Threshold effects of inelastic neutrino-dark-matter scattering in AGN spikes

High-energy neutrinos from active galactic nuclei provide a powerful probe of dark matter (DM) spikes surrounding supermassive black holes. While elastic neutrino-DM scattering mediated by a light particle produces little net attenuation due to regeneration, inelastic scattering, $νχ_1\toνχ_2$, introduces a nonzero minimum energy loss, $E_{\rm th}$. We show that this threshold prevents the near-cancellation between attenuation and regeneration, producing a distinctive spectral deficit above $E_{\rm th}$ accompanied by a pile-up below it. Solving the full neutrino cascade, including repeated scattering and secondary neutrinos from excited-state decays, we demonstrate that this feature persists for both narrow and broad energy-transfer kernels. Using public IceCube data for NGC 1068 and an unbinned likelihood analysis with a freely fitted intrinsic power-law spectrum, we derive upper limits on the asymptotic optical depth of $τ_0<26-39$ for a narrow kernel (kernel width $ε=0.01$) over $E_{\rm th}=0.24-3.0$ TeV and $τ_0<20-26$ for a broad kernel ($ε=0.36$) over $E_{\rm th}=0.24-2.4$ TeV, at the nominal $90\%$ confidence level. An intrinsic high-energy cutoff weakens the narrow-kernel limits by factors of $2.2-7.4$ and the broad-kernel limits by factors of $2.0-5.0$. We further investigate DM velocity effects and combine NGC 1068 with CGCG 420-015 and TXS 0506+056 to extend the threshold reach. In a vector-mediator realization, an optimistic adiabatic DM spike can yield constraints on the neutrino coupling competitive with laboratory bounds for broad energy-transfer kernels.