$\sum m_\nu < 0.13$ eV (95% c. l.) and $\sum m_\nu < 0.16$ eV (99% c. l.) provides a robust measurement that relies on mostly linear scales. With CMB data alone, informed by ACT lensing, we find that the universe is consistent with spatial flatness and requires a dark energy component.

We have only considered a subset of interesting model extensions here. Our publicly released likelihoods encapsulate linear scales of the total matter density field primarily over the redshift range $z = 0.5–5$. A variety of follow-up investigations will be of interest, including those that combine with galaxy lensing and clustering covering a range of redshifts and scales, possibly fitting these measurements jointly with models that look for nonstandard dark matter physics and modifications of general relativity. An exciting near-term prospect is an exclusion of the inverted hierarchy of massive neutrinos; for example, improved BAO data from the ongoing DESI (DESI Collaboration et al. 2016) will significantly reduce the degeneracy of our $\sum m_\nu$ measurement with the matter density $\Omega_m$ (Allison et al. 2015).

The publicly released mass maps can be used for a variety of cross-correlations; those with galaxy surveys, for example, can produce improved constraints on local primordial non-Gaussianity $f_{\text{NL}}$ (Schmittfull & Seljak 2018; McCarthy et al. 2023), as well as constraints on the amplitude of structure as a function of redshift $\sigma_8(z)$ (e.g., White et al. 2022). The mass maps can be combined with measurements of the tSZ and kSZ effects along with X-ray measurements to study the thermodynamics of galaxy formation and evolution by supplementing electron pressure, density, and temperature measurements with gravitational mass on arcminute scales (Battaglia et al. 2017; Bolliet et al. 2023). They can also be used to study the nonlinear universe, providing an unbiased view of the distribution of voids and filaments (e.g., He et al. 2018; Raghunathan et al. 2020).

ACT completed observations in 2022, but several possibilities lie ahead for significantly improved mass maps and cosmological constraints. In particular, we will explore the fidelity of roughly 50% of ACT data collected (mostly during the daytime) that were not used in this analysis. Data at lower frequencies and at 220 GHz can be used to enhance the foreground cleaning, which, in combination with hybrid mitigation strategies (Darwish et al. 2022), may allow us to use higher multipoles in the CMB lensing reconstruction. Other areas of exploration include (1) optimal filtering of ACT maps that accounts for noise nonideologies (Mirmelstein et al. 2019), (2) CMB-map-level combination with Planck data, (3) improved accuracy and precision of the lensing signal at the location of galaxy clusters (Hu et al. 2007), and (4) improved compact-object treatment allowing for less aggressive masking of the galaxy, thus enabling larger sky coverage of the mass map.

Looking further ahead, the Simons Observatory (SO Collaboration 2019), under construction at the same site as ACT, will significantly improve the sensitivity of CMB maps. This will enable subpercent constraints on the amplitude of matter fluctuations and a wide variety of cosmological and astrophysical science goals.

## Acknowledgments

We are grateful to Marika Asgari, Federico Bianchini, Julien Carron, Chihway Chang, Antony Lewis, Emily Longley, Hironao Miyatake, Jessie Muir, Luca Pagano, Kimmy Wu,

and Joe Zuntz for help with various aspects of the external codes and data sets used here. We are especially grateful to Xiangchong Li and the HSC team for making a consistent analysis of their Y3 results available to us. Some of the results in this paper have been derived using the healpy (Zonca et al. 2019) and HEALPix (Górski et al. 2005) packages. This research made use of Astropy (http://www.astropy.org), a community-developed core Python package for Astronomy (Astropy Collaboration et al. 2013, 2018). We also acknowledge use of the matplotlib (Hunter 2007) package and the Python Image Library for producing plots in this paper, use of the Boltzmann code CAMB (Lewis et al. 2000) for calculating theory spectra, and use of the GetDist (Lewis 2019), Cobaya (Torrado & Lewis 2021), and CosmoSIS (Zuntz et al. 2015) software for likelihood analysis and sampling. We acknowledge work done by the Simons Observatory Pipeline and Analysis Working Groups in developing open-source software used in this paper.

Support for ACT was through the U.S. National Science Foundation through awards AST-0408698, AST-0965625, and AST-1440226 for the ACT project, as well as awards PHY-0355328, PHY-0855887, and PHY-1214379. Funding was also provided by Princeton University, the University of Pennsylvania, and a Canada Foundation for Innovation (CFI) award to UBC. ACT operated in the Parque Astronómico Atacama in northern Chile under the auspices of the Agencia Nacional de Investigación y Desarrollo (ANID). The development of multichroic detectors and lenses was supported by NASA grants NNX13AE56G and NNX14AB58G. Detector research at NIST was supported by the NIST Innovations in Measurement Science program.

Computing was performed using the Princeton Research Computing resources at Princeton University, the Niagara supercomputer at the SciNet HPC Consortium, and the Symmetry cluster at the Perimeter Institute. SciNet is funded by the CFI under the auspices of Compute Canada, the Government of Ontario, the Ontario Research FundResearch Excellence, and the University of Toronto. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Industry Canada and by the Province of Ontario through the Ministry of Colleges and Universities. This research also used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under contract No. DE-AC02-05CH11231 using NERSC award HEP-ERCAPmp107.

