<figure><img src="image_9.png" /><figcaption>**Fig. 8.** Cumulative number counts as shown in Fig. 7 in comparison with the models from Zavala et al. (2021), Bisigello et al. (2021; i.e. the SPRITZ simulation), and Gkogkou et al. (2023; i.e. the SIDES model). The shaded regions of the SIDES curves indicate one standard deviation due to cosmic variance for an area corresponding to our coverage in each band.</figcaption></figure>

Thanks to the sensitivity of ALMA and the large areal coverage offered by the combination of all ALMA archive projects in the COSMOS and GOODS-S fields, the number counts inferred here bridge the flux density gap between single-dish observations, which constrain only the bright end of the number counts, and pencil beam ALMA blind surveys, which constrain the faint end of the number counts (albeit with large uncertainties due to cosmic variance). In the flux density ranges in common, our number counts are overall in good agreement with those retrieved from the literature. Finally, down to the flux density limits of our number counts, about 40%, 34%, 18%, and 4% of the CIB is resolved into discrete sources in Bands 7, 6, 4, and 3, respectively. Thus, while the emission of DSFGs is well constrained at 0.9–1.3 mm wavelengths, it remains poorly understood at longer wavelengths. However, as ALMA observations are far less limited by confusion noise than single-dish facilities, future deeper observations with ALMA will undoubtedly allow us to resolve an ever larger fraction of the CIB at these wavelengths.

The homogeneous measurements of the number counts in ALMA Bands 3, 4, 6, and 7 over such broad flux density ranges offer unique constraints for galaxy evolution models. We thus compared our number counts with predictions from the models of Zavala et al. (2021), SPRITZ (Bisigello et al. 2021), and SIDES (Béthermin et al. 2017; Gkogkou et al. 2023). We find that our results are in very good agreement with the predictions from the model of Zavala et al. (2021). The SPRITZ model is consistent with our number counts in Bands 6 and 7, but its predictions are significantly higher than the observations at longer wavelengths (i.e. Bands 3 and 4). Our number counts are overall in good agreement with those predicted by the SIDES simulation, even thought this one was not explicitly fitted to any measured (sub)millimetre number counts. However, this simulation tends to slightly underestimate the number of bright sources in Bands 4 and 7.

We have shown that it is possible to infer number counts from the heterogeneous target-driven $A^3\text{COSMOS}$ database. This method naturally has limitations; however, the effects on the number counts are negligible compared to the effects of cosmic variance. Our analysis therefore profits from the large areal coverage of the $A^3\text{COSMOS}$ database and the availability of blind surveys within it. Future updates will allow us to decrease the cosmic variance and further improve the accuracy of the measured number counts.

**Acknowledgements.** S.A. would like to thank Laura Bisigello and Jorge Zavala for providing the curves of their respective models. F.B. and S.A. gratefully acknowledge the Collaborative Research Center 1601 (SFB 1601 sub-project C2) funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 500700252. I.D. acknowledges support from INAF Minigrant "Harnessing the power of VLBA towards a census of AGN and star formation at high redshift". This paper makes use of the following ALMA data: ADS/JAO.ALMA#2011.0.00064.S, ADS/JAO.ALMA#2011.0.00097.S, ADS/JAO.ALMA#2011.0.00539.S, ADS/JAO.ALMA#2012.1.00076.S, ADS/JAO.ALMA#2012.1.00307.S, ADS/JAO.ALMA#2012.1.00523.S, ADS/JAO.ALMA#2012.1.00775.S, ADS/JAO.ALMA#2012.1.00919.S, ADS/JAO.ALMA#2012.1.00978.S, ADS/JAO.ALMA#2013.1.00034.S, ADS/JAO.ALMA#2013.1.00118.S, ADS/JAO.ALMA#2013.1.00146.S, ADS/JAO.ALMA#2013.1.00171.S, ADS/JAO.ALMA#2013.1.00208.S, ADS/JAO.ALMA#2013.1.00276.S, ADS/JAO.ALMA#2013.1.00668.S, ADS/JAO.ALMA#2013.1.00786.S, ADS/JAO.ALMA#2013.1.00836.S, ADS/JAO.ALMA#2013.1.00914.S, ADS/JAO.ALMA#2013.1.01271.S, ADS/JAO.ALMA#2015.1.00026.S, ADS/JAO.ALMA#2015.1.00040.S, ADS/JAO.ALMA#2015.1.00098.S, ADS/JAO.ALMA#2015.1.00173.S, ADS/JAO.ALMA#2012.1.00323.S, ADS/JAO.ALMA#2012.1.00536.S, ADS/JAO.ALMA#2012.1.00869.S, ADS/JAO.ALMA#2012.1.00952.S, ADS/JAO.ALMA#2012.1.00983.S, ADS/JAO.ALMA#2013.1.00092.S, ADS/JAO.ALMA#2013.1.00139.S, ADS/JAO.ALMA#2013.1.00151.S, ADS/JAO.ALMA#2013.1.00205.S, ADS/JAO.ALMA#2013.1.00250.S, ADS/JAO.ALMA#2013.1.00470.S, ADS/JAO.ALMA#2013.1.00718.S, ADS/JAO.ALMA#2013.1.00815.S, ADS/JAO.ALMA#2013.1.00884.S, ADS/JAO.ALMA#2013.1.01258.S, ADS/JAO.ALMA#2013.1.01292.S, ADS/JAO.ALMA#2015.1.00039.S, ADS/JAO.ALMA#2015.1.00055.S, ADS/JAO.ALMA#2015.1.00122.S,
<figure><img src="image_9.png" /><figcaption>**Fig. 8.** Cumulative number counts as shown in Fig. 7 in comparison with the models from Zavala et al. (2021), Bisigello et al. (2021; i.e. the SPRITZ simulation), and Gkogkou et al. (2023; i.e. the SIDES model). The shaded regions of the SIDES curves indicate one standard deviation due to cosmic variance for an area corresponding to our coverage in each band.</figcaption></figure>

