Exploratory calculation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi mathvariant="normal">L</mml:mi> </mml:msub> <mml:mo stretchy="false">→</mml:mo> <mml:msup> <mml:mi>μ</mml:mi>

We compute the complex, long-distance two-photon-exchange amplitude which contributes to the rare <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>K</a:mi> <a:mi mathvariant="normal">L</a:mi> </a:msub> <a:mo stretchy="false">→</a:mo> <a:msup> <a:mi>μ</a:mi> <a:mo>+</a:mo> </a:msup> <a:msup> <a:mi>μ</a:mi> <a:mo>−</a:mo> </a:msup> </a:math> decay from lattice QCD. We use a <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msup> <e:mn>24</e:mn> <e:mn>3</e:mn> </e:msup> <e:mo>×</e:mo> <e:mn>64</e:mn> </e:math> physical-pion-mass gauge field ensemble at an inverse lattice spacing of 1.023 GeV and a <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mrow> <g:msub> <g:mrow> <g:mi>QED</g:mi> </g:mrow> <g:mrow> <g:mi>∞</g:mi> </g:mrow> </g:msub> </g:mrow> </g:math> -based formalism. Our implementation strategies for all five non-SU(3)-flavor-suppressed diagram topologies are given in detail. We achieve a 25% statistical precision on the dispersive part of this long-distance amplitude. This calculation is carried out with <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mrow> <i:mn>2</i:mn> <i:mo>+</i:mo> <i:mn>1</i:mn> </i:mrow> </i:math> quark flavors and therefore requires the addition of counterterms to compensate for the absence of the Glashow-Iliopoulos-Maiani mechanism. These counterterms are not included in the current calculation and will be the subject of a second paper. Although a direct comparison to experiment cannot yet be made because of those omitted counterterms, the present exploratory calculation allows one to identify principal sources of statistical uncertainty in this calculation. The precision of our results is limited by the reconstruction of the physical contribution of the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>η</k:mi> </k:math> intermediate state, for which various strategies are tested and compared.