M.M. and A.L. acknowledge support from NASA grant 21-ATP21-0145. B.D.S., F.J.Q., B.B., I.A.-C., G.S.F., N.M., and D.H. acknowledge support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No. 851274). B.D.S. further acknowledges support from an STFC Ernest Rutherford Fellowship. E.C., B.B., I.H., and H.T.J. acknowledge support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No. 849169). J.C.H. acknowledges support from NSF grant AST-2108536, NASA grants 21-ATP21-0129 and 22-ADAP22-0145, DOE grant DE-SC00233966, the Sloan Foundation, and the Simons Foundation. C.S. acknowledges support from the Agencia Nacional de Investigación y Desarrollo (ANID) through FONDECYT grant
No. 11191125 and BASAL project FB210003. R.D. acknowledges support from ANID BASAL project FB210003. A.D.H. acknowledges support from the Sutton Family Chair in Science, Christianity and Cultures and from the Faculty of Arts and Science, University of Toronto. J.D., Z.A., and E.S. acknowledge support from NSF grant AST-2108126. K.M. acknowledges support from the National Research Foundation of South Africa. A.M. and N.S. acknowledge support from NSF award No. AST-1907657. I.A.-C. acknowledges support from Fundación Mauricio y Carlota Botton. L.P. acknowledges support from the Misrahi and Wilkinson funds. M.H.i. acknowledges support from the National Research Foundation of South Africa (grant No. 137975). S.N. acknowledges support from a grant from the Simons Foundation (CCA 918271, PBL). C.H.-C. acknowledges FONDECYT Postdoc fellowship 322025. A.C. acknowledges support from the STFC (grant Nos. ST/N000927/1, ST/S000623/1 and ST/X006387/1). R.D. acknowledges support from the NSF Graduate Research Fellowship Program under grant No. DGE-2039656. O.D. acknowledges support from the SNSF Eccellenza Professorial Fellowship (No. 186879). O.D. acknowledges support from the SNSF Eccellenza Professorial Fellowship (No. 186879). C.S. acknowledges support from the Agencia Nacional de Investigación y Desarrollo (ANID) through FONDECYT grant No. 11191125 and BASAL project FB210003. T.N. acknowledges support from JSPS KAKENHI (grant Nos. JP20H05859 and JP22K03682) and World Premier International Research Center Initiative (WPI), MEXT, Japan. A.V.E. acknowledges support from NASA grants 22-ADAP22-0149 and 22-ADAP22-0150.

# Appendix A

## Lensing (Four-point) Likelihood and Theory

In this appendix, we describe in more detail the components of our lensing likelihood. We approximate this as being Gaussian in the band powers of the estimated lensing power spectrum $\hat{C}_{L_b}^{\kappa\kappa}$:

$$-2 \ln \mathcal{L} = \sum_{bb'} [\hat{C}_{L_b}^{\kappa\kappa} - C_{L_b}^{\kappa\kappa}(\theta)] C_{bb'}^{-1} [\hat{C}_{L_{b'}}^{\kappa\kappa} - C_{L_{b'}}^{\kappa\kappa}(\theta)], \quad (A1)$$

where $C_{L_b}^{\kappa\kappa}$ is the theory lensing convergence power spectrum evaluated with cosmological parameters $\theta$ and $C_{bb'}$ is the baseline covariance matrix for the binned spectrum, obtained from realistic sky simulations and detailed in Qu et al. (2024). When combining the lensing likelihood with that for the CMB anisotropy power spectra, we ignore the covariance between the measured lensing and anisotropy spectra, as these are negligible for DR6 noise sensitivities (Schmittfull et al. 2013; Peloton et al. 2017).

The reconstructed CMB lensing power spectrum depends on the four-point function of the CMB fields and, thus, quadratically on the CMB anisotropy power spectra. We normalize the estimated lensing power spectrum with a fiducial choice of CMB power spectra, but we account for the cosmology dependence of the true normalization (and of one of the bias corrections) in the likelihood analysis. For joint constraints with CMB anisotropy spectra, we correct the normalization at each point in parameter space as discussed below. For cosmology runs that do not include information from the primary CMB, we effectively marginalize over realizations of the CMB power spectrum in the normalization,

informed by current constraints. Specifically, we obtain 1000 posterior samples from the ACT DR4 + Planck primary CMB chains and propagate these to the covariance matrix as described in Appendix B of Qu et al. (2024). This step is done consistently to both the ACT and the NPIPE parts of the covariance matrix.

A fiducial cosmology $\theta_0$ is assumed in various steps of the lensing measurement. This includes the calculation of the normalization $\mathcal{R}_{L_b}^{-1}$ and $N_L^{-1}$ bias. To account for the dependence on $\theta_0$, the theoretical lensing power spectrum at each sampled point $\theta$ needs to be corrected as

$$C_{L_b}^{\kappa\kappa,\text{th}}(\theta) = \frac{[\mathcal{R}_{L_b}^{-1}(\theta_0)]^2}{[\mathcal{R}_{L_b}^{-1}(\theta)]^2} C_{L_b}^{\kappa\kappa}(\theta) - N_{L_b}^{1}(\theta_0) + N_{L_b}^{1}(\theta). \quad (A2)$$

Fully calculating the above for each point in the sampled parameter space is unfeasible, and hence we follow the approach of Planck Collaboration et al. (2016c), Sherwin et al. (2017), and Planck Collaboration et al. (2020d) and forward-model the linearized corrections to the theory spectrum due to the parameter deviations from the fiducial cosmology. For small deviations, expanding the normalization and $N_{L_b}^1$ around $\theta$ leads to

$$C_{L_b}^{\kappa\kappa,\text{th}}(\theta) \approx C_{L_b}^{\kappa\kappa}(\theta) + 2 \frac{d \ln \mathcal{R}_{L_b}(\theta_0)}{dC_{\ell'}^j} \\ \times [C_{\ell'}^j(\theta) - C_{\ell'}^j(\theta_0)] C_{L_b}^{\kappa\kappa}(\theta_0) + \frac{dN_{L_b}^1}{dC_{\ell'}^j} [C_{\ell'}^j(\theta) - C_{\ell'}^j(\theta_0)] \\ + \frac{dN_{L_b}^1}{dC_{L_b}^{\kappa\kappa}} [C_{L_b}^{\kappa\kappa}(\theta) - C_{L_b}^{\kappa\kappa}(\theta_0)]. \quad (A3)$$

