by MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and Astrophysics of the University of Tübingen, with the support of DLR and the Max Planck Society. The Argelander Institute for Astronomy of the University of Bonn and the Ludwig Maximilians Universität Munich also participated in the science preparation for eROSITA. The eROSITA data shown here were processed using the eSASS software system developed by the German eROSITA consortium. V.G., E.B., A.L., C.G., S.Z., and X.Z. acknowledge financial support from the European Research Council (ERC) Consolidator Grant under the European Union's Horizon 2020 research and innovation program (grant agreement COG DarkQuest No 101002585). N.C. was financially supported by CNES. T. Schrabback and F.K.I. acknowledge support from the German Federal Ministry for Economic Affairs and Energy (BMWi) provided through DLR under projects 50OR2002, 50OR2106, and 50OR2302, as well as the support provided by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under grant 415537506. V.G. also thanks Margherita Ghirardini for her invaluable joyful support. The Legacy Surveys consist of three individual and complementary projects: the Dark Energy Camera Legacy Survey (DECaLS; Proposal ID #2014B-0404; PIs: David Schlegel and Arjun Dey), the Beijing-Arizona Sky Survey (BASS; NOAO Prop. ID #2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy Survey (MzLS; Prop. ID #2016A-0453; PI: Arjun Dey). DECaLS, BASS and MzLS together include data obtained, respectively, at the Blanco telescope, Cerro Tololo Inter-American Observatory, NSF's NOIRLab; the Bok telescope, Steward Observatory, University of Arizona; and the Mayall telescope, Kitt Peak National Observatory, NOIRLab. Pipeline processing and analyses of the data were supported by NOIRLab and the Lawrence Berkeley National Laboratory (LBNL). The Legacy Surveys project is honored to be permitted to conduct astronomical research on Iolkam Du'ag (Kitt Peak), a mountain with particular significance to the Tohono O'odham Nation. Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology FacilitiesCouncil of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e Tecnológico and the Ministério da Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft, and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energéticas, Medioambientes y Tecnológicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciències de l'Espai (IEEC/CSIC), the Institut de Física d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universität München and the associated Excellence Cluster Universe, the University of Michigan, the National Optical Astronomy Observatory, the University of Nottingham, The Ohio State University, the OzDES Membership Consortium, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A&M University. Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 177.A-3016, 177.A-3017, 177.A-3018 and 179.A-2004, and on data products produced by the KiDS consortium. The KiDS production team acknowledges support from: Deutsche Forschungsgemeinschaft, ERC, NOVA and NWO-M grants; Target; the University of Padova, and the University Federico II (Naples). N.O. and S.M. acknowledge JSPS KAKENHI Grant Number JP19KK0076. The Hyper Suprime-Cam (HSC) collaboration includes the astronomical communities of Japan and Taiwan, and Princeton University. The HSC instrumentation and software were developed by the National Astronomical Observatory of Japan (NAOJ), the Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High Energy Accelerator Research Organization (KEK), the Academia Sinica Institute for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University. Funding was contributed by the FIRST program from the Japanese Cabinet Office, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the Japan Society for the Promotion of Science (JSPS), Japan Science and Technology Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK, ASIAA, and Princeton University. This paper makes use of software developed for the Large Synoptic Survey Telescope. We thank the LSST Project for making their code available as free software at http://dm.lsst.org This paper is based [in part] on data collected at the Subaru Telescope and retrieved from the HSC data archive system, which is operated

by Subaru Telescope and Astronomy Data Center (ADC) at NAOJ. Data analysis was in part carried out with the cooperation of Center for Computational Astrophysics (CfCA), NAOJ. We are honored and grateful for the opportunity of observing the Universe from Maunakea, which has the cultural, historical and natural significance in Hawaii. I.C. acknowledges the support from the National Science and Technology Council in Taiwan (Grant NSTC 111-2112-M-006-037-MY3). This work made use of the following Python software packages: SciPy<sup>23</sup> (Virtanen et al. 2020), Matplotlib<sup>24</sup> (Hunter 2007), Astropy<sup>25</sup> (Astropy Collaboration 2022), NumPy<sup>26</sup> (Harris et al. 2019), CAMB (Lewis & Challinor 2011), pyCCL<sup>27</sup> (Chisari et al. 2019), GPY<sup>28</sup> (GPy 2012), climin<sup>29</sup> (Bayer et al. 2015), ultranest<sup>30</sup> (Buchner 2021).

