<table><caption>**Table 1.** Number of lensed stars expected with and without imposing the HD limit.</caption><thead><tr><th>Filter</th><th>Depth</th><th>no HD<sub>δ=0</sub></th><th>HD<sub>δ=0</sub></th><th>HD<sub>δ=1</sub></th><th>HD<sub>δ=-1</sub></th></tr></thead><tbody><tr><td>F814W</td><td>27.5</td><td>2.1</td><td>0.18</td><td>0.15</td><td>0.13</td></tr><tr><td>F814W</td><td>28.5</td><td>4.6</td><td>0.9</td><td>0.8</td><td>0.6</td></tr><tr><td>F200LP</td><td>29</td><td>5.6</td><td>1.0</td><td>1.2</td><td>0.5</td></tr><tr><td>F200W</td><td>29</td><td>17.5</td><td>4.6</td><td>1.5</td><td>10.5</td></tr></tbody></table>

**Notes.** Numbers represent the total of the temperature–magnification diagrams in Figs. 15, 16, and 18. The slope $\delta$ in Eq. (9) controls the relative abundance of red SGs vs. blue SGs. F814W and F200LP are HST filters, while F200W is a JWST filter.

<figure><img src="image_19.png" /><figcaption>**Fig. 17.** Cumulative number of detected stars in JWST’s F200W filter, at depth 29 mag, and for stars with $T = 3500$ K. SG stars with this temperature are often less massive than implied by the abscissa. The colors and line styles are the same as in Fig. 11. The HD limit is marked by HD and a red vertical dotted line in the plot. Cool SG stars beyond the HD limit are not observed in our local universe, but if they exist in larger numbers at $z \approx 1$, JWST has the best chance to see them in regions of the Spock arc with magnification factors $\mu \approx 100$ (light-blue and yellow curves). The black dotted line represents the differential function $dN/dM$ of the black solid curve (corresponding to the largest magnification factor, $\mu \approx 1000$), and it shows a turnover at $\sim 20 M_\odot$.</figcaption></figure>

## 8.2. Predictions for JWST

It is natural to expect the number of discovered lensed stars at cosmological distances will increase with JWST observations. Owing to its sensitivity to longer wavelengths, JWST will extend the detections to lower-temperature stars, making possible the detection of red SGs with temperatures $T \approx 3500$ K. The first such example is Quyllur, a red SG at $z = 2.1878$ setting the record for the first SG star detected by JWST beyond $z = 1$ (Diego et al. 2023). Other lensed stars have been reported by JWST in recent months (Chen et al. 2022; Mahler et al. 2023; Pascale et al. 2022; Meena et al. 2023b).

First we consider an observation at 29 AB mag depth. We also consider a cool red SG star, which is more likely to be detected in the IR by JWST. The expected number of detections with JWST in the Spock arc is shown in Fig. 17, which corresponds to the F200W NIRCam filter and assumes one can detect stars above AB magnitude 29 in this filter. In this case, JWST will be detecting mostly red SGs, and preferably in lower-magnification regions. At $z \approx 1$, the peak of the emission of a star with $T = 3500$ K falls in the F200W filter, so naturally this filter selects those stars at that redshift.

m<sub>AB</sub>=29 @ F200W (JWST)

<figure><img src="image_20.png" /><figcaption>**Fig. 18.** Similar to Fig. 15 but for JWST’s F200W filter and limiting AB magnitude 29.</figcaption></figure>

One interesting result from Fig. 17 is that the differential number of detections in the bin with the largest magnification (dotted black curve) starts to drop with mass following the shape of the IMF. This phenomenon takes place when the minimum magnification allowed by microlensing fluctuations times the luminosity of the star is above $L_{min}$. In this case, all stars more luminous than this will be detected and the number of detections traces directly the IMF. Even deeper observations with JWST will repeat the same phenomenon at larger distances from the critical curve, allowing one to probe different portions of the IMF at different separations from the critical curve. Hence, JWST will detect any red SG brighter than $\sim 10^5 L_\odot$ that is closer than 0.1 pc from the caustic. This corresponds to an area of $\sim 50 \text{ pc}^2$ in the Spock galaxy; hence, a lack of detection of SGs in this regime implies a density $\rho_\mu < 0.02 \text{ pc}^{-2}$.

When considering the dependence with temperature, as shown in Fig. 18, we predict a substantial number of red SGs to be detected by JWST at moderate magnification factors $\mu \approx 67$. These would be spread along the Spock arc as a population of unresolved sources, and show occasional fluctuations in flux as they move across the network of microcaustics. However, as in previous cases, most of the predicted detections in this bin lie above the HD limit (see Fig. 17), so if these stars do not exist, the number of events in the reddest bins should be reduced accordingly. Lack of a significant number of detections by JWST, or relatively low numbers of red stars spread over the arc, would set a limit on the maximum luminosity of red SG stars at $z \approx 1$, similar to the HD limit established at $z \approx 0$. On the contrary, a significant number of stars detected at relatively low magnification factors would imply that red SGs can be intrinsically more luminous at $z \approx 1$ than at $z \approx 0$.

## 9. Discussion

In Sect. 5 we presented the expected number of lensed galaxies at $z = 6$ and $z = 9$. Recent work based on JWST on cluster fields
<table><caption>**Table 1.** Number of lensed stars expected with and without imposing the HD limit.</caption><thead><tr><th>Filter</th><th>Depth</th><th>no HD<sub>δ=0</sub></th><th>HD<sub>δ=0</sub></th><th>HD<sub>δ=1</sub></th><th>HD<sub>δ=-1</sub></th></tr></thead><tbody><tr><td>F814W</td><td>27.5</td><td>2.1</td><td>0.18</td><td>0.15</td><td>0.13</td></tr><tr><td>F814W</td><td>28.5</td><td>4.6</td><td>0.9</td><td>0.8</td><td>0.6</td></tr><tr><td>F200LP</td><td>29</td><td>5.6</td><td>1.0</td><td>1.2</td><td>0.5</td></tr><tr><td>F200W</td><td>29</td><td>17.5</td><td>4.6</td><td>1.5</td><td>10.5</td></tr></tbody></table>

**Notes.** Numbers represent the total of the temperature–magnification diagrams in Figs. 15, 16, and 18. The slope $\delta$ in Eq. (9) controls the relative abundance of red SGs vs. blue SGs. F814W and F200LP are HST filters, while F200W is a JWST filter.

<figure><img src="image_18.png" /><figcaption>**Fig. 17.** Cumulative number of detected stars in JWST’s F200W filter, at depth 29 mag, and for stars with $T = 3500$ K. SG stars with this temperature are often less massive than implied by the abscissa. The colors and line styles are the same as in Fig. 11. The HD limit is marked by HD and a red vertical dotted line in the plot. Cool SG stars beyond the HD limit are not observed in our local universe, but if they exist in larger numbers at $z \approx 1$, JWST has the best chance to see them in regions of the Spock arc with magnification factors $\mu \approx 100$ (light-blue and yellow curves). The black dotted line represents the differential function $dN/dM$ of the black solid curve (corresponding to the largest magnification factor, $\mu \approx 1000$), and it shows a turnover at $\sim 20 M_\odot$.</figcaption></figure>

## 8.2. Predictions for JWST

It is natural to expect the number of discovered lensed stars at cosmological distances will increase with JWST observations. Owing to its sensitivity to longer wavelengths, JWST will extend the detections to lower-temperature stars, making possible the detection of red SGs with temperatures $T \approx 3500$ K. The first such example is Quyllur, a red SG at $z = 2.1878$ setting the record for the first SG star detected by JWST beyond $z = 1$ (Diego et al. 2023). Other lensed stars have been reported by JWST in recent months (Chen et al. 2022; Mahler et al. 2023; Pascale et al. 2022; Meena et al. 2023b).

First we consider an observation at 29 AB mag depth. We also consider a cool red SG star, which is more likely to be detected in the IR by JWST. The expected number of detections with JWST in the Spock arc is shown in Fig. 17, which corresponds to the F200W NIRCam filter and assumes one can detect stars above AB magnitude 29 in this filter. In this case, JWST will be detecting mostly red SGs, and preferably in lower-magnification regions. At $z \approx 1$, the peak of the emission of a star with $T = 3500$ K falls in the F200W filter, so naturally this filter selects those stars at that redshift.

$m_{AB}=29$ @ F200W (JWST)

<figure><img src="image_19.png" /><figcaption>**Fig. 18.** Similar to Fig. 15 but for JWST’s F200W filter and limiting AB magnitude 29.</figcaption></figure>

One interesting result from Fig. 17 is that the differential number of detections in the bin with the largest magnification (dotted black curve) starts to drop with mass following the shape of the IMF. This phenomenon takes place when the minimum magnification allowed by microlensing fluctuations times the luminosity of the star is above $L_{min}$. In this case, all stars more luminous than this will be detected and the number of detections traces directly the IMF. Even deeper observations with JWST will repeat the same phenomenon at larger distances from the critical curve, allowing one to probe different portions of the IMF at different separations from the critical curve. Hence, JWST will detect any red SG brighter than $\sim 10^5 L_\odot$ that is closer than 0.1 pc from the caustic. This corresponds to an area of $\sim 50 \text{ pc}^2$ in the Spock galaxy; hence, a lack of detection of SGs in this regime implies a density $\rho_\mu < 0.02 \text{ pc}^{-2}$.

When considering the dependence with temperature, as shown in Fig. 18, we predict a substantial number of red SGs to be detected by JWST at moderate magnification factors $\mu \approx 67$. These would be spread along the Spock arc as a population of unresolved sources, and show occasional fluctuations in flux as they move across the network of microcaustics. However, as in previous cases, most of the predicted detections in this bin lie above the HD limit (see Fig. 17), so if these stars do not exist, the number of events in the reddest bins should be reduced accordingly. Lack of a significant number of detections by JWST, or relatively low numbers of red stars spread over the arc, would set a limit on the maximum luminosity of red SG stars at $z \approx 1$, similar to the HD limit established at $z \approx 0$. On the contrary, a significant number of stars detected at relatively low magnification factors would imply that red SGs can be intrinsically more luminous at $z \approx 1$ than at $z \approx 0$.

## 9. Discussion

In Sect. 5 we presented the expected number of lensed galaxies at $z = 6$ and $z = 9$. Recent work based on JWST on cluster fields
appear to show a deficit of galaxies beyond redshift 5 when com-
pared with parallel or blank fields (Bhatawdekar et al., in prep.).
The deficit at z = 6 can be easily understood as a depletion effect
owing to the shallower than 2 slope of the luminosity function
at the faint end. We predict that a similar depletion effect must
exist for z = 6 galaxies multiply lensed by MACS0416. How-
ever, for z = 9 (and probably also for redshifts beyond this one),
we predict that ~3 times more galaxies should be found in the
cluster field than in blank fields (ignoring cosmic variance). This
prediction is based on recently constrained slopes for the lumi-
nosity function that find this slope to be steeper than α = 2. If
the slope is instead shallower than α = 2, a depletion effect, sim-
ilar to the one at z = 6, should be taking place. Simply counting
z > 9 galaxies around clusters, and comparing with the census
of similar galaxies in blank fields, is a simple way of probing
the slope of the faint end of the luminosity function. The ratio
N_cluster/N_field can then be directly related to the magnification,
but also to the slope α (Umetsu et al. 2015; Umetsu 2020).

Regarding the microlensing events identified in the Spock galaxy, our findings are consistent with observations only if the HD limit is still valid at *z* = 1. Assuming this limit is valid at this redshift, and accounting for different temperatures, we expect that ~0.5 lensed stars should be detectable per pointing with HST's F814W filter that reaches 28.5 mag, as shown in Table 1. Focusing on the SE event (S1) reported by Rodney et al. (2018), and on the deeper observations from their FrontierSN program, their observations cover a period of ~30 days, with a very good cadence of ~1 observation per day except for two blind periods of ~3 and ~9 days for which there are no observations. In this period one event is reported with a duration of ~4 days, although both the beginning and end of the bright phase of the event are missed since they fall in the blind periods mentioned above. We adopt 6 days as a conservative duration for the event (above the detection threshold) – that is, one day before and one day after the observed window. This is at the low end of the estimated duration of these events, as discussed at the end of Sect. 7. Hence, the success rate of events per pointing near a critical curve is 6/30 = 0.2. This is only a factor of a few below our expected number of 0.9 events per pointing, and is easily accommodated by small changes (consistent with the photometry) in the IMF derived from the SED model (black line in Fig. 9), or by parsec-scale variations in the distribution of SG stars in this portion of the arc. Hence, the predicted rate of 0.9 events per observation is reasonably consistent with the observed rate of 0.2 stars detected per pointing in the FrontierSN program. We discuss later how the predicted rate matches better the observed one when modifying the *B/R* SG ratio. Similarly, the F1 detection in the Flashlights program corresponds to a success rate of 1/2 = 0.5 (two observations are carried out for each cluster in the Flashlights program, and F1 was seen in only one of them), which is perfectly consistent with our estimate of 0.5 lensed stars per observation in the *F200LP* filter. Our findings are also consistent with the fact that S1 and F1 are found in regions where the macromodel predicts a magnification *μ* ≈ 150–300 for each counterimage, and the hypothesis that S1 and F1 are counterimages of one another, but are detected when each one is experiencing a microlensing event at different epochs.

As discussed above, the two events S2 and F2 can be eas-
ily interpreted in a similar way, but only if a second critical
curve passes near S2 and F2 (see, for instance, the critical curve
from the model of Bergamini et al. (2023) in Fig. 7, that predicts
the critical curve passing closer to S2 and F2). Our current lens
model does not produce such a second critical curve, but small
variations in the lens model should easily produce the needed

critical curve (as already shown in Rodney et al. 2018). If no
critical curve passes near S2 and F2, then these stars must be
incredibly luminous in order to be detected at such low magni-
fication factors. Their luminosities would imply the existence of
stars beyond the HD limit and would create an internal incon-
sistency with our own predictions on S1 and F1, since if the
HD limit evolves at this redshift, we would expect significantly
more lensed stars than detected near S1 and F1. We consider this
possibility unlikely and instead blame the lower accuracy of our
lens model in this portion of the arc. Future improved lens mod-
els are needed in order to better constrain that second possible
critical curve crossing. JWST data will provide further lensing
constraints for the lens model.

In summary, when comparing our results with observations,
our predictions exceed the number of observed lensed stars if
we do not set an upper limit to the luminosity of stars with tem-
peratures T < 15 000 K. On the other hand, our predictions are
in reasonable agreement with the observations if one considers
that SG stars at z = 1 are also less luminous than the HD limit
established at z ≈ 0. Hence, our results are consistent with the
hypothesis that the HD limit does not evolve between z = 0 and
z = 1, and that the abundance of SG stars at z = 1 is consistent
with local measurements.

