Convergence of heavy-flavor parton distribution functions at small <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>x</mml:mi> </mml:math> and its impact on future <mml:math xmlns:mml="http://www.w3.org/1998/Math/Mat

The small- <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>x</a:mi> </a:math> regime remains a key testing ground for perturbative QCD and future lepton-hadron collider physics. We demonstrate that the long-standing <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mo>∼</c:mo> <c:mn>30</c:mn> <c:mo>%</c:mo> </c:math> spread among global next-to-next-to-leading-order parton distribution functions (PDFs) for heavy-flavor parton densities at small <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>x</e:mi> </e:math> has now converged below <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mo>≈</g:mo> <g:mn>17</g:mn> <g:mo>%</g:mo> </g:math> . Using the latest 4.0, 18, and 20 sets, we quantify a maximal relative deviation of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi mathvariant="normal">Δ</i:mi> <i:mi>max</i:mi> </i:msub> <i:mo>≃</i:mo> <i:mn>16.8</i:mn> <i:mo>%</i:mo> </i:math> at <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mi>x</l:mi> <l:mo>=</l:mo> <l:msup> <l:mn>10</l:mn> <l:mrow> <l:mo>−</l:mo> <l:mn>6</l:mn> </l:mrow> </l:msup> </l:math> and <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mi>Q</n:mi> <n:mo>=</n:mo> <n:mn>100</n:mn> <n:mtext> </n:mtext> <n:mtext> </n:mtext> <n:mi>GeV</n:mi> </n:math> . This convergence, further stabilized by consistent <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:msub> <p:mi>α</p:mi> <p:mi>s</p:mi> </p:msub> </p:math> and scale variations, marks a new precision frontier for the heavy-flavor sector. We evaluate the phenomenological impact for heavy-quark electroproduction at future <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:mi>e</r:mi> <r:mi>p</r:mi> </r:math> colliders (LHeC and FCC–eh), showing that PDF-driven cross section uncertainties for <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:msup> <t:mi>e</t:mi> <t:mo>−</t:mo> </t:msup> <t:mi>p</t:mi> <t:mo stretchy="false">→</t:mo> <t:msup> <t:mi>e</t:mi> <t:mo>−</t:mo> </t:msup> <t:mi>c</t:mi> <t:mover accent="true"> <t:mi>c</t:mi> <t:mo stretchy="false">¯</t:mo> </t:mover> <t:mi>X</t:mi> </t:math> and <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:msup> <y:mi>e</y:mi> <y:mo>−</y:mo> </y:msup> <y:mi>p</y:mi> <y:mo stretchy="false">→</y:mo> <y:msup> <y:mi>e</y:mi> <y:mo>−</y:mo> </y:msup> <y:mi>b</y:mi> <y:mover accent="true"> <y:mi>b</y:mi> <y:mo stretchy="false">¯</y:mo> </y:mover> <y:mi>X</y:mi> </y:math> have been reduced by nearly a factor of 2 compared to 2013 estimates.