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## Appendix A: Mixture model

We start by employing our mixture model on the first term of Equation (6), by explicitly dividing the three populations that comprise our sample; clusters, RSs, and AGNs, as:

$$ \frac{dN_{tot}}{d\hat{C}_R d\hat{z} d\hat{H}_i} P(I|\hat{C}_R, \hat{z}, \hat{H}_i) = \frac{dN_C}{d\hat{C}_R d\hat{z} d\hat{H}_i} P(I|\hat{C}_R, \hat{z}, \hat{H}_i) \\ + \frac{N_C}{(1 - f_{RS} - f_{AGN})} \left( f_{RS} P(\hat{C}_R, \hat{z}, \hat{H}_i|RS) \right. \\ \left. + f_{AGN} P(\hat{C}_R, \hat{z}, \hat{H}_i|AGN) \right) \qquad (A.1) $$

where $P(\hat{C}_R, \hat{z}, \hat{H}_i|RS)$ and $P(\hat{C}_R, \hat{z}, \hat{H}_i|AGN)$ are the probability density function (PDF) for the contaminants that represent RSs and AGNs, respectively, while the first term, $\frac{dN_C}{d\hat{C}_R d\hat{z} d\hat{H}_i} P(I|\hat{C}_R, \hat{z}, \hat{H}_i)$, represents the distribution of our selection observable for the detected clusters.

On the second term of Equation (1), we use the relation between these three populations when computing the total number of detected objects as,

$$ N_{tot} = \frac{N_C}{1 - f_{RS} - f_{AGN}} \qquad (A.2) $$

where

$$ N_C = \iiint \frac{dN_C}{d\hat{C}_R d\hat{z} d\hat{\lambda} d\hat{H}_i} P(I|\hat{C}_R, \hat{z}, \hat{H}_i) \Theta(\hat{\lambda} > 3) d\hat{C}_R d\hat{z} d\hat{\lambda} d\hat{H}_i \qquad (A.3) $$

On the third term in Equation (1), when using the optical information, we write

$$ P(\hat{\lambda}|\hat{C}_R, \hat{z}, \hat{H}_i, I) = \left( 1 - f_{RS} \frac{P(\hat{C}_R, \hat{z}, \hat{H}_i|RS, I)}{P(\hat{C}_R, \hat{z}, \hat{H}_i, I)} \right. \\ - f_{AGN} \frac{P(\hat{C}_R, \hat{z}, \hat{H}_i|AGN, I)}{P(\hat{C}_R, \hat{z}, \hat{H}_i, I)} \\ \left. \times P(\hat{\lambda}|\hat{C}_R, \hat{z}, \hat{H}_i, I, C) \right. \\ + f_{RS} \frac{P(\hat{\lambda}, \hat{z}|RS, I) P(\hat{C}_R, \hat{H}_i|RS, I)}{P(\hat{C}_R, \hat{z}, \hat{H}_i, I)} \\ + f_{AGN} \frac{P(\hat{\lambda}, \hat{z}|AGN, I) P(\hat{C}_R, \hat{H}_i|AGN, I)}{P(\hat{C}_R, \hat{z}, \hat{H}_i, I)} \qquad (A.4) $$

Finally, on the last term in Equation (1), when using the weak lensing information;

$$ P(\hat{g}_i|\hat{C}_R, \hat{z}, \hat{H}_i, I) = \left( 1 - f_{RS} \frac{P(\hat{C}_R, \hat{z}, \hat{H}_i|RS, I)}{P(\hat{C}_R, \hat{z}, \hat{H}_i, I)} \right. \\ - f_{AGN} \frac{P(\hat{C}_R, \hat{z}, \hat{H}_i|AGN, I)}{P(\hat{C}_R, \hat{z}, \hat{H}_i, I)} \\ \left. \times P(\hat{g}_i|\hat{C}_R, \hat{z}, \hat{H}_i, I, C) \right. \\ + f_{RS} \frac{P(\hat{g}_i|RS, I) P(\hat{z}|RS, I) P(\hat{C}_R, \hat{H}_i|RS, I)}{P(\hat{C}_R, \hat{z}, \hat{H}_i, I)} \\ + f_{AGN} \frac{P(\hat{g}_i|\hat{z}, AGN, I) P(\hat{z}|AGN, I) P(\hat{C}_R, \hat{H}_i|AGN, I)}{P(\hat{C}_R, \hat{z}, \hat{H}_i, I)} \qquad (A.5) $$