In the results discussed up to this point, we have assumed
δ = 0 in Eq. (9). This can be considered as a strong assumption
since it implicitly assumes that the ratio of blue SGs to red SGs
( of the same luminosity) is equal to 1. For old galaxies blue SGs
are expected to be very rare, while red SGs represent the evolved
state of less massive (and longer lived) intermediate-mass MS
stars. In this sense, we can consider δ as a parameterization of
the age of the galaxy. From the SED fit to the observed photom-
etry, and the blue color of the galaxy, we infer that the Spock
galaxy is relatively young (see Sect. 6.3), or it has sufficient star
formation to support the existence of blue SGs. So our choice
of δ = 0 is, in principle, well motivated. If blue SGs are more
common than red SGs, then δ > 0. For values of δ = 1, the
numbers for each temperature bin in Figs. 15, 16, and 18 need
to be weighted not by a factor 1/5 as done earlier, but by a factor
W = T^δ / ∫ T^δ = 0.04, 0.13, 0.19, 0.25, and 0.38 for the tem-
perature bins T = 3500 K, 10 000 K, 15 000 K, 20 000 K, and
30 000 K, respectively. While for δ = -1 the weights are 0.53,
0.19, 0.13, 0.09, and 0.06 for the same respective bins. Obvi-
ously, W = 0.2 for all temperatures when δ = 0. The weights
above assume all luminous stars are distributed in the five tem-
perature bins. This is a simplification since in a real situation
there may be stars with higher temperatures. In this work we are
assuming the number of stars above 30 000 K is negligible. This
approximation is appropriate if the *B/R* ratio is small (such as in
the *δ* = -1 case), but not so much if the *B/R* > 1.

For δ = 1, and after imposing the HD limit, the number of
predicted detections remains more or less constant in the F814W
filter with respect to the case δ = 0, but it increases in the bluer
HST's F200LP filter in Table 1. For the IR JWST filter F200W,
we observe a significant reduction. This filter is well suited for
cool stars at z = 1, hence is very sensitive to their abundance.
For δ = -1 (more red SGs than blue SGs), the situation reverses,
and we predict a large number of detections in F200W. The cur-
rent statistics based on HST data do not allow us to discriminate
between different values of δ, but future observations with JWST
should allow us to favor one scenario over the rest.

Interestingly, the results discussed above are to first order
insensitive to changes in the micromodel. As shown in Fig. 10,
considering a smaller value of Σ\* has a minor impact on the prob-
ability of high-magnification events. The smaller Σ\* = 5 M⊙ pc⁻²
appear to show a deficit of galaxies beyond redshift 5 when com-
pared with parallel or blank fields (Bhatawdekar et al., in prep.).
The deficit at z = 6 can be easily understood as a depletion effect
owing to the shallower than 2 slope of the luminosity function
at the faint end. We predict that a similar depletion effect must
exist for z = 6 galaxies multiply lensed by MACS0416. How-
ever, for z = 9 (and probably also for redshifts beyond this one),
we predict that ~3 times more galaxies should be found in the
cluster field than in blank fields (ignoring cosmic variance). This
prediction is based on recently constrained slopes for the lumi-
nosity function that find this slope to be steeper than α = 2. If
the slope is instead shallower than α = 2, a depletion effect, sim-
ilar to the one at z = 6, should be taking place. Simply counting
z > 9 galaxies around clusters, and comparing with the census
of similar galaxies in blank fields, is a simple way of probing
the slope of the faint end of the luminosity function. The ratio
N_cluster/N_field can then be directly related to the magnification,
but also to the slope α (Umetsu et al. 2015; Umetsu 2020).

Regarding the microlensing events identified in the Spock galaxy, our findings are consistent with observations only if the HD limit is still valid at *z* = 1. Assuming this limit is valid at this redshift, and accounting for different temperatures, we expect that ~0.5 lensed stars should be detectable per pointing with HST's F814W filter that reaches 28.5 mag, as shown in Table 1. Focusing on the SE event (S1) reported by Rodney et al. (2018), and on the deeper observations from their FrontierSN program, their observations cover a period of ~30 days, with a very good cadence of ~1 observation per day except for two blind periods of ~3 and ~9 days for which there are no observations. In this period one event is reported with a duration of ~4 days, although both the beginning and end of the bright phase of the event are missed since they fall in the blind periods mentioned above. We adopt 6 days as a conservative duration for the event (above the detection threshold) – that is, one day before and one day after the observed window. This is at the low end of the estimated duration of these events, as discussed at the end of Sect. 7. Hence, the success rate of events per pointing near a critical curve is 6/30 = 0.2. This is only a factor of a few below our expected number of 0.9 events per pointing, and is easily accommodated by small changes (consistent with the photometry) in the IMF derived from the SED model (black line in Fig. 9), or by parsec-scale variations in the distribution of SG stars in this portion of the arc. Hence, the predicted rate of 0.9 events per observation is reasonably consistent with the observed rate of 0.2 stars detected per pointing in the FrontierSN program. We discuss later how the predicted rate matches better the observed one when modifying the *B/R* SG ratio. Similarly, the F1 detection in the Flashlights program corresponds to a success rate of 1/2 = 0.5 (two observations are carried out for each cluster in the Flashlights program, and F1 was seen in only one of them), which is perfectly consistent with our estimate of 0.5 lensed stars per observation in the *F200LP* filter. Our findings are also consistent with the fact that S1 and F1 are found in regions where the macromodel predicts a magnification *μ* ≈ 150–300 for each counterimage, and the hypothesis that S1 and F1 are counterimages of one another, but are detected when each one is experiencing a microlensing event at different epochs.

As discussed above, the two events S2 and F2 can be eas-
ily interpreted in a similar way, but only if a second critical
curve passes near S2 and F2 (see, for instance, the critical curve
from the model of Bergamini et al. (2023) in Fig. 7, that predicts
the critical curve passing closer to S2 and F2). Our current lens
model does not produce such a second critical curve, but small
variations in the lens model should easily produce the needed

critical curve (as already shown in Rodney et al. 2018). If no
critical curve passes near S2 and F2, then these stars must be
incredibly luminous in order to be detected at such low magni-
fication factors. Their luminosities would imply the existence of
stars beyond the HD limit and would create an internal incon-
sistency with our own predictions on S1 and F1, since if the
HD limit evolves at this redshift, we would expect significantly
more lensed stars than detected near S1 and F1. We consider this
possibility unlikely and instead blame the lower accuracy of our
lens model in this portion of the arc. Future improved lens mod-
els are needed in order to better constrain that second possible
critical curve crossing. JWST data will provide further lensing
constraints for the lens model.

In summary, when comparing our results with observations,
our predictions exceed the number of observed lensed stars if
we do not set an upper limit to the luminosity of stars with tem-
peratures T < 15 000 K. On the other hand, our predictions are
in reasonable agreement with the observations if one considers
that SG stars at z = 1 are also less luminous than the HD limit
established at z ≈ 0. Hence, our results are consistent with the
hypothesis that the HD limit does not evolve between z = 0 and
z = 1, and that the abundance of SG stars at z = 1 is consistent
with local measurements.

In the results discussed up to this point, we have assumed
δ = 0 in Eq. (9). This can be considered as a strong assumption
since it implicitly assumes that the ratio of blue SGs to red SGs
(or the same luminosity) is equal to 1. For old galaxies blue SGs
are expected to be very rare, while red SGs represent the evolved
state of less massive (and longer lived) intermediate-mass MS
stars. In this sense, we can consider δ as a parameterization of
the age of the galaxy. From the SED fit to the observed photom-
etry, and the blue color of the galaxy, we infer that the Spock
galaxy is relatively young (see Sect. 6.3), or it has sufficient star
formation to support the existence of blue SGs. So our choice
of δ = 0 is, in principle, well motivated. If blue SGs are more
common than red SGs, then δ > 0. For values of δ = 1, the
numbers for each temperature bin in Figs. 15, 16, and 18 need
to be weighted not by a factor 1/5 as done earlier, but by a factor
W = T^δ / ∫ T^δ = 0.04, 0.13, 0.19, 0.25, and 0.38 for the tem-
perature bins T = 3500 K, 10 000 K, 15 000 K, 20 000 K, and
30 000 K, respectively. While for δ = -1 the weights are 0.53,
0.19, 0.13, 0.09, and 0.06 for the same respective bins. Obvi-
ously, W = 0.2 for all temperatures when δ = 0. The weights
above assume all luminous stars are distributed in the five tem-
perature bins. This is a simplification since in a real situation
there may be stars with higher temperatures. In this work we are
assuming the number of stars above 30 000 K is negligible. This
approximation is appropriate if the *B/R* ratio is small (such as in
the *δ* = -1 case), but not so much if the *B/R* > 1.

For δ = 1, and after imposing the HD limit, the number of
predicted detections remains more or less constant in the F814W
filter with respect to the case δ = 0, but it increases in the bluer
HST's F200LP filter in Table 1. For the IR JWST filter F200W,
we observe a significant reduction. This filter is well suited for
cool stars at z = 1, hence is very sensitive to their abundance.
For δ = -1 (more red SGs than blue SGs), the situation reverses,
and we predict a large number of detections in F200W. The cur-
rent statistics based on HST data do not allow us to discriminate
between different values of δ, but future observations with JWST
should allow us to favor one scenario over the rest.

Interestingly, the results discussed above are to first order
insensitive to changes in the micromodel. As shown in Fig. 10,
considering a smaller value of Σ\* has a minor impact on the prob-
ability of high-magnification events. The smaller Σ\* = 5 M⊙ pc⁻²
results in a narrower PDF for the magnification, but its median value peaks at larger magnifications, and the tail of extreme magnification still follows the canonical $\mu^{-2}$ power law past $\mu = 1000$. These two effects compensate the smaller abundance of microlenses, especially at large magnification values.

These results are derived without taking into account specific models of stellar evolution. In particular, the relatively short lifetimes of SG stars are not being considered in this work. Instead, we have assumed that these stars must exist in the Spock galaxy to be able to interpret the observations. Our results show that the abundance of these stars is well approximated by the abundance predicted by the IMF of stars with the corresponding mass to produce such luminosities (assuming the MS $M-L$ relation). This reasoning can be turned around to conclude that in order to have such an abundance of luminous stars, a relatively recent episode of star formation must have taken place, which is consistent with the young age derived for the Spock galaxy from SED fitting.

The results derived above assume there are no stars beyond the HD limit. Under that assumption, we find that our results are roughly consistent with the observations (that is, ~0.5 observed stars per pointing). Given the relatively small area that is being amplified in the Spock arc, we also consider the possibility that stars beyond the HD limit may exist, but their surface number density is smaller than what was assumed in our previous calculations. Otherwise these stars would have been detected by the FrontierSN or Flashlights programs, preferentially at larger distances from the critical curve. Since the observed stars appear near the expected position of the critical curves (true at least for S1 and F1, and likely true also for S2 and F2, assuming a second critical curve passes between them), we can conclude that if stars with luminosities beyond the HD limit exist in this small portion of the Spock galaxy, their number density must be smaller than what was assumed in our calculations above.

The strongest constraint on the abundance of stars beyond the HD limit comes from the regions with smallest magnification, as shown in Figs. 12 and 14. Our model assumed a surface number density of ~8000 stars kpc$^{-2}$ with luminosities beyond $6 \times 10^5 L_{\odot}$. From the results in Table 1, and adopting the case $\delta = 0$, the number of expected observed stars per pointing, and beyond the HD limit is simply the difference between Column 3 (labeled “No HD$_{\delta=0}$”), and Cols. 4–6 – that is, ~3.5 for the FrontierSN program or ~4.5 for the Flashlights program. These numbers need to be reduced by approximately a factor 20 for these stars to be rare enough to not contribute significantly to the number of detections farther away from the critical curve. This translates into a surface number density of <400 stars kpc$^{-2}$ with luminosities beyond the HD limit. If we consider the bins with magnifications 67 and 135, where these very bright stars are more likely to be found, the area in the source plane with magnification in this range is $A \approx 530 \times 20 \approx 0.01$ kpc$^2$. This means the number of stars beyond the HD limit in this region can be up to ~4. These stars remain in general undetected without the aid of microlensing provided they are less luminous than $L_{bol} < 3 \times 10^7 L_{\odot}$, as estimated earlier. Similar arguments can be used, but this time applied to the smaller area associated with the larger-magnification bins, and assuming that the stars responsible for the observed events are less luminous than the HD limit. Our model contains ~9000 stars kpc$^{-2}$ with bolometric luminosities between ~$10^5 L_{\odot}$ and ~$6 \times 10^5 L_{\odot}$. As discussed above, this number density (equivalent to ~1 SG star per $10 \times 10 \text{ pc}^2$) reproduces the observations from the FrontierSN and Flashlights programs reasonably well. Although high, this number density is consistent with the fact that SG stars are often found in large concentrations of stars (Davies et al. 2007).

The abundance of SG stars in the Spock galaxy can be better constrained with future observations of this arc. If we consider the farthest bin in Fig. 11 at $d \approx 16$ pc from the caustic (yellow bin, or total magnification $\mu \approx 70$, equivalent to regions in the image plane where $\mu \approx 35$ for each counterimage), this bin is highly sensitive to the presence of the most luminous stars (as shown more clearly in Fig. 12). By simply mapping how far from the critical curve lensed stars are observed, one can constrain the abundance of the most massive and luminous stars, since at relatively large distances from the caustic, only the most luminous stars are detectable.

Future JWST observations will improve the statistics on the number of detected lensed stars, offering us a privileged view into the massive end of the IMF of the Spock galaxy. Once enough SG stars are detected by JWST, an interesting application is to reverse engineer the tip of the red giant branch (TRGB) distance indicator and use it instead to constrain the magnification. If the TRGB holds at this redshift, we can use the observed luminosities of the brightest red stars to infer their true un lensed luminosity. This gives a direct constraint on the magnification which can be used to improve on the lens models. Current models typically disagree by a factor of 2–3 in the magnification near the critical curves. If the maximum bolometric luminosity for red SGs is still ~$6 \times 10^5 L_{\odot}$, JWST will be able to see these bright stars and use them as distance indicators to constrain $\mu$. There is of course the issue of microlensing that may introduce uncertainty in the magnification, but monitoring these stars should provide accurate measurements of the mode of the magnification. These stars can also show variability, although this is expected to be different than the $1/\sqrt{t}$ variability predicted by lensing (where $t$ is time to the maximum magnification). Frequent monitoring of these stars should easily distinguish between intrinsic variability and a microlensing event. Since JWST will be able to see red SGs farther away from the critical curve, at these separations the mode of the magnification is a good tracer of the macromodel magnification at that particular position.

Another interesting property that can be checked with future observations is to estimate the presence of luminous blue companions to red SGs. From local measurements, ~20% of the red SGs are expected to have such companions (Neugent et al. 2020a). Monitoring of microcaustic events can unambiguously reveal companion stars provided that this companion is bright enough so it can be observed when the companion crosses a microcaustic. These events would show distinctive chromatic features since the stars in the pair would have different temperatures (and colors) which manifest themselves when each star is crossing a microcaustic.

