TFPT scientific source excerpt
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000128 | \begin{theorem}[Joint discrete admissible sector from the boundary primitive kernel]
000129 | \label{thm:joint-discrete-admissible-sector}
000130 | \label{thm:discrete-sector-from-primitive-kernel}
000131 | For every admissible boundary primitive kernel $\Tker^\partial$, the compatible discrete datum
000132 | \[
000133 | \mathfrak D_{\mathrm{disc}}(\Tker^\partial)
000134 | \]
000135 | is a singleton. Its unique element is
000136 | \[
000137 | d^\star_{\mathrm{disc}}
000138 | :=
000139 | \Bigl(
000140 | E_3\oplus E_2,\,
000141 | Y,\,
000142 | S^+,\,
000143 | X_f^\circ,\,
000144 | [\nabla_F^\star],\,
000145 | c_1(L_2),\,
000146 | c_1(L_3),\,
000147 | N_{\mathrm{fam}},\,
000148 | \theta_{\mathrm{eff}}
000149 | \Bigr)
000150 | \]
000151 | with
000152 | \[
000153 | X_f^\circ\cong\mathbb P^1\setminus\mu_4,
000154 | \qquad
000155 | N_{\mathrm{fam}}=3,
000156 | \qquad
000157 | (g,b,s)=(5,3,2),
000158 | \qquad
000159 | \dim S^+=16,
000160 | \]
000161 | \[
000162 | \begin{aligned}
000163 | S^+&=\Lambda^{\mathrm{even}}(E_3\oplus E_2),\\
000164 | Y&=\mathrm{diag}\!\left(-\tfrac13,-\tfrac13,-\tfrac13,\tfrac12,\tfrac12\right),\\
000165 | (c_1(L_2),c_1(L_3))&=(1,0),\qquad
000166 | \theta_{\mathrm{eff}}=0.
000167 | \end{aligned}
000168 | \]
000169 | Equivalently, carrier data, family data, determinant data, and the effective strong angle are fixed
000170 | jointly from the same boundary primitive kernel.
000171 | \end{theorem}