To estimate the overall total contamination in the sample, we use the PDF of count rate and sky position information, that is: $P(\hat{C}_R, \hat{H}_i|AGN, I)$, and $P(\hat{C}_R, \hat{H}_i|RS, I)$ of the AGN and RS populations, obtained from our realistic eRASS1 simulations (Seppi et al. 2022; Comparat et al. 2020). We obtain the AGN and RS PDF in redshift and richness (Kluge et al. 2024), i.e., $P(\hat{\lambda}, \hat{z}|AGN, I)$ from the eROMaPPer run on locations of the eRASS1 point sources (Kluge et al. 2024; Merloni et al. 2024) and $P(\hat{\lambda}, \hat{z}|RS, I)$ from the eROMaPPer run at random lines of sight.

Finally, we use the predicted AGN shear profiles reported in the Comparat et al. (2023). In contrast, we assume that the shear profile is zero at all radii for the RS component. Therefore the shear signal is consistent with a noise fluctuation, as elaborated in Grandis et al. (2024), Sect. 5.1.1.
# Appendix B: Additional tests on systematics

## Appendix B.1: Relation between scatter and normalization

The reported normalization parameters for the X-ray count rate to mass scaling relation reported in Chiu et al. (2022, 2023) are in tension with our results. In particular, the difference in normalization is due to the observed degeneracy between the normalization and the scatter. Chiu et al. (2022, 2023) assume an informative prior on the count rate mass scatter, $\sigma_X = 0.30 \pm 0.08$ derived by Grandis et al. (2019) from the luminosity and temperature scaling relations measured by Bulbul et al. (2019). This choice artificially breaks the amplitude scatter degeneracy. We show this in the following Fig. B.1 and the selection of the sample (Bahar et al. 2022). For a more detailed comparison of the scaling relation parameters, we refer to Kleinebreil et al. (2024) and Grandis et al. (2024).

<figure><img src="image_16.png" /></figure>

Fig. B.1: Degeneracy between scatter and amplitude of the count rate to mass scaling relation. We show our obtained value and the comparison with individual mass calibration from DES, KiDS, and HSC as reported in Kleinebreil et al. (2024); Grandis et al. (2024). The previous measurements from the eFEDS survey presented in Chiu et al. (2022) and Chiu et al. (2023) are shown with crosses.

## Appendix B.2: Relation between cosmology and scatter parameters

In the following figures, Fig. B.2 and B.3, we show the posterior and correlation between the cosmology parameters in the fit and the X-ray scaling relation parameters.

<figure><img src="image_17.png" /></figure>

Fig. B.2: Degeneracies between cosmological parameters and X-ray count rate scaling relation parameters. It is interesting to point out that the scaling relation parameters affect the degeneracy between $\Omega_m$ and $\sigma_8$.

<figure><img src="image_18.png" /></figure>

Fig. B.3: Correlation between fitted parameters.
### Appendix B.3: $\Omega_m$ vs $\sigma_8$ in the several models fitted to our dataset

In this paper, we have fitted for several different cosmological models. All of these extensions have in common the starting point in the standard vanilla ΛCDM cosmological model. We have fitted the ΛCDM, and then we have allowed several parameters to be free. Particularly dark energy equation of state $w$ and summed neutrino masses $\sum m_\nu$. Finally, for comparison purposes with ΛCDM fit in Planck analysis (Planck Collaboration VI 2020), we have also fitted the ΛCDM model with $\sum m_\nu$ fixed to 0.06 eV. It is important to show that these extra free parameters do not significantly affect the base robust parameters we are fitting and constraining in the ΛCDM: $\Omega_m$ vs $\sigma_8$. We show this test in Fig. B.4.

<figure><img src="image_19.png" /></figure>

Fig. B.4: Posterior on $\Omega_m-\sigma_8$ from the ΛCDM, $w$CDM, $w$CDM, $w\nu$CDM, and ΛCDM with $\sum m_\nu = 0.06$ eV fits from our eRASS1 cluster dataset. We see the nice overlap in the parameter space that indicates that our fit in the $\Omega_m-\sigma_8$ plane is stable against modification to the standard ΛCDM cosmological model.

