Probing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>C</mml:mi> <mml:mi>P</mml:mi> </mml:math> -violating Higgs-gauge couplings with Higgsstrahlung at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" disp

We investigate the sensitivity of a future high-luminosity <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mi>e</a:mi> <a:mo>−</a:mo> </a:msup> <a:msup> <a:mi>e</a:mi> <a:mo>+</a:mo> </a:msup> </a:math> collider operating at <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msqrt> <c:mi>s</c:mi> </c:msqrt> <c:mo>=</c:mo> <c:mn>250</c:mn> <c:mtext> </c:mtext> <c:mtext> </c:mtext> <c:mi>GeV</c:mi> </c:math> to <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>C</e:mi> <e:mi>P</e:mi> </e:math> -violating and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>C</g:mi> <g:mi>P</g:mi> </g:math> -conserving anomalous <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>h</i:mi> <i:mi>V</i:mi> <i:mi>V</i:mi> </i:math> interactions via the Higgsstrahlung process. The effects of new physics are parametrized in the Standard Model effective field theory (SMEFT) framework through six dimension-6 operators modifying the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>h</k:mi> <k:mi>V</k:mi> <k:mi>V</k:mi> </k:math> vertices. Using polarized beams and exploiting polarization and spin correlation asymmetries reconstructed from Higgs decay products, we perform a comprehensive analysis across the three dominant decay modes, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>h</m:mi> <m:mo stretchy="false">→</m:mo> <m:mi>b</m:mi> <m:mover accent="true"> <m:mi>b</m:mi> <m:mo stretchy="false">¯</m:mo> </m:mover> </m:math> , <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:mi>W</r:mi> <r:msup> <r:mi>W</r:mi> <r:mo>⋆</r:mo> </r:msup> </r:math> , and <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mi>Z</t:mi> <t:msup> <t:mi>Z</t:mi> <t:mo>⋆</t:mo> </t:msup> </t:math> . The <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:mi>h</v:mi> <v:mo stretchy="false">→</v:mo> <v:mi>W</v:mi> <v:msup> <v:mi>W</v:mi> <v:mo>⋆</v:mo> </v:msup> </v:math> channel exhibits the highest sensitivity to <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:msub> <y:mi mathvariant="script">O</y:mi> <y:mrow> <y:mi>H</y:mi> <y:mi>W</y:mi> </y:mrow> </y:msub> </y:math> and <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"> <bb:msub> <bb:mi mathvariant="script">O</bb:mi> <bb:mrow> <bb:mi>H</bb:mi> <bb:mover accent="true"> <bb:mi>W</bb:mi> <bb:mo stretchy="false">˜</bb:mo> </bb:mover> </bb:mrow> </bb:msub> </bb:math> , while the <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"> <gb:mi>b</gb:mi> <gb:mover accent="true"> <gb:mi>b</gb:mi> <gb:mo stretchy="false">¯</gb:mo> </gb:mover> </gb:math> mode constrains the remaining operators with high statistical precision. Sensitivity studies incorporating luminosity scaling and systematic uncertainties show that the projected bounds improve significantly with increasing integrated luminosity, but saturate once experimental systematics exceed the few-percent level. These results highlight the crucial role of spin-based observables and beam polarization in achieving subpercent precision on SMEFT coefficients at future lepton colliders.