Constraints on Pion Superradiance and SF-QED Neutrino Production in Relativistic Magnetar Magnetospheres

We examine whether collective pion superradiance and strong-field QED (SF-QED) photon-to-neutrino conversion could produce multi-PeV neutrinos in the equatorial current sheet of a rapidly rotating magnetar (Ω ≥ 700 Hz). The proposed chain runs from centrifugally confined baryonic plasma, through proton–proton production of neutral pions, stimulated π⁰→γγ emission, and γ→νν̄ conversion near the light cylinder. Using an assumed sheet density of 3×10²⁰ cm⁻³ and Lorentz factor γ ~ 10⁶, we find a steady-state pion density of about 9×10¹⁵ cm⁻³ and a single-pass gain depth of about 10⁻⁴, bounded by ≲ 10⁻³ for any cross-section convention. The cooperative growth rate lies at least 10⁷ below the pion decay rate, and the gain depth approaches unity only in a corner case requiring the maximum confinable proton density at the stellar surface. Surviving magnetic pair production requires photon–field alignment that, with the Standard Model γ→νν̄ rate, gives a conversion probability ≲ 10⁻²² for 1 PeV photons. Dipole spin-down at 700 Hz and 10¹⁴ G takes about 1.6 days, and crustal strain energy (about 2×10⁴⁵ erg) cannot by itself eject material beyond the corotation radius. We conclude that collective pion superradiance is not viable under the parameters examined and that SF-QED conversion is strongly suppressed. The paper is offered as a negative baseline for cooperative PeV neutrino production models. All estimates are order-of-magnitude.