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000122 | \subsection{Bosonic relative index and Higgs selection}
000123 | 
000124 | \begin{theorem}[Carrier-forced determinant classes on the seam]
000125 | \label{thm:determinant-class-higgs-selection}
000126 | Let
000127 | \[
000128 | L_2:=\det E_2,
000129 | \qquad
000130 | L_3:=\det E_3.
000131 | \]
000132 | Assume the total seam twist on the oriented double cover is one unit. Then the determinant holonomy classes satisfy
000133 | \[
000134 | [\mathrm{Hol}_\Sigma(L_2)]=1\in\pi_1(U(1)),
000135 | \qquad
000136 | [\mathrm{Hol}_\Sigma(L_3)]=0.
000137 | \]
000138 | Equivalently, for the asymptotic bosonic mass fields on the retained branch,
000139 | \[
000140 | \mathrm{wind}(\det\Phi_2)=1,
000141 | \qquad
000142 | \mathrm{wind}(\det\Phi_3)=0.
000143 | \]
000144 | The doublet winding is forced by the spin lift on the double cover, whereas center neutrality on the admissible branch forces determinant triviality in the triplet sector.
000145 | \end{theorem}
000146 | 
000147 | \begin{proof}[Proof sketch]
000148 | The oriented double cover already forces a half-angle lift on the rank-two weak sector. Passing to the determinant line therefore gives one full winding in $L_2$. By contrast, the admissible branch requires center neutrality in the color sector, so the determinant line of $E_3$ is trivial on the retained branch. One unit of total seam twist can therefore be realized only by the doublet determinant class.
000149 | \end{proof}