The corrections involving changes in the CMB 2-point spectra (second and third terms on the RHS above) are not included for runs that combine with the ACT DR4 CMB 2-point likelihood. This is motivated following Appendix B of Qu et al. (2024) where it is noted that the ACT DR6 maps used in this work are calibrated against Planck maps, and the calibration uncertainty in this process is significantly smaller than that for the maps involved in the ACT DR4 2-point CMB cosmology analysis. We have checked that an MCMC run that includes the above terms returns parameter constraints similar to those when the corrections are excluded, if the chain point spectra $C_{\ell'}^j(\theta)$ are rescaled through calibration of $C_{\ell'}^{TT}(\theta)$ against Planck in the multipole range $1000 < \ell < 2000$. This procedure closely approximates what was done for the ACT DR6 data. For theory predictions, we use the Einstein-Boltzmann code CAMB (Lewis et al. 2000; ver. 1.3.6)<fnref n="85" /> with sufficiently high accuracy $lmax = 4000$, $lens\_margin = 1250$, $lens\_potential\_accuracy = 4$, Accuracy-Boost = 1, $lSampleBoost = 1$, and $lAccuracy-Boost = 1$. While these are lower than recommended in McCarthy et al. (2022)—see Appendix A of Hill et al. (2022) for the importance to current-generation CMB surveys—the evaluation time is significantly lower while being of sufficient accuracy given the precision of our measurement. We use the mead2016 nonlinear matter power spectrum prescription (Mead et al. 2015, 2016) with the default parameters

<footnote n="85">⁸⁵ https://camb.info/</footnote>
No. 11191125 and BASAL project FB210003. R.D. acknowledges support from ANID BASAL project FB210003. A.D.H. acknowledges support from the Sutton Family Chair in Science, Christianity and Cultures and from the Faculty of Arts and Science, University of Toronto. J.D., Z.A., and E.S. acknowledge support from NSF grant AST-2108126. K.M. acknowledges support from the National Research Foundation of South Africa. A.M. and N.S. acknowledge support from NSF award No. AST-1907657. I.A.-C. acknowledges support from Fundación Mauricio y Carlota Botton. L.P. acknowledges support from the Misrahi and Wilkinson funds. M.H.i. acknowledges support from the National Research Foundation of South Africa (grant No. 137975). S.N. acknowledges support from a grant from the Simons Foundation (CCA 918271, PBL). C.H.-C. acknowledges FONDECYT Postdoc fellowship 322025. A.C. acknowledges support from the STFC (grant Nos. ST/N000927/1, ST/S000623/1 and ST/X006387/1). R.D. acknowledges support from the NSF Graduate Research Fellowship Program under grant No. DGE-2039656. O.D. acknowledges support from the SNSF Eccellenza Professorial Fellowship (No. 186879). O.D. acknowledges support from the SNSF Eccellenza Professorial Fellowship (No. 186879). C.S. acknowledges support from the Agencia Nacional de Investigación y Desarrollo (ANID) through FONDECYT grant No. 11191125 and BASAL project FB210003. T.N. acknowledges support from JSPS KAKENHI (grant Nos. JP20H05859 and JP22K03682) and World Premier International Research Center Initiative (WPI), MEXT, Japan. A.V.E. acknowledges support from NASA grants 22-ADAP22-0149 and 22-ADAP22-0150.

# Appendix A

## Lensing (Four-point) Likelihood and Theory

In this appendix, we describe in more detail the components of our lensing likelihood. We approximate this as being Gaussian in the band powers of the estimated lensing power spectrum $\hat{C}_{L_b}^{\kappa\kappa}$:

$$-2 \ln \mathcal{L} = \sum_{bb'} [\hat{C}_{L_b}^{\kappa\kappa} - C_{L_b}^{\kappa\kappa}(\theta)] C_{bb'}^{-1} [\hat{C}_{L_{b'}}^{\kappa\kappa} - C_{L_{b'}}^{\kappa\kappa}(\theta)], \quad (A1)$$

where $C_{L_b}^{\kappa\kappa}$ is the theory lensing convergence power spectrum evaluated with cosmological parameters $\theta$ and $C_{bb'}$ is the baseline covariance matrix for the binned spectrum, obtained from realistic sky simulations and detailed in Qu et al. (2024). When combining the lensing likelihood with that for the CMB anisotropy power spectra, we ignore the covariance between the measured lensing and anisotropy spectra, as these are negligible for DR6 noise sensitivities (Schmittfull et al. 2013; Peloton et al. 2017).

The reconstructed CMB lensing power spectrum depends on the four-point function of the CMB fields and, thus, quadratically on the CMB anisotropy power spectra. We normalize the estimated lensing power spectrum with a fiducial choice of CMB power spectra, but we account for the cosmology dependence of the true normalization (and of one of the bias corrections) in the likelihood analysis. For joint constraints with CMB anisotropy spectra, we correct the normalization at each point in parameter space as discussed below. For cosmology runs that do not include information from the primary CMB, we effectively marginalize over realizations of the CMB power spectrum in the normalization,

informed by current constraints. Specifically, we obtain 1000 posterior samples from the ACT DR4 + Planck primary CMB chains and propagate these to the covariance matrix as described in Appendix B of Qu et al. (2024). This step is done consistently to both the ACT and the NPIPE parts of the covariance matrix.

