Kaon gluon parton distribution and momentum fraction from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:m

We present a high-statistics lattice-QCD determination of the kaon gluon parton distribution function and gluon momentum fraction. We use clover valence fermion action to take 1,296,640 kaon-correlator measurements on a highly improved staggered quark ensemble with <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>a</a:mi><a:mo>≈</a:mo><a:mn>0.12</a:mn><a:mtext> </a:mtext><a:mtext> </a:mtext><a:mi>fm</a:mi></a:math> and 310-MeV pion mass generated by the MILC Collaboration. A detailed investigation into the impact of gauge-link smearing on the gluonic matrix elements indicates that five steps of hypercubic smearing offer an effective balance between signal quality and preservation of long-distance physics. We report a nonperturbatively renormalized kaon gluon momentum fraction of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mo stretchy="false">⟨</c:mo><c:mi>x</c:mi><c:msubsup><c:mo stretchy="false">⟩</c:mo><c:mi>g</c:mi><c:mrow><c:mover accent="true"><c:mi>MS</c:mi><c:mo stretchy="true">¯</c:mo></c:mover><c:mo>,</c:mo><c:mi>K</c:mi></c:mrow></c:msubsup><c:mo>=</c:mo><c:mn>0.557</c:mn><c:mo stretchy="false">(</c:mo><c:mn>18</c:mn><c:msub><c:mo stretchy="false">)</c:mo><c:mrow><c:mi>stat</c:mi></c:mrow></c:msub><c:mo stretchy="false">(</c:mo><c:mn>24</c:mn><c:msub><c:mo stretchy="false">)</c:mo><c:mi>NPR</c:mi></c:msub><c:mo stretchy="false">(</c:mo><c:mn>56</c:mn><c:msub><c:mo stretchy="false">)</c:mo><c:mi>mixing</c:mi></c:msub></c:math> at <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>μ</o:mi><o:mo>=</o:mo><o:mn>2</o:mn><o:mtext> </o:mtext><o:mtext> </o:mtext><o:mi>GeV</o:mi></o:math> in the <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mover accent="true"><q:mi>MS</q:mi><q:mo stretchy="true">¯</q:mo></q:mover></q:math> scheme. Using reduced pseudo-Ioffe-time-distribution matrix elements and pseudo-parton-distribution-function (PDF) matching, we extract the kaon gluon PDF and compare with the prediction from the Dyson-Schwinger equation and with the pion PDF obtained from the same ensemble.