | 3 | K22 |
A2537-C49 |  $\cdots$ | $1.69^{+0.22}_{-0.17}$ | $0.79\pm 0.20$ | $13.04\pm 0.17$ | 3.7${}_{-1.5}^{+1.5}$ | 3 | K22 |
M0035-C41 |  $\cdots$ | $2.40^{+0.15}_{-0.15}$ | $0.88\pm 0.15$ | $12.35\pm 0.25$ | 2.0${}_{-0.8}^{+0.8}$ | 4 | App.B | No counterparts in HST
M0035-C68 |  $\cdots$ | $4.04^{+0.21}_{-0.23}$ | $0.99\pm 0.15$ | $12.06\pm 0.35$ | 2.8${}_{-1.1}^{+1.1}$ | 3 | K22 |
M0035-C94 | 0.359 | $0.28^{+0.12}_{-0.05}$ | $0.37\pm 0.06$ | $10.99\pm 0.20$ | 1.0 | 1 | App.A | BCG
AS1063-C15 | 1.427 | $1.35^{+0.13}_{-0.02}$ | $1.42\pm 0.13$ | $12.20\pm 0.02$ | 2.3${}_{-0.5}^{+0.5}$ | 1 | T15,S14 |
AS1063-C17 | 1.438 | $1.55^{+0.11}_{-0.07}$ | $1.37\pm 0.13$ | $12.70\pm 0.16$ | 2.5${}_{-0.5}^{+0.5}$ | 1 | T15,S14 |
AS1063-C147 | 0.609 | $0.63^{+0.10}_{-0.10}$ | $0.78\pm 0.08$ | $12.06\pm 0.04$ | 4.1${}_{-0.8}^{+0.8}$ | 1 | G12 |
AS1063-C222 |  $\cdots$ | $3.97^{+0.18}_{-0.38}$ | $7.73\pm 0.13$ | $13.08\pm 0.05$ | 2.0${}_{-0.4}^{+0.4}$ | 3 | K22 |
Prior Catalog ($N=$39)
AC0102-C128 |  $\cdots$ | $2.90^{+0.09}_{-0.31}$ | $0.49\pm 0.13$ | $12.88\pm 0.06$ | 3.0${}_{-1.2}^{+1.2}$ | 3 | K22 |
P171-C161 |  $\cdots$ | $1.46^{+1.10}_{-0.62}$ | $0.75\pm 0.10$ | $11.78\pm 0.61$ | 2.4${}_{-1.0}^{+1.0}$ | 3 | K22 |
M1115-C33 |  $\cdots$ | $0.69^{+0.05}_{-0.02}$ | $0.69\pm 0.14$ | $10.95\pm 0.17$ | 1.4${}_{-0.4}^{+0.5}$ | 3 | K22 |
SM0723-C98 |  $\cdots$ | $2.01^{+0.10}_{-0.09}$ | $0.45\pm 0.20$ | $11.69\pm 0.36$ | 2.7${}_{-1.1}^{+1.1}$ | 3 | K22 |
M1931-C42 |  $\cdots$ | $0.32^{+0.44}_{-0.00}$ | $0.37\pm 0.06$ | $10.60\pm 0.15$ | 1.0${}_{-0.0}^{+21.9}$ | 3 | K22 |
M1931-C69 |  $\cdots$ | $3.23^{+0.25}_{-0.43}$ | $0.22\pm 0.07$ | $12.57\pm 0.17$ | 3.3${}_{-1.3}^{+1.3}$ | 3 | K22 |
R0949-C14 |  $\cdots$ | $1.45^{+0.14}_{-0.14}$ | $0.23\pm 0.05$ | $11.27\pm 0.62$ | 2.2${}_{-0.9}^{+0.9}$ | 4 | App.B | No counterparts in HST
R0949-C113 |  $\cdots$ | $0.39^{+0.03}_{-0.15}$ | $0.57\pm 0.19$ | $11.19\pm 0.01$ | 1.0${}_{-0.0}^{+0.4}$ | 3 | K22 |
R0949-C119 |  $\cdots$ | $0.56^{+0.08}_{-0.05}$ | $0.46\pm 0.09$ | $11.27\pm 0.16$ | 1.5${}_{-0.5}^{+0.6}$ | 3 | K22 |
M0416-C120 | 2.448 | $9.62^{+0.07}_{-0.06}$ | $0.23\pm 0.08$ | $11.45\pm 0.16$ | 1.7${}_{-0.3}^{+0.3}$ | 1 | App.A |
M0416-C138 |  $\cdots$ | $2.50^{+0.04}_{-0.57}$ | $0.31\pm 0.08$ | $12.31\pm 0.15$ | 1.4${}_{-0.3}^{+0.3}$ | 3 | K22 |
M0416-C156 |  $\cdots$ | $1.58^{+0.10}_{-0.10}$ | $0.25\pm 0.08$ | $12.05\pm 0.15$ | 1.5${}_{-0.3}^{+0.3}$ | 3 | K22 |
AS295-C9 |  $\cdots$ | $2.00^{+0.19}_{-0.19}$ | $0.57\pm 0.13$ | $11.63\pm 0.59$ | 5.0${}_{-2.0}^{+2.0}$ | 4 | App.B | No counterparts in HST
AS295-C269 |  $\cdots$ | $2.75^{+0.36}_{-0.36}$ | $1.97\pm 0.33$ | $12.70\pm 0.24$ | 2.0${}_{-0.8}^{+0.8}$ | 4 | App.B | Outside of HST/WFC3
M1149-C27 |  $\cdots$ | $4.61^{+0.27}_{-0.08}$ | $0.55\pm 0.16$ | $12.87\pm 0.17$ | 1.6${}_{-0.3}^{+0.3}$ | 3 | K22 |
M1149-C95 |  $\cdots$ | $1.53^{+0.02}_{-0.07}$ | $0.75\pm 0.17$ | $11.78\pm 0.04$ | 1.4${}_{-0.3}^{+0.3}$ | 3 | K22 |
A2744-C17 |  $\cdots$ | $2.87^{+0.94}_{-0.04}$ | $0.53\pm 0.14$ | $12.34\pm 0.16$ | 2.1${}_{-0.4}^{+0.4}$ | 3 | K22 |
A2744-C47 |  $\cdots$ | $1.55^{+0.14}_{-0.09}$ | $0.58\pm 0.20$ | $11.92\pm 0.15$ | 2.0${}_{-0.4}^{+0.4}$ | 3 | K22 |
A2744-C176 |  $\cdots$ | $0.44^{+0.02}_{-0.03}$ | $0.62\pm 0.11$ | $10.71\pm 0.06$ | 1.3${}_{-0.3}^{+0.3}$ | 3 | K22 |
A2744-C178 | 0.943 | $1.11^{+0.00}_{-0.10}$ | $0.54\pm 0.09$ | $11.70\pm 0.01$ | 1.9${}_{-0.4}^{+0.4}$ | 1 | W15 |
A2744-C227 | 2.582 | $2.58^{+0.06}_{-0.12}$ | $0.36\pm 0.14$ | $12.80\pm 0.16$ | 1.7${}_{-0.3}^{+0.3}$ | 1 | W15 |
M1423-C52 |  $\cdots$ | $0.97^{+0.21}_{-0.05}$ | $0.23\pm 0.08$ | $11.57\pm 0.16$ | 15.2${}_{-11.7}^{+18.3}$ | 3 | K22 |
M1423-C76 |  $\cdots$ | $4.33^{+0.03}_{-3.32}$ | $0.45\pm 0.08$ | $12.86\pm 0.12$ | 19.0${}_{-18.0}^{+11.4}$ | 3 | K22 |
M2129-C62 | 1.480 | $1.45^{+0.09}_{-0.07}$ | $0.27\pm 0.06$ | $12.03\pm 0.17$ | 7.4${}_{-3.7}^{+3.7}$ | 1 | T15,S14 |
A3192-C83 |  $\cdots$ | $0.63^{+0.03}_{-0.04}$ | $0.34\pm 0.09$ | $11.24\pm 0.38$ | 0.2${}_{-0.7}^{+0.1}$ | 3 | K22 |
A3192-C154 |  $\cdots$ | $3.33^{+0.03}_{-0.87}$ | $0.83\pm 0.19$ | $12.41\pm 0.19$ | 1.7${}_{-0.7}^{+0.7}$ | 3 | K22 |
M1206-C38 | 1.488 | $1.15^{+0.11}_{-0.01}$ | $0.27\pm 0.09$ | $11.93\pm 0.19$ | 3.2${}_{-1.3}^{+1.3}$ | 1 | B13 |
M1206-C84 | 0.476 | $0.51^{+0.04}_{-0.03}$ | $0.40\pm 0.07$ | $11.29\pm 0.02$ | 1.1${}_{-0.1}^{+0.5}$ | 1 | B13 |
R2211-C164 |  $\cdots$ | $4.92^{+0.18}_{-0.09}$ | $0.38\pm 0.09$ | $11.38\pm 0.52$ | 2.0${}_{-0.8}^{+0.8}$ | 3 | K22 |
R0032-C63 |  $\cdots$ | $3.0^{+0.5}_{-0.5}$ | $0.33\pm 0.09$ | $11.36\pm 0.55$ | 3.3${}_{-1.3}^{+1.3}$ | 2 | A20 |  $z_{\rm phot}$ \+ Lens model
R0032-C162 |  $\cdots$ | $0.41^{+0.08}_{-0.06}$ | $0.39\pm 0.09$ | $11.06\pm 0.24$ | 1.3${}_{-0.3}^{+8.4}$ | 3 | K22 |
R0032-C245 |  $\cdots$ | $1.50^{+0.20}_{-0.20}$ | $0.36\pm 0.09$ | $12.75\pm 0.05$ | 15.3${}_{-14.3}^{+70.7}$ | 5 |  $\cdots$ | No counterparts in HST
A383-C24 |  $\cdots$ | $0.70^{+0.04}_{-0.02}$ | $0.33\pm 0.09$ | $11.73\pm 0.05$ | 4.8${}_{-1.9}^{+1.9}$ | 3 | K22 |
A383-C50 | 1.010 | $0.90^{+0.14}_{-0.02}$ | $0.36\pm 0.09$ | $11.72\pm 0.15$ | 53.1${}_{-42.5}^{+42.5}$ | 1 | S04 |
A370-C27 |  $\cdots$ | $3.26^{+0.13}_{-0.03}$ | $0.57\pm 0.09$ | $12.40\pm 0.18$ | 13.7${}_{-8.2}^{+8.2}$ | 3 | K22 |
A370-C172 |  $\cdots$ | $0.24^{+0.03}_{-0.03}$ | $0.57\pm 0.19$ | $10.90\pm 0.00$ | 1.0${}_{-0.0}^{+0.0}$ | 3 | K22 |
A2537-C6 |  $\cdots$ | $0.45^{+0.04}_{-0.12}$ | $0.29\pm 0.09$ | $10.92\pm 0.00$ | 1.5${}_{-0.5}^{+0.6}$ | 3 | K22 |
A2537-C24 |  $\cdots$ | $3.2^{+0.5}_{-0.5}$ | $0.40\pm 0.08$ | $13.37\pm 0.00$ | 20.5${}_{-12.9}^{+12.9}$ | 2 | Ce18 |  $z_{\rm phot}$ \+ Lens model
M0035-C33 |  $\cdots$ | $2.0^{+0.5}_{-0.5}$ | $0.18\pm 0.06$ | $12.81\pm 0.03$ | 3.8${}_{-1.6}^{+1.9}$ | 2 |  $\cdots$ |  $z_{\rm phot}$ \+ Lens model