A fiducial cosmology $\theta_0$ is assumed in various steps of the lensing measurement. This includes the calculation of the normalization $\mathcal{R}_{L_b}^{-1}$ and $N_L^{-1}$ bias. To account for the dependence on $\theta_0$, the theoretical lensing power spectrum at each sampled point $\theta$ needs to be corrected as

$$C_{L_b}^{\kappa\kappa,\text{th}}(\theta) = \frac{[\mathcal{R}_{L_b}^{-1}(\theta_0)]^2}{[\mathcal{R}_{L_b}^{-1}(\theta)]^2} C_{L_b}^{\kappa\kappa}(\theta) - N_{L_b}^{1}(\theta_0) + N_{L_b}^{1}(\theta). \quad (A2)$$

Fully calculating the above for each point in the sampled parameter space is unfeasible, and hence we follow the approach of Planck Collaboration et al. (2016c), Sherwin et al. (2017), and Planck Collaboration et al. (2020d) and forward-model the linearized corrections to the theory spectrum due to the parameter deviations from the fiducial cosmology. For small deviations, expanding the normalization and $N_{L_b}^1$ around $\theta$ leads to

$$C_{L_b}^{\kappa\kappa,\text{th}}(\theta) \approx C_{L_b}^{\kappa\kappa}(\theta) + 2 \frac{d \ln \mathcal{R}_{L_b}(\theta_0)}{dC_{\ell'}^j} \\ \times [C_{\ell'}^j(\theta) - C_{\ell'}^j(\theta_0)] C_{L_b}^{\kappa\kappa}(\theta_0) + \frac{dN_{L_b}^1}{dC_{\ell'}^j} [C_{\ell'}^j(\theta) - C_{\ell'}^j(\theta_0)] \\ + \frac{dN_{L_b}^1}{dC_{L_b}^{\kappa\kappa}} [C_{L_b}^{\kappa\kappa}(\theta) - C_{L_b}^{\kappa\kappa}(\theta_0)]. \quad (A3)$$

The corrections involving changes in the CMB 2-point spectra (second and third terms on the RHS above) are not included for runs that combine with the ACT DR4 CMB 2-point likelihood. This is motivated following Appendix B of Qu et al. (2024) where it is noted that the ACT DR6 maps used in this work are calibrated against Planck maps, and the calibration uncertainty in this process is significantly smaller than that for the maps involved in the ACT DR4 2-point CMB cosmology analysis. We have checked that an MCMC run that includes the above terms returns parameter constraints similar to those when the corrections are excluded, if the chain point spectra $C_{\ell'}^j(\theta)$ are rescaled through calibration of $C_{\ell'}^{TT}(\theta)$ against Planck in the multipole range $1000 < \ell < 2000$. This procedure closely approximates what was done for the ACT DR6 data. For theory predictions, we use the Einstein-Boltzmann code CAMB (Lewis et al. 2000; ver. 1.3.6)<fnref n="85" /> with sufficiently high accuracy $l_{max} = 4000$, $lens\_margin = 1250$, $lens\_potential\_accuracy = 4$, Accuracy-Boost = 1, $lSampleBoost = 1$, and $lAccuracy-Boost = 1$. While these are lower than recommended in McCarthy et al. (2022)—see Appendix A of Hill et al. (2022) for the importance to current-generation CMB surveys—the evaluation time is significantly lower while being of sufficient accuracy given the precision of our measurement. We use the mead2016 nonlinear matter power spectrum prescription (Mead et al. 2015, 2016) with the default parameters

<footnote n="85">⁸⁵ https://camb.info/</footnote>
$HMCode_A\_baryon = 3.13$ and $HMCode_\_eta\_baryon = 0.603$. Since our measurement mainly probes linear scales, this choice and baryonic feedback effects do not matter at current sensitivities, which can be explicitly seen in our analysis in Figure 9. For runs that include the ACT DR4 CMB anisotropy likelihood (see Appendix B), power spectra are calculated out to $l_{max} = 7000$. We have confirmed that $\chi^2$ values from this likelihood only differ by 0.04% when using accuracy settings from McCarthy et al. (2022), and so we do not use higher accuracy settings for ACT DR4.

We perform our Markov Chain Monte Carlo (MCMC) inference using the `Cobaya` package (Torrado & Lewis 2021) with the Metropolis-Hastings (MH) sampler with adaptive covariance learning, and we run our chains until the Gelman-Rubin criterion (Gelman & Rubin 1992) for chain variances falls below $R-1 = 0.01$, except in cases where the curvature density is varied, where we only require a threshold of $R-1 = 0.02$.

## Appendix B
### CMB Anisotropy (Two-point) Likelihoods

While our baseline constraint on structure growth only uses the ACT (and in some cases Planck) gravitational lensing reconstruction measurements in this work, we sometimes use information from the primary CMB anisotropies themselves either for comparison or in combination with the lensing measurement. CMB experiments like Planck and ACT produce maps of the temperature (*T*) and polarization anisotropies (*E*-mode and *B*-mode). The angular power spectra of these maps (*TT*, *TE*, *EE*) provide information mainly on the primary anisotropies of the CMB, which depend on the early universe (redshifts $z>1100$). They, however, also are screened by reionization ($z \simeq 8$) and therefore have a dependence on the optical depth to that epoch, $\tau$, and pick up secondary anisotropies like lensing. Reionization produces a suppression of the power spectra (as well as enhanced low-$\ell$ polarization), and the lensing effect induces smearing of the CMB acoustic peaks and a transfer of power from large to small scales. While the anisotropy power spectra measurements still mainly provide an early-universe extrapolation of late-time parameters like $\sigma_8$, in some cases we marginalize over an $A_{\text{lens}}$ parameter that frees up the amplitude of the lensing-induced peak smearing. This isolates the early-universe information so as to allow comparison with the late-time CMB lensing reconstruction (through the CMB four-point function) and with galaxy lensing.