Thanks to the sensitivity of ALMA and the large areal coverage offered by the combination of all ALMA archive projects in the COSMOS and GOODS-S fields, the number counts inferred here bridge the flux density gap between single-dish observations, which constrain only the bright end of the number counts, and pencil beam ALMA blind surveys, which constrain the faint end of the number counts (albeit with large uncertainties due to cosmic variance). In the flux density ranges in common, our number counts are overall in good agreement with those retrieved from the literature. Finally, down to the flux density limits of our number counts, about 40%, 34%, 18%, and 4% of the CIB is resolved into discrete sources in Bands 7, 6, 4, and 3, respectively. Thus, while the emission of DSFGs is well constrained at 0.9–1.3 mm wavelengths, it remains poorly understood at longer wavelengths. However, as ALMA observations are far less limited by confusion noise than single-dish facilities, future deeper observations with ALMA will undoubtedly allow us to resolve an ever larger fraction of the CIB at these wavelengths.

The homogeneous measurements of the number counts in ALMA Bands 3, 4, 6, and 7 over such broad flux density ranges offer unique constraints for galaxy evolution models. We thus compared our number counts with predictions from the models of Zavala et al. (2021), SPRITZ (Bisigello et al. 2021), and SIDES (Béthermin et al. 2017; Gkogkou et al. 2023). We find that our results are in very good agreement with the predictions from the model of Zavala et al. (2021). The SPRITZ model is consistent with our number counts in Bands 6 and 7, but its predictions are significantly higher than the observations at longer wavelengths (i.e. Bands 3 and 4). Our number counts are overall in good agreement with those predicted by the SIDES simulation, even thought this one was not explicitly fitted to any measured (sub)millimetre number counts. However, this simulation tends to slightly underestimate the number of bright sources in Bands 4 and 7.

We have shown that it is possible to infer number counts from the heterogeneous target-driven $A^3\text{COSMOS}$ database. This method naturally has limitations; however, the effects on the number counts are negligible compared to the effects of cosmic variance. Our analysis therefore profits from the large areal coverage of the $A^3\text{COSMOS}$ database and the availability of blind surveys within it. Future updates will allow us to decrease the cosmic variance and further improve the accuracy of the measured number counts.

*Acknowledgements.* S.A. would like to thank Laura Bisigello and Jorge Zavala for providing the curves of their respective models. F.B. and S.A. gratefully acknowledge the Collaborative Research Center 1601 (SFB 1601 sub-project C2) funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 500700252. I.D. acknowledges support from INAF Minigrant "Harnessing the power of VLBA towards a census of AGN and star formation at high redshift". This paper makes use of the following ALMA data: ADS/JAO.ALMA#2011.0.00064.S, ADS/JAO.ALMA#2011.0.00097.S, ADS/JAO.ALMA#2011.0.00539.S, ADS/JAO.ALMA#2012.1.00076.S, ADS/JAO.ALMA#2012.1.00307.S, ADS/JAO.ALMA#2012.1.00523.S, ADS/JAO.ALMA#2012.1.00775.S, ADS/JAO.ALMA#2012.1.00919.S, ADS/JAO.ALMA#2012.1.00978.S, ADS/JAO.ALMA#2013.1.00034.S, ADS/JAO.ALMA#2013.1.00118.S, ADS/JAO.ALMA#2013.1.00146.S, ADS/JAO.ALMA#2013.1.00171.S, ADS/JAO.ALMA#2013.1.00208.S, ADS/JAO.ALMA#2013.1.00276.S, ADS/JAO.ALMA#2013.1.00668.S, ADS/JAO.ALMA#2013.1.00786.S, ADS/JAO.ALMA#2013.1.00836.S, ADS/JAO.ALMA#2013.1.00914.S, ADS/JAO.ALMA#2013.1.01271.S, ADS/JAO.ALMA#2015.1.00026.S, ADS/JAO.ALMA#2015.1.00040.S, ADS/JAO.ALMA#2015.1.00098.S, ADS/JAO.ALMA#2015.1.000742.S, ADS/JAO.ALMA#2012.1.00173.S, ADS/JAO.ALMA#2012.1.00323.S, ADS/JAO.ALMA#2012.1.00536.S, ADS/JAO.ALMA#2012.1.00869.S, ADS/JAO.ALMA#2012.1.00952.S, ADS/JAO.ALMA#2012.1.00983.S, ADS/JAO.ALMA#2013.1.00092.S, ADS/JAO.ALMA#2013.1.00139.S, ADS/JAO.ALMA#2013.1.00151.S, ADS/JAO.ALMA#2013.1.00205.S, ADS/JAO.ALMA#2013.1.00250.S, ADS/JAO.ALMA#2013.1.00470.S, ADS/JAO.ALMA#2013.1.00718.S, ADS/JAO.ALMA#2013.1.00815.S, ADS/JAO.ALMA#2013.1.00884.S, ADS/JAO.ALMA#2013.1.01258.S, ADS/JAO.ALMA#2013.1.01292.S, ADS/JAO.ALMA#2015.1.00039.S, ADS/JAO.ALMA#2015.1.00055.S, ADS/JAO.ALMA#2015.1.00122.S,
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ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.