# 10. Conclusions

We have presented an updated model of the cluster MACS0416, based on the hybrid algorithm WSLAP+, taking advantage of new weak-lensing measurements over the extended region covered by the BUFFALO program, as well as new spectroscopically confirmed multiply lensed galaxies confirmed by MUSE data. Our new lens model contains $4.9 \times 10^{14} M_{\odot}$ in the central 1 Mpc. The mass adopts the same bimodal distribution found in earlier work, but we find the mass contained in each substructure to be almost identical, suggesting a mass ratio 1:1 for the structures in this cluster.

Using this lens model, we predict the number of expected multiply lensed galaxies to be detected with JWST at $z=6$ and $z=9$, using the latest luminosity functions at high redshift
results in a narrower PDF for the magnification, but its median value peaks at larger magnifications, and the tail of extreme magnification still follows the canonical $\mu^{-2}$ power law past $\mu = 1000$. These two effects compensate the smaller abundance of microlenses, especially at large magnification values.

These results are derived without taking into account specific models of stellar evolution. In particular, the relatively short lifetimes of SG stars are not being considered in this work. Instead, we have assumed that these stars must exist in the Spock galaxy to be able to interpret the observations. Our results show that the abundance of these stars is well approximated by the abundance predicted by the IMF of stars with the corresponding mass to produce such luminosities (assuming the MS $M-L$ relation). This reasoning can be turned around to conclude that in order to have such an abundance of luminous stars, a relatively recent episode of star formation must have taken place, which is consistent with the young age derived for the Spock galaxy from SED fitting.

The results derived above assume there are no stars beyond the HD limit. Under that assumption, we find that our results are roughly consistent with the observations (that is, ~0.5 observed stars per pointing). Given the relatively small area that is being amplified in the Spock arc, we also consider the possibility that stars beyond the HD limit may exist, but their surface number density is smaller than what was assumed in our previous calculations. Otherwise these stars would have been detected by the FrontierSN or Flashlights programs, preferentially at larger distances from the critical curve. Since the observed stars appear near the expected position of the critical curves (true at least for S1 and F1, and likely true also for S2 and F2, assuming a second critical curve passes between them), we can conclude that if stars with luminosities beyond the HD limit exist in this small portion of the Spock galaxy, their number density must be smaller than what was assumed in our calculations above.

The strongest constraint on the abundance of stars beyond the HD limit comes from the regions with smallest magnification, as shown in Figs. 12 and 14. Our model assumed a surface number density of ~8000 stars kpc$^{-2}$ with luminosities beyond $6 \times 10^5 L_{\odot}$. From the results in Table 1, and adopting the case $\delta = 0$, the number of expected observed stars per pointing, and beyond the HD limit is simply the difference between Column 3 (labeled “No HD$_{\delta=0}$”), and Cols. 4–6 – that is, ~3.5 for the FrontierSN program or ~4.5 for the Flashlights program. These numbers need to be reduced by approximately a factor 20 for these stars to be rare enough to not contribute significantly to the number of detections farther away from the critical curve. This translates into a surface number density of <400 stars kpc$^{-2}$ with luminosities beyond the HD limit. If we consider the bins with magnifications 67 and 135, where these very bright stars are more likely to be found, the area in the source plane with magnification in this range is $A \approx 530 \times 20 \approx 0.01$ kpc$^2$. This means the number of stars beyond the HD limit in this region can be up to ~4. These stars remain in general undetected without the aid of microlensing provided they are less luminous than $L_{bol} < 3 \times 10^7 L_{\odot}$, as estimated earlier. Similar arguments can be used, but this time applied to the smaller area associated with the larger-magnification bins, and assuming that the stars responsible for the observed events are less luminous than the HD limit. Our model contains ~9000 stars kpc$^{-2}$ with bolometric luminosities between ~$10^5 L_{\odot}$ and ~$6 \times 10^5 L_{\odot}$. As discussed above, this number density (equivalent to ~1 SG star per $10 \times 10 \text{ pc}^2$) reproduces the observations from the FrontierSN and Flashlights programs reasonably well. Although high, this number density is consistent with the fact that SG stars are often found in large concentrations of stars (Davies et al. 2007).

The abundance of SG stars in the Spock galaxy can be better constrained with future observations of this arc. If we consider the farthest bin in Fig. 11 at $d \approx 16$ pc from the caustic (yellow bin, or total magnification $\mu \approx 70$, equivalent to regions in the image plane where $\mu \approx 35$ for each counterimage), this bin is highly sensitive to the presence of the most luminous stars (as shown more clearly in Fig. 12). By simply mapping how far from the critical curve lensed stars are observed, one can constrain the abundance of the most massive and luminous stars, since at relatively large distances from the caustic, only the most luminous stars are detectable.

Future JWST observations will improve the statistics on the number of detected lensed stars, offering us a privileged view into the massive end of the IMF of the Spock galaxy. Once enough SG stars are detected by JWST, an interesting application is to reverse engineer the tip of the red giant branch (TRGB) distance indicator and use it instead to constrain the magnification. If the TRGB holds at this redshift, we can use the observed luminosities of the brightest red stars to infer their true un lensed luminosity. This gives a direct constraint on the magnification which can be used to improve on the lens models. Current models typically disagree by a factor of 2–3 in the magnification near the critical curves. If the maximum bolometric luminosity for red SGs is still ~$6 \times 10^5 L_{\odot}$, JWST will be able to see these bright stars and use them as distance indicators to constrain $\mu$. There is of course the issue of microlensing that may introduce uncertainty in the magnification, but monitoring these stars should provide accurate measurements of the mode of the magnification. These stars can also show variability, although this is expected to be different than the $1/\sqrt{t}$ variability predicted by lensing (where $t$ is time to the maximum magnification). Frequent monitoring of these stars should easily distinguish between intrinsic variability and a microlensing event. Since JWST will be able to see red SGs farther away from the critical curve, at these separations the mode of the magnification is a good tracer of the macromodel magnification at that particular position.

Another interesting property that can be checked with future observations is to estimate the presence of luminous blue companions to red SGs. From local measurements, ~20% of the red SGs are expected to have such companions (Neugent et al. 2020a). Monitoring of microcaustic events can unambiguously reveal companion stars provided that this companion is bright enough so it can be observed when the companion crosses a microcaustic. These events would show distinctive chromatic features since the stars in the pair would have different temperatures (and colors) which manifest themselves when each star is crossing a microcaustic.

## 10. Conclusions

We have presented an updated model of the cluster MACS0416, based on the hybrid algorithm WSLAP+, taking advantage of new weak-lensing measurements over the extended region covered by the BUFFALO program, as well as new spectroscopically confirmed multiply lensed galaxies confirmed by MUSE data. Our new lens model contains $4.9 \times 10^{14} M_{\odot}$ in the central 1 Mpc. The mass adopts the same bimodal distribution found in earlier work, but we find the mass contained in each substructure to be almost identical, suggesting a mass ratio 1:1 for the structures in this cluster.

Using this lens model, we predict the number of expected multiply lensed galaxies to be detected with JWST at $z=6$ and $z=9$, using the latest luminosity functions at high redshift
derived from recent work based on JWST observations. For $z = 6$ we predict the usual depletion effect where, despite the increased effective depth provided by lensing, we expect fewer galaxies near the cluster center than in the parallel field, where the magnification is $\mu \approx 1$. However, at $z = 9$ the situation is reversed and we predict a larger number of $z = 9$ galaxies detected in the cluster field than in the parallel field. This is a direct consequence of the steeper luminosity function at $z = 9$ than at $z = 6$ in the faint end.

We focus our attention on four transients identified in one of the strongly lensed arcs in MACS0416, dubbed the Spock arc. Although other interpretations are still possible for these transients (Rodney et al. 2018; Kelly et al. 2022), we demonstrate that they are consistent with the hypothesis that they are due to microlensing events of SG stars at $z = 1.0054$. These microlensing events are expected to be frequent in this arc. We consider two of the HST filters (and depths) to match the real observations, the red filter F814W (and observations at 28.5 AB mag depth) and the wider, but bluer, filter F200LP (and a depth of 29 mag). From the lack of detection of lensed stars in most observations of this arc, we set an upper limit to the luminosity of detectable stars, ruling out the existence of SG stars with $L > 3 \times 10^6 L_\odot$ in the portion of the arc that is being multiply lensed. We assume the stars that are being lensed all belong to the SG type, with luminosities at least $\sim 10^5 L_\odot$. These stars can have a wide range of temperatures. We consider temperatures from $T = 3500$ K to $T = 30000$ K. The abundance of these stars is originally constrained by a model based on the Kroupa IMF that we fit to the observed photometry in different bands using BAGPIPES. We find that the predicted number of observed lensed stars matches the observations if the HD limit has no significant evolution between $z = 0$ and $z = 1$. Alternatively, stars beyond the HD limit may still exist in this portion of the galaxy, but their number density must be <400 stars kpc$^{-2}$ in order for their probability of being observed to be sufficiently small. On the other hand, if the stars responsible for the observed events are less luminous than the HD limit but with bolometric luminosities greater than $10^5 L_\odot$, their surface number density must be $\sim 9000$ SG stars kpc$^{-2}$.

The four observed events can be fully described as microlensing events of SG stars. In particular, the NW S2 event is consistent with the microlensing of a blue SG star (assuming a critical curve passes nearby), while the SE event is consistent with the microlensing of a red SG star (see Fig. 1). For F1 and F2, the lack of color information does not allow us to constrain the type of star being lensed, but the filter in this case is better suited for hot stars than for red stars. The HD limit can also be connected to metallicity. We find that the metallicity of the Spock galaxy is likely supersolar. In this scenario, massive stars would lose mass more efficiently through radiation-driven stellar wind, tightening the HD limit.

An alternative way of testing for the HD limit is by measuring the separation from the microlensed events to the critical curve. Stars with luminosities beyond the HD limit can be found at larger separations since they require smaller magnification factors in order to be detected. The Spock arc is not ideal for this purpose because there is still significant uncertainty on the position of the critical curve (or even the number of them) that is (are) crossing the arc. However, under the assumption that at least one of them (between S1 and F1) is well reproduced by our lens model, we also find good consistency between the observed distance to the critical curve of the microlensed events and the existence of an HD limit.

We also make predictions for future JWST observations of this cluster. We find that in the F200W filter ($\sim 2$ µm),

JWST detects $\sim 5$ SG stars per observation spread along the Spock arc.

We have shown how there is a high variability in the number of expected detected stars depending on the depth of the observation (or redshift of the background source), the filter used to perform the observation, the temperature of the lensed stars, and obviously the luminosity of the stars and their age. Each particular arc and observation needs to be modeled specifically. Other arcs in the same cluster (as well as in other clusters) have similar events (see Kelly et al. 2022; Meena et al. 2023a, for events discovered as part of the Flashlights program). The JWST predictions made in this work are consistent with recent findings based on actual JWST observations of this arc by Yan et al. (2023), and they were made public after submission of this work. In particular, our predictions imply the validity of the HD limit at $z = 1$ based on the observed JWST detection rate of approximately one event per pointing in this arc reported by Yan et al. (2023). Future observations of this cluster with JWST are expected to detect even more events that will allow better constraints on the abundance of SG stars in the Spock galaxy at $z \approx 1$.

**Acknowledgements.** The authors wish to thank Roberta Humphreys and the anonymous referee for useful comments and feedback. J.M.D. acknowledges the support of project PGC2018-101814-B-100 (MCIU/AEI/MINECO/FEDER, UE) Ministerio de Ciencia, Investigación y Universidades. This project was funded by the Agencia Estatal de Investigación, Unidad de Excelencia María de Maeztu, ref. MDM-2017-0765. A.N., M.J., and DJ.L. are supported by the UK Research and Innovation (UKRI) Future Leaders Fellowship (FLF), "Using Cosmic Beasts to Uncover the Nature of Dark Matter" (grant MR/S017216/1). D.J.L. is also supported by STFC grants ST/T000244/1 and ST/W002612/1. A.A. has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 101024195 - ROSEAU. M.L. acknowledges CNRS and CNES. P.K. is supported by NSF grant AST-1908823, and NASA/STScI grants GO-15936 and GO-16278. A.Z. acknowledges support by grant 2020750 from the US-Israel Binalional Science Foundation (BSF) and grant 2109066 from the US National Science Foundation (NSF), and by the Ministry of Science & Technology, Israel. A.V.F. is grateful for financial support from the Christopher R. Redlich Fund and numerous individual donors. S.K.L. and J.L. are supported by the Collaborative Research Fund under grant C6017-20G, which is issued by Research Grants Council of Hong Kong S.A.R. This research is based on observations made with the NASA/ESA Hubble Space Telescope and obtained from the Space Telescope Science Institute (STScI), which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. These observations are associated with program GO-15117 (PI: C. Steinhardt).

## References

* Bacon, R., Accardo, M., Adjali, L., et al. 2010, SPIE Conf. Ser., 7735, 773508

* Bergamini, P., Grillo, C., Rosati, P., et al. 2023, A&A, 674, A79

* Bertin, E., & Arnouts, S. 1996, A&AS, 117, 393

* Bessell, M., & Murphy, S. 2012, PASP, 124, 140

* Bowler, R. A. A., Dunlop, J. S., McLure, R. J., et al. 2015, MNRAS, 452, 1817

* Caminha, G. B., Grillo, C., Rosati, P., et al. 2017, A&A, 600, A90

* Carnall, A. C., McLure, R. J., Dunlop, J. S., & Davé, R. 2018, MNRAS, 480, 4379

* Chen, Y., Bressan, A., Girardi, L., et al. 2015, MNRAS, 452, 1068

* Chen, W., Kelly, P. L., Diego, J. M., et al. 2019, ApJ, 881, 8

* Chen, W., Kelly, P. L., Treu, T., et al. 2022, ApJ, 940, L54

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derived from recent work based on JWST observations. For $z = 6$ we predict the usual depletion effect where, despite the increased effective depth provided by lensing, we expect fewer galaxies near the cluster center than in the parallel field, where the magnification is $\mu \approx 1$. However, at $z = 9$ the situation is reversed and we predict a larger number of $z = 9$ galaxies detected in the cluster field than in the parallel field. This is a direct consequence of the steeper luminosity function at $z = 9$ than at $z = 6$ in the faint end.

We focus our attention on four transients identified in one of the strongly lensed arcs in MACS0416, dubbed the Spock arc. Although other interpretations are still possible for these transients (Rodney et al. 2018; Kelly et al. 2022), we demonstrate that they are consistent with the hypothesis that they are due to microlensing events of SG stars at $z = 1.0054$. These microlensing events are expected to be frequent in this arc. We consider two of the HST filters (and depths) to match the real observations, the red filter F814W (and observations at 28.5 AB mag depth) and the wider, but bluer, filter F200LP (and a depth of 29 mag). From the lack of detection of lensed stars in most observations of this arc, we set an upper limit to the luminosity of detectable stars, ruling out the existence of SG stars with $L > 3 \times 10^6 L_\odot$ in the portion of the arc that is being multiply lensed. We assume the stars that are being lensed all belong to the SG type, with luminosities at least $\sim 10^5 L_\odot$. These stars can have a wide range of temperatures. We consider temperatures from $T = 3500$ K to $T = 30000$ K. The abundance of these stars is originally constrained by a model based on the Kroupa IMF that we fit to the observed photometry in different bands using BAGPIPES. We find that the predicted number of observed lensed stars matches the observations if the HD limit has no significant evolution between $z = 0$ and $z = 1$. Alternatively, stars beyond the HD limit may still exist in this portion of the galaxy, but their number density must be <400 stars kpc$^{-2}$ in order for their probability of being observed to be sufficiently small. On the other hand, if the stars responsible for the observed events are less luminous than the HD limit but with bolometric luminosities greater than $10^5 L_\odot$, their surface number density must be $\sim 9000$ SG stars kpc$^{-2}$.

