Parameter degeneracy and information loss in inverse flavor mapping for high-energy astrophysical neutrinos

Given a high-energy astrophysical neutrino flux, its source flavor ratios $η= \{η^{}_e, η^{}_μ, η^{}_τ\}$ are correlated with the ones $f = \{f^{}_e, f^{}_μ, f^{}_τ\}$ measured at a neutrino telescope via $f = P η$, where the elements of $P$ consist of four lepton flavor mixing parameters. But reali