Field-level inference of cosmological initial conditions from galaxy surveys requires a forward model that is simultaneously accurate in the non-linear regime, computationally efficient, and fully differentiable. Traditional N-body simulations are accurate but computationally prohibitive for iterative inference, while approximate solvers like Lagrangian Perturbation Theory (LPT) fail to capture the knotty halo-forming dynamics of the cosmic web at late times. We introduce the \textit{Lagrangian Neural Cellular Automaton} (LNCA), a hybrid deep learning framework that can be applied to emulate structure formation as a local, iterative dynamical process on a comoving lattice. Unlike standard Eulerian Convolutional Neural Networks (CNNs) which map fixed density fields, the LNCA operates in the Lagrangian frame, advecting the computational graph itself to follow the flow of mass. By training the network to learn only the \textit{residual} displacement corrections to the Zeldovich approximation, we achieve high-fidelity emulation of the non-linear physics while guaranteeing accuracy at large scales. We further constrain our model to produce complete trajectories, not just final states, by adopting an equivariant cellular automaton architecture, which recurrently iterates on its internal states to yield a dynamic history. The resulting model is strictly local, translationally and rotationally equivariant, and naturally supports continuous time integration, making it a reliable differentiable forward model for reconstructing the initial conditions of the universe from lightcone data. Our trained model supports percent-level precision in the power and cross spectra well into the non-linear regime ($k \lesssim 0.5 \, h \text{Mpc}^{-1}$), while requiring $\sim10^4$ times fewer learned parameters than comparable models which take the form of an interpretable internal dynamic rule set.
Changing-look (CL) AGNs trace rapid changes in nuclear activity, but their connection to host galaxy properties remains unclear. We present a study of the host galaxies of 105 CL AGNs previously selected by comparing DESI and SDSS data. We apply a two-epoch spectrophotometric decomposition to the DESI and SDSS spectra of the 105 objects. Meanwhile, HSC images are used to constrain their varying AGN components and non-varying stellar population components. We find that 79 of the 105 (75.2%) CL AGN hosts are quiescent galaxies, and 31/105 (29.5%) also show post-starburst signatures. We focus on 82 CL AGNs with extended host emission in the HSC images and compare them with extended quasars at similar redshift and stellar mass. Their star formation activity, Balmer absorption, and quiescent fractions are broadly consistent with those of the comparison quasars, although post-starburst hosts are more common among the CL AGNs. Our CL AGNs with extended host emission are more often quiescent than those with compact morphology, but this difference is not apparent after matching in redshift and stellar mass. The $\mathrm{O\, \small II}$ and $\mathrm{O\, \small III}$ narrow lines show no population-wide response to the continuum and broad line changes, consistent with the slower response expected from the narrow line region. Together, these results favor changes in the central supermassive black hole accretion rate as the main origin of the CL transitions.