The four observed events can be fully described as microlensing events of SG stars. In particular, the NW S2 event is consistent with the microlensing of a blue SG star (assuming a critical curve passes nearby), while the SE event is consistent with the microlensing of a red SG star (see Fig. 1). For F1 and F2, the lack of color information does not allow us to constrain the type of star being lensed, but the filter in this case is better suited for hot stars than for red stars. The HD limit can also be connected to metallicity. We find that the metallicity of the Spock galaxy is likely supersolar. In this scenario, massive stars would lose mass more efficiently through radiation-driven stellar wind, tightening the HD limit.

An alternative way of testing for the HD limit is by measuring the separation from the microlensed events to the critical curve. Stars with luminosities beyond the HD limit can be found at larger separations since they require smaller magnification factors in order to be detected. The Spock arc is not ideal for this purpose because there is still significant uncertainty on the position of the critical curve (or even the number of them) that is (are) crossing the arc. However, under the assumption that at least one of them (between S1 and F1) is well reproduced by our lens model, we also find good consistency between the observed distance to the critical curve of the microlensed events and the existence of an HD limit.

We also make predictions for future JWST observations of this cluster. We find that in the F200W filter ($\sim 2$ µm),

JWST detects $\sim 5$ SG stars per observation spread along the Spock arc.

We have shown how there is a high variability in the number of expected detected stars depending on the depth of the observation (or redshift of the background source), the filter used to perform the observation, the temperature of the lensed stars, and obviously the luminosity of the stars and their age. Each particular arc and observation needs to be modeled specifically. Other arcs in the same cluster (as well as in other clusters) have similar events (see Kelly et al. 2022; Meena et al. 2023a, for events discovered as part of the Flashlights program). The JWST predictions made in this work are consistent with recent findings based on actual JWST observations of this arc by Yan et al. (2023), and they were made public after submission of this work. In particular, our predictions imply the validity of the HD limit at $z = 1$ based on the observed JWST detection rate of approximately one event per pointing in this arc reported by Yan et al. (2023). Future observations of this cluster with JWST are expected to detect even more events that will allow better constraints on the abundance of SG stars in the Spock galaxy at $z \approx 1$.

**Acknowledgements.** The authors wish to thank Roberta Humphreys and the anonymous referee for useful comments and feedback. J.M.D. acknowledges the support of project PGC2018-101814-B-100 (MCIU/AEI/MINECO/FEDER, UE) Ministerio de Ciencia, Investigación y Universidades. This project was funded by the Agencia Estatal de Investigación, Unidad de Excelencia María de Maeztu, ref. MDM-2017-0765. A.N., M.J., and DJ.L. are supported by the UK Research and Innovation (UKRI) Future Leaders Fellowship (FLF), "Using Cosmic Beasts to Uncover the Nature of Dark Matter" (grant MR/S017216/1). D.J.L. is also supported by STFC grants ST/T000244/1 and ST/W002612/1. A.A. has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 101024195 - ROSEAU. M.L. acknowledges CNRS and CNES. P.K. is supported by NSF grant AST-1908823, and NASA/STScI grants GO-15936 and GO-16278. A.Z. acknowledges support by grant 2020750 from the US-Israel Binalional Science Foundation (BSF) and grant 2109066 from the US National Science Foundation (NSF), and by the Ministry of Science & Technology, Israel. A.V.F. is grateful for financial support from the Christopher R. Redlich Fund and numerous individual donors. S.K.L. and J.L. are supported by the Collaborative Research Fund under grant C6017-20G, which is issued by Research Grants Council of Hong Kong S.A.R. This research is based on observations made with the NASA/ESA Hubble Space Telescope and obtained from the Space Telescope Science Institute (STScI), which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. These observations are associated with program GO-15117 (PI: C. Steinhardt).

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## Appendix A: WSLAP+

To optimize the lens model we use the code WSLAP+ (Diego et al. 2005, 2007; Sendra et al. 2014; Diego et al. 2016). A lens model derived using WSLAP+ is considered a hybrid type of model as it combines a free-form decomposition of the lens plane for the smooth large-scale component with a small-scale contribution from the member galaxies. The code combines weak and strong lensing in a natural way, with changes made in the large-scale and small-scale components impacting equally the strong- and weak-lensing observables. Details can be found in the references above. Here we give a brief description of the method.

We start with the classic definition of the lens equation

$$ \beta = \theta - \alpha(\theta, \Sigma), \qquad (A.1) $$

where $\theta$ is the observed position of the source, $\alpha$ is the deflection angle, $\Sigma(\theta)$ is the unknown surface mass density of the cluster at the position $\theta$, and $\beta$ is the unknown position of the background source. The optimization of the WSLAP+ solution takes advantage of the fact that the lens equation can be expressed as a linear function of the surface mass density, $\Sigma$. WSLAP+ parameterizes $\Sigma$ as a linear superposition of functions, which translates into $\alpha(\theta, \Sigma)$ also being linear in $\Sigma$.

In WSLAP+, $\Sigma$ is described by the combination of two components: (i) a smooth component (usually parameterized as superposition of Gaussians) corresponding to the free-form part of the model, or large-scale cluster potential, and (ii) a compact component that accounts for the mass associated with individual galaxies in the cluster. For the smooth component, we use Gaussian functions defined over a grid of points. A Gaussian function is simple, and enables fast computation of the deflection field, but also provides a good compromise between the desired compactness and smoothness of the basis function. The grid configuration can be defined as regular (all grid points have the same size) or irregular (grid points near the center are in general smaller). Adopting a regular grid is similar to a flat prior in the mass distribution, while an irregular grid can be interpreted as a model with a prior on the mass distribution with higher mass density assigned to smaller cells. For this work, we use a dynamical (irregular) grid with increased resolution in the central region, where strong-lensing constraints are present. In particular, we consider a grid with 745 points. The distribution of grid points is obtained from a Monte Carlo realization of the mass density previously obtained with a regular grid. The final distribution of points is more concentrated around the two brightest cluster galaxies (BCGs), with the density of grid points being smallest in the outskirts of the $6 \times 6$ arcmin$^2$ region.

For the compact component, we directly adopt the light distribution in HST's F160W band around the brightest member elliptical galaxies in the cluster, and spectroscopically confirmed (by MUSE) members. For each galaxy, we assign a mass proportional to its surface brightness. This mass is the only free parameter, and it is later readjusted as part of the optimization process. The number of parameters connected with the compact component depends on the number of adopted layers. Each layer

contains a number of member galaxies. The minimum number of layers is 1, corresponding to the case where all galaxies are placed in the same layer — that is, they are all assumed to have the same mass-to-light ratio. In this case, the single layer is proportional to the light distribution of all member galaxies, and is assigned a fiducial mass accounting for the total mass of member galaxies. For each layer there is one extra parameter which accounts for the renormalization constant multiplying the map of the mass distribution. This renormalization parameter is later optimized by WSLAP+. For the particular case of MACS0416, we use four layers. The first two layers contain the two main BCGs, the third layer contains red-sequence member galaxies, and the fourth layer contains spectroscopically confirmed member galaxies which are not in the red sequence and hence may have a different mass-to-light ratio.

As shown by Diego et al. (2005, 2007), the strong- and weak-lensing problem can be expressed as a system of linear equations that can be represented in a compact form,

$$ \Theta = \Gamma X, \qquad (A.2) $$

where the measured strong- and weak-lensing observables are arranged in $\Theta$ of dimension $N_{\Theta} = 2N_{sl} + 2N_{wl}$ (where $N_{sl}$ and $N_{wl}$ are the strong- and weak-lensing observables, each contributing with two constraints). The unknown surface mass density and source positions are in the array $X$ of dimension

$$ N_X = N_c + N_l + 2N_s. \qquad (A.3) $$

The array $X$ also contains the $N_l = 4$ unknown renormalization factors associated with the four layers. $N_c$ is the number of grid points (or cells) that we use to divide the field of view, and $N_s$ is the number of background sources being strongly lensed (each source contributes with two variables, $\beta_x$ and $\beta_y$). Finally, the matrix $\Gamma$ is known (for a given grid configuration and fiducial galaxy deflection field) and has dimension $N_{\Theta} \times N_X$.

The solution, $X$, of the system of Eq. (A.2) is found after minimizing a quadratic function of $X$ (derived from the system of Eq. (A.2) as described by Diego et al. 2005). A detailed discussion of the algorithm used to solve the system of linear equations is given by Diego et al. (2005). For a discussion of its convergence and performance (based on simulated data), we refer the reader to Sendra et al. (2014).

## Appendix B: Arc positions and redshifts

Strong lensing constraints used to derive the lens model. $ID_2$ and $z_2$ are the IDs and redshifts in Bergamini et al. (2023). In general, all redshifts agree to the subpercent level, except for our system (ID = 42) where Bergamini et al. (2023) derives a redshift 14.5% larger. Arcs marked with † are from Bergamini et al. (2023), and are not included in our set of constraints since they were published after the model was derived. Arcs that have no $ID_2$ nor $z_2$ in the table are systems identified with BUFFALO and MUSE data that are included in our model, but are not included by Bergamini et al. (2023). These are systems ID = 33, 38, 40, 43, 44, 46, 50, 53, 54, 55, 70, and 72.
# Appendix A: WSLAP+

To optimize the lens model we use the code WSLAP+ (Diego et al. 2005, 2007; Sendra et al. 2014; Diego et al. 2016). A lens model derived using WSLAP+ is considered a hybrid type of model as it combines a free-form decomposition of the lens plane for the smooth large-scale component with a small-scale contribution from the member galaxies. The code combines weak and strong lensing in a natural way, with changes made in the large-scale and small-scale components impacting equally the strong- and weak-lensing observables. Details can be found in the references above. Here we give a brief description of the method.

We start with the classic definition of the lens equation

$$ \beta = \theta - \alpha(\theta, \Sigma), \qquad (A.1) $$

where $\theta$ is the observed position of the source, $\alpha$ is the deflection angle, $\Sigma(\theta)$ is the unknown surface mass density of the cluster at the position $\theta$, and $\beta$ is the unknown position of the background source. The optimization of the WSLAP+ solution takes advantage of the fact that the lens equation can be expressed as a linear function of the surface mass density, $\Sigma$. WSLAP+ parameterizes $\Sigma$ as a linear superposition of functions, which translates into $\alpha(\theta, \Sigma)$ also being linear in $\Sigma$.

In WSLAP+, $\Sigma$ is described by the combination of two components: (i) a smooth component (usually parameterized as superposition of Gaussians) corresponding to the free-form part of the model, or large-scale cluster potential, and (ii) a compact component that accounts for the mass associated with individual galaxies in the cluster. For the smooth component, we use Gaussian functions defined over a grid of points. A Gaussian function is simple, and enables fast computation of the deflection field, but also provides a good compromise between the desired compactness and smoothness of the basis function. The grid configuration can be defined as regular (all grid points have the same size) or irregular (grid points near the center are in general smaller). Adopting a regular grid is similar to a flat prior in the mass distribution, while an irregular grid can be interpreted as a model with a prior on the mass distribution with higher mass density assigned to smaller cells. For this work, we use a dynamical (irregular) grid with increased resolution in the central region, where strong-lensing constraints are present. In particular, we consider a grid with 745 points. The distribution of grid points is obtained from a Monte Carlo realization of the mass density previously obtained with a regular grid. The final distribution of points is more concentrated around the two brightest cluster galaxies (BCGs), with the density of grid points being smallest in the outskirts of the 6 × 6 arcmin² region.

For the compact component, we directly adopt the light distribution in HST's F160W band around the brightest member elliptical galaxies in the cluster, and spectroscopically confirmed (by MUSE) members. For each galaxy, we assign a mass proportional to its surface brightness. This mass is the only free parameter, and it is later readjusted as part of the optimization process. The number of parameters connected with the compact component depends on the number of adopted layers. Each layer

contains a number of member galaxies. The minimum number of layers is 1, corresponding to the case where all galaxies are placed in the same layer — that is, they are all assumed to have the same mass-to-light ratio. In this case, the single layer is proportional to the light distribution of all member galaxies, and is assigned a fiducial mass accounting for the total mass of member galaxies. For each layer there is one extra parameter which accounts for the renormalization constant multiplying the map of the mass distribution. This renormalization parameter is later optimized by WSLAP+. For the particular case of MACS0416, we use four layers. The first two layers contain the two main BCGs, the third layer contains red-sequence member galaxies, and the fourth layer contains spectroscopically confirmed member galaxies which are not in the red sequence and hence may have a different mass-to-light ratio.

As shown by Diego et al. (2005, 2007), the strong- and weak-lensing problem can be expressed as a system of linear equations that can be represented in a compact form,

$$ \Theta = \Gamma X, \qquad (A.2) $$

where the measured strong- and weak-lensing observables are arranged in $\Theta$ of dimension $N_{\Theta} = 2N_{sl} + 2N_{wl}$ (where $N_{sl}$ and $N_{wl}$ are the strong- and weak-lensing observables, each contributing with two constraints). The unknown surface mass density and source positions are in the array $X$ of dimension

$$ N_X = N_c + N_l + 2N_s. \qquad (A.3) $$

The array $X$ also contains the $N_l = 4$ unknown renormalization factors associated with the four layers. $N_c$ is the number of grid points (or cells) that we use to divide the field of view, and $N_s$ is the number of background sources being strongly lensed (each source contributes with two variables, $\beta_x$ and $\beta_y$). Finally, the matrix $\Gamma$ is known (for a given grid configuration and fiducial galaxy deflection field) and has dimension $N_{\Theta} \times N_X$.

The solution, $X$, of the system of Eq. (A.2) is found after minimizing a quadratic function of $X$ (derived from the system of Eq. (A.2) as described by Diego et al. 2005). A detailed discussion of the algorithm used to solve the system of linear equations is given by Diego et al. (2005). For a discussion of its convergence and performance (based on simulated data), we refer the reader to Sendra et al. (2014).

