Context. The detection of synchrotron radiation from the intergalactic medium (IGM) that pervades the filaments of the Cosmic Web constitutes an upcoming frontier to test physical models of astrophysical shocks and their radiation mechanisms, trace the missing baryons, and constrain magnetogenesis – the origin and evolution of extragalactic magnetic fields. Aims. The first synchrotron detections of the IGM within filaments have recently been claimed. Now is the time to develop a rigorous statistical framework to predict sky regions with the strongest signal and to move from mere detection to inference, that is to say identifying the most plausible physical models and parameter values from observations. Methods. Current theory posits that the filament IGM lights up through shocks that originate from large-scale structure formation. With Bayesian inference, we generated a probability distribution on the set of specific intensity functions that represent our view of the merger- and accretion-shocked synchrotron Cosmic Web (MASSCW). We combined the Bayesian Origin Reconstruction from Galaxies (BORG) Sloan Digital Sky Survey (SDSS) total matter density posterior, which is based on spectroscopic observations of galaxies within SDSS DR7, snapshots of Enzo magnetohydrodynamics (MHD) cosmological simulations, a Gaussian random field (GRF), and a ray tracing approach to arrive at the result. Results. We present a physics-based prediction of the MASSCW signal, including principled uncertainty quantification, for a quarter of the sky and up to cosmological redshift z_max = 0.2. The super-Mpc 3D resolution of the current implementation limits the resolution of the predicted 2D imagery, so that individual merger and accretion shocks are not resolved. The MASSCW prior can be used to identify the most promising fields to target with low-frequency radio telescopes and to conduct actual detection experiments. We furthermore calculated a probability distribution for the flux-density–weighted mean (i.e. sky-averaged) redshift z̄̄ of the MASSCW signal up to z_max, and found a median of z̄̄ = 0.077. We constructed a low-parametric analytic model that produces a similar distribution for z̄̄, with a median of z̄̄ = 0.072. Extrapolating the model, we were able to calculate z̄̄ for all large-scale structure in the Universe (including what lies beyond z_max) and show that, if one only considers filaments, z̄̄ depends on virtually one parameter. As case studies, we finally explore the predictions of our MASSCW specific intensity function prior in the vicinity of three galaxy clusters, the Hercules Cluster, the Coma Cluster, and Abell 2199, and in three deep Low-frequency Array (LOFAR) High-band Antennae (HBA) fields, the Lockman Hole, Abell 2255, and the Ursa Major Supercluster. Conclusions. We describe and implement a novel, flexible, and principled framework for predicting the low-frequency, low-resolution specific intensity function of the Cosmic Web due to merger and accretion shocks that arise during large-scale structure formation. The predictions guide Local Universe searches for filamentary baryons through half of the Northern Sky. Once cosmological simulations of alternative emission mechanisms have matured, our approach can be extended to predict additional physical pathways that contribute to the elusive synchrotron Cosmic Web signal.