As our baseline for CMB anisotropies, we use data measuring the two-point function from Planck. For the low-$\ell$ temperature component, we use the likelihood at $\ell < 30$ derived from the PR3 maps (Planck Collaboration et al. 2020b). For the high-$\ell$ temperature and polarization, we use the likelihood for *TT*, *TE*, and *EE* presented in Rosenberg et al. (2022), derived from the NPIPE maps (Planck Collaboration et al. 2020e) using the CamSpec likelihood. This gives consistent results to PR3 (Planck Collaboration et al. 2020b; Efstathiou & Gratton 2021), with around 10% more constraining power.

To include information from Planck's large-scale polarization data that constrains the optical depth to reionization, we use the likelihood estimated in Pagano et al. (2020) from the Sroll2 maps, sampling from the Sroll likelihood released with PR3 but updating the data with Sroll2.<fnref n="86" />

We also form a second independent combination of two-point function data by combining the WMAP 9 yr likelihood with the ACT DR4 likelihood, for *TT*, *TE*, and *EE*. For WMAP we use the Python implementation of the $\ell > 23$ likelihood, `pyWMAP`.<fnref n="87" /> For ACT we use the DR4 foreground-marginalized `pyactlite` likelihood software.<fnref n="88" /> In this case, we discard the large-scale WMAP polarization data, keeping the information on the optical depth from the Planck Sroll2 likelihood. In some cases we also test the effect of using the Planck PR3 high-$\ell$ likelihood in place of the NPIPE likelihood and of approximating the optical depth with a Gaussian distribution with mean and error shifted compared to the Sroll2 measurement.

## Appendix C
### Reanalysis of Galaxy Weak Lensing

We perform our galaxy weak-lensing analysis (cosmic shear) parameter inference using the CosmoSIS (Zuntz et al. 2015) framework. To facilitate a consistent comparison, the reanalysis here departs from the published works from KiDS and DES in the following ways: (1) we choose the cosmological parameterization from Table 1 (i.e., we sample in $\ln(10^{10}A_s)$, $100\theta_{\text{MC}}$, $\Omega_c h^2$, $\Omega_b h^2$ instead of $A_s$, $\Omega_c$, $\Omega_b$, $H_0$); (2) we choose the priors from Table 1, most notably a broader prior on $H_0$ and a sharper prior on $n_s$; (3) we have minor differences in the version and accuracy of the CAMB Boltzmann code (see Appendix A); and (4) we sample using the MH sampler instead of a nested sampler. These choices match those from Planck analyses (Planck Collaboration et al. 2016c, 2020d). For the MH sampling, we use adaptive covariance learning through an interface with the Cobaya package (Torrado & Lewis 2021) and run our chains until the Gelman-Rubin criterion (Gelman & Rubin 1992) for chain variances falls below $R-1 = 0.05$. We use the tensiometer package<fnref n="89" /> (Raveri et al. 2020; Raveri & Doux 2021) to load CosmoSIS outputs into getdist (Lewis 2019); the latter is used throughout this work to obtain marginalized 1D and 2D densities from MCMC samples. All reported tensions in this work use a Gaussian metric, i.e., the difference in the mean of the marginalized posteriors divided by the quadrature sum of the 68% confidence limits for the parameter of interest. The HSC reanalyses shown here were provided by the HSC team. They were run with the same priors and parameterization as above, with the same combination of galaxy BAO, but differ in the Boltzmann codes and sampling techniques.

In Figure 15, we compare the constraints from the reanalysis (blue) with those in the literature (orange). The “galaxy lensing” constraints all only include cosmic shear measurements, whereas “3 × 2 pt” measurements also include galaxy clustering and galaxy-galaxy lensing. Our DES-Y3 reanalysis constraints on $\sigma_8(\Omega_m/0.3)^{0.5}$ that include BAO are in agreement with those from the DES-Y3 galaxy-lensing-alone analysis in Amon et al. (2022) and Secco et al. (2022), the Fourier variant of the former (Doux et al. 2022), and the DES-Y3 3 × 2 pt analysis in Abbott et al. (2022). Similarly, our KiDS-1000 reanalysis constraints that include BAO are in agreement with those from the galaxy lensing reanalysis in Longley et al. (2023; whose framework we follow, including for scale cuts),

<footnote n="86">⁸⁶ https://web.fe.infn.it/~pagano/low_ell_data%20sets/sroll2/</footnote>

<footnote n="87">⁸⁷ https://github.com/HTJense/pyWMAP</footnote>

<footnote n="88">⁸⁸ https://github.com/ACTCollaboration/pyactlike</footnote>

<footnote n="89">⁸⁹ https://github.com/mraveri/tensiometer/</footnote>
$HMCode_A\_baryon = 3.13$ and $HMCode_\_eta\_baryon = 0.603$. Since our measurement mainly probes linear scales, this choice and baryonic feedback effects do not matter at current sensitivities, which can be explicitly seen in our analysis in Figure 9. For runs that include the ACT DR4 CMB anisotropy likelihood (see Appendix B), power spectra are calculated out to $l_{max} = 7000$. We have confirmed that $\chi^2$ values from this likelihood only differ by 0.04% when using accuracy settings from McCarthy et al. (2022), and so we do not use higher accuracy settings for ACT DR4.