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ADS/JAO.ALMA#2015.1.00137.S,
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ADS/JAO.ALMA#2018.A.00022.S,
ADS/JAO.ALMA#2019.1.00074.S,

ADS/JAO.ALMA#2015.1.00207.S,
ADS/JAO.ALMA#2015.1.00228.S,
ADS/JAO.ALMA#2015.1.00299.S,
ADS/JAO.ALMA#2015.1.00388.S,
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ADS/JAO.ALMA#2016.1.01155.S,
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ADS/JAO.ALMA#2016.1.01604.S,
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ADS/JAO.ALMA#2018.1.00216.S,
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ADS/JAO.ALMA#2018.1.00251.S,
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ADS/JAO.ALMA#2018.1.01136.S,
ADS/JAO.ALMA#2018.1.01281.S,
ADS/JAO.ALMA#2018.1.01521.S,
ADS/JAO.ALMA#2018.1.01551.S,
ADS/JAO.ALMA#2018.1.01605.S,
ADS/JAO.ALMA#2018.1.01739.S,
ADS/JAO.ALMA#2018.1.01841.S,
ADS/JAO.ALMA#2018.1.01871.S,
ADS/JAO.ALMA#2018.A.00037.S,
ADS/JAO.ALMA#2019.1.00102.S,

ADS/JAO.ALMA#2019.1.00151.S,
ADS/JAO.ALMA#2019.1.00399.S,
ADS/JAO.ALMA#2019.1.00477.S,
ADS/JAO.ALMA#2019.1.00678.S,
ADS/JAO.ALMA#2019.1.00909.S,
ADS/JAO.ALMA#2019.1.01075.S,
ADS/JAO.ALMA#2019.1.01142.S,
ADS/JAO.ALMA#2019.1.01286.S,
ADS/JAO.ALMA#2019.1.01491.S,
ADS/JAO.ALMA#2019.1.01528.S,
ADS/JAO.ALMA#2019.1.01600.S,
ADS/JAO.ALMA#2019.1.01634.L,
ADS/JAO.ALMA#2019.1.01722.S,
ADS/JAO.ALMA#2019.2.00118.S,
ADS/JAO.ALMA#2019.2.00246.S,

ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.

ADS/JAO.ALMA#2019.1.00397.S,
ADS/JAO.ALMA#2019.1.00459.S,
ADS/JAO.ALMA#2019.1.00652.S,
ADS/JAO.ALMA#2019.1.00702.S,
ADS/JAO.ALMA#2019.1.00964.S,
ADS/JAO.ALMA#2019.1.01127.S,
ADS/JAO.ALMA#2019.1.01201.S,
ADS/JAO.ALMA#2019.1.01329.S,
ADS/JAO.ALMA#2019.1.01524.S,
ADS/JAO.ALMA#2019.1.01537.S,
ADS/JAO.ALMA#2019.1.01615.S,
ADS/JAO.ALMA#2019.1.01702.S,
ADS/JAO.ALMA#2019.1.01832.S,
ADS/JAO.ALMA#2019.2.00143.S,
ADS/JAO.ALMA#2019.A.00015.S.

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# Appendix A: Counterpart association

Here, we provide an overview of the counterpart association method and the machine learning (ML) model applied in the process. For a detailed description we refer the reader to the work of Liu et al. (2019a, their Sect. 4.2 and Appendix D).

To assess the accuracy of the association between priors and extracted ALMA sources, we made use of several ancillary images from optical and near-infrared instruments covering the COSMOS and GOODS-S fields, as listed at the end of Sects. 2.2 and 2.3, respectively. The following parameters were then measured for each ALMA source-counterpart pair:

- The angular separation between the positions of the ALMA source and the counterpart, normalised by the angular radius of the ALMA source ('Sep.');

- The signal-to-noise ratio of the total flux of the ALMA source (S/N<sub>ALMA</sub>);

- The signal-to-noise ratio measured in the counterpart images at the positions of the ALMA source (S/N<sub>S</sub>) and at the reference position of the counterpart source (S/N<sub>Ref.</sub>), determined via an aperture photometry approach;

- The amount of extended emission in the counterpart images in-between the ALMA and counterpart source positions, determined by measuring the flux in a series of apertures placed between the respective positions.

To assess the reliability of the association based on these parameters, Liu et al. (2019a) trained a supervised ML model using the Python package `Scikit-learn` (Pedregosa et al. 2011) with the Gaussian Process classifier. The training data for the model consisted of a $A^3\text{COSMOS}$ subsample of ~600 ran-

domly selected sources which had been manually classified as either robust or spurious counterpart. The accuracy of the trained model was then assessed by running it on a validation sample of another ~400 manually classified sources not contained in the training data. They found an agreement between the manual and ML classification for ~97% of the sources in the validation sample. The same accuracy was also obtained for the combined sample of ~1000 sources.

