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000142 | \subsection{Bosonic relative index and Higgs selection}
000143 | 
000144 | \begin{theorem}[Carrier-forced determinant classes on the seam]
000145 | \label{thm:determinant-class-higgs-selection}
000146 | Let
000147 | \[
000148 | L_2:=\det E_2,
000149 | \qquad
000150 | L_3:=\det E_3.
000151 | \]
000152 | Assume the total seam twist on the oriented double cover is one unit. Then the determinant holonomy classes satisfy
000153 | \[
000154 | [\mathrm{Hol}_\Sigma(L_2)]=1\in\pi_1(U(1)),
000155 | \qquad
000156 | [\mathrm{Hol}_\Sigma(L_3)]=0.
000157 | \]
000158 | Equivalently, for the asymptotic bosonic mass fields on the retained branch,
000159 | \[
000160 | \mathrm{wind}(\det\Phi_2)=1,
000161 | \qquad
000162 | \mathrm{wind}(\det\Phi_3)=0.
000163 | \]
000164 | 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.
000165 | \end{theorem}
000166 | 
000167 | \begin{proof}[Proof sketch]
000168 | 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.
000169 | \end{proof}