We perform our Markov Chain Monte Carlo (MCMC) inference using the `Cobaya` package (Torrado & Lewis 2021) with the Metropolis-Hastings (MH) sampler with adaptive covariance learning, and we run our chains until the Gelman-Rubin criterion (Gelman & Rubin 1992) for chain variances falls below $R-1 = 0.01$, except in cases where the curvature density is varied, where we only require a threshold of $R-1 = 0.02$.

# Appendix B
## CMB Anisotropy (Two-point) Likelihoods

While our baseline constraint on structure growth only uses the ACT (and in some cases Planck) gravitational lensing reconstruction measurements in this work, we sometimes use information from the primary CMB anisotropies themselves either for comparison or in combination with the lensing measurement. CMB experiments like Planck and ACT produce maps of the temperature (*T*) and polarization anisotropies (*E*-mode and *B*-mode). The angular power spectra of these maps (*TT*, *TE*, *EE*) provide information mainly on the primary anisotropies of the CMB, which depend on the early universe (redshifts $z>1100$). They, however, also are screened by reionization ($z \simeq 8$) and therefore have a dependence on the optical depth to that epoch, $\tau$, and pick up secondary anisotropies like lensing. Reionization produces a suppression of the power spectra (as well as enhanced low-$\ell$ polarization), and the lensing effect induces smearing of the CMB acoustic peaks and a transfer of power from large to small scales. While the anisotropy power spectra measurements still mainly provide an early-universe extrapolation of late-time parameters like $\sigma_8$, in some cases we marginalize over an $A_{\text{lens}}$ parameter that frees up the amplitude of the lensing-induced peak smearing. This isolates the early-universe information so as to allow comparison with the late-time CMB lensing reconstruction (through the CMB four-point function) and with galaxy lensing.

As our baseline for CMB anisotropies, we use data measuring the two-point function from Planck. For the low-$\ell$ temperature component, we use the likelihood at $\ell < 30$ derived from the PR3 maps (Planck Collaboration et al. 2020b). For the high-$\ell$ temperature and polarization, we use the likelihood for *TT*, *TE*, and *EE* presented in Rosenberg et al. (2022), derived from the NPIPE maps (Planck Collaboration et al. 2020e) using the CamSpec likelihood. This gives consistent results to PR3 (Planck Collaboration et al. 2020b; Efstathiou & Gratton 2021), with around 10% more constraining power.

To include information from Planck's large-scale polarization data that constrains the optical depth to reionization, we use the likelihood estimated in Pagano et al. (2020) from the Sroll2 maps, sampling from the Sroll likelihood released with PR3 but updating the data with Sroll2.<fnref n="86" />

We also form a second independent combination of two-point function data by combining the WMAP 9 yr likelihood with the ACT DR4 likelihood, for *TT*, *TE*, and *EE*. For WMAP we use the Python implementation of the $\ell > 23$ likelihood, `pyWMAP`.<fnref n="87" /> For ACT we use the DR4 foreground-marginalized `pyactlite` likelihood software.<fnref n="88" /> In this case, we discard the large-scale WMAP polarization data, keeping the information on the optical depth from the Planck Sroll2 likelihood. In some cases we also test the effect of using the Planck PR3 high-$\ell$ likelihood in place of the NPIPE likelihood and of approximating the optical depth with a Gaussian distribution with mean and error shifted compared to the Sroll2 measurement.

# Appendix C
## Reanalysis of Galaxy Weak Lensing

We perform our galaxy weak-lensing analysis (cosmic shear) parameter inference using the CosmoSIS (Zuntz et al. 2015) framework. To facilitate a consistent comparison, the reanalysis here departs from the published works from KiDS and DES in the following ways: (1) we choose the cosmological parameterization from Table 1 (i.e., we sample in $\ln(10^{10}A_s)$, $100\theta_{\text{MC}}$, $\Omega_c h^2$, $\Omega_b h^2$ instead of $A_s$, $\Omega_c$, $\Omega_b$, $H_0$); (2) we choose the priors from Table 1, most notably a broader prior on $H_0$ and a sharper prior on $n_s$; (3) we have minor differences in the version and accuracy of the CAMB Boltzmann code (see Appendix A); and (4) we sample using the MH sampler instead of a nested sampler. These choices match those from Planck analyses (Planck Collaboration et al. 2016c, 2020d). For the MH sampling, we use adaptive covariance learning through an interface with the Cobaya package (Torrado & Lewis 2021) and run our chains until the Gelman-Rubin criterion (Gelman & Rubin 1992) for chain variances falls below $R-1 = 0.05$. We use the tensiometer package<fnref n="89" /> (Raveri et al. 2020; Raveri & Doux 2021) to load CosmoSIS outputs into getdist (Lewis 2019); the latter is used throughout this work to obtain marginalized 1D and 2D densities from MCMC samples. All reported tensions in this work use a Gaussian metric, i.e., the difference in the mean of the marginalized posteriors divided by the quadrature sum of the 68% confidence limits for the parameter of interest. The HSC reanalyses shown here were provided by the HSC team. They were run with the same priors and parameterization as above, with the same combination of galaxy BAO, but differ in the Boltzmann codes and sampling techniques.