### Appendix B.4: Tests of the effect on $\Omega_m$ vs $\sigma_8$ from our modeling choices

In this paper, we have applied several cuts to the datasets, for instance, $0.1 < z < 0.8$, EXT_LIKE> 6, we have chosen a specific halo mass function in the Tinker et al. (2008), and we have employed the mixture model. The following two figures show the effect of our constraints on $\Omega_m$ vs $\sigma_8$ in ΛCDM cosmological model. In Fig. B.5 in Appendix B, we show the impact of not applying the mixture model, cutting the data in redshift at 0.6 instead of 0.8, and changing the EXT_LIKE cut to a slightly more conservative value of 10. We see changes within the $1\sigma$ level for all these cases. In Fig B.6, we show the effect of changing the halo mass function from Tinker et al. (2008) to either Tinker et al. (2010) or Despali et al. (2016). We further tested the effects of systematic uncertainties on the halo mass function by applying the formalism introduced in Costanzi et al. (2019b) and used in the literature (Dark Energy Survey Collaboration 2020; Bocquet et al. 2024) to correct Tinker et al. (2008) model adding two nuisance parameters, $q_{HMF}$ and $s_{HMF}$, which allow for extra degrees of freedom for both normalization and slope of the halo mass function. We used the following priors on these parameters: $q_{HMF} \sim N(1, 0.035)$ and $s_{HMF} \sim N(0, 0.02)$ as in the analysis of

Bocquet et al. (2024). We can see that $\Omega_m$ is unchanged, while $\sigma_8$ is lowered by around $1\sigma$. This indicates that change in mass function does not significantly impact our constraints; however, in future eROSITA cosmological analysis, we will need to pay much more care and attention to our modeling choice regarding the halo mass function.

<figure><img src="image_20.png" /></figure>

Fig. B.5: Comparison of our main ΛCDM results (orange) with several tests: switching off the mixture model likelihood of Sect. 4.2 (blue), restricting the redshift range $z < 0.6$.

<figure><img src="image_21.png" /></figure>

Fig. B.6: Robustness of our main ΛCDM results (orange) against changes in the halo mass function: Tinker et al. (2008) (blue), and Despali et al. (2016) (green), and 5.5% systematics uncertainty added to the halo mass function of Tinker et al. (2008) following the prescription of (Costanzi et al. 2019b; Dark Energy Survey Collaboration 2020)
## Appendix C: Constraints from neutrino summed masses to the mass of the lightest

A constraint on the summed neutrino mass will shed light on the mass of the lightest neutrino eigenstate. Obviously, this also depends on the neutrino mass hierarchy as:

$$ \sum m_{\nu} = \begin{cases} m_L + \sqrt{m_L^2 + \Delta m_{21}^2} + \sqrt{m_L^2 + \Delta m_{31}^2} & \text{Normal Hierarchy} \\ m_L + \sqrt{m_L^2 + \Delta m_{32}^2} + \sqrt{m_L^2 + \Delta m_{31}^2} & \text{Inverted Hierarchy} \end{cases} \quad (C.1) $$

where $m_L$ is the mass of the lightest neutrino eigenstate. We reproduce Fig. 3 in Qian & Vogel (2015) with updated parameters from NuFIT (Esteban et al. 2020) in Fig. C.1. In the same figure, we also show with horizontal dashed lines the constraints on summed neutrino masses for the combination of eROSITA with Planck CMB and lower allowed limit from neutrino oscillation experiments (Tanabashi et al. 2018).

<figure><img src="image_22.png" /><figcaption>Fig. C.1: Current estimates of the relation between the mass of the lightest neutrino eigenstate and the summed neutrino masses (solid lines). The horizontal dashed lines and the shaded area represent the eRASS1 and CMB combined estimate. The vertical dash-dotted line represents the lightest neutrino value corresponding to our summed mass estimate. Finally, the vertical dotted line corresponds to the $1\sigma$ uncertainty around the lightest neutrino mass estimate. We use blue and orange to distinguish between normal and inverted hierarchy. It is remarkable to show that although the difference between the models, the estimation for the lightest neutrino mass is almost identical.</figcaption></figure>
## Appendix D: Optical and X-ray completeness

