We investigate the cosmological information in higher-order statistics of the cosmic microwave background (CMB) lensing convergence field for a near-term experiment with noise properties similar to the Simons Observatory (SO). Using a fully field-level forward-modeling pipeline based on ray-traced simulations from the massivenus suite and realistic SO-like CMB lensing reconstruction, we naturally include nonlinear structure formation, post-Born effects, and higher-order reconstruction noise. We measure several non-Gaussian statistics, including Minkowski functionals, peak and minima counts, moments, and wavelet-scattering coefficients. We train Gaussian-process emulators to model each statistic’s dependence on the matter density fraction ${\mathrm{\Omega }}_m$, the scalar power spectrum amplitude $A_s$, and the neutrino mass sum $M_{\nu }$. We quantify the relative information gain these statistics provide beyond the lensing power spectrum and identify which are most robust to reconstruction noise. We find that morphology-based statistics, particularly Minkowski functionals and peak/minima counts, offer significant complementary constraining power: combining all non-Gaussian statistics with the power spectrum yields reductions of 40% and 38% in the marginalized uncertainties on ${\mathrm{\Omega }}_m$ and $A_s$, respectively, and a 70% reduction in the one-sided uncertainty on $M_{\nu }$. These gains remain non-negligible even when the power spectrum is extended to larger scales and combined with primary CMB and BAO data, with Minkowski functionals providing an additional 11% improvement in $\sigma (M_{\nu })$ and 35% in $\sigma ({\mathrm{\Omega }}_m)$ beyond the extended power spectrum. By contrast, moments and wavelet-scattering coefficients provide more limited gains at SO noise levels. Our results highlight the potential of non-Gaussian statistics to enhance cosmological constraints from SO and future CMB surveys.

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• Cosmic microwave background
• Cosmological parameters
• Cosmology
• Large scale structure of the Universe