# Appendix B: Arc positions and redshifts

Strong lensing constraints used to derive the lens model. $ID_2$ and $z_2$ are the IDs and redshifts in Bergamini et al. (2023). In general, all redshifts agree to the subpercent level, except for our system (ID = 42) where Bergamini et al. (2023) derives a redshift 14.5% larger. Arcs marked with † are from Bergamini et al. (2023), and are not included in our set of constraints since they were published after the model was derived. Arcs that have no $ID_2$ nor $z_2$ in the table are systems identified with BUFFALO and MUSE data that are included in our model, but are not included by Bergamini et al. (2023). These are systems ID = 33, 38, 40, 43, 44, 46, 50, 53, 54, 55, 70, and 72.
<table><caption>**Table B.1.** Full strong lensing dataset.</caption><thead><tr><th>ID</th><th>RA</th><th>DEC</th><th>z<sub>MUSE</sub></th><th>ID<sub>2</sub></th><th>z<sub>2</sub></th></tr></thead><tbody><tr><td>1</td><td>64.0408675</td><td>-24.0616447</td><td>1.896</td><td>5.1c</td><td>1.893</td></tr><tr><td>1</td><td>64.0435396</td><td>-24.0635339</td><td>-</td><td>5.1b</td><td>1.893</td></tr><tr><td>1</td><td>64.0474367</td><td>-24.0686997</td><td>-</td><td>5.1a</td><td>1.893</td></tr><tr><td>2</td><td>64.0411592</td><td>-24.0618356</td><td>1.895</td><td>5.2c</td><td>1.893</td></tr><tr><td>2</td><td>64.0431533</td><td>-24.0631203</td><td>-</td><td>5.2b</td><td>1.893</td></tr><tr><td>2</td><td>64.0475325</td><td>-24.0688519</td><td>-</td><td>5.4a</td><td>1.893</td></tr><tr><td>3</td><td>64.0308421</td><td>-24.0671572</td><td>1.9894</td><td>15.1c</td><td>1.990</td></tr><tr><td>3</td><td>64.0353283</td><td>-24.0710203</td><td>-</td><td>15.1b</td><td>1.990</td></tr><tr><td>3</td><td>64.0418754</td><td>-24.0757592</td><td>-</td><td>15.1a</td><td>1.990</td></tr><tr><td>4</td><td>64.0308492</td><td>-24.0672517</td><td>1.99</td><td>15.2c</td><td>1.990</td></tr><tr><td>4</td><td>64.0352429</td><td>-24.0710325</td><td>-</td><td>15.2b</td><td>1.990</td></tr><tr><td>4</td><td>64.0419121</td><td>-24.0758594</td><td>-</td><td>15.2a</td><td>1.990</td></tr><tr><td>5</td><td>64.0324804</td><td>-24.0684442</td><td>2.0948</td><td>16.1c</td><td>2.095</td></tr><tr><td>5</td><td>64.0327321</td><td>-24.0686961</td><td>-</td><td>16.1b</td><td>2.095</td></tr><tr><td>5</td><td>64.0335933</td><td>-24.0694772</td><td>-</td><td>16.1a</td><td>2.095</td></tr><tr><td>5</td><td>64.0436363</td><td>-24.0769933</td><td>-</td><td>-</td><td>-</td></tr><tr><td>6</td><td>64.0398687</td><td>-24.0631192</td><td>2.0881</td><td>7c</td><td>2.085</td></tr><tr><td>6</td><td>64.0407412</td><td>-24.0636225</td><td>-</td><td>7b</td><td>2.085</td></tr><tr><td>6</td><td>64.0471708</td><td>-24.0711372</td><td>-</td><td>7a</td><td>2.085</td></tr><tr><td>7</td><td>64.0261300</td><td>-24.0772783</td><td>2.2982</td><td>29c</td><td>2.298</td></tr><tr><td>7</td><td>64.0284838</td><td>-24.0797714</td><td>-</td><td>29b</td><td>2.298</td></tr><tr><td>7</td><td>64.0367787</td><td>-24.0839397</td><td>-</td><td>29a</td><td>2.298</td></tr><tr><td>8</td><td>64.0393337</td><td>-24.0704250</td><td>1.0054</td><td>13a</td><td>1.005</td></tr><tr><td>8</td><td>64.0383658</td><td>-24.0697531</td><td>-</td><td>13b</td><td>1.005</td></tr><tr><td>8</td><td>64.0343062</td><td>-24.0660503</td><td>-</td><td>13c</td><td>1.005</td></tr><tr><td>9</td><td>64.0276462</td><td>-24.0727036</td><td>3.2175</td><td>20.1c</td><td>3.222</td></tr><tr><td>9</td><td>64.0322367</td><td>-24.0751339</td><td>-</td><td>20.1b</td><td>3.222</td></tr><tr><td>9</td><td>64.0404263</td><td>-24.0815053</td><td>-</td><td>20.1a</td><td>3.222</td></tr><tr><td>10</td><td>64.0263242</td><td>-24.0743661</td><td>1.6333</td><td>24c</td><td>1.637</td></tr><tr><td>10</td><td>64.0311150</td><td>-24.0789806</td><td>-</td><td>24b</td><td>1.637</td></tr><tr><td>10</td><td>64.0359133</td><td>-24.0813494</td><td>-</td><td>24a</td><td>1.637</td></tr><tr><td>11</td><td>64.0241529</td><td>-24.0809203</td><td>1.9656</td><td>36c</td><td>1.964</td></tr><tr><td>11</td><td>64.0284129</td><td>-24.0845719</td><td>-</td><td>36b</td><td>1.964</td></tr><tr><td>11</td><td>64.0316896</td><td>-24.0857892</td><td>-</td><td>36a</td><td>1.964</td></tr><tr><td>12</td><td>64.0235546</td><td>-24.0817486</td><td>2.2182</td><td>37c</td><td>2.218</td></tr><tr><td>12</td><td>64.0286679</td><td>-24.0860025</td><td>-</td><td>37b</td><td>2.218</td></tr><tr><td>12</td><td>64.0298979</td><td>-24.0863944</td><td>-</td><td>37a</td><td>2.218</td></tr><tr><td>13</td><td>64.0344096</td><td>-24.0637628</td><td>2.0910</td><td>10c</td><td>2.093</td></tr><tr><td>13</td><td>64.0396529</td><td>-24.0666592</td><td>-</td><td>10b</td><td>2.0943</td></tr><tr><td>13</td><td>64.0446321</td><td>-24.0721236</td><td>-</td><td>10a</td><td>2.093</td></tr><tr><td>14</td><td>64.0465296</td><td>-24.0604303</td><td>3.2355</td><td>1c</td><td>3.238</td></tr><tr><td>14</td><td>64.0470354</td><td>-24.0608272</td><td>-</td><td>1b</td><td>3.238</td></tr><tr><td>14</td><td>64.0491533</td><td>-24.0628947</td><td>-</td><td>1a</td><td>3.238</td></tr><tr><td>15</td><td>64.0422746</td><td>-24.0605783</td><td>2.1067</td><td>4c</td><td>2.107</td></tr><tr><td>15</td><td>64.0475379</td><td>-24.0660783</td><td>-</td><td>4b</td><td>2.107</td></tr><tr><td>15</td><td>64.0481908</td><td>-24.0669914</td><td>-</td><td>4a</td><td>2.107</td></tr><tr><td>16</td><td>64.0365775</td><td>-24.0670578</td><td>0.9397</td><td>12.1c</td><td>0.940</td></tr><tr><td>16</td><td>64.0369150</td><td>-24.0674903</td><td>-</td><td>12.1b</td><td>0.940</td></tr><tr><td>16</td><td>64.0409596</td><td>-24.0712164</td><td>-</td><td>-</td><td>-</td></tr><tr><td>17</td><td>64.0230608</td><td>-24.0772986</td><td>5.3659</td><td>32c</td><td>5.365</td></tr><tr><td>17</td><td>64.0284821</td><td>-24.0830294</td><td>-</td><td>32b</td><td>5.365</td></tr><tr><td>17</td><td>64.0351254</td><td>-24.0855331</td><td>-</td><td>32a</td><td>5.365</td></tr><tr><td>18</td><td>64.0293633</td><td>-24.0733550</td><td>5.1060</td><td>21.1c</td><td>5.106</td></tr><tr><td>18</td><td>64.0308662</td><td>-24.0742067</td><td>-</td><td>21.1b</td><td>5.106</td></tr><tr><td>19</td><td>64.0250233</td><td>-24.0750497</td><td>3.4909</td><td>27c</td><td>3.492</td></tr><tr><td>19</td><td>64.0294879</td><td>-24.0799181</td><td>-</td><td>27b</td><td>3.492</td></tr><tr><td>19</td><td>64.0375504</td><td>-24.0836828</td><td>-</td><td>27a</td><td>3.492</td></tr></tbody></table>

