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Determination of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:math> -even fraction of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>D</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy="false">→</mml:mo><mml:msubsup><mml:mi>K</mml:mi><mml:mi>S</mml:mi><mml:mn>0</mml:mn></mml:msubsup><mml:msup><mml:mi>π</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:…
Quantum-correlated $D\overline{D}$ pairs collected by the BESIII experiment at the $\ensuremath{\psi}(3770)$ resonance corresponding to an integrated luminosity of $2.93\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ are used to study the ${D}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ decay mode. The $CP$-even fraction of ${D}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ decays is determined to be $0.235\ifmmode\pm\else\textpm\fi{}0.010\ifmmode\pm\else\textpm\fi{}0.002$, where the first uncertainty is statistical and the second is …