The Diffuse Supernova Neutrino Background: The Distance to Detection, Priced

The diffuse supernova neutrino background (DSNB) flux is computed from a star-formation-rate-based core-collapse history and explicit per-supernova spectra, with the failed-supernova fraction as a free parameter. Central prediction (failed fraction 0.2): 0.71 cm-2 s-1 above 17.3 MeV, a factor 3.8 below the published Super-Kamiokande limit of 2.7 cm-2 s-1. Hyper-Kamiokande reaches 3 sigma by counting in 4.4 years (821 kt yr), 2.7-8.6 years across failed fractions 0.4-0. A sensitivity run identifies background level and efficiency as controlling the detection date, and spectral temperature and failed fraction as the dominant astrophysical uncertainties.