<table><caption>**Table B.1.** cont.</caption><thead><tr><th>ID</th><th>RA</th><th>DEC</th><th>z<sub>MUSE</sub></th><th>ID<sub>2</sub></th><th>z<sub>2</sub></th></tr></thead><tbody><tr><td>20</td><td>64.0227721</td><td>-24.0746250</td><td>3.4406</td><td>30c</td><td>3.440</td></tr><tr><td>20</td><td>64.0313208</td><td>-24.0819283</td><td>-</td><td>30b</td><td>3.440</td></tr><tr><td>20</td><td>64.0337146</td><td>-24.0832186</td><td>-</td><td>30a</td><td>3.440</td></tr><tr><td>21</td><td>64.0416350</td><td>-24.0600322</td><td>3.2885</td><td>3c</td><td>3.290</td></tr><tr><td>21</td><td>64.0453321</td><td>-24.0627914</td><td>-</td><td>3b</td><td>2.290</td></tr><tr><td>21</td><td>64.0493079</td><td>-24.0682117</td><td>-</td><td>3a</td><td>3.290</td></tr><tr><td>22†</td><td>64.0383500</td><td>-24.0841261</td><td>3.253</td><td>28a</td><td>3.253</td></tr><tr><td>22</td><td>64.0264042</td><td>-24.0767289</td><td>3.2526</td><td>28c</td><td>3.253</td></tr><tr><td>22</td><td>64.0284067</td><td>-24.0790322</td><td>-</td><td>28b</td><td>3.253</td></tr><tr><td>23</td><td>64.0234888</td><td>-24.0761606</td><td>4.1218</td><td>31c</td><td>4.122</td></tr><tr><td>23</td><td>64.0293129</td><td>-24.0818383</td><td>-</td><td>31b</td><td>4.122</td></tr><tr><td>23</td><td>64.0355646</td><td>-24.0847072</td><td>-</td><td>31a</td><td>4.122</td></tr><tr><td>24</td><td>64.0269333</td><td>-24.0699764</td><td>3.8696</td><td>-</td><td>-</td></tr><tr><td>24</td><td>64.0340133</td><td>-24.0745969</td><td>3.8710</td><td>18b</td><td>3.871</td></tr><tr><td>24</td><td>64.0402579</td><td>-24.0798950</td><td>-</td><td>18a</td><td>3.871</td></tr><tr><td>25</td><td>64.0266083</td><td>-24.0705319</td><td>4.1032</td><td>19.1c</td><td>4.103</td></tr><tr><td>25</td><td>64.0337633</td><td>-24.0748514</td><td>-</td><td>19.1b</td><td>4.103</td></tr><tr><td>25</td><td>64.0401517</td><td>-24.0803403</td><td>-</td><td>19.1a</td><td>4.103</td></tr><tr><td>26†</td><td>64.0468408</td><td>-24.0753850</td><td>3.292</td><td>11a</td><td>3.292</td></tr><tr><td>26</td><td>64.0353038</td><td>-24.0647647</td><td>3.2922</td><td>11c</td><td>3.292</td></tr><tr><td>26</td><td>64.0385875</td><td>-24.0659972</td><td>-</td><td>11b</td><td>3.292</td></tr><tr><td>27</td><td>64.0252696</td><td>-24.0736092</td><td>3.0773</td><td>26c</td><td>3.081</td></tr><tr><td>27</td><td>64.0305596</td><td>-24.0792475</td><td>-</td><td>26b</td><td>3.081</td></tr><tr><td>27</td><td>64.0377979</td><td>-24.0824158</td><td>-</td><td>26a</td><td>3.081</td></tr><tr><td>28</td><td>64.0255200</td><td>-24.0736789</td><td>3.1103</td><td>25c</td><td>3.111</td></tr><tr><td>28</td><td>64.0304467</td><td>-24.0790472</td><td>-</td><td>25b</td><td>3.111</td></tr><tr><td>28</td><td>64.0381496</td><td>-24.0824269</td><td>-</td><td>25a</td><td>3.111</td></tr><tr><td>29</td><td>64.0345663</td><td>-24.0669883</td><td>3.2215</td><td>14.1a</td><td>3.221</td></tr><tr><td>29</td><td>64.0342617</td><td>-24.0665167</td><td>-</td><td>14.1b</td><td>3.221</td></tr><tr><td>29</td><td>64.0340392</td><td>-24.0669400</td><td>3.2240</td><td>14.1d</td><td>3.221</td></tr><tr><td>29</td><td>64.0461275</td><td>-24.0768133</td><td>-</td><td>14.1f</td><td>3.221</td></tr><tr><td>29</td><td>64.0340812</td><td>-24.0664786</td><td>3.2215</td><td>14.1c</td><td>3.221</td></tr><tr><td>30</td><td>64.0401537</td><td>-24.0667703</td><td>3.2882</td><td>9.1b</td><td>3.292</td></tr><tr><td>30</td><td>64.0451842</td><td>-24.0723725</td><td>-</td><td>9.1a</td><td>3.292</td></tr><tr><td>31</td><td>64.0436462</td><td>-24.0590344</td><td>6.1452</td><td>2.1c</td><td>6.145</td></tr><tr><td>31</td><td>64.0509446</td><td>-24.0665642</td><td>-</td><td>2.1a</td><td>6.145</td></tr><tr><td>31</td><td>64.0479096</td><td>-24.0620906</td><td>6.1480</td><td>2.1b</td><td>6.145</td></tr><tr><td>32</td><td>64.0377496</td><td>-24.0607753</td><td>3.6065</td><td>6c</td><td>3.607</td></tr><tr><td>32</td><td>64.0437075</td><td>-24.0644186</td><td>-</td><td>6b</td><td>3.607</td></tr><tr><td>32</td><td>64.0478788</td><td>-24.0701956</td><td>-</td><td>6a</td><td>3.607</td></tr><tr><td>33</td><td>64.0272008</td><td>-24.0736031</td><td>3.9230</td><td>-</td><td>-</td></tr><tr><td>33</td><td>64.0310679</td><td>-24.0772083</td><td>-</td><td>-</td><td>-</td></tr><tr><td>33</td><td>64.0312446</td><td>-24.0770633</td><td>-</td><td>-</td><td>-</td></tr><tr><td>34</td><td>64.0247004</td><td>-24.0714867</td><td>2.5420</td><td>-</td><td>-</td></tr><tr><td>34</td><td>64.0327150</td><td>-24.0785378</td><td>2.5425</td><td>23b</td><td>2.542</td></tr><tr><td>34</td><td>64.0357392</td><td>-24.0799517</td><td>-</td><td>23a</td><td>2.542</td></tr><tr><td>35</td><td>64.0309054</td><td>-24.0837350</td><td>5.9729</td><td>33b</td><td>5.973</td></tr><tr><td>35</td><td>64.0321100</td><td>-24.0842606</td><td>-</td><td>33a</td><td>5.973</td></tr><tr><td>36</td><td>64.0238171</td><td>-24.0785033</td><td>3.9228</td><td>34c</td><td>3.923</td></tr><tr><td>36</td><td>64.0277029</td><td>-24.0826383</td><td>-</td><td>34b</td><td>3.923</td></tr><tr><td>37</td><td>64.0222117</td><td>-24.0773086</td><td>5.6380</td><td>35c</td><td>5.638</td></tr><tr><td>37</td><td>64.0287462</td><td>-24.0842464</td><td>-</td><td>35b</td><td>5.638</td></tr><tr><td>37</td><td>64.0338062</td><td>-24.0857294</td><td>-</td><td>35a</td><td>5.638</td></tr><tr><td>38</td><td>64.0269388</td><td>-24.0757711</td><td>2.3355</td><td>-</td><td>-</td></tr><tr><td>38</td><td>64.0295250</td><td>-24.0786211</td><td>2.3340</td><td>-</td><td>-</td></tr><tr><td>38</td><td>64.0383083</td><td>-24.0830053</td><td>-</td><td>-</td><td>-</td></tr><tr><td>39</td><td>64.0272463</td><td>-24.0682528</td><td>3.9680</td><td>17c</td><td>3.966</td></tr></tbody></table>
<table><caption>**Table B.1.** Full strong lensing dataset.</caption><thead><tr><th>ID</th><th>RA</th><th>DEC</th><th>z<sub>MUSE</sub></th><th>ID<sub>2</sub></th><th>z<sub>2</sub></th></tr></thead><tbody><tr><td>1</td><td>64.0408675</td><td>-24.0616447</td><td>1.896</td><td>5.1c</td><td>1.893</td></tr><tr><td>1</td><td>64.0435396</td><td>-24.0635339</td><td>-</td><td>5.1b</td><td>1.893</td></tr><tr><td>1</td><td>64.0474367</td><td>-24.0686997</td><td>-</td><td>5.1a</td><td>1.893</td></tr><tr><td>2</td><td>64.0411592</td><td>-24.0618356</td><td>1.895</td><td>5.2c</td><td>1.893</td></tr><tr><td>2</td><td>64.0431533</td><td>-24.0631203</td><td>-</td><td>5.2b</td><td>1.893</td></tr><tr><td>2</td><td>64.0475325</td><td>-24.0688519</td><td>-</td><td>5.4a</td><td>1.893</td></tr><tr><td>3</td><td>64.0308421</td><td>-24.0671572</td><td>1.9894</td><td>15.1c</td><td>1.990</td></tr><tr><td>3</td><td>64.0353283</td><td>-24.0710203</td><td>-</td><td>15.1b</td><td>1.990</td></tr><tr><td>3</td><td>64.0418754</td><td>-24.0757592</td><td>-</td><td>15.1a</td><td>1.990</td></tr><tr><td>4</td><td>64.0308492</td><td>-24.0672517</td><td>1.99</td><td>15.2c</td><td>1.990</td></tr><tr><td>4</td><td>64.0352429</td><td>-24.0710325</td><td>-</td><td>15.2b</td><td>1.990</td></tr><tr><td>4</td><td>64.0419121</td><td>-24.0758594</td><td>-</td><td>15.2a</td><td>1.990</td></tr><tr><td>5</td><td>64.0324804</td><td>-24.0684442</td><td>2.0948</td><td>16.1c</td><td>2.095</td></tr><tr><td>5</td><td>64.0327321</td><td>-24.0686961</td><td>-</td><td>16.1b</td><td>2.095</td></tr><tr><td>5</td><td>64.0335933</td><td>-24.0694772</td><td>-</td><td>16.1a</td><td>2.095</td></tr><tr><td>5</td><td>64.0436363</td><td>-24.0769933</td><td>-</td><td>-</td><td>-</td></tr><tr><td>6</td><td>64.0398687</td><td>-24.0631192</td><td>2.0881</td><td>7c</td><td>2.085</td></tr><tr><td>6</td><td>64.0407412</td><td>-24.0636225</td><td>-</td><td>7b</td><td>2.085</td></tr><tr><td>6</td><td>64.0471708</td><td>-24.0711372</td><td>-</td><td>7a</td><td>2.085</td></tr><tr><td>7</td><td>64.0261300</td><td>-24.0772783</td><td>2.2982</td><td>29c</td><td>2.298</td></tr><tr><td>7</td><td>64.0284838</td><td>-24.0797714</td><td>-</td><td>29b</td><td>2.298</td></tr><tr><td>7</td><td>64.0367787</td><td>-24.0839397</td><td>-</td><td>29a</td><td>2.298</td></tr><tr><td>8</td><td>64.0393337</td><td>-24.0704250</td><td>1.0054</td><td>13a</td><td>1.005</td></tr><tr><td>8</td><td>64.0383658</td><td>-24.0697531</td><td>-</td><td>13b</td><td>1.005</td></tr><tr><td>8</td><td>64.0343062</td><td>-24.0660503</td><td>-</td><td>13c</td><td>1.005</td></tr><tr><td>9</td><td>64.0276462</td><td>-24.0727036</td><td>3.2175</td><td>20.1c</td><td>3.222</td></tr><tr><td>9</td><td>64.0322367</td><td>-24.0751339</td><td>-</td><td>20.1b</td><td>3.222</td></tr><tr><td>9</td><td>64.0404263</td><td>-24.0815053</td><td>-</td><td>20.1a</td><td>3.222</td></tr><tr><td>10</td><td>64.0263242</td><td>-24.0743661</td><td>1.6333</td><td>24c</td><td>1.637</td></tr><tr><td>10</td><td>64.0311150</td><td>-24.0789806</td><td>-</td><td>24b</td><td>1.637</td></tr><tr><td>10</td><td>64.0359133</td><td>-24.0813494</td><td>-</td><td>24a</td><td>1.637</td></tr><tr><td>11</td><td>64.0241529</td><td>-24.0809203</td><td>1.9656</td><td>36c</td><td>1.964</td></tr><tr><td>11</td><td>64.0284129</td><td>-24.0845719</td><td>-</td><td>36b</td><td>1.964</td></tr><tr><td>11</td><td>64.0316896</td><td>-24.0857892</td><td>-</td><td>36a</td><td>1.964</td></tr><tr><td>12</td><td>64.0235546</td><td>-24.0817486</td><td>2.2182</td><td>37c</td><td>2.218</td></tr><tr><td>12</td><td>64.0286679</td><td>-24.0860025</td><td>-</td><td>37b</td><td>2.218</td></tr><tr><td>12</td><td>64.0298979</td><td>-24.0863944</td><td>-</td><td>37a</td><td>2.218</td></tr><tr><td>13</td><td>64.0344096</td><td>-24.0637628</td><td>2.0910</td><td>10c</td><td>2.093</td></tr><tr><td>13</td><td>64.0396529</td><td>-24.0666592</td><td>-</td><td>10b</td><td>2.0943</td></tr><tr><td>13</td><td>64.0446321</td><td>-24.0721236</td><td>-</td><td>10a</td><td>2.093</td></tr><tr><td>14</td><td>64.0465296</td><td>-24.0604303</td><td>3.2355</td><td>1c</td><td>3.238</td></tr><tr><td>14</td><td>64.0470354</td><td>-24.0608272</td><td>-</td><td>1b</td><td>3.238</td></tr><tr><td>14</td><td>64.0491533</td><td>-24.0628947</td><td>-</td><td>1a</td><td>3.238</td></tr><tr><td>15</td><td>64.0422746</td><td>-24.0605783</td><td>2.1067</td><td>4c</td><td>2.107</td></tr><tr><td>15</td><td>64.0475379</td><td>-24.0660783</td><td>-</td><td>4b</td><td>2.107</td></tr><tr><td>15</td><td>64.0481908</td><td>-24.0669914</td><td>-</td><td>4a</td><td>2.107</td></tr><tr><td>16</td><td>64.0365775</td><td>-24.0670578</td><td>0.9397</td><td>12.1c</td><td>0.940</td></tr><tr><td>16</td><td>64.0369150</td><td>-24.0674903</td><td>-</td><td>12.1b</td><td>0.940</td></tr><tr><td>16</td><td>64.0409596</td><td>-24.0712164</td><td>-</td><td>-</td><td>-</td></tr><tr><td>17</td><td>64.0230608</td><td>-24.0772986</td><td>5.3659</td><td>32c</td><td>5.365</td></tr><tr><td>17</td><td>64.0284821</td><td>-24.0830294</td><td>-</td><td>32b</td><td>5.365</td></tr><tr><td>17</td><td>64.0351254</td><td>-24.0855331</td><td>-</td><td>32a</td><td>5.365</td></tr><tr><td>18</td><td>64.0293633</td><td>-24.0733550</td><td>5.1060</td><td>21.1c</td><td>5.106</td></tr><tr><td>18</td><td>64.0308662</td><td>-24.0742067</td><td>-</td><td>21.1b</td><td>5.106</td></tr><tr><td>19</td><td>64.0250233</td><td>-24.0750497</td><td>3.4909</td><td>27c</td><td>3.492</td></tr><tr><td>19</td><td>64.0294879</td><td>-24.0799181</td><td>-</td><td>27b</td><td>3.492</td></tr><tr><td>19</td><td>64.0375504</td><td>-24.0836828</td><td>-</td><td>27a</td><td>3.492</td></tr></tbody></table>

