Time-domain Radio-loudness of Active Galactic Nuclei: Intermittency, Memory, and Jet Escape

The classical radio-loudness parameter $R \equiv f_\nu(5\,\mathrm{GHz})/f_\nu(4400\,\text{\AA})$ divides a radio flux density by an optical/UV accretion tracer, but the two terms do not probe the same clock. The radio numerator can blend compact-core emission from the current engine, lobe and relic plasma left by earlier jet episodes, and host-galaxy synchrotron emission. We introduce a time-domain radio-loudness (TDRL) description that keeps these contributions separate. The radio numerator is written as compact-core and extended-lobe terms, with recovered fractions set by observing frequency, angular resolution, and surface-brightness sensitivity. For a single intermittently jetted AGN population, a two-state duty cycle filtered by exponential lobe fading gives an exact stationary Beta distribution for the normalized extended-radio response. Its mean is $\fduty$, while its variance scales as $(1+\chi_\nu)^{-1}$, where $\chi_\nu\equiv\taunu/t_{\rm switch}$. In this reference limit, the familiar GHz valley near the classical radio-loud/radio-quiet boundary can arise from short radio memory alone, without requiring two intrinsic engine classes. Metre-wave surveys that recover diffuse emission and subtract the host contribution should therefore progressively fill the valley. In the $(\Rcore,\Rlobe)$ plane, a core--lobe mismatch index separates triggering, sustained, and remnant phases, provided orientation-dependent core beaming is modelled or controlled. A complementary two-barrier picture, involving horizon-threading magnetic flux and jet escape through the nuclear medium, separates jet launching from the formation of large-scale radio structure. This view makes radio loudness a probe of jet duty cycle, radio memory, and escape through the host environment.