$\Lambda_{\rm s}$CDM cosmology from a type-II minimally modified gravity

We integrate $\Lambda_{\rm s}$CDM, a promising scenario for alleviating cosmological tensions, into VCDM, a type-II minimally modified gravity. This promotes the scenario to a fully predictive model (dubbed $\Lambda_{\rm s}$VCDM) that specifies the cosmological evolution self-consistently, including through the late-time AdS-to-dS transition epoch. In this theory, an auxiliary scalar field generates an effective cosmological constant with either a constant or a linear potential. This allows an abrupt mirror AdS-to-dS transition via a piecewise-linear potential with a sudden slope change. To remove the associated sudden singularity and ensure stable evolution, we smooth the junction using a blended sigmoid interpolant, obtaining rapid but continuous transitions. We identify two qualitatively distinct smooth mirror AdS-to-dS realisations of $\Lambda_{\rm s}$: (i) an agitated transition, in which the potential interpolates between equal-magnitude AdS and dS plateaus and $\Lambda_{\rm s}$ develops a central bump; and (ii) a quiescent transition, in which the potential remains continuous but changes slope across the transition layer, so that $\Lambda_{\rm s}(a)$ can remain monotone, with possible shallow shoulders, and a central bump is not automatic. Depending on type and sharpness, a finite-width transition can induce a transient accelerated-expansion interval ($\ddot a>0$) around $z\sim 1.5-2$, in addition to present-day acceleration, and, if the background enters a region where $V_{,\phi\phi}>2/3$, a nested super-acceleration episode. These distinct transient histories can imprint signatures on background and perturbation evolution. Our construction enables a self-consistent observational assessment of smooth $\Lambda_{\rm s}$CDM realisations and motivates multi-probe analyses to test transition dynamics and reassess cosmological tensions.