<figure><img src="image_23.png" /><figcaption>Fig. D.1: Optical (blue), X-ray (orange), and total (green) completeness computed as a function of mass from the best-fit values. The three panels show different redshift, 0.3, 0.5, and 0.15, and in each panel, a solid line corresponds to an exposure of 150 s, dashed to 500 s, and dotted to 50 s. The shaded area around each line represents the 1σ uncertainty from resampling the best fitting chain 1000 times.</figcaption></figure>

As we have chosen to pursue a forward modeling approach, the completeness is not pre-determined, but it is modeled on the fly by propagating the selection function in count rate and the selection function in richness to halo mass via the observable mass relations and their correlated scatters. Following the approach proposed in Grandis et al. (2020) for modeling the total selection function of X-ray-selected, optically confirmed clusters, we computed the completeness curves as a function of mass and redshift for optical and X-ray selection functions. For the optical, we have:

$$P(I|M, z) = \iint \Theta(\hat{\lambda} > 3) P(\hat{\lambda}|\lambda) P(\lambda|M, z) d\lambda d\hat{\lambda}, \quad (D.1)$$

while for the X-ray, we have

$$P(I|M, z, \hat{H}_i) = \iint P(I|C_R, z, \hat{H}_i) P(C_R|M, z) dC_R. \quad (D.2)$$

We computed both of these as a function of mass and displayed them in Figure D.1. In practice, the two incompleteness functions are allowed to correlate by the unknown correlation coefficient $\rho_{\lambda,C_R}$, which we also sample on the fly. As appears from the figure, the optical cut $\hat{\lambda} > 3$ only plays a minor role for low redshift, high exposure time regions, where it leads to a minimal extra incompleteness for the few $3 \times 10^{13} M_{\odot}$ halos that up-scatter into our sample. Computing the total number of expected objects with and without completeness indicates that this effect would reduce the total number of expected detections by 1.1%, if not modeled.
Appendix E: $\sigma_8$

$\sigma_8$ is defined as the amplitude of linear matter fluctuations in a spherical top hat of radius $R = 8h^{-1}$ Mpc at redshift $z = 0$, i.e.,

$$ \sigma_8^2 = \int_0^\infty \Delta^2(k, z=0) \left( \frac{3j_1(kR)}{kR} \right)^2 d \log k \quad (E.1) $$

where $j_1(kR)$ is the spherical Bessel function of the first kind of order one, and $\Delta^2(k, z = 0)$ is the power of linear matter fluctuation in logarithmic bins of wave number $k$ at redshift $z = 0$,

$$ \begin{aligned} \Delta^2(k, z=0) &= \frac{k^3 P(k, z=0)}{2\pi^2} \\ &= \frac{A_s}{(\Omega_m h^2)^2} \frac{4}{25} \left(\frac{k}{k_{\text{piv}}}\right)^{n_s-1} \left(\frac{k}{100}\right)^4 g^2(z=0) T^2(k) \end{aligned} \quad (E.2) $$

where $g(z)$ is the growth suppression factor (see Equation 10 in Huterer 2023) and $T(k)$ is the linear transfer function. It is therefore clear that $\sigma_8$ is correlated with the combination of cosmological parameters $\frac{\sqrt{A_s}}{\Omega_m h^2}$.
## Appendix F: Systematics on scaling relation modeling

The measured intrinsic scatter in count rate is surprisingly large, $\sigma_X = 0.98^{+0.05}_{-0.04}$, see in Table 2, compared with the value of ~0.4 reported in WtG (Mantz et al. 2016), or the value of 0.33 reported in eFEDS (Chiu et al. 2022). Given that our cluster sample spans a much larger range in mass and redshift than previous samples, this may indicate mass- and/or redshift-dependence of the scatter. Here, we explore a more complex model of the scatter and its impact on scaling relation and cosmological parameters. We note that this test was performed after unblinding and that we consider the model presented in Sect. 6 as our main analysis.

Thus, we model the scatter, allowing mass and redshift trends as

$$ \sigma_X(M, z) = \sigma_{X,0} + \sigma_{X,z} \log \frac{1+z}{1+z_p} + \sigma_{X,M} \log \frac{M}{M_p} \quad (F.1) $$

for the X-ray count rate, and

$$ \sigma_{\lambda}(M, z) = \sigma_{\lambda,0} + \sigma_{\lambda,z} \log \frac{1+z}{1+z_p} + \sigma_{\lambda,M} \log \frac{M}{M_p} \quad (F.2) $$

for the richness. Finally, we further unfreeze $s_M$ parameter, see Equation (30), enabling the scatter of the WL bias to depend on mass, as it was already redshift dependent.