Note. — (1) Source ID. (2) Spectroscopic redshift (Section 3.5). (3)
Photometric redshift (Section 3.5). (4) Observed flux density, including the
primary beam correction and systematic offsets through the MC simulations
(Section 3.3), where the uncertainties are propagated. For the multiple-image
system, we show an observed flux density calculated by multiplying the average
values of the intrinsic flux densities and magnifications. (5) Magnification
factor based on our fidcual lens model, where the error is evaluated from the
propagation of the redshift uncertainty and the systematic uncertainty from
different lens models (Section 3.6). (6) Redshift category. 1: $z_{\rm spec}$,
2: $z_{\rm phot}$ \+ multiple images used in the construction of the lens
model, 3: $z_{\rm phot}$ with eazy, 4: $z_{\rm phot}$ with the composite SED
from 707 dusty galaxies in AS2UDS for the sources without HST
counterparts/data (Appendix B), 5: $z=3.0\pm 2.0$ assumed for four sources
with no counterparts in all bands other than ALMA). (7): Reference: B13
(Biviano et al. 2013), C21 (Caputi et al. 2021), Ca19 (Caminha et al. 2019),
Ce18 (Cerny et al. 2018), D17 (Dessauges-Zavadsky et al. 2017), E09 (Ebeling
et al. 2009), E17 (Ebeling et al. 2017), F21 (Fujimoto et al. 2021), G12
(Gómez et al. 2012), K22 (Kokorev et al. 2022), L21 (Laporte et al. 2021) R21
(Richard et al. 2021), MA22 (Muñoz Arancibia et al. 2022), S04 (Sand et al.
2004), S14 (Schmidt et al. 2014), S99 (Soucail et al. 1999), T15 (Treu et al.
2015), W15 (Wang et al. 2015), W12 (Wold et al. 2012),

