If neutrinos are a significant contributor to the matter density of the universe, then they should have ∼ eV mass and cluster in galactic (super) cluster halos, and possibly in galactic halos as well. It was noted in the early 1980's that cosmic ray neutrinos with energy within δE/E R = Γ Z /M Z ∼ 3% of the peak energy E R = 4 (eV/m ν ) × 10 21 eV will annihilate on the nonrelativistic relic antineutrinos (and vice versa) to produce the Z-boson with an enhanced, resonant cross section of O(G F ) ∼ 10 −32 cm 2 . The result of the resonant neutrino annihilation is a hadronic Z-burst 70% of the time, which contains, on average, thirty photons and 2.7 nucleons with energies near or above the GZK cutoff energy of 5 × 10 19 eV. These photons and nucleons produced within our Supergalactic halo may easily propagate to earth and initiate super-GZK air showers.Here we show that the probability for each neutrino flavor at its resonant energy to annihilate within the halo of our Supergalactic cluster is likely within an order of magnitude of 1%, with the exact value depending on unknown aspects of neutrino mixing and relic neutrino clustering. The absolute lower bound in a hot Big Bang universe for the probability to annihilate within a 50 Mpc radius (roughly a nucleon propagation distance) of earth is 0.036%. From fragmentation data for Z-decay, we estimate that the nucleons are more energetic than the photons by a factor ∼ 10.Several tests of the hypothesis are indicated. 1 The Cosmic Ray Puzzle Above 10 20 eV The recent discoveries by the AGASA[1], Fly's Eye[2], Haverah Park[3], and Yakutsk[4]collaborations of air shower events with energies above the Greisen-Zatsepin-Kuzmin (GZK) cutoff of ∼ 5×10 19 eV presents an outstanding puzzle in ultrahigh-energy cosmic-ray physics.It was anticipated that the highest-energy cosmic primaries would be protons from outside the galaxy, perhaps produced in active galactic nuclei (AGNs). It was also anticipated that the highest energies for protons arriving at earth would be ∼ 5 × 10 19 eV [5]. The origin of this GZK cutoff is degradation of the proton energy by the resonant scattering process p + γ 2.7K → ∆ * → N + π when the proton is above the resonant threshold for ∆ * production; γ 2.7K denotes a photon in the 2.7K cosmic background radiation. For every mean free path ∼ 6 Mpc of travel, the proton loses 20% of its energy on average [6]. A proton produced at its cosmic source with an initial energy E p will on average arrive at earth with only a fraction ∼ (0.8) D/6 Mpc of its original energy. Since AGNs are hundreds of megaparsecs away, the energy requirement at an AGN source for a proton which arrives at earth with a super-GZK energy is unrealistically high [7]. Of course, proton energy is not lost significantly if the highest energy protons come from a rather nearby source, < ∼ 50 to 100 Mpc [8]. However, no AGN sources are known to exist within 100 Mpc of earth. Hence, the observation of air shower events above 5 × 10 19 eV challenges standard theory. 1 In principle, there exists...