= Solution
For this dump scheme, there is no practical parent-sign or lifetime selection that isolates only the relatively rare <Ds meson> tau channel: other charmed <hadrons> promptly produce <electron neutrinos> and <muon neutrinos> by semileptonic decay. Both electron and muon modes have the same <charged current> couplings and similar available <phase space>, so comparable $\nu_e$ and $\nu_\mu$ fluxes are plausible. The <tau lepton> also decays into electron and muon channels. A rare tau-producing channel cannot remove the abundant prompt backgrounds, and its two-step chain emits both tau-flavour signs.
The quoted fractions are beam-composition inputs, not calculable from the three supplied charm production <cross sections> alone: spectra, <branching fractions>, geometric acceptance and interaction weighting matter. Impossibility of a pure tau source is a practical statement about this production method, not a fundamental prohibition on selecting tau flavour at an oscillation maximum.
Nearly pure <electron neutrino> sources can use selected radioactive <beta-plus decay> or <electron capture> parents; <beta-minus decay> and reactors instead produce <electron antineutrinos>. A boosted radioactive-ion beam gives directional electron-flavour flux. Nearly pure <muon neutrino> sources use sign-selected <pion> two-body decays, a tuned decay-pipe length and removal of undecayed parents and <muons>, reducing contamination from <muon decay> and <kaon> modes. Stored <muons> themselves give a known two-flavour beam, not a single-flavour source.
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