In Figure 15, we compare the constraints from the reanalysis (blue) with those in the literature (orange). The “galaxy lensing” constraints all only include cosmic shear measurements, whereas “3 × 2 pt” measurements also include galaxy clustering and galaxy-galaxy lensing. Our DES-Y3 reanalysis constraints on $\sigma_8(\Omega_m/0.3)^{0.5}$ that include BAO are in agreement with those from the DES-Y3 galaxy-lensing-alone analysis in Amon et al. (2022) and Secco et al. (2022), the Fourier variant of the former (Doux et al. 2022), and the DES-Y3 3 × 2 pt analysis in Abbott et al. (2022). Similarly, our KiDS-1000 reanalysis constraints that include BAO are in agreement with those from the galaxy lensing reanalysis in Longley et al. (2023; whose framework we follow, including for scale cuts),

<footnote n="86">⁸⁶ https://web.fe.infn.it/~pagano/low_ell_data%20sets/sroll2/</footnote>

<footnote n="87">⁸⁷ https://github.com/HTJense/pyWMAP</footnote>

<footnote n="88">⁸⁸ https://github.com/ACTCollaboration/pyactlike</footnote>

<footnote n="89">⁸⁹ https://github.com/mraveri/tensiometer/</footnote>
<figure><img src="image_18.png" /><figcaption>**Figure 15.** Comparison of $\sigma_8(\Omega_m/0.3)^{0.5}$ constraints from a consistent reanalysis of galaxy lensing (blue) with results from the literature (orange).</figcaption></figure>

the galaxy lensing analysis by the KiDS Collaboration (Asgari et al. 2021) and its Fourier variant (Loureiro et al. 2022), and the 3 × 2 pt analysis by the KiDS Collaboration (Heymans et al. 2021). The HSC-Y3 results are consistent with those from Dalal et al. (2023) and Li et al. (2023). Apart from the differences outlined above (including our choices of priors), it should be noted that some of the constraints reported in the literature do not use the marginalized mean and standard error as we do, but might report quantities such as the multivariate maximum posterior (MAP) and its credible interval calculated using its projected joint highest posterior density (PJ-HPD; e.g., Asgari et al. 2021).

## Appendix D

### Parameter Dependence of CMB and Galaxy Lensing

CMB lensing constraints on parameters arise from two different ranges of scales. First, on small scales, the CMB lensing power spectrum primarily probes the high-k power-law tail of the matter power spectrum in projection; this implies that CMB lensing parameter constraints can be well approximated by a parameter combination $\sigma_8^\alpha \Omega_m^\beta h^\gamma$, where $\alpha, \beta, \gamma$ are constants (Planck Collaboration et al. 2016c; Baxter & Sherwin 2021). On the other hand, much of our CMB lensing power spectrum constraining power also arises from intermediate and large scales, where, due to projection of the matter power spectrum near the peak, the lensing spectrum deviates from this high-L power law, providing a different sensitivity to the matter–radiation equality multipole $L_{eq} \sim (\Omega_m^{0.6} h)$. Therefore, considering the 3D $\sigma_8, \Omega_m, h$ parameter space, the two constraints arising from CMB lensing power spectrum constraints define two surfaces; their intersection implies that the CMB lensing power spectrum constraints define a line in this space. Now we can easily explain why the constraint on $\sigma_8$ when combining with BAO (as seen in Figure 16) is so tight:

BAO defines another surface in this space, so that the intersection of the BAO and lensing constraints is a point (or a small region in parameter space; see left panel of Figure 17).

In contrast, galaxy lensing generally does not probe the large-scale regime of scales approaching the matter power spectrum peak; effectively, it only provides one small-scale constraint within the $\sigma_8, \Omega_m, h$ space, defining a single surface. Adding the BAO data, which define a different surface, the intersection gives a line-shaped constraint (instead of a point as for the CMB lensing and BAO combination; see right panel of Figure 17). This explains why the $\sigma_8$ constraint is much broader and shows a significant degeneracy with the matter density.

<figure><img src="image_19.png" /><figcaption>**Figure 16.** Constraints in the $\sigma_8 - \Omega_m$ plane when combining ACT CMB lensing (red) or DES galaxy lensing (blue) with galaxy BAO. The posteriors in the absence of BAO are shown in lighter shades and are constrained well roughly along the $\sigma_8(\Omega_m/0.3)^{0.25}$ and $\sigma_8(\Omega_m/0.3)^{0.5}$ directions for CMB and galaxy lensing, respectively.</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>**Figure 15.** Comparison of $\sigma_8(\Omega_m/0.3)^{0.5}$ constraints from a consistent reanalysis of galaxy lensing (blue) with results from the literature (orange).</figcaption></figure>

the galaxy lensing analysis by the KiDS Collaboration (Asgari et al. 2021) and its Fourier variant (Loureiro et al. 2022), and the 3 × 2 pt analysis by the KiDS Collaboration (Heymans et al. 2021). The HSC-Y3 results are consistent with those from Dalal et al. (2023) and Li et al. (2023). Apart from the differences outlined above (including our choices of priors), it should be noted that some of the constraints reported in the literature do not use the marginalized mean and standard error as we do, but might report quantities such as the multivariate maximum posterior (MAP) and its credible interval calculated using its projected joint highest posterior density (PJ-HPD; e.g., Asgari et al. 2021).