For our $A^3\text{COSMOS}$ database update, we applied this trained model to determine the reliability of our counterpart association for the newer COSMOS data, using the same counterpart images as Liu et al. (2019a), only replacing the Spitzer/IRAC 3.6 µm image by a newer image from Moneti et al. (2022) (see Sect. 2.2). However, naturally, new counterpart images and most importantly different filters had to be used for GOODS-S, making the model not applicable to $A^3\text{GOODSS}$. The number of sources in our current version of the $A^3\text{GOODSS}$ prior source catalogue (i.e. 568) approximately corresponds to the size of the initial training dataset of the ML model. We therefore classified all $A^3\text{GOODSS}$ sources manually, creating the training dataset for the ML model in preparation for the future updates of the $A^3\text{GOODSS}$ database.

# Appendix B: $A^3\text{COSMOS}$ and $A^3\text{GOODSS}$ sky coverage by band

Here, we show the spatial sky coverage of our $A^3\text{COSMOS}$ and $A^3\text{GOODSS}$ databases, as displayed cumulatively in Fig. 1, separately for each ALMA band. The values of the individual coverages are also listed in Table 1.
# Appendix A: Counterpart association

Here, we provide an overview of the counterpart association method and the machine learning (ML) model applied in the process. For a detailed description we refer the reader to the work of Liu et al. (2019a, their Sect. 4.2 and Appendix D).

To assess the accuracy of the association between priors and extracted ALMA sources, we made use of several ancillary images from optical and near-infrared instruments covering the COSMOS and GOODS-S fields, as listed at the end of Sects. 2.2 and 2.3, respectively. The following parameters were then measured for each ALMA source-counterpart pair:

- The angular separation between the positions of the ALMA source and the counterpart, normalised by the angular radius of the ALMA source ('Sep.');

- The signal-to-noise ratio of the total flux of the ALMA source (S/N<sub>ALMA</sub>);

- The signal-to-noise ratio measured in the counterpart images at the positions of the ALMA source (S/N<sub>S</sub>) and at the reference position of the counterpart source (S/N<sub>Ref.</sub>), determined via an aperture photometry approach;

- The amount of extended emission in the counterpart images in-between the ALMA and counterpart source positions, determined by measuring the flux in a series of apertures placed between the respective positions.

To assess the reliability of the association based on these parameters, Liu et al. (2019a) trained a supervised ML model using the Python package `Scikit-learn` (Pedregosa et al. 2011) with the Gaussian Process classifier. The training data for the model consisted of a $A^3\text{COSMOS}$ subsample of ~600 ran-

domly selected sources which had been manually classified as either robust or spurious counterpart. The accuracy of the trained model was then assessed by running it on a validation sample of another ~400 manually classified sources not contained in the training data. They found an agreement between the manual and ML classification for ~97% of the sources in the validation sample. The same accuracy was also obtained for the combined sample of ~1000 sources.

For our $A^3\text{COSMOS}$ database update, we applied this trained model to determine the reliability of our counterpart association for the newer COSMOS data, using the same counterpart images as Liu et al. (2019a), only replacing the *Spitzer*/IRAC 3.6 µm image by a newer image from Moneti et al. (2022) (see Sect. 2.2). However, naturally, new counterpart images and most importantly different filters had to be used for GOODS-S, making the model not applicable to $A^3\text{GOODSS}$. The number of sources in our current version of the $A^3\text{GOODSS}$ prior source catalogue (i.e. 568) approximately corresponds to the size of the initial training dataset of the ML model. We therefore classified all $A^3\text{GOODSS}$ sources manually, creating the training dataset for the ML model in preparation for the future updates of the $A^3\text{GOODSS}$ database.

# Appendix B: $A^3\text{COSMOS}$ and $A^3\text{GOODSS}$ sky coverage by band

Here, we show the spatial sky coverage of our $A^3\text{COSMOS}$ and $A^3\text{GOODSS}$ databases, as displayed cumulatively in Fig. 1, separately for each ALMA band. The values of the individual coverages are also listed in Table 1.
<figure><img src="image_10.png" /><figcaption>**Fig. B.1.** Spatial sky coverage of A$^3$COSMOS as shown in the top panel of Fig. 1, and split into individual ALMA bands, as indicated in the top left of each panel. The dashed coloured lines indicate the approximate outlines of the CANDELS (yellow; Grogin et al. 2011), COSMOS HST/ACS (orange; Koekemoer et al. 2007), and S2COSMOS (red; Simpson et al. 2019) fields.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Fig. B.1.** Spatial sky coverage of A$^3$COSMOS as shown in the top panel of Fig. 1, and split into individual ALMA bands, as indicated in the top left of each panel. The dashed coloured lines indicate the approximate outlines of the CANDELS (yellow; Grogin et al. 2011), COSMOS HST/ACS (orange; Koekemoer et al. 2007), and S2COSMOS (red; Simpson et al. 2019) fields.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Fig. B.2.** Spatial sky coverage of A³GOODSS as shown in the bottom panel of Fig. 1, and split into individual ALMA bands, as indicated in the top left of each panel. The dashed coloured lines indicate the approximate outlines of the Hubble Ultra Deep Field (yellow; Beckwith et al. 2006), CANDELS (orange; as in Guo et al. 2013), and Hubble Legacy Fields (red; Illingworth et al. 2016).</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Fig. B.2.** Spatial sky coverage of A³GOODSS as shown in the bottom panel of Fig. 1, and split into individual ALMA bands, as indicated in the top left of each panel. The dashed coloured lines indicate the approximate outlines of the Hubble Ultra Deep Field (yellow; Beckwith et al. 2006), CANDELS (orange; as in Guo et al. 2013), and Hubble Legacy Fields (red; Illingworth et al. 2016).</figcaption></figure>
## Appendix C: Simulating observations in SIDES