<table><caption>**Table B.1.** cont.</caption><thead><tr><th>ID</th><th>RA</th><th>DEC</th><th>z<sub>MUSE</sub></th><th>ID<sub>2</sub></th><th>z<sub>2</sub></th></tr></thead><tbody><tr><td>20</td><td>64.0227721</td><td>-24.0746250</td><td>3.4406</td><td>30c</td><td>3.440</td></tr><tr><td>20</td><td>64.0313208</td><td>-24.0819283</td><td>-</td><td>30b</td><td>3.440</td></tr><tr><td>20</td><td>64.0337146</td><td>-24.0832186</td><td>-</td><td>30a</td><td>3.440</td></tr><tr><td>21</td><td>64.0416350</td><td>-24.0600322</td><td>3.2885</td><td>3c</td><td>3.290</td></tr><tr><td>21</td><td>64.0453321</td><td>-24.0627914</td><td>-</td><td>3b</td><td>2.290</td></tr><tr><td>21</td><td>64.0493079</td><td>-24.0682117</td><td>-</td><td>3a</td><td>3.290</td></tr><tr><td>22†</td><td>64.0383500</td><td>-24.0841261</td><td>3.253</td><td>28a</td><td>3.253</td></tr><tr><td>22</td><td>64.0264042</td><td>-24.0767289</td><td>3.2526</td><td>28c</td><td>3.253</td></tr><tr><td>22</td><td>64.0284067</td><td>-24.0790322</td><td>-</td><td>28b</td><td>3.253</td></tr><tr><td>23</td><td>64.0234888</td><td>-24.0761606</td><td>4.1218</td><td>31c</td><td>4.122</td></tr><tr><td>23</td><td>64.0293129</td><td>-24.0818383</td><td>-</td><td>31b</td><td>4.122</td></tr><tr><td>23</td><td>64.0355646</td><td>-24.0847072</td><td>-</td><td>31a</td><td>4.122</td></tr><tr><td>24</td><td>64.0269333</td><td>-24.0699764</td><td>3.8696</td><td>-</td><td>-</td></tr><tr><td>24</td><td>64.0340133</td><td>-24.0745969</td><td>3.8710</td><td>18b</td><td>3.871</td></tr><tr><td>24</td><td>64.0402579</td><td>-24.0798950</td><td>-</td><td>18a</td><td>3.871</td></tr><tr><td>25</td><td>64.0266083</td><td>-24.0705319</td><td>4.1032</td><td>19.1c</td><td>4.103</td></tr><tr><td>25</td><td>64.0337633</td><td>-24.0748514</td><td>-</td><td>19.1b</td><td>4.103</td></tr><tr><td>25</td><td>64.0401517</td><td>-24.0803403</td><td>-</td><td>19.1a</td><td>4.103</td></tr><tr><td>26†</td><td>64.0468408</td><td>-24.0753850</td><td>3.292</td><td>11a</td><td>3.292</td></tr><tr><td>26</td><td>64.0353038</td><td>-24.0647647</td><td>3.2922</td><td>11c</td><td>3.292</td></tr><tr><td>26</td><td>64.0385875</td><td>-24.0659972</td><td>-</td><td>11b</td><td>3.292</td></tr><tr><td>27</td><td>64.0252696</td><td>-24.0736092</td><td>3.0773</td><td>26c</td><td>3.081</td></tr><tr><td>27</td><td>64.0305596</td><td>-24.0792475</td><td>-</td><td>26b</td><td>3.081</td></tr><tr><td>27</td><td>64.0377979</td><td>-24.0824158</td><td>-</td><td>26a</td><td>3.081</td></tr><tr><td>28</td><td>64.0255200</td><td>-24.0736789</td><td>3.1103</td><td>25c</td><td>3.111</td></tr><tr><td>28</td><td>64.0304467</td><td>-24.0790472</td><td>-</td><td>25b</td><td>3.111</td></tr><tr><td>28</td><td>64.0381496</td><td>-24.0824269</td><td>-</td><td>25a</td><td>3.111</td></tr><tr><td>29</td><td>64.0345663</td><td>-24.0669883</td><td>3.2215</td><td>14.1a</td><td>3.221</td></tr><tr><td>29</td><td>64.0342617</td><td>-24.0665167</td><td>-</td><td>14.1b</td><td>3.221</td></tr><tr><td>29</td><td>64.0340392</td><td>-24.0669400</td><td>3.2240</td><td>14.1d</td><td>3.221</td></tr><tr><td>29</td><td>64.0461275</td><td>-24.0768133</td><td>-</td><td>14.1f</td><td>3.221</td></tr><tr><td>29</td><td>64.0340812</td><td>-24.0664786</td><td>3.2215</td><td>14.1c</td><td>3.221</td></tr><tr><td>30</td><td>64.0401537</td><td>-24.0667703</td><td>3.2882</td><td>9.1b</td><td>3.292</td></tr><tr><td>30</td><td>64.0451842</td><td>-24.0723725</td><td>-</td><td>9.1a</td><td>3.292</td></tr><tr><td>31</td><td>64.0436462</td><td>-24.0590344</td><td>6.1452</td><td>2.1c</td><td>6.145</td></tr><tr><td>31</td><td>64.0509446</td><td>-24.0665642</td><td>-</td><td>2.1a</td><td>6.145</td></tr><tr><td>31</td><td>64.0479096</td><td>-24.0620906</td><td>6.1480</td><td>2.1b</td><td>6.145</td></tr><tr><td>32</td><td>64.0377496</td><td>-24.0607753</td><td>3.6065</td><td>6c</td><td>3.607</td></tr><tr><td>32</td><td>64.0437075</td><td>-24.0644186</td><td>-</td><td>6b</td><td>3.607</td></tr><tr><td>32</td><td>64.0478788</td><td>-24.0701956</td><td>-</td><td>6a</td><td>3.607</td></tr><tr><td>33</td><td>64.0272008</td><td>-24.0736031</td><td>3.9230</td><td>-</td><td>-</td></tr><tr><td>33</td><td>64.0310679</td><td>-24.0772083</td><td>-</td><td>-</td><td>-</td></tr><tr><td>33</td><td>64.0312446</td><td>-24.0770633</td><td>-</td><td>-</td><td>-</td></tr><tr><td>34</td><td>64.0247004</td><td>-24.0714867</td><td>2.5420</td><td>-</td><td>-</td></tr><tr><td>34</td><td>64.0327150</td><td>-24.0785378</td><td>2.5425</td><td>23b</td><td>2.542</td></tr><tr><td>34</td><td>64.0357392</td><td>-24.0799517</td><td>-</td><td>23a</td><td>2.542</td></tr><tr><td>35</td><td>64.0309054</td><td>-24.0837350</td><td>5.9729</td><td>33b</td><td>5.973</td></tr><tr><td>35</td><td>64.0321100</td><td>-24.0842606</td><td>-</td><td>33a</td><td>5.973</td></tr><tr><td>36</td><td>64.0238171</td><td>-24.0785033</td><td>3.9228</td><td>34c</td><td>3.923</td></tr><tr><td>36</td><td>64.0277029</td><td>-24.0826383</td><td>-</td><td>34b</td><td>3.923</td></tr><tr><td>37</td><td>64.0222117</td><td>-24.0773086</td><td>5.6380</td><td>35c</td><td>5.638</td></tr><tr><td>37</td><td>64.0287462</td><td>-24.0842464</td><td>-</td><td>35b</td><td>5.638</td></tr><tr><td>37</td><td>64.0338062</td><td>-24.0857294</td><td>-</td><td>35a</td><td>5.638</td></tr><tr><td>38</td><td>64.0269388</td><td>-24.0757711</td><td>2.3355</td><td>-</td><td>-</td></tr><tr><td>38</td><td>64.0295250</td><td>-24.0786211</td><td>2.3340</td><td>-</td><td>-</td></tr><tr><td>38</td><td>64.0383083</td><td>-24.0830053</td><td>-</td><td>-</td><td>-</td></tr><tr><td>39</td><td>64.0272463</td><td>-24.0682528</td><td>3.9680</td><td>17c</td><td>3.966</td></tr></tbody></table>
<table><caption>**Table B.1. cont.**</caption><thead><tr><th>ID</th><th>RA</th><th>DEC</th><th>z<sub>MUSE</sub></th><th>ID<sub>2</sub></th><th>z<sub>2</sub></th></tr></thead><tbody><tr><td>39</td><td>64.0351867</td><td>-24.0738864</td><td>-</td><td>17b</td><td>3.996</td></tr><tr><td>39</td><td>64.0405671</td><td>-24.0784167</td><td>-</td><td>17a</td><td>3.966</td></tr><tr><td>40</td><td>64.0251204</td><td>-24.0833403</td><td>2.2210</td><td>-</td><td>-</td></tr><tr><td>40</td><td>64.0261767</td><td>-24.0842808</td><td>-</td><td>-</td><td>-</td></tr><tr><td>40</td><td>64.0309450</td><td>-24.0867550</td><td>-</td><td>-</td><td>-</td></tr><tr><td>41</td><td>64.0409938</td><td>-24.0629858</td><td>2.2430</td><td>203c</td><td>2.245</td></tr><tr><td>41</td><td>64.0411571</td><td>-24.0630958</td><td>-</td><td>203b</td><td>2.245</td></tr><tr><td>42</td><td>64.0366754</td><td>-24.0661542</td><td>1.8270</td><td>202.2c</td><td>2.091</td></tr><tr><td>42</td><td>64.0369308</td><td>-24.0663756</td><td>-</td><td>202.2b</td><td>2.091</td></tr><tr><td>43</td><td>64.0291179</td><td>-24.0667792</td><td>1.9530</td><td>-</td><td>-</td></tr><tr><td>43</td><td>64.0375096</td><td>-24.0731747</td><td>-</td><td>-</td><td>-</td></tr><tr><td>43</td><td>64.0386171</td><td>-24.0739053</td><td>-</td><td>-</td><td>-</td></tr><tr><td>44</td><td>64.0238937</td><td>-24.0776206</td><td>1.8178</td><td>-</td><td>-</td></tr><tr><td>44</td><td>64.0305688</td><td>-24.0827056</td><td>-</td><td>-</td><td>-</td></tr><tr><td>44</td><td>64.0325083</td><td>-24.0837817</td><td>-</td><td>-</td><td>-</td></tr><tr><td>45</td><td>64.0377446</td><td>-24.0610592</td><td>4.1138</td><td>103c</td><td>4.115</td></tr><tr><td>45</td><td>64.0429642</td><td>-24.0639314</td><td>-</td><td>103b</td><td>4.115</td></tr><tr><td>45</td><td>64.0482517</td><td>-24.0709208</td><td>-</td><td>103a</td><td>4.115</td></tr><tr><td>46</td><td>64.0258396</td><td>-24.0722631</td><td>2.2800</td><td>-</td><td>-</td></tr><tr><td>46</td><td>64.0329717</td><td>-24.0770292</td><td>-</td><td>-</td><td>-</td></tr><tr><td>46</td><td>64.0377117</td><td>-24.0805028</td><td>-</td><td>-</td><td>-</td></tr><tr><td>47</td><td>64.0362746</td><td>-24.0606817</td><td>2.9220</td><td>107c</td><td>2.921</td></tr><tr><td>47</td><td>64.0448367</td><td>-24.0667225</td><td>-</td><td>107b</td><td>2.921</td></tr><tr><td>47</td><td>64.0461029</td><td>-24.0688289</td><td>-</td><td>107a</td><td>2.921</td></tr><tr><td>48</td><td>64.0400717</td><td>-24.0666814</td><td>3.2883</td><td>9.2b</td><td>3.292</td></tr><tr><td>48</td><td>64.0454287</td><td>-24.0725575</td><td>-</td><td>9.2a</td><td>3.292</td></tr><tr><td>49</td><td>64.0343333</td><td>-24.0630317</td><td>2.2810</td><td>8c</td><td>2.282</td></tr><tr><td>49</td><td>64.0405508</td><td>-24.0663597</td><td>-</td><td>8b</td><td>2.282</td></tr><tr><td>49</td><td>64.0447008</td><td>-24.0715192</td><td>-</td><td>8a</td><td>2.282</td></tr><tr><td>50</td><td>64.0268658</td><td>-24.0692417</td><td>3.2195</td><td>-</td><td>-</td></tr><tr><td>50</td><td>64.0348675</td><td>-24.0746150</td><td>-</td><td>-</td><td>-</td></tr><tr><td>50</td><td>64.0397671</td><td>-24.0788958</td><td>-</td><td>-</td><td>-</td></tr><tr><td>51</td><td>64.0378375</td><td>-24.0598592</td><td>4.1150</td><td>106c</td><td>4.116</td></tr><tr><td>51</td><td>64.0459412</td><td>-24.0658472</td><td>-</td><td>106b</td><td>4.116</td></tr><tr><td>51</td><td>64.0477708</td><td>-24.0686717</td><td>-</td><td>106a</td><td>4.116</td></tr><tr><td>52</td><td>64.0239225</td><td>-24.0750281</td><td>3.2910</td><td>204c</td><td>3.291</td></tr><tr><td>52</td><td>64.0301767</td><td>-24.0809514</td><td>-</td><td>204b</td><td>3.291</td></tr><tr><td>52</td><td>64.0357446</td><td>-24.0836114</td><td>-</td><td>204a</td><td>3.291</td></tr><tr><td>53</td><td>64.0262950</td><td>-24.0760453</td><td>2.9259</td><td>-</td><td>-</td></tr><tr><td>53</td><td>64.0289225</td><td>-24.0791542</td><td>-</td><td>-</td><td>-</td></tr><tr><td>54</td><td>64.0267154</td><td>-24.0747042</td><td>2.2300</td><td>-</td><td>-</td></tr><tr><td>54</td><td>64.0302275</td><td>-24.0786061</td><td>-</td><td>-</td><td>-</td></tr><tr><td>54</td><td>64.0381763</td><td>-24.0823422</td><td>-</td><td>-</td><td>-</td></tr><tr><td>55</td><td>64.0263417</td><td>-24.0687342</td><td>5.1060</td><td>-</td><td>-</td></tr><tr><td>55</td><td>64.0349887</td><td>-24.0746083</td><td>-</td><td>-</td><td>-</td></tr><tr><td>55</td><td>64.0401338</td><td>-24.0791333</td><td>-</td><td>-</td><td>-</td></tr><tr><td>56</td><td>64.0366617</td><td>-24.0633144</td><td>4.3000</td><td>101c</td><td>4.299</td></tr><tr><td>56</td><td>64.0397071</td><td>-24.0642781</td><td>-</td><td>101b</td><td>4.299</td></tr><tr><td>56</td><td>64.0480804</td><td>-24.0742222</td><td>-</td><td>101a</td><td>4.299</td></tr><tr><td>57</td><td>64.0256154</td><td>-24.0713078</td><td>4.5018</td><td>207c</td><td>4.502</td></tr></tbody></table>

