The possibility exists that cosmic-ray neutrinos with energies of ∼ 10 20 eV interact in the atmosphere with a cross section at the millibarn level, giving rise to some of the highest-energy air showers. In a specific dynamical model which can give rise to such a cross section, we show that there can be a small effect upon the extraction of the effective value of sin 2 θW from νµ-hadron, neutral-current cross section data at NuTeV.The possibility that neutrinos, with energies of the order of 10 20 eV, have stronger interactions with hadrons than those which are given by electroweak theory, has been considered, specifically in connection with the puzzling observation of the highest energy cosmic-ray air shower events.[1, 2] Such energetic neutrinos can propagate from sources at cosmological distances (red shifts > ∼ 1), undisturbed by the cosmic microwave background radiation, which photons largely prevent protons of > ∼ 0.6 × 10 20 eV from arriving here. [3,4] Recently [5] we have explicitly calculated a σ νµ of ∼ 10 −29 cm 2 at E νµ ∼ 10 20 eV, in a specific dynamical model which involves the hypothesis of a strong, parity-conserving interaction of ν µ with a hypothetical, point-like component of a π 0 , leading to a very massive neutral lepton L, with m L ∼ 2 × 10 6 GeV. F 1 The essential physical idea is that of neutrino structure at very small distances (< 10 −18 cm), due to hadron-like coupling to L. For this m L , E νµ must reach ∼ 2 × 10 21 eV for production of real L. However, the amplitude for the diagram in Fig. (1), which involves a virtual L, results in a cross section above the electroweak σ νµ ∼ 10 −31 cm 2 , already at E νµ ∼ 10 20 eV.[5] For a somewhat smaller m L ∼ 0.3 × 10 6 GeV, the production of real L is possible for E νµ above about 0.5 × 10 20 eV, which is close to where the GZK cut-off [3,4] is located[1, 2] for protons coming from cosmological distances. In this case σ νµ can be of the order of millibarns at E νµ ∼ 10 20 eV. Then, a dip at the GZK cut-off, followed by a "bump-structure"[10] near to 10 20 eV could occur in the air-showerThe possibility of an analogous hypothetical structure (with a charged pion) for the muon, can lead to a positive-definite, unusual contribution to (gµ − 2)/2 at the level of 10 −9 , for this value of m L . [5,6] However, in addition to the squared coupling, the magnitude is rather sensitive to possible damping at the structure vertex for virtual momentum near to m L , where the integral gets its main contribution. At present, the significance of a possible experimental deviation [7] from expectation is reduced by the correction of a long-standing sign error in the calculation of a "standard" theoretical contribution. [8,9] In this paper, we consider explicitly, only neutrino, effective neutral-current couplings.