PRIMAger General Observer programs: a {\Large $\pi$}-sr Infrared Survey and other General Observer wide-field programs

The PRobe far-Infrared Mission for Astrophysics (PRIMA) is a cryogenically-cooled, far-infrared (far-IR) observatory expected to begin serving the astronomical community by early 2030. The mission features two advanced instruments: PRIMAger and FIRESS. PRIMAger will operate across the mid- to far-IR spectrum, covering wavelengths from approximately 25 to 260 $\mu$m. It will offer hyperspectral imaging in medium resolution bands (R $\sim$ 8, using a linear variable filter) from 25 to 80 $\mu$m, and broad band (R $\sim$ 4) photometric and polarimetric imaging in four bands spanning 80 to 260 $\mu$m. The capabilities of PRIMAger will enable a broad range of unique scientific programs, accessible through General Observer (GO) projects. In this paper, we present and define a PRIMAger survey over 25\% of the sky, called $\pi$-IR survey. This survey would exploit PRIMAger's hyperspectral and polarimetric modes to collect data on about 8 $\times$ 10$^{6}$ galaxies to z $\sim$ 4. The R=8 spectral resolution of the PRIMAger Hyperspectral Imaging (PHI) filters will enable users to study the emission of polycyclic aromatic hydrocarbon (PAH). A large sample of galaxies will be observed with the polarimetric bands of PRIMAger, allowing unique statistical information for galaxies to be harvested for the first time.