<table><caption>**Table B.1. cont.**</caption><thead><tr><th>ID</th><th>RA</th><th>DEC</th><th>z<sub>MUSE</sub></th><th>ID<sub>2</sub></th><th>z<sub>2</sub></th></tr></thead><tbody><tr><td>57</td><td>64.0331017</td><td>-24.0760244</td><td>-</td><td>207b</td><td>4.502</td></tr><tr><td>57</td><td>64.0393000</td><td>-24.0813358</td><td>-</td><td>207a</td><td>4.502</td></tr><tr><td>58†</td><td>64.0235629</td><td>-24.0731711</td><td>3.715</td><td>205c</td><td>3.715</td></tr><tr><td>58</td><td>64.0311558</td><td>-24.0803381</td><td>3.7153</td><td>205b</td><td>3.715</td></tr><tr><td>58</td><td>64.0360404</td><td>-24.0825781</td><td>-</td><td>205a</td><td>3.715</td></tr><tr><td>59</td><td>64.0311383</td><td>-24.0643317</td><td>4.0690</td><td>-</td><td>-</td></tr><tr><td>59</td><td>64.0373113</td><td>-24.0697069</td><td>-</td><td>104b</td><td>4.070</td></tr><tr><td>59</td><td>64.0439900</td><td>-24.0750847</td><td>-</td><td>104a</td><td>4.070</td></tr><tr><td>60†</td><td>64.0493708</td><td>-24.0710067</td><td>6.145</td><td>112a</td><td>6.145</td></tr><tr><td>60</td><td>64.0389625</td><td>-24.0606803</td><td>6.1470</td><td>112c</td><td>6.145</td></tr><tr><td>60</td><td>64.0433679</td><td>-24.0629836</td><td>-</td><td>112b</td><td>6.145</td></tr><tr><td>61†</td><td>64.0508671</td><td>-24.0664261</td><td>6.145</td><td>2.2a</td><td>6.145</td></tr><tr><td>61</td><td>64.0434875</td><td>-24.0589444</td><td>6.1480</td><td>2.2c</td><td>6.145</td></tr><tr><td>61</td><td>64.0482517</td><td>-24.0624331</td><td>6.1452</td><td>2.2b</td><td>6.145</td></tr><tr><td>62</td><td>64.0335671</td><td>-24.0650447</td><td>4.6090</td><td>108c</td><td>4.607</td></tr><tr><td>62</td><td>64.0369179</td><td>-24.0680367</td><td>-</td><td>-</td><td>-</td></tr><tr><td>62†</td><td>64.0366592</td><td>-24.0680269</td><td>4.607</td><td>108b alt</td><td>4.607</td></tr><tr><td>62</td><td>64.0465858</td><td>-24.0762047</td><td>4.6090</td><td>108a</td><td>4.607</td></tr><tr><td>63</td><td>64.0333479</td><td>-24.0674986</td><td>1.1472</td><td>201c</td><td>1.147</td></tr><tr><td>63</td><td>64.0404400</td><td>-24.0730881</td><td>-</td><td>201a</td><td>1.147</td></tr><tr><td>64</td><td>64.0338125</td><td>-24.0657972</td><td>4.0707</td><td>105c</td><td>4.071</td></tr><tr><td>64</td><td>64.0360046</td><td>-24.0678972</td><td>-</td><td>105b</td><td>4.071</td></tr><tr><td>64</td><td>64.0464792</td><td>-24.0767511</td><td>-</td><td>105a</td><td>4.071</td></tr><tr><td>65</td><td>64.0392596</td><td>-24.0698083</td><td>4.3000</td><td>110b</td><td>4.298</td></tr><tr><td>65</td><td>64.0427467</td><td>-24.0721897</td><td>-</td><td>110a</td><td>4.298</td></tr><tr><td>66</td><td>64.0407246</td><td>-24.0591806</td><td>5.0996</td><td>209c</td><td>5.100</td></tr><tr><td>66</td><td>64.0468725</td><td>-24.0638661</td><td>-</td><td>209b</td><td>5.100</td></tr><tr><td>66</td><td>64.0496512</td><td>-24.0680561</td><td>-</td><td>209a</td><td>5.0996</td></tr><tr><td>67</td><td>64.0399400</td><td>-24.0669239</td><td>5.9980</td><td>113b</td><td>5.995</td></tr><tr><td>67</td><td>64.0459558</td><td>-24.0740589</td><td>-</td><td>113a</td><td>5.995</td></tr><tr><td>68</td><td>64.0364992</td><td>-24.0622758</td><td>6.0664</td><td>102c</td><td>6.064</td></tr><tr><td>68</td><td>64.0411162</td><td>-24.0641319</td><td>-</td><td>102b</td><td>6.064</td></tr><tr><td>68</td><td>64.0484808</td><td>-24.0735939</td><td>-</td><td>102a</td><td>6.064</td></tr><tr><td>69</td><td>64.0378317</td><td>-24.0688725</td><td>2.9911</td><td>109b</td><td>2.989</td></tr><tr><td>69</td><td>64.0438154</td><td>-24.0737064</td><td>-</td><td>109a</td><td>2.989</td></tr><tr><td>70</td><td>64.0289792</td><td>-24.0755694</td><td>3.0750</td><td>-</td><td>-</td></tr><tr><td>70</td><td>64.0296996</td><td>-24.0762108</td><td>-</td><td>-</td><td>-</td></tr><tr><td>71</td><td>64.0247200</td><td>-24.0771819</td><td>4.5300</td><td>208c</td><td>4.530</td></tr><tr><td>71</td><td>64.0277733</td><td>-24.0806775</td><td>-</td><td>208b</td><td>4.530</td></tr><tr><td>72</td><td>64.0308846</td><td>-24.0837572</td><td>5.9730</td><td>-</td><td>-</td></tr><tr><td>72</td><td>64.0320288</td><td>-24.0842872</td><td>-</td><td>-</td><td>-</td></tr><tr><td>73†</td><td>64.0455825</td><td>-24.0727003</td><td>3.2920</td><td>9.3a</td><td>3.29</td></tr><tr><td>73†</td><td>64.0400004</td><td>-24.0666450</td><td>3.2920</td><td>9.3b</td><td>3.29</td></tr><tr><td>74†</td><td>64.0507508</td><td>-24.0658519</td><td>6.1490</td><td>210a</td><td>6.149</td></tr><tr><td>74†</td><td>64.0472208</td><td>-24.0611967</td><td>6.1490</td><td>210b</td><td>6.149</td></tr><tr><td>74†</td><td>64.0445658</td><td>-24.0593369</td><td>6.1490</td><td>210c</td><td>6.149</td></tr><tr><td>75†</td><td>64.0414950</td><td>-24.0626300</td><td>6.1490</td><td>210.4b</td><td>6.149</td></tr><tr><td>75†</td><td>64.0401050</td><td>-24.0618739</td><td>6.1490</td><td>210.4c</td><td>6.149</td></tr><tr><td>76†</td><td>64.0458967</td><td>-24.0602003</td><td>6.6290</td><td>211b</td><td>6.629</td></tr><tr><td>76†</td><td>64.0456046</td><td>-24.0600436</td><td>6.6290</td><td>211c</td><td>6.629</td></tr></tbody></table>
<table><caption>Table B.1. cont.</caption><thead><tr><th>ID</th><th>RA</th><th>DEC</th><th>z<sub>MUSE</sub></th><th>ID<sub>2</sub></th><th>z<sub>2</sub></th></tr></thead><tbody><tr><td>39</td><td>64.0351867</td><td>-24.0738864</td><td>-</td><td>17b</td><td>3.996</td></tr><tr><td>39</td><td>64.0405671</td><td>-24.0784167</td><td>-</td><td>17a</td><td>3.966</td></tr><tr><td>40</td><td>64.0251204</td><td>-24.0833403</td><td>2.2210</td><td>-</td><td>-</td></tr><tr><td>40</td><td>64.0261767</td><td>-24.0842808</td><td>-</td><td>-</td><td>-</td></tr><tr><td>40</td><td>64.0309450</td><td>-24.0867550</td><td>-</td><td>-</td><td>-</td></tr><tr><td>41</td><td>64.0409938</td><td>-24.0629858</td><td>2.2430</td><td>203c</td><td>2.245</td></tr><tr><td>41</td><td>64.0411571</td><td>-24.0630958</td><td>-</td><td>203b</td><td>2.245</td></tr><tr><td>42</td><td>64.0366754</td><td>-24.0661542</td><td>1.8270</td><td>202.2c</td><td>2.091</td></tr><tr><td>42</td><td>64.0369308</td><td>-24.0663756</td><td>-</td><td>202.2b</td><td>2.091</td></tr><tr><td>43</td><td>64.0291179</td><td>-24.0667792</td><td>1.9530</td><td>-</td><td>-</td></tr><tr><td>43</td><td>64.0375096</td><td>-24.0731747</td><td>-</td><td>-</td><td>-</td></tr><tr><td>43</td><td>64.0386171</td><td>-24.0739053</td><td>-</td><td>-</td><td>-</td></tr><tr><td>44</td><td>64.0238937</td><td>-24.0776206</td><td>1.8178</td><td>-</td><td>-</td></tr><tr><td>44</td><td>64.0305688</td><td>-24.0827056</td><td>-</td><td>-</td><td>-</td></tr><tr><td>44</td><td>64.0325083</td><td>-24.0837817</td><td>-</td><td>-</td><td>-</td></tr><tr><td>45</td><td>64.0377446</td><td>-24.0610592</td><td>4.1138</td><td>103c</td><td>4.115</td></tr><tr><td>45</td><td>64.0429642</td><td>-24.0639314</td><td>-</td><td>103b</td><td>4.115</td></tr><tr><td>45</td><td>64.0482517</td><td>-24.0709208</td><td>-</td><td>103a</td><td>4.115</td></tr><tr><td>46</td><td>64.0258396</td><td>-24.0722631</td><td>2.2800</td><td>-</td><td>-</td></tr><tr><td>46</td><td>64.0329717</td><td>-24.0770292</td><td>-</td><td>-</td><td>-</td></tr><tr><td>46</td><td>64.0377117</td><td>-24.0805028</td><td>-</td><td>-</td><td>-</td></tr><tr><td>47</td><td>64.0362746</td><td>-24.0606817</td><td>2.9220</td><td>107c</td><td>2.921</td></tr><tr><td>47</td><td>64.0448367</td><td>-24.0667225</td><td>-</td><td>107b</td><td>2.921</td></tr><tr><td>47</td><td>64.0461029</td><td>-24.0688289</td><td>-</td><td>107a</td><td>2.921</td></tr><tr><td>48</td><td>64.0400717</td><td>-24.0666814</td><td>3.2883</td><td>9.2b</td><td>3.292</td></tr><tr><td>48</td><td>64.0454287</td><td>-24.0725575</td><td>-</td><td>9.2a</td><td>3.292</td></tr><tr><td>49</td><td>64.0343333</td><td>-24.0630317</td><td>2.2810</td><td>8c</td><td>2.282</td></tr><tr><td>49</td><td>64.0405508</td><td>-24.0663597</td><td>-</td><td>8b</td><td>2.282</td></tr><tr><td>49</td><td>64.0447008</td><td>-24.0715192</td><td>-</td><td>8a</td><td>2.282</td></tr><tr><td>50</td><td>64.0268658</td><td>-24.0692417</td><td>3.2195</td><td>-</td><td>-</td></tr><tr><td>50</td><td>64.0348675</td><td>-24.0746150</td><td>-</td><td>-</td><td>-</td></tr><tr><td>50</td><td>64.0397671</td><td>-24.0788958</td><td>-</td><td>-</td><td>-</td></tr><tr><td>51</td><td>64.0378375</td><td>-24.0598592</td><td>4.1150</td><td>106c</td><td>4.116</td></tr><tr><td>51</td><td>64.0459412</td><td>-24.0658472</td><td>-</td><td>106b</td><td>4.116</td></tr><tr><td>51</td><td>64.0477708</td><td>-24.0686717</td><td>-</td><td>106a</td><td>4.116</td></tr><tr><td>52</td><td>64.0239225</td><td>-24.0750281</td><td>3.2910</td><td>204c</td><td>3.291</td></tr><tr><td>52</td><td>64.0301767</td><td>-24.0809514</td><td>-</td><td>204b</td><td>3.291</td></tr><tr><td>52</td><td>64.0357446</td><td>-24.0836114</td><td>-</td><td>204a</td><td>3.291</td></tr><tr><td>53</td><td>64.0262950</td><td>-24.0760453</td><td>2.9259</td><td>-</td><td>-</td></tr><tr><td>53</td><td>64.0289225</td><td>-24.0791542</td><td>-</td><td>-</td><td>-</td></tr><tr><td>54</td><td>64.0267154</td><td>-24.0747042</td><td>2.2300</td><td>-</td><td>-</td></tr><tr><td>54</td><td>64.0302275</td><td>-24.0786061</td><td>-</td><td>-</td><td>-</td></tr><tr><td>54</td><td>64.0381763</td><td>-24.0823422</td><td>-</td><td>-</td><td>-</td></tr><tr><td>55</td><td>64.0263417</td><td>-24.0687342</td><td>5.1060</td><td>-</td><td>-</td></tr><tr><td>55</td><td>64.0349887</td><td>-24.0746083</td><td>-</td><td>-</td><td>-</td></tr><tr><td>55</td><td>64.0401338</td><td>-24.0791333</td><td>-</td><td>-</td><td>-</td></tr><tr><td>56</td><td>64.0366617</td><td>-24.0633144</td><td>4.3000</td><td>101c</td><td>4.299</td></tr><tr><td>56</td><td>64.0397071</td><td>-24.0642781</td><td>-</td><td>101b</td><td>4.299</td></tr><tr><td>56</td><td>64.0480804</td><td>-24.0742222</td><td>-</td><td>101a</td><td>4.299</td></tr><tr><td>57</td><td>64.0256154</td><td>-24.0713078</td><td>4.5018</td><td>207c</td><td>4.502</td></tr></tbody></table>

<table><caption>Table B.1. cont.</caption><thead><tr><th>ID</th><th>RA</th><th>DEC</th><th>z<sub>MUSE</sub></th><th>ID<sub>2</sub></th><th>z<sub>2</sub></th></tr></thead><tbody><tr><td>57</td><td>64.0331017</td><td>-24.0760244</td><td>-</td><td>207b</td><td>4.502</td></tr><tr><td>57</td><td>64.0393000</td><td>-24.0813358</td><td>-</td><td>207a</td><td>4.502</td></tr><tr><td>58†</td><td>64.0235629</td><td>-24.0731711</td><td>3.715</td><td>205c</td><td>3.715</td></tr><tr><td>58</td><td>64.0311558</td><td>-24.0803381</td><td>3.7153</td><td>205b</td><td>3.715</td></tr><tr><td>58</td><td>64.0360404</td><td>-24.0825781</td><td>-</td><td>205a</td><td>3.715</td></tr><tr><td>59</td><td>64.0311383</td><td>-24.0643317</td><td>4.0690</td><td>-</td><td>-</td></tr><tr><td>59</td><td>64.0373113</td><td>-24.0697069</td><td>-</td><td>104b</td><td>4.070</td></tr><tr><td>59</td><td>64.0439900</td><td>-24.0750847</td><td>-</td><td>104a</td><td>4.070</td></tr><tr><td>60†</td><td>64.0493708</td><td>-24.0710067</td><td>6.145</td><td>112a</td><td>6.145</td></tr><tr><td>60</td><td>64.0389625</td><td>-24.0606803</td><td>6.1470</td><td>112c</td><td>6.145</td></tr><tr><td>60</td><td>64.0433679</td><td>-24.0629836</td><td>-</td><td>112b</td><td>6.145</td></tr><tr><td>61†</td><td>64.0508671</td><td>-24.0664261</td><td>6.145</td><td>2.2a</td><td>6.145</td></tr><tr><td>61</td><td>64.0434875</td><td>-24.0589444</td><td>6.1480</td><td>2.2c</td><td>6.145</td></tr><tr><td>61</td><td>64.0482517</td><td>-24.0624331</td><td>6.1452</td><td>2.2b</td><td>6.145</td></tr><tr><td>62</td><td>64.0335671</td><td>-24.0650447</td><td>4.6090</td><td>108c</td><td>4.607</td></tr><tr><td>62</td><td>64.0369179</td><td>-24.0680367</td><td>-</td><td>-</td><td>-</td></tr><tr><td>62†</td><td>64.0366592</td><td>-24.0680269</td><td>4.607</td><td>108b alt</td><td>4.607</td></tr><tr><td>62</td><td>64.0465858</td><td>-24.0762047</td><td>4.6090</td><td>108a</td><td>4.607</td></tr><tr><td>63</td><td>64.0333479</td><td>-24.0674986</td><td>1.1472</td><td>201c</td><td>1.147</td></tr><tr><td>63</td><td>64.0404400</td><td>-24.0730881</td><td>-</td><td>201a</td><td>1.147</td></tr><tr><td>64</td><td>64.0338125</td><td>-24.0657972</td><td>4.0707</td><td>105c</td><td>4.071</td></tr><tr><td>64</td><td>64.0360046</td><td>-24.0678972</td><td>-</td><td>105b</td><td>4.071</td></tr><tr><td>64</td><td>64.0464792</td><td>-24.0767511</td><td>-</td><td>105a</td><td>4.071</td></tr><tr><td>65</td><td>64.0392596</td><td>-24.0698083</td><td>4.3000</td><td>110b</td><td>4.298</td></tr><tr><td>65</td><td>64.0427467</td><td>-24.0721897</td><td>-</td><td>110a</td><td>4.298</td></tr><tr><td>66</td><td>64.0407246</td><td>-24.0591806</td><td>5.0996</td><td>209c</td><td>5.100</td></tr><tr><td>66</td><td>64.0468725</td><td>-24.0638661</td><td>-</td><td>209b</td><td>5.100</td></tr><tr><td>66</td><td>64.0496512</td><td>-24.0680561</td><td>-</td><td>209a</td><td>5.0996</td></tr><tr><td>67</td><td>64.0399400</td><td>-24.0669239</td><td>5.9980</td><td>113b</td><td>5.995</td></tr><tr><td>67</td><td>64.0459558</td><td>-24.0740589</td><td>-</td><td>113a</td><td>5.995</td></tr><tr><td>68</td><td>64.0364992</td><td>-24.0622758</td><td>6.0664</td><td>102c</td><td>6.064</td></tr><tr><td>68</td><td>64.0411162</td><td>-24.0641319</td><td>-</td><td>102b</td><td>6.064</td></tr><tr><td>68</td><td>64.0484808</td><td>-24.0735939</td><td>-</td><td>102a</td><td>6.064</td></tr><tr><td>69</td><td>64.0378317</td><td>-24.0688725</td><td>2.9911</td><td>109b</td><td>2.989</td></tr><tr><td>69</td><td>64.0438154</td><td>-24.0737064</td><td>-</td><td>109a</td><td>2.989</td></tr><tr><td>70</td><td>64.0289792</td><td>-24.0755694</td><td>3.0750</td><td>-</td><td>-</td></tr><tr><td>70</td><td>64.0296996</td><td>-24.0762108</td><td>-</td><td>-</td><td>-</td></tr><tr><td>71</td><td>64.0247200</td><td>-24.0771819</td><td>4.5300</td><td>208c</td><td>4.530</td></tr><tr><td>71</td><td>64.0277733</td><td>-24.0806775</td><td>-</td><td>208b</td><td>4.530</td></tr><tr><td>72</td><td>64.0308846</td><td>-24.0837572</td><td>5.9730</td><td>-</td><td>-</td></tr><tr><td>72</td><td>64.0320288</td><td>-24.0842872</td><td>-</td><td>-</td><td>-</td></tr><tr><td>73†</td><td>64.0455825</td><td>-24.0727003</td><td>3.2920</td><td>9.3a</td><td>3.29</td></tr><tr><td>73†</td><td>64.0400004</td><td>-24.0666450</td><td>3.2920</td><td>9.3b</td><td>3.29</td></tr><tr><td>74†</td><td>64.0507508</td><td>-24.0658519</td><td>6.1490</td><td>210a</td><td>6.149</td></tr><tr><td>74†</td><td>64.0472208</td><td>-24.0611967</td><td>6.1490</td><td>210b</td><td>6.149</td></tr><tr><td>74†</td><td>64.0445658</td><td>-24.0593369</td><td>6.1490</td><td>210c</td><td>6.149</td></tr><tr><td>75†</td><td>64.0414950</td><td>-24.0626300</td><td>6.1490</td><td>210.4b</td><td>6.149</td></tr><tr><td>75†</td><td>64.0401050</td><td>-24.0618739</td><td>6.1490</td><td>210.4c</td><td>6.149</td></tr><tr><td>76†</td><td>64.0458967</td><td>-24.0602003</td><td>6.6290</td><td>211b</td><td>6.629</td></tr><tr><td>76†</td><td>64.0456046</td><td>-24.0600436</td><td>6.6290</td><td>211c</td><td>6.629</td></tr></tbody></table>