We further decided to allow a mass trend in the mass slope of the count rate to mass scaling relation. We thus modify the scaling relations as in Sect. 6, changing Equation (25) as follows:

$$ \left\langle \log \frac{C_R}{C_{R,p}} \middle| M, z \right\rangle = \log A_X + b_X(M, z) \log \frac{M}{M_p} + e_X(z), \quad (F.3) $$

where $e_X(z)$ remains as of Equation (27), while the mass-redshift dependent slope of the scaling relation is expressed by the term $b_X(M, z)$ as:

$$ b_X(z) = B_X + F_X \log \frac{1+z}{1+z_p} + C_X \log \frac{M}{M_p}. \quad (F.4) $$

Here, $C_X$ allows the mass-dependent slope.

It is clear that this test introduces six extra free parameters, and therefore, assesses the level of the systematics of the model choices we have made throughout this paper. What we find is quite interesting. In fact, we discover a degeneracy between mixture model parameters, $f_{AGN}$ and $f_{RS}$, and the scatter evolution and mass dependence parameters. This degeneracy can very well be explained physically. The population that occupies the low-count-rate, low-richness can be comprised of both contaminants that our detection and confirmation pipeline is not able to distinguish from a galaxy cluster or low-mass groups of galaxies that upscatter in the scaling relation by, for instance, because of the presence of AGNs in its core (Bulbul et al. 2022). Our simulations (Comparat et al. 2020; Seppi et al. 2022) suggest that low-count-rate, low-richness entries in our catalog are due to AGN and RS contaminants. However, only a chance overlap between AGNs and clusters was introduced in eROSITA digital twin, while AGNs inside group and cluster-sized halos were not introduced. However, the situation in the real extended source cosmology catalog might be somewhat different.

Allowing the scatter to change as a function of mass and redshift effectively enables the degeneracy between these two populations mentioned above, and indeed, the chains prefer a non-negligible fraction of low-mass halos that upscatter in the scaling relations. Even though these changes do not affect the value of $S_8 = 0.866 \pm 0.012$, and the $\Omega_m$ and $\sigma_8$ change by about $1\sigma$ but still remain fully consistent with Planck CMB (Planck Collaboration VI 2020), we remain very cautious regarding the physical interpretation of the corresponding best fitting parameters in this model. In fact, despite the several improvements over the consistency with literature results on the value of the scatter at the pivot mass, now $0.66 \pm 0.04$, we have several indications that this model physically fails in determining some parameters:

- The slope of the X-ray count rate scaling relation becomes significantly higher, $2.0 \pm 0.04$, much higher than previously reported in the literature for X-ray luminosity to mass scaling relation, see Sect. 6.1.

- The mass trend of the scatter is very strong, $-0.37 \pm 0.02$, with suspiciously high precision, quite higher than what is expected from numerical simulations (~0.1 Pop et al. 2022), and with significant implication for high-mass clusters (> $10^{15} M_{\odot}$) having below 10% scatter in their luminosity to mass scaling relation, much lower than what was previously reported in dedicated studies of high-mass clusters, e.g. 40% in WtG (Mantz et al. 2016).

- Mixture model parameters $f_{AGN}$ and $f_{RS}$ are forced to 0 value. This would imply that our cleaning procedure has reduced the fraction of contaminants to nearly zero. However, we know from our simulations (Comparat et al. 2020; Seppi et al. 2022) and optical confirmation (Kluge et al. 2024) that a remaining contamination fraction of the order of 5% should be left in the cosmology sample.

It is thus clear that despite the intriguing results of this run, further dedicated studies are needed to physically understand what is causing the observed parameters, currently not consistent with our best understanding of the eROSITA sample. Further multi-wavelength studies of the eROSITA cluster population are clearly necessary; for instance, if these populations are indeed AGN, they should have characteristic infrared emission (Salvato et al. 2018, 2022). In our upcoming scaling relation paper (Pacaud et al. in preparation), we will investigate several aspects of the population modeling of galaxy clusters through scaling relation between mass and observables, with particular attention to the aspects discussed in this appendix. In our future eROSITA cosmological analysis, we will use these results to update our population modeling.