In order to 'observe' mock sources in each of our 58 realisations of a 2 deg² sky field made from SIDES, we created a set of mock pointings. For this, we used the observational properties (i.e. areal coverage and sensitivity profile) of each precisely targeted pointing used in our real number counts analysis. Since the SIDES simulation does not include source size information we, however, assumed an angular resolution of 1 arcsec (which corresponds to the mean angular resolution in our A³COSMOS database) for all mock pointings and that all mock sources are point sources at this resolution.

These mock pointings were then placed in each of the 58 fields, centred on mock target sources according to the respective observational bias mode (i.e. Bias, Superbias or Random, see Sect. 3.3). We selected sources from the SIDES catalogue in each respective field as mock targets based on their flux density (except the Random mode in which the pointings were distributed randomly). In the Superbias mode, all the brightest sources in a given field were chosen as targets. In the Bias mode, mock target sources were drawn randomly based on their flux density, with a probability according to the flux density distribution of the real target sources in all precisely targeted pointings in our database in the ALMA band in consideration. Figure C.1 shows the flux density distribution of our chosen mock targets in the Bias and Superbias modes, compared to that of the real A³COSMOS targets in Band 6. The Bias and real target distributions are almost identical by construction, with the majority of targets being in the intermediate to higher flux density range (~0.5 – 4 mJy). In contrast, Superbias targets are exclusively bright sources (≥2 mJy). In both the Bias and Superbias modes, the choice of which mock pointing to assign to which mock target was made to ensure that every mock target is detectable in its assigned mock pointing (as it is the case in our set of real pointings).

The source extraction from these mock observations followed the same constraint as our real source extraction, that is, only sources with $S/N_{peak} > 5.4$ were considered as detected, but assuming that above this threshold $C_{compl.} = 1$ and $C_{contam.} = 0$. Here, $S/N_{peak}$ is calculated from the sensitivity of a mock pointing at the position of the mock source covered and the flux density of that source as given in the SIDES catalogue (i.e. $S_{peak} = S_{total}$, as all mock sources are assumed to be point sources).

For the corrected mock number counts, a central circular region around the phase centre of each mock pointing was disregarded during source extraction. In addition, in order to allow for a weighting of mock sources by redshift proximity to their

<table><caption>**Table C.1.** Fractions of sources detected within precisely targeted A³COSMOS pointings that have a prior spectroscopic, photometric, or no prior redshift, split into sources within a radius of 1 arcsec around the phase centre (Targets) and sources further away from the phase centre than 1 arcsec (Serendipitous).</caption><thead><tr><th></th><th>Targets</th><th>Serendipitous</th></tr></thead><tbody><tr><td>Spec. z</td><td>40.66%</td><td>9.23%</td></tr><tr><td>Photo. z</td><td>54.58%</td><td>73.85%</td></tr><tr><td>No prior z</td><td>4.76%</td><td>16.92%</td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>**Fig. C.1.** Band 6 flux density distribution of targets in the simulation compared to the real targets of precisely targeted pointings in the A³COSMOS database. Orange with circles indicates Bias mode; green with squares is Superbias mode; and blue with diagonal lines is for real targets.</figcaption></figure>

respectively mock targets (as in our real analysis), we assigned to each detected mock source a mock redshift uncertainty. For this, we randomly assigned each detected mock (serendipitous) source and each mock target to either a photometric, spectroscopic, or no redshift, based on the fractions of our real sources (see Table C.1). We then assigned to each mock source and mock target a redshift randomly picked within a Gaussian PDF centred on their original SIDES redshift, $z$, with a width of 0.001(1 + $z$) for spectroscopic or 0.06(1 + $z$) for photometric redshifts. These assigned redshifts were then used for weighting as described in Sect. 3.1.2.
## Appendix C: Simulating observations in SIDES

In order to 'observe' mock sources in each of our 58 realisations of a 2 deg² sky field made from SIDES, we created a set of mock pointings. For this, we used the observational properties (i.e. areal coverage and sensitivity profile) of each precisely targeted pointing used in our real number counts analysis. Since the SIDES simulation does not include source size information we, however, assumed an angular resolution of 1 arcsec (which corresponds to the mean angular resolution in our A³COSMOS database) for all mock pointings and that all mock sources are point sources at this resolution.

