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Abstract

In core-collapse supernovae or compact binary merger remnants, neutrino-neutrino refraction can spawn fast pair conversion of the type $\nu_e \bar\nu_e \leftrightarrow \nu_x \bar\nu_x$ (with $x=\mu, \tau$), governed by the angle-dependent density matrices of flavor lepton number. All angle modes evolve coherently for a homogeneous and axially symmetric two-flavor system, and the nonlinear equations of motion are equivalent to those of a gyroscopic pendulum. Our main innovation is to identify the real part of the eigenfrequency resulting from the linear normal-mode analysis of the neutrino system as the "pendulum spin." This is the elusive characteristic of the lepton-number angle distribution that determines the depth of conversion, without solving the nonlinear evolution equations.