Global determination of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> <mml:msub> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>u</mml:mi> <mml:mi>s</mml:mi> </

In light of ongoing issues with the first-row unitarity test of the Cabibbo-Kobayashi-Maskawa matrix, we showcase a global fit to measurements of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>K</a:mi> <a:mo stretchy="false">→</a:mo> <a:mo>ℓ</a:mo> <a:mi>ν</a:mi> </a:math> , <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mi>K</d:mi> <d:mo stretchy="false">→</d:mo> <d:mi>π</d:mi> <d:mo>ℓ</d:mo> <d:mi>ν</d:mi> </d:math> , <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>τ</g:mi> <g:mo stretchy="false">→</g:mo> <g:mi>K</g:mi> <g:mi>ν</g:mi> </g:math> , and <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mi>τ</j:mi> <j:mo stretchy="false">→</j:mo> <j:mi>K</j:mi> <j:mi>π</j:mi> <j:mi>ν</j:mi> </j:math> decays for the first time. Fitting the semileptonic and three-body <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>τ</m:mi> </m:math> decay data simultaneously becomes computationally feasible because we employ a simple form factor parametrization for the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>K</o:mi> <o:mi>π</o:mi> </o:math> form factor that manifestly connects the semileptonic and pair-production regions. We find good agreement with the data and use our analyses to infer <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mo stretchy="false">|</q:mo> <q:msub> <q:mi>V</q:mi> <q:mrow> <q:mi>u</q:mi> <q:mi>s</q:mi> </q:mrow> </q:msub> <q:mo stretchy="false">|</q:mo> </q:math> and the parameters for the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>K</u:mi> <u:mi>π</u:mi> </u:math> form factors. Our result for <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:mo stretchy="false">|</w:mo> <w:msub> <w:mi>V</w:mi> <w:mrow> <w:mi>u</w:mi> <w:mi>s</w:mi> </w:mrow> </w:msub> <w:mo stretchy="false">|</w:mo> </w:math> sits close to the results extracted from exclusive <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:msub> <ab:mi>K</ab:mi> <ab:mrow> <ab:mo>ℓ</ab:mo> <ab:mn>2</ab:mn> </ab:mrow> </ab:msub> </ab:math> and somewhat above the <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:msub> <cb:mi>K</cb:mi> <cb:mrow> <cb:mo>ℓ</cb:mo> <cb:mn>3</cb:mn> </cb:mrow> </cb:msub> </cb:math> decay and inclusive <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mi>τ</eb:mi> </eb:math> decay result. Moreover, our results for the <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"> <gb:mi>K</gb:mi> <gb:mo stretchy="false">→</gb:mo> <gb:mi>π</gb:mi> </gb:math> vector form factor at zero momentum transfer are compatible with lattice QCD determinations, despite not using lattice QCD inputs for this quantity in our fit of the hadronic matrix elements. We are further able to determine some of the pole parameters of the individual scalar and vector <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"> <jb:mi>K</jb:mi> <jb:mi>π</jb:mi> </jb:math> resonances with masses below the <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"> <lb:mi>τ</lb:mi> </lb:math> mass. We caution that our results account only for short-distance electromagnetic corrections but not long-distance contributions; this is due to the lack of a consistent description of long-distance corrections to the <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" display="inline"> <nb:mi>K</nb:mi> <nb:mi>π</nb:mi> </nb:math> matrix elements both above and below threshold for an arbitrary model of the form factors.