These mock pointings were then placed in each of the 58 fields, centred on mock target sources according to the respective observational bias mode (i.e. Bias, Superbias or Random, see Sect. 3.3). We selected sources from the SIDES catalogue in each respective field as mock targets based on their flux density (except the Random mode in which the pointings were distributed randomly). In the Superbias mode, all the brightest sources in a given field were chosen as targets. In the Bias mode, mock target sources were drawn randomly based on their flux density, with a probability according to the flux density distribution of the real target sources in all precisely targeted pointings in our database in the ALMA band in consideration. Figure C.1 shows the flux density distribution of our chosen mock targets in the Bias and Superbias modes, compared to that of the real A³COSMOS targets in Band 6. The Bias and real target distributions are almost identical by construction, with the majority of targets being in the intermediate to higher flux density range (~0.5 – 4 mJy). In contrast, Superbias targets are exclusively bright sources (≥2 mJy). In both the Bias and Superbias modes, the choice of which mock pointing to assign to which mock target was made to ensure that every mock target is detectable in its assigned mock pointing (as it is the case in our set of real pointings).

The source extraction from these mock observations followed the same constraint as our real source extraction, that is, only sources with S/N<sub>peak</sub> > 5.4 were considered as detected, but assuming that above this threshold C<sub>compl.</sub> = 1 and C<sub>contam.</sub> = 0. Here, S/N<sub>peak</sub> is calculated from the sensitivity of a mock pointing at the position of the mock source covered and the flux density of that source as given in the SIDES catalogue (i.e. S<sub>peak</sub> = S<sub>total</sub>, as all mock sources are assumed to be point sources).

For the corrected mock number counts, a central circular region around the phase centre of each mock pointing was disregarded during source extraction. In addition, in order to allow for a weighting of mock sources by redshift proximity to their

<table><caption>**Table C.1.** Fractions of sources detected within precisely targeted A³COSMOS pointings that have a prior spectroscopic, photometric, or no prior redshift, split into sources within a radius of 1 arcsec around the phase centre (Targets) and sources further away from the phase centre than 1 arcsec (Serendipitous).</caption><thead><tr><th></th><th>Targets</th><th>Serendipitous</th></tr></thead><tbody><tr><td>Spec. z</td><td>40.66%</td><td>9.23%</td></tr><tr><td>Photo. z</td><td>54.58%</td><td>73.85%</td></tr><tr><td>No prior z</td><td>4.76%</td><td>16.92%</td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>**Fig. C.1.** Band 6 flux density distribution of targets in the simulation compared to the real targets of precisely targeted pointings in the A³COSMOS database. Orange with circles indicates Bias mode; green with squares is Superbias mode; and blue with diagonal lines is for real targets.</figcaption></figure>

respective mock targets (as in our real analysis), we assigned to each detected mock source a mock redshift uncertainty. For this, we randomly assigned each detected mock (serendipitous) source and each mock target to either a photometric, spectroscopic, or no redshift, based on the fractions of our real sources (see Table C.1). We then assigned to each mock source and mock target a redshift randomly picked within a Gaussian PDF centred on their original SIDES redshift, z, with a width of 0.001(1 + z) for spectroscopic or 0.06(1 + z) for photometric redshifts. These assigned redshifts were then used for weighting as described in Sect. 3.1.2.
## Appendix D: Selecting the radius of the central mask

We wanted to determine the optimal size of the mask put on the centre of our precisely targeted pointings. To do so, we ran our simulation (see Sect. 3.3) with several mask radii of increasing size and compared the retrieved corrected number counts ('Output') to the underlying source distribution in SIDES ('Input'). The results are shown in Fig. D.1 for Band 6 with radii from 1 arcsec to 4 arcsec. The lighter shaded region indicates the 1 $\sigma$ scatter between realisations (i.e. the cosmic variance). The darker shaded region indicates the error on the mean over all realisations (i.e. $\sigma/\sqrt{58}$). For clarity, the uncertainties are shown only for the Bias line. While we see a slight convergence, there is no significant overall improvement with increasing radius. The excess of the Bias over the Input number counts at the bright end remains consistently at ~6%. The deviation, however, is far smaller than the cosmic variance in all flux density bins at any radius. Cosmic variance introduces an uncertainty of ~15-20% in the intermediate flux density ranges and ~50% and ~90% at the faint and bright end, respectively. From a simulation viewpoint, increasing the radius of the mask above 1 arcsec therefore offers no additional benefit.

However, these conclusions are only meaningful if the clustering properties in the simulations are an accurate representation of reality. To test the applicability on our $A^3\text{COSMOS}$ database, we therefore investigated the influence of varying the mask radius on the inferred number counts. Figure D.2 shows the number counts inferred using our method using different mask radii up to 9 arcsec as coloured lines. In Bands 3, 4, and 6 there is no significant change in the number counts within the uncertainties when increasing the radius beyond 1 arcsec. In Band 7, however, there is a slight trend towards lower number counts with increasing radius, but with a converging behaviour. For $r \gtrsim 4$ arcsec the number counts are consistent within their uncertainties in all flux density bins. This is most likely due to a higher contamination from close-by neighbours of the target source, are not caught as well in this band as in other bands by our redshift weighting method (see Sect. 3.1.2). Band 7 is the band in which the lowest fraction of serendipitous sources in close angular proximity (i.e. < 4 arcsec) to the target have a redshift counterpart in our ancillary catalogues. This reduces our possibility to perform redshift weighting of these sources. Thus, while redshift weighting excludes most close neighbours due to clustering in Bands 3, 4, and 6, it is not sufficient in Band 7. We therefore had to choose a larger exclusion radius in this band. In brief, we chose for our central mask a radius of 1 arcsec in Bands 3, 4, and 6, and a radius of 4 arcsec for Band 7.