# Appendix D

## Parameter Dependence of CMB and Galaxy Lensing

CMB lensing constraints on parameters arise from two different ranges of scales. First, on small scales, the CMB lensing power spectrum primarily probes the high-k power-law tail of the matter power spectrum in projection; this implies that CMB lensing parameter constraints can be well approximated by a parameter combination $\sigma_8^\alpha \Omega_m^\beta h^\gamma$, where $\alpha, \beta, \gamma$ are constants (Planck Collaboration et al. 2016c; Baxter & Sherwin 2021). On the other hand, much of our CMB lensing power spectrum constraining power also arises from intermediate and large scales, where, due to projection of the matter power spectrum near the peak, the lensing spectrum deviates from this high-L power law, providing a different sensitivity to the matter–radiation equality multipole $L_{eq} \sim (\Omega_m^{0.6} h)$. Therefore, considering the 3D $\sigma_8, \Omega_m, h$ parameter space, the two constraints arising from CMB lensing power spectrum constraints define two surfaces; their intersection implies that the CMB lensing power spectrum constraints define a line in this space. Now we can easily explain why the constraint on $\sigma_8$ when combining with BAO (as seen in Figure 16) is so tight:

BAO defines another surface in this space, so that the intersection of the BAO and lensing constraints is a point (or a small region in parameter space; see left panel of Figure 17).

In contrast, galaxy lensing generally does not probe the large-scale regime of scales approaching the matter power spectrum peak; effectively, it only provides one small-scale constraint within the $\sigma_8, \Omega_m, h$ space, defining a single surface. Adding the BAO data, which define a different surface, the intersection gives a line-shaped constraint (instead of a point as for the CMB lensing and BAO combination; see right panel of Figure 17). This explains why the $\sigma_8$ constraint is much broader and shows a significant degeneracy with the matter density.

<figure><img src="image_20.png" /><figcaption>**Figure 16.** Constraints in the $\sigma_8 - \Omega_m$ plane when combining ACT CMB lensing (red) or DES galaxy lensing (blue) with galaxy BAO. The posteriors in the absence of BAO are shown in lighter shades and are constrained well roughly along the $\sigma_8(\Omega_m/0.3)^{0.25}$ and $\sigma_8(\Omega_m/0.3)^{0.5}$ directions for CMB and galaxy lensing, respectively.</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>**Figure 17.** Distribution of MCMC samples for weak lensing in the $\sigma_8-\Omega_m-H_0$ space. Left: CMB lensing. Galaxy BAO samples are shown in gray; their density does not depend on $\sigma_8$. Due to the large range of scales probed by CMB lensing, shown here for ACT DR6 (blue in 3D; purple in projection), they form a line in this space. The intersection of ACT CMB lensing with BAO (red in 3D; orange in projection) provides a tight constraint on $\sigma_8$. Right: galaxy lensing. In contrast, the galaxy weak-lensing samples define a surface owing to their not probing the large-scale regime, shown here for DES-Y3 for illustration (blue in 3D; purple in projection). The intersection with BAO (red in 3D; orange in projection) provides weaker constraints on $\sigma_8$.</figcaption></figure>

We have verified this explanation with a simple exercise: we perform an analysis on mock CMB lensing data, artificially adjusting the errors to vary the scales from which the information originates, while holding the total signal-to-noise ratio constant. When we shift the mock CMB lensing information only to arise from small scales, $L > 2000$, the shapes of the parameter constraint contours and the constraints on $\sigma_8$ show a close resemblance to the constraints from the combination of galaxy weak lensing and BAO.

## ORCID iDs

Mathew S. Madhavacheril https://orcid.org/0000-0001-6740-5350

Peter Doze https://orcid.org/0000-0002-2408-0813

Kirsten R. Hall https://orcid.org/0000-0002-4176-845X

Matt Hilton https://orcid.org/0000-0002-8490-8117

Kenda Knowles https://orcid.org/0000-0002-8452-0825

Kavilan Moodley https://orcid.org/0000-0001-6606-7142

Tony Mroczkowski https://orcid.org/0000-0003-3816-5372

David N. Spergel https://orcid.org/0000-0002-5151-0006

Edward J. Wollack https://orcid.org/0000-0002-7567-4451

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<figure><img src="image_21.png" /><figcaption>**Figure 17.** Distribution of MCMC samples for weak lensing in the $\sigma_8-\Omega_m-H_0$ space. Left: CMB lensing. Galaxy BAO samples are shown in gray; their density does not depend on $\sigma_8$. Due to the large range of scales probed by CMB lensing, shown here for ACT DR6 (blue in 3D; purple in projection), they form a line in this space. The intersection of ACT CMB lensing with BAO (red in 3D; orange in projection) provides a tight constraint on $\sigma_8$. Right: galaxy lensing. In contrast, the galaxy weak-lensing samples define a surface owing to their not probing the large-scale regime, shown here for DES-Y3 for illustration (blue in 3D; purple in projection). The intersection with BAO (red in 3D; orange in projection) provides weaker constraints on $\sigma_8$.</figcaption></figure>

We have verified this explanation with a simple exercise: we perform an analysis on mock CMB lensing data, artificially adjusting the errors to vary the scales from which the information originates, while holding the total signal-to-noise ratio constant. When we shift the mock CMB lensing information only to arise from small scales, $L > 2000$, the shapes of the parameter constraint contours and the constraints on $\sigma_8$ show a close resemblance to the constraints from the combination of galaxy weak lensing and BAO.

## ORCID iDs

Mathew S. Madhavacheril https://orcid.org/0000-0001-6740-5350

Peter Doze https://orcid.org/0000-0002-2408-0813

Kirsten R. Hall https://orcid.org/0000-0002-4176-845X

Matt Hilton https://orcid.org/0000-0002-8490-8117

Kenda Knowles https://orcid.org/0000-0002-8452-0825

Kavilan Moodley https://orcid.org/0000-0001-6606-7142

Tony Mroczkowski https://orcid.org/0000-0003-3816-5372

David N. Spergel https://orcid.org/0000-0002-5151-0006

Edward J. Wollack https://orcid.org/0000-0002-7567-4451

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