In Table 12 and 13, we summarize the observed flux density, source redshift,
and the magnification factor for the ALCS 180 sources.

In Figure 26, we also summarize the image cutouts of ALMA/Band6, HST/F814W,
HST/F160W, and IRAC/ch1 for the ALCS 180 sources.

Figure 25:  Same as Figure 1, but for the remaining 32 lensing clusters and
the sky positions of the ALCS sources. ALMA Band 6, HST/F160W, and IRAC/ch1
maps are presented from left to right. The green solid and dashed circles show
the ALCS sources in the primary and secondary samples, respectively. The white
lines denote the magnification curve of $\mu=200$ at $z=2$ with our fiducial
lens model. The solid and dashed cyan lines indicate the PB sensitivity at 50%
and 30%, respectively.

Figure 25 (continued)

Figure 25 (continued)

Figure 25 (continued)

Figure 25 (continued)

Figure 25 (continued)

Figure 25 (continued)

Figure 25 (continued)

Figure 26:  $10^{\prime\prime}\times 10^{\prime\prime}$ image cutouts – ALMA
1.2-mm (natural-weighted), ALMA 1.2-mm ($uv$-tapered, HST/F814W, HST/F160W,
and IRAC/ch1 from left to right – for 141 ALCS sources in the blind catalog.
The contours indicate the intensity at 3$\sigma$, 4$\sigma$, and 5$\sigma$
levels.

Figure 26 (continued)

Figure 26 (continued)

Figure 26 (continued)

Figure 26 (continued)

Figure 26 (continued)

Figure 27:  Same as Figure 26, but for 39 ALCS sources in the prior catalog.

Figure 27 (continued)

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