<p>For a finite cyclic group <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40062_2025_368_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(C_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>, we identify Greenlees’ equivariant connective K-theory <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40062_2025_368_Article_IEq2.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">\(kU_{C_n}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <msub> <mi>U</mi> <msub> <mi>C</mi> <mi>n</mi> </msub> </msub> </mrow> </math></EquationSource> </InlineEquation> as an <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40062_2025_368_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">\(RO(C_n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>O</mi> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-graded localization of the actual connective cover of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40062_2025_368_Article_IEq4.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(KU_{C_n}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <msub> <mi>U</mi> <msub> <mi>C</mi> <mi>n</mi> </msub> </msub> </mrow> </math></EquationSource> </InlineEquation>.</p>