A measurement of the time-dependent ratio of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi>D</a:mi><a:mn>0</a:mn></a:msup><a:mo stretchy="false">→</a:mo><a:mrow><a:msup><a:mi>K</a:mi><a:mo>+</a:mo></a:msup><a:msup><a:mi>π</a:mi><a:mo>−</a:mo></a:msup></a:mrow></a:math> to <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mrow><d:mrow><d:msup><d:mrow><d:mover accent="true"><d:mrow><d:mi>D</d:mi></d:mrow><d:mrow><d:mo stretchy="false">¯</d:mo></d:mrow></d:mover></d:mrow><d:mrow><d:mn>0</d:mn></d:mrow></d:msup></d:mrow><d:mo stretchy="false">→</d:mo><d:mrow><d:msup><d:mrow><d:mi>K</d:mi></d:mrow><d:mrow><d:mo>+</d:mo></d:mrow></d:msup><d:msup><d:mrow><d:mi>π</d:mi></d:mrow><d:mrow><d:mo>−</d:mo></d:mrow></d:msup></d:mrow></d:mrow></d:math> decay rates is reported. The analysis uses a sample of proton-proton collisions corresponding to an integrated luminosity of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mrow><i:mn>6</i:mn><i:mtext> </i:mtext><i:mtext> </i:mtext><i:msup><i:mrow><i:mi>fb</i:mi></i:mrow><i:mrow><i:mo>−</i:mo><i:mn>1</i:mn></i:mrow></i:msup></i:mrow></i:math> recorded by the LHCb experiment from 2015 through 2018 at a center-of-mass energy of 13 TeV. The <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msup><k:mi>D</k:mi><k:mn>0</k:mn></k:msup></k:math> meson is required to originate from a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mrow><m:mrow><m:msup><m:mrow><m:mi>D</m:mi></m:mrow><m:mrow><m:mo>*</m:mo></m:mrow></m:msup><m:mo stretchy="false">(</m:mo><m:mn>2010</m:mn><m:msup><m:mrow><m:mo stretchy="false">)</m:mo></m:mrow><m:mrow><m:mo>+</m:mo></m:mrow></m:msup></m:mrow><m:mo stretchy="false">→</m:mo><m:mrow><m:msup><m:mrow><m:mi>D</m:mi></m:mrow><m:mrow><m:mn>0</m:mn></m:mrow></m:msup><m:msup><m:mrow><m:mi>π</m:mi></m:mrow><m:mrow><m:mo>+</m:mo></m:mrow></m:msup></m:mrow></m:mrow></m:math> decay, such that its flavor at production is inferred from the charge of the accompanying pion. The measurement is performed simultaneously for the <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:msup><r:mi>K</r:mi><r:mo>+</r:mo></r:msup></r:math> <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:msup><t:mi>π</t:mi><t:mo>−</t:mo></t:msup></t:math> and <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:msup><v:mi>K</v:mi><v:mo>−</v:mo></v:msup></v:math> <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:msup><x:mi>π</x:mi><x:mo>+</x:mo></x:msup></x:math> final states, allowing both mixing and <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:mi>C</z:mi><z:mi>P</z:mi></z:math>-violation parameters to be determined. The value of the ratio of the decay rates at production is determined to be <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:mrow><bb:msub><bb:mrow><bb:mi>R</bb:mi></bb:mrow><bb:mrow><bb:mi>K</bb:mi><bb:mi>π</bb:mi></bb:mrow></bb:msub><bb:mo>=</bb:mo><bb:mo stretchy="false">(</bb:mo><bb:mn>343.1</bb:mn><bb:mo>±</bb:mo><bb:mn>2.0</bb:mn><bb:mo stretchy="false">)</bb:mo><bb:mo>×</bb:mo><bb:msup><bb:mrow><bb:mn>10</bb:mn></bb:mrow><bb:mrow><bb:mo>−</bb:mo><bb:mn>5</bb:mn></bb:mrow></bb:msup></bb:mrow></bb:math>. The mixing parameters are measured to be <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"><fb:msub><fb:mi>c</fb:mi><fb:mrow><fb:mi>K</fb:mi><fb:mi>π</fb:mi></fb:mrow></fb:msub><fb:mo>=</fb:mo><fb:mo stretchy="false">(</fb:mo><fb:mn>51.4</fb:mn><fb:mo>±</fb:mo><fb:mn>3.5</fb:mn><fb:mo stretchy="false">)</fb:mo><fb:mo>×</fb:mo><fb:msup><fb:mn>10</fb:mn><fb:mrow><fb:mo>−</fb:mo><fb:mn>4</fb:mn></fb:mrow></fb:msup></fb:math> and <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"><jb:mrow><jb:msubsup><jb:mrow><jb:mi>c</jb:mi></jb:mrow><jb:mrow><jb:mi>K</jb:mi><jb:mi>π</jb:mi></jb:mrow><jb:mrow><jb:mo>′</jb:mo></jb:mrow></jb:msubsup><jb:mo>=</jb:mo><jb:mo stretchy="false">(</jb:mo><jb:mn>13</jb:mn><jb:mo>±</jb:mo><jb:mn>4</jb:mn><jb:mo stretchy="false">)</jb:mo><jb:mo>×</jb:mo><jb:msup><jb:mrow><jb:mn>10</jb:mn></jb:mrow><jb:mrow><jb:mo>−</jb:mo><jb:mn>6</jb:mn></jb:mrow></jb:msup></jb:mrow></jb:math>, where <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" display="inline"><nb:msqrt><nb:msub><nb:mi>R</nb:mi><nb:mrow><nb:mi>K</nb:mi><nb:mi>π</nb:mi></nb:mrow></nb:msub></nb:msqrt><nb:msub><nb:mi>c</nb:mi><nb:mrow><nb:mi>K</nb:mi><nb:mi>π</nb:mi></nb:mrow></nb:msub></nb:math> is the linear coefficient of the expansion of the ratio as a function of decay time in units of the <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"><pb:msup><pb:mi>D</pb:mi><pb:mn>0</pb:mn></pb:msup></pb:math> lifetime, and <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" display="inline"><rb:mrow><rb:msubsup><rb:mrow><rb:mi>c</rb:mi></rb:mrow><rb:mrow><rb:mi>K</rb:mi><rb:mi>π</rb:mi></rb:mrow><rb:mrow><rb:mo>′</rb:mo></rb:mrow></rb:msubsup></rb:mrow></rb:math> is the quadratic coefficient, both averaged between the <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"><tb:msup><tb:mi>K</tb:mi><tb:mo>+</tb:mo></tb:msup></tb:math> <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML" display="inline"><vb:msup><vb:mi>π</vb:mi><vb:mo>−</vb:mo></vb:msup></vb:math> and <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML" display="inline"><xb:msup><xb:mi>K</xb:mi><xb:mo>−</xb:mo></xb:msup></xb:math> <zb:math xmlns:zb="http://www.w3.org/1998/Math/MathML" display="inline"><zb:msup><zb:mi>π</zb:mi><zb:mo>+</zb:mo></zb:msup></zb:math> final states. The precision is improved relative to the previous best measurement by approximately 60%. No evidence for <bc:math xmlns:bc="http://www.w3.org/1998/Math/MathML" display="inline"><bc:mi>C</bc:mi><bc:mi>P</bc:mi></bc:math> violation is found. © 2025 CERN, for the LHCb Collaboration 2025 CERN