<figure><img src="image_13.png" /></figure>

**Fig. D.1.** Simulated differential number counts in ALMA Band 6, retrieved with the corrected Bias and Superbias modes for four different central mask radii, divided by the underlying Input number counts. Shown here is the mean over ten run-throughs of the simulation (i.e. 580 realisations). The light shaded regions show the standard deviation due to cosmic variance; the dark shaded regions show the error on the mean (i.e. $\sigma_{\text{cos.var.}}/\sqrt{580}$). The errors are of approximately the same size for the Bias and Superbias modes, but for clarity they are only shown for the Bias mode.
## Appendix D: Selecting the radius of the central mask

We wanted to determine the optimal size of the mask put on the centre of our precisely targeted pointings. To do so, we ran our simulation (see Sect. 3.3) with several mask radii of increasing size and compared the retrieved corrected number counts ('Output') to the underlying source distribution in SIDES ('Input'). The results are shown in Fig. D.1 for Band 6 with radii from 1 arcsec to 4 arcsec. The lighter shaded region indicates the 1 $\sigma$ scatter between realisations (i.e. the cosmic variance). The darker shaded region indicates the error on the mean over all realisations (i.e. $\sigma/\sqrt{58}$). For clarity, the uncertainties are shown only for the Bias line. While we see a slight convergence, there is no significant overall improvement with increasing radius. The excess of the Bias over the Input number counts at the bright end remains consistently at ~6%. The deviation, however, is far smaller than the cosmic variance in all flux density bins at any radius. Cosmic variance introduces an uncertainty of ~15-20% in the intermediate flux density ranges and ~50% and ~90% at the faint and bright end, respectively. From a simulation viewpoint, increasing the radius of the mask above 1 arcsec therefore offers no additional benefit.

However, these conclusions are only meaningful if the clustering properties in the simulations are an accurate representation of reality. To test the applicability on our $A^3\text{COSMOS}$ database, we therefore investigated the influence of varying the mask radius on the inferred number counts. Figure D.2 shows the number counts inferred using our method using different mask radii up to 9 arcsec as coloured lines. In Bands 3, 4, and 6 there is no significant change in the number counts within the uncertainties when increasing the radius beyond 1 arcsec. In Band 7, however, there is a slight trend towards lower number counts with increasing radius, but with a converging behaviour. For $r \gtrsim 4$ arcsec the number counts are consistent within their uncertainties in all flux density bins. This is most likely due to a higher contamination from close-by neighbours of the target source, are not caught as well in this band as in other bands by our redshift weighting method (see Sect. 3.1.2). Band 7 is the band in which the lowest fraction of serendipitous sources in close angular proximity (i.e. < 4 arcsec) to the target have a redshift counterpart in our ancillary catalogues. This reduces our possibility to perform redshift weighting of these sources. Thus, while redshift weighting excludes most close neighbours due to clustering in Bands 3, 4, and 6, it is not sufficient in Band 7. We therefore had to choose a larger exclusion radius in this band. In brief, we chose for our central mask a radius of 1 arcsec in Bands 3, 4, and 6, and a radius of 4 arcsec for Band 7.

<figure><img src="image_13.png" /></figure>

**Fig. D.1.** Simulated differential number counts in ALMA Band 6, retrieved with the corrected Bias and Superbias modes for four different central mask radii, divided by the underlying Input number counts. Shown here is the mean over ten run-throughs of the simulation (i.e. 580 realisations). The light shaded regions show the standard deviation due to cosmic variance; the dark shaded regions show the error on the mean (i.e. $\sigma_{\text{cos.var.}}/\sqrt{580}$). The errors are of approximately the same size for the Bias and Superbias modes, but for clarity they are only shown for the Bias mode.
<figure><img src="image_14.png" /><figcaption>**Fig. D.2.** Differential continuum number counts computed from the $A^3\text{COSMOS}$ database in four ALMA bands (as labelled in the top left corner of each panel) for nine different radii of the mask around the phase centre of single pointings. Uncertainties are shown as vertical coloured lines in the respective bins. The dotted grey line shows the number counts using all data in the $A^3\text{COSMOS}$ database and applying no corrections. Error bars are not shown for the uncorrected number counts.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**Fig. D.2.** Differential continuum number counts computed from the $A^3\text{COSMOS}$ database in four ALMA bands (as labelled in the top left corner of each panel) for nine different radii of the mask around the phase centre of single pointings. Uncertainties are shown as vertical coloured lines in the respective bins. The dotted grey line shows the number counts using all data in the $A^3\text{COSMOS}$ database and applying no corrections. Error bars are not shown for the uncorrected number counts.</figcaption></figure>
## Appendix E: Comparing blind surveys and single pointings

In order to check the validity of our selection, masking and weighting methods on single pointings, we computed the number counts from our A³COSMOS database for four different cases. The first is using the entire database, that is, all images and sources, applying no selection, masking or weighting. This way we see the full effect of the observational biases in the data. The second case is computing the number counts exclusively from blind surveys in our database and the sources detected therein. Since blind surveys do not have selection biases by design, no masking or weighting by redshift is necessary to infer accurate number counts. The third case is using only single pointings with a central target, applying our masking and weighting method, so we can see the extent to which our method constitutes an improvement. The fourth case is using blind surveys combined with the selected and masked single pointings, which represents our final results as shown in Fig. 7.

