Universality Classes of Interacting Dark Energy from Spontaneous Symmetry Breaking
Phenomenological models of interacting dark energy (IDE) often treat the late-time activation history of the dark sector coupling as an independent function. We show that in conformally coupled scalar--tensor theories, this freedom is constrained by the local restoring structure of the symmetry-breaking potential. Within the adiabatic tracking regime, the coupling evolution satisfies $n=3/p$, where $p$ is the restoring order near the broken minimum thereby organizing distinct symmetry-breaking potentials such as quartic, Coleman--Weinberg, and axion-like forms into a common asymptotic dynamical class ($p=1$, $n=3$). We test this framework using Planck~2018 CMB lensing, RSD, and supernova data. Current observations provide only limited discrimination between the predicted activation classes and yield no statistically significant evidence for a nonzero interaction with $|\beta_0|\lesssim0.26$ at $95%$ credibility. The rigid asymptotic implementation ($n=3$) is strongly disfavored by the combined geometric and growth constraints indicating that the observable coupling history cannot be identified directly with its asymptotic attractor form. In the heavy-scalar adiabatic regime, the modifications to the growth rate $f(z)$ and growth factor $D(z)$ are of opposite sign throughout $0\le z\le2$, suppressing the net deviation in $f\sigma_8(z)$ to $\Delta f\sigma_8/f\sigma_8\lesssim0.3%$ across the posterior. Standard growth-rate measurements therefore have limited sensitivity to this class of models, shifting the observational focus toward probes that constrain $f(z)$ and $D(z)$ independently. Taken together, these results establish a dynamical classification of late-time IDE activation histories and clarify how finite-redshift observables are related to the asymptotic attractor structure and the local restoring properties of the underlying scalar potential.