The comparison of these four cases is shown in Fig. E.1. In all bands, there is a significant improvement over the uncorrected number counts, especially towards higher flux densities, which is primarily due to brighter sources being targeted preferentially. In Bands 3, 4, and 6, the blind survey, single pointing, and combined number counts are consistent with each other within their range of uncertainties. This makes us confident of the validity of our selection and correction method for single pointings to retrieve accurate number counts. Blind survey and combined number counts are not shown for Band 7, as there are no blind surveys available for this band within the A³COSMOS and A³GOODSS databases. The only noteworthy difference that can be seen between the blind survey and the combined number counts appears in one flux density bin in Band 6 (i.e. 0.35 – 0.67 mJy). While still within the errors, the blind survey count in that bin is by a factor 2 higher than the combined. However, this disagreement can entirely be attributed to cosmic variance and small number statistics.

We conclude from this comparison that our selection and correction method for single pointings is valid, as it yields consistent results with blind surveys.

<figure><img src="image_15.png" /></figure>

**Fig. E.1.** Differential continuum number counts computed from the A³COSMOS database in four ALMA bands (as labelled in the top left corner of each panel). The number counts are measured for four different sets of data. The grey dotted line indicates use of all images and blindly extracted sources contained within the entire database, applying no selection or correction; the blue dash-dotted line is for the use of images only from ALMA blind surveys and sources found therein; the orange dashed line is for the use of only precisely targeted single pointings, after masking out a central circular region of 1 arcsec (4 arcsec in Band 7) radius and weighting the contribution of each detected source outside the masked region by the proximity of its redshift to that of the respective target (see Sect. 3.1); the black solid line is for both blind surveys and single pointings with targets, after masking the phase centre and weighting sources by redshift. As the single pointing line overlaps with other lines in several bins, it has been slightly shifted to improve visibility. In Band 3, it is increased by 5%, in Band 4 and 6 it is lowered by 5%. Error bars are shown for each bin of all lines except the uncorrected case, in the colour of the respective line. No blind surveys exist in our database for Band 7, and therefore no number counts are shown for that and the combined case.
## Appendix E: Comparing blind surveys and single pointings

In order to check the validity of our selection, masking and weighting methods on single pointings, we computed the number counts from our A³COSMOS database for four different cases. The first is using the entire database, that is, all images and sources, applying no selection, masking or weighting. This way we see the full effect of the observational biases in the data. The second case is computing the number counts exclusively from blind surveys in our database and the sources detected therein. Since blind surveys do not have selection biases by design, no masking or weighting by redshift is necessary to infer accurate number counts. The third case is using only single pointings with a central target, applying our masking and weighting method, so we can see the extent to which our method constitutes an improvement. The fourth case is using blind surveys combined with the selected and masked single pointings, which represents our final results as shown in Fig. 7.

The comparison of these four cases is shown in Fig. E.1. In all bands, there is a significant improvement over the uncorrected number counts, especially towards higher flux densities, which is primarily due to brighter sources being targeted preferentially. In Bands 3, 4, and 6, the blind survey, single pointing, and combined number counts are consistent with each other within their range of uncertainties. This makes us confident of the validity of our selection and correction method for single pointings to retrieve accurate number counts. Blind survey and combined number counts are not shown for Band 7, as there are no blind surveys available for this band within the A³COSMOS and A³GOODSS databases. The only noteworthy difference that can be seen between the blind survey and the combined number counts appears in one flux density bin in Band 6 (i.e. 0.35 – 0.67 mJy). While still within the errors, the blind survey count in that bin is by a factor 2 higher than the combined. However, this disagreement can entirely be attributed to cosmic variance and small number statistics.

We conclude from this comparison that our selection and correction method for single pointings is valid, as it yields consistent results with blind surveys.

<figure><img src="image_15.png" /></figure>

**Fig. E.1.** Differential continuum number counts computed from the A³COSMOS database in four ALMA bands (as labelled in the top left corner of each panel). The number counts are measured for four different sets of data. The grey dotted line indicates use of all images and blindly extracted sources contained within the entire database, applying no selection or correction; the blue dash-dotted line is for the use of images only from ALMA blind surveys and sources found therein; the orange dashed line is for the use of only precisely targeted single pointings, after masking out a central circular region of 1 arcsec (4 arcsec in Band 7) radius and weighting the contribution of each detected source outside the masked region by the proximity of its redshift to that of the respective target (see Sect. 3.1); the black solid line is for both blind surveys and single pointings with targets, after masking the phase centre and weighting sources by redshift. As the single pointing line overlaps with other lines in several bins, it has been slightly shifted to improve visibility. In Band 3, it is increased by 5%, in Band 4 and 6 it is lowered by 5%. Error bars are shown for each bin of all lines except the uncorrected case, in the colour of the respective line. No blind surveys exist in our database for Band 7, and